The Gan-Gross-Prasad period of Klingen Eisenstein families over unitary groups

Ruichen Xu Academy of Mathematics and Systems Science, Chinese Academy of Science, No. 55 Zhongguancun East Road, Beijing, 100190, China. [email protected]

Abstract.

In this article, we compute the Gan-Gross-Prasad period integral of Klingen Eisenstein series over the unitary group $\mathrm{U}(m+1,n+1)$ with a cuspidal automorphic form over $\mathrm{U}(m+1,n)$ , and show that it is related to certain special Rankin-Selberg $L$ -values. We $p$ -adically interpolate these Gan-Gross-Prasad period integrals as the Klingen Eisenstein series and the cuspidal automorphic form vary in Hida families. As a byproduct, we obtain a $p$ -adic $L$ -function of Rankin-Selberg type over $\mathrm{U}(m,n)\times\mathrm{U}(m+1,n)$ . The ultimate motivation is to show the $p$ -primitive property of Klingen Eisenstein series over unitary groups, by computing such Gan-Gross-Prasad period integrals, and this article is a starting point of this project. The $p$ -primitivity of Eisenstein series is an essential property in the automorphic method in Iwasawa theory.

1. Introduction

Let $p$ be an odd prime. In Iwasawa theory, people study the mysterious relations between special values of $L$ -functions and arithmetic objects as they vary in $p$ -adic families. Such relations are formulated as Iwasawa main conjectures. Among the successful attempts on Iwasawa main conjectures, one divisibility of them, namely “the lower bound of Selmer groups”, are often proved by the machinery of Eisenstein congruences.

When running Eisenstein congruences, the most challenging part is to verify the “ $p$ -primitivity” of the Eisenstein series and their $p$ -adic families (i.e. Eisenstein families), which can be regarded as certain “modulo $p$ nonvanishing” property of Eisenstein families. In 1990s, Wiles [Wil90] proved the Iwasawa main conjecture for $\mathrm{GL}_{1}$ over totally real field by running Eisenstein congruences over $\mathrm{GL}_{2}$ , where the Eisenstein series have explicit $q$ -expansions. Later on, Skinner and Urban [SU14] proved one-side divisibility of Iwasawa main conjecture for modular forms that are ordinary at $p$ . They used the Klingen Eisenstein series over the unitary group $\mathrm{U}(2,2)$ , and the $p$ -primitive property is proved by computing the Fourier coefficients of them. Hsieh [Hsi14] proved the Iwasawa main conjectures for $\mathrm{GL}_{1}$ over CM fields using the Klingen Eisenstein series over $\mathrm{U}(2,1)$ , using the Fourier-Jacobi coefficient of Klingen Eisenstein family. Wan [Wan20] considered the complement of the case in [SU14], that is, the Rankin-Selberg product of a modular form $f$ and an ordinary CM form whose weight is higher than $f$ , using Klingen Eisenstein series over another rank four unitary group $\mathrm{U}(3,1)$ . The highlight is that Wan’s approach removed the ordinary condition of $f$ at $p$ , yet still keeps the “ordinary nature” of the problem. The $p$ -primitivity of the Klingen Eisenstein series there is also proved by computing their Fourier-Jacobi coefficients. ¹¹1The aforementioned results on Iwasawa main conjectures depend on various technical assumptions, which we did not specify.

In this article, we start the project of investigating the $p$ -primitivity of Klingen Eisenstein series over unitary groups of general signatures $\mathrm{U}(m+1,n+1)$ (constructed explicitly in [Wan15]), by computing the Gan-Gross-Prasad period of them with cuspidal automorphic forms over $\mathrm{U}(m+1,n)$ . Now we summarize the main result of this article, starting with introducing necessary notations.

Let $\mathcal{K}/\mathcal{F}$ be a CM extension, that is, $\mathcal{F}$ is a totally real number field and $\mathcal{K}$ is an imaginary quadratic extension of $\mathcal{F}$ . Let $p$ be an odd prime that is unramified in $\mathcal{F}$ and each prime of $\mathcal{F}$ above $p$ splits in $\mathcal{K}$ . Throughout, we fix an isomorphism $\iota_{p}:\overline{\mathbb{Q}}_{p}\simeq\mathbb{C}$ . Let $\mathcal{K}_{\infty}/\mathcal{K}$ be the composition of all $\mathbb{Z}_{p}$ -extensions of $\mathcal{K}$ , with Galois group $\Gamma_{\mathcal{K}}$ . Let $m\geq n\geq 0$ be integers, we put $\mathrm{U}(m,n)$ as the unitary group of signature $(m,n)$ , that is, the unitary group associated to the skew-Hermitian matrix

\begin{bmatrix}&&\mathbf{1}_{n}\\ &\vartheta&\\ -\mathbf{1}_{n}\end{bmatrix},

where $\vartheta$ is a diagonal matrix such that $i^{-1}\vartheta$ is totally positive definite, and we assume $\mathrm{U}(m,n)(\mathcal{F}_{v})$ is quasi-split for every finite place $v$ of $\mathcal{F}$ . Throughout, we assume that the multiplicity one theorem holds for automorphic representations over any unitary groups $\mathrm{U}(m,n)$ (see Section 2.3).

1.1. Main result: automorphic computations

Let $\sigma$ be an irreducible tempered unitary cuspidal representation of $H:=\mathrm{U}(m,n)$ and $\chi$ be a Hecke character over $\mathcal{K}$ . Let $\Phi$ be a cuspidal automorphic form in $\sigma$ . We can construct a Klingen Eisenstein series by pulling back from a Siegel Eisenstein series $E^{\mathrm{Sieg}}_{\chi,s}(-)$ on the large unitary group $G^{\blacklozenge}:=\mathrm{U}(m+n+1,m+n+1)$ with the cuspidal automorphic form $\Phi$ . We denote the Klingen Eisenstein series by

E^{\mathrm{Kling}}(F^{\heartsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;-);-)\quad\text{ over }G^{\heartsuit}:=\mathrm{U}(m+1,n+1),

where $f^{\mathrm{Sieg}}_{s,\chi}(-)$ is the Siegel Eisenstein section defining the Siegel Eisenstein series. The operator $F^{\heartsuit}(-)$ is the so-called pullback integral.

Let $\pi$ be an irreducible tempered unitary cuspidal representation of $G:=\mathrm{U}(m+1,n)$ . The main theme of this article is to compute and $p$ -adically interpolate the following Gan-Gross-Prasad (GGP) period integral of $E^{\mathrm{Kling}}(-)$ over $G^{\heartsuit}$ with a cuspidal automorphic form $\Psi$ in $\pi$ ,

\mathcal{P}^{\mathrm{Kling}}(\Phi,\Psi,\chi,s):=\int_{G(\mathcal{F})\backslash G(\mathbb{A}_{\mathcal{F}})}E^{\mathrm{Kling}}((f^{\mathrm{Sieg}}_{s,\chi},\Phi;-);\jmath^{\flat}(g))\Psi(g)\,\mathrm{d}g,

where $\jmath^{\flat}:G\hookrightarrow G^{\heartsuit}$ is a canonical embedding of unitary groups. By the cuspidality of $\Psi$ , this converges absolutely for those values of $s$ at which the Klingen Eisenstein series is defined.

The first main result of this article is to unfold this period integral into local integrals and show that the local $L$ -factors appear at unramified places of $\mathcal{F}$ . We write $\mathcal{P}^{\mathrm{Kling}}(\Phi,\Psi,\chi,s)$ as $\mathcal{P}_{\Phi,\Psi}^{\mathrm{Kling}}$ for simplicity, and let

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$\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}$ be the set of finite places $v$ of $\mathcal{F}$ such that $\sigma_{v},\pi_{v}$ and the local extension $\mathcal{K}_{v}/\mathcal{F}_{v}$ are all unramified. The complement of $\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}$ in the set of all places of $\mathcal{F}$ is denoted by $\mathscr{V}_{\mathcal{F}}^{\mathrm{bad}}$ , and let
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$\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ be the set of finite places $v$ of $\mathcal{F}$ such that $\sigma_{v},\pi_{v},\chi_{v}$ and the local extension $\mathcal{K}_{v}/\mathcal{F}_{v}$ are all unramified. The complement of $\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ in the set of all places of $\mathcal{F}$ is denoted by $\mathscr{S}_{\mathcal{F}}^{\mathrm{bad}}$ .

Then we state our first main theorem.

Theorem A (Theorem 3.7).

We choose Siegel Eisenstein sections at $v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ as spherical sections $f^{\mathrm{Sieg},\mathrm{sph}}_{v,s,\chi}$ in the construction of the Klingen Eisenstein series. Then

\dfrac{(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}}{\left|\left|{\Phi}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\Psi}\right|\right|_{\pi,\mathrm{Pet}}^{2}}=\dfrac{1}{2^{\varkappa_{\sigma}+\varkappa_{\pi}}}\cdot\mathscr{L}_{\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}}(\sigma\times\pi)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v},\chi_{v}\right)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v}^{\vee},\chi_{v}\right)\\ \times\prod_{v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{bad}}}\mathscr{I}_{v}(\sigma_{v},\pi_{v})\prod_{v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}d_{N+1,v}(s,\chi_{v})^{-1}\prod_{v\in\mathscr{S}_{\mathcal{F}}^{\mathrm{bad}}}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v}^{\vee}),

Here the local $L$ -factors are

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local Rankin-Selberg $L$ -factors

\mathscr{L}_{\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}}(\sigma\times\pi):=\prod_{v\in\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(\dfrac{L(\frac{1}{2},\sigma_{v}\times\pi_{v})}{L(1,\sigma_{v},\operatorname{Ad})L(1,\pi_{v},\operatorname{Ad})}\prod_{i=1}^{m+n+1}L(i,\epsilon_{\mathcal{K}_{v}/\mathcal{F}_{v}}^{i})\right),

with $L(s,\sigma_{v}\times\pi_{v})$ is the local $L$ -factor of the Rankin-Selberg product $\mathrm{BC}(\sigma)\times\mathrm{BC}(\pi)$ , where $\mathrm{BC}(-)$ is the base change to an automorphic representation of general linear groups over $\mathcal{K}$ , and

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local doubling $L$ -factors

L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v}^{\vee},\chi_{v}\right):=\prod_{v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}L_{v}(s,\mathrm{BC}(\pi_{v})\otimes\chi_{v}\circ\det),

where the right hand side is the standard local Godement-Jacquet $L$ -factor and $\mathrm{BC}(\pi_{v})$ is the local base change from $G(\mathcal{F}_{v})$ to $\mathrm{GL}_{m+n+1}(\mathcal{K}_{v})$ .

Here $\mathscr{I}_{v}$ ’s are local Ichino-Ikeda integrals and $\mathscr{Z}_{v}^{\diamondsuit}$ ’s are local doubling integrals. Other unspecified factors will be defined in the text.

The next task is to compute the local Ichino-Ikeda integrals at bad places explicitly. We require $\sigma$ and $\pi$ are regularly ordinary at places of $\mathcal{F}$ above $p$ (see Definition 5.10). We put some further technical assumptions to make the calculation simpler.

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At archimedean places, we require $\sigma_{\infty}$ is of constant scalar weight, with the weights of $\sigma_{\infty}$ and $\pi_{\infty}$ satisfying the “Gan-Gross-Prasad weight interlacing property”.
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We require $\mathcal{K}/\mathcal{F}$ is unramified at any finite places of $\mathcal{F}$ .
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The local representations $\sigma_{v}$ and $\pi_{v}$ are “ramified disjointly” (i.e. they cannot be both ramified at any places), and these places where either $\sigma_{v}$ or $\pi_{v}$ ramifies shall split in $\mathcal{K}$ .
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For $p$ -adic places $v$ of $\mathcal{F}$ , the local representations $(\sigma_{v},\chi_{v})$ satisfy the generic assumption in [Wan15, Definition 4.42].

They are recorded as Assumption 3.8, 3.9, 3.10, 3.11, 6.1, 6.2 more explicitly in the text.

Theorem B (Theorem 6.3).

Under aforementioned assumptions, we have

\dfrac{(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}}{\left|\left|{\Phi}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\Psi}\right|\right|_{\pi,\mathrm{Pet}}^{2}}=\dfrac{1}{2^{\varkappa_{\sigma}+\varkappa_{\pi}}}\cdot\mathscr{L}^{\mathscr{V}_{\mathcal{F}}^{(p)}}(\sigma\times\pi)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v},\chi_{v}\right)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v}^{\vee},\chi_{v}\right)\\ \times\prod_{v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}}\zeta_{v}C_{\sigma_{v},\boldsymbol{\psi}_{v}}C_{\sigma_{v}^{\vee},\boldsymbol{\psi}_{v}^{-1}}\mathcal{B}_{\pi_{v}}^{\mathrm{ess}}\mathcal{B}_{\sigma_{v}}^{\mathrm{ess}}\prod_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}\zeta_{v}\Delta_{\beta}^{2}\cdot\mathfrak{G}(\kappa,\breve{\underline{\mu}},\breve{\underline{\lambda}})\mathfrak{G}(\kappa,\breve{\check{\underline{\mu}}},\breve{\check{\underline{\lambda}}})\mathcal{C}_{\sigma}^{\operatorname{ord}}\mathcal{C}_{\pi}^{\operatorname{ord}}\\ \times\prod_{v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}d_{N+1,v}(s,\chi_{v})^{-1}\prod_{v\in\mathscr{S}_{\mathcal{F}}^{\mathrm{bad}}}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v}^{\vee}).

Here $\mathscr{L}^{\mathscr{V}_{\mathcal{F}}^{(p)}}(\sigma\times\pi)$ is the product of all corresponding local factors except for those in $\mathscr{V}_{\mathcal{F}}^{(p)}$ , and the second line arises from the local Ichino-Ikeda integrals. They are defined explicitly in the text which we shall not record here, but instead remark that the unspecified local factors involve only explicit local Gauss sums, local critical adjoint $L$ -values, the local inner product of canonically chosen local vectors in local representations.

We present these theorems in Part I of this article.

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In Section 2, we introduce some backgrounds on unitary groups and their embeddings, then on Eisenstein series over unitary groups. We recall the doubling methods, both in the manner of Piatetski-Shapiro and Rallis and that of Garrett. These facts are definitely well-known but we offer proofs for some results due to the lack of references.
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In Section 3, we define the Gan-Gross-Prasad period integral of Klingen Eisenstein series with cuspidal automorphic forms, unfold them and relate them to local $L$ -factors. There will be a more detailed sketch of our approach at the beginning of Section 3.
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From Section 4 to Section 6, we calculate the local Ichino-Ikeda integrals at bad places.

1.2. Main result: $p$ -adic interpolations

The next step is to $p$ -adically interpolate the Gan-Gross-Prasad period integrals defined above, when the Klingen Eisenstein series $E^{\mathrm{Kling}}$ and the cuspidal automorphic form $\Psi$ vary in $p$ -adic families.

Let $L/\mathbb{Q}_{p}$ be a sufficiently large finite extension, with $\mathcal{O}_{L}$ being its ring of integers, and let $\Lambda_{m,n}^{\circ}:=\mathcal{O}_{L}\llbracket{T_{m+n}(1+p\mathbb{Z}_{p})}\rrbracket$ be the weight algebra for Hida families over the unitary group ${\mathrm{GU}}(m,n)$ , where $T_{m+n}$ is the diagonal torus in the general linear group $\mathrm{GL}_{m+n}$ . Let $\mathbb{I}$ (resp. $\mathbb{J}$ ) be a normal domain over (resp. $\Lambda_{m+1,n}^{\circ}$ ), which is also a finite algebra over $\Lambda_{m,n}^{\circ}$ (resp. $\Lambda_{m+1,n}^{\circ}$ ). Consider $\mathbf{f}$ (resp. $\mathbf{g}$ ), an $\mathbb{I}$ -adic (resp. $\mathbb{J}$ -adic) Hida family of tempered cuspidal ordinary eigenforms on ${\mathrm{GU}}(m,n)$ (resp. ${\mathrm{GU}}(m+1,n)$ ) of branching character $\eta_{0}$ . Let $\chi_{0}$ be a finite order Hecke character $\mathcal{K}^{\times}\backslash\mathbb{K}^{\times}\rightarrow\mathbb{C}^{\times}$ whose conductors at primes above $p$ divides $(p)$ . We collect these inputs as a tuple

\mathbf{G}:=(L,\mathbb{I},\mathbb{J},\mathbf{f},\mathbf{g},\chi_{0},\eta_{0}),

called a $p$ -adic Gan-Gross-Prasad datum (see Definition 9.1). Given such a datum $\mathbf{G}$ , we have a corresponding weight space ²²2Here $(-)^{{\mathrm{ur}}}$ is an “unramified extension of coefficient rings”, see Definition 8.3 for details.

\mathcal{X}_{\mathbf{G}}:=\mathcal{X}_{\mathbf{D}}\times\mathcal{Y}_{\mathbf{G}}:=\operatorname{Spec}\mathbb{I}^{{\mathrm{ur}}}\llbracket{\Gamma_{\mathcal{K}}}\rrbracket(\overline{\mathbb{Q}}_{p}^{\times})\times\operatorname{Spec}\mathbb{J}^{{\mathrm{ur}}}(\overline{\mathbb{Q}}_{p}^{\times}).

and we $p$ -adically interpolate the Gan-Gross-Prasad period integral on a Zariski dense subset $\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ of $\mathcal{X}_{\mathbf{G}}$ , consisting of “classical generic admissible points” (see Definition 9.2). Then the next main result of this article is as follows.

Theorem C (Theorem 9.3).

Notations being as above, then there exists an element

\mathbf{P}_{\mathbf{G}}\in(\mathbb{I}^{{\mathrm{ur}}}\llbracket{\Gamma_{\mathcal{K}}}\rrbracket\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}}))\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}}),

such that for any “classical generic admissible points” $\mathsf{P}\times\mathsf{Q}\in\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ , we have

(\mathsf{P}\times\mathsf{Q})(\mathbf{P}_{\mathbf{G}})=B(s_{\mathsf{P}},\chi_{\mathsf{P}})\mathcal{P}^{\mathrm{Kling}}(\mathbf{f}_{\mathsf{P}},\mathbf{g}_{\mathsf{Q}},\chi_{\mathsf{P}},s_{\mathsf{P}}),

with right-hand-side the GGP period integral of Klingen Eisenstein series with a cuspidal automorphic form, normalized by an explicit factor $B(s_{\mathsf{P}},\chi_{\mathsf{P}})$ defined in (8.25).

The main technique is to apply appropriate Hecke projectors introduced by Wan in [Wan19] to the family of Klingen Eisenstein series. When $n=0$ , an alternative approach is to use the $p$ -adic interpolation of Petersson inner products in Hida families. The method of Hecke projectors may date back to the series of works by Hida [Hid85, Hid88, Hid91], where he constructed $p$ -adic $L$ -functions for the Rankin-Selberg product of two modular forms and their Hida families.

When Hida families $\mathbf{f}$ and $\mathbf{g}$ satisfy the aforementioned ramification conditions at classical points together with certain multiplicity one property (Assumption 9.4), combining with the automorphic computations (Theorem B), we have a more explicit interpolation formula relating $\mathbf{P}_{\mathbf{G}}^{2}$ to critial $L$ -values at points $\mathsf{P}\times\mathsf{Q}\in\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ , which is Theorem 9.5 in the text.

In [Wan15], Wan constructed $p$ -adic $L$ -functions $\mathbf{L}_{\mathbf{g},\chi_{0}}$ for Hida families $\mathbf{g}$ over unitary groups. Comparing the interpolation formulas, a byproduct of our construction is an imprimitive $p$ -adic $L$ -function for the Rankin-Selberg product of Hida families $\mathbf{f}$ and $\mathbf{g}$ over $\mathrm{U}(m,n)\times\mathrm{U}(m+1,n)$ .

Theorem D (Theorem 9.7).

Notations being as above with the aforementioned ramification conditions, then there exists an element

\mathbf{L}_{\mathbf{f},\mathbf{g}}\in\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}})

such that for any $\mathsf{P}\times\mathsf{Q}$ in a certain Zariski dense subset of $\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ with valid multiplicity one theorems, we have $(\mathsf{P}\times\mathsf{Q})(\mathbf{L}_{\mathbf{f},\mathbf{g}})$ equals

\dfrac{1}{2^{\varkappa_{\sigma_{\mathbf{f}_{\mathsf{P}}}}+\varkappa_{\pi_{\mathbf{g}_{\mathsf{Q}}}}}}\times\left|\left|{\mathbf{f}_{\mathsf{P}}}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\mathbf{g}_{\mathsf{Q}}}\right|\right|_{\mathbf{g}_{\mathsf{Q}},\mathrm{Pet}}^{2}\cdot\prod_{v\in\mathscr{S}_{\mathbf{f}}\cup\mathscr{S}_{\mathbf{g}}}\zeta_{v}C_{\sigma_{\mathbf{f}_{\mathsf{P}},v},\boldsymbol{\psi}_{v}}C_{\sigma_{\mathbf{f}_{\mathsf{P}},v}^{\vee},\boldsymbol{\psi}_{v}^{-1}}\mathcal{B}_{\pi_{\mathbf{g}_{\mathsf{Q}},v}}^{\mathrm{ess}}\mathcal{B}_{\sigma_{\mathbf{f}_{\mathsf{P}},v}}^{\mathrm{ess}}\\ \times\prod_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}\zeta_{v}\Delta_{v,\beta_{v}}^{2}\cdot\mathfrak{G}(\kappa_{v},\breve{\underline{\mu}_{v,\mathsf{P}}},\breve{\underline{\lambda}_{v,\mathsf{Q}}})\mathfrak{G}(\kappa_{v},\breve{\check{\underline{\mu}_{v,\mathsf{P}}}},\breve{\check{\underline{\lambda}_{v,\mathsf{Q}}}})\mathcal{C}_{\sigma_{\mathbf{f}_{\mathsf{P}}}}^{\operatorname{ord}}\mathcal{C}_{\pi_{\mathbf{g}_{\mathsf{Q}}}}^{\operatorname{ord}}\cdot\mathscr{L}^{\mathscr{V}_{\mathcal{F}}^{(p)}}(\sigma_{\mathbf{f}_{\mathsf{P}}}\times\pi_{\mathbf{g}_{\mathsf{Q}}}).

where $\mathscr{S}_{\mathbf{f}}$ and $\mathscr{S}_{\mathbf{g}}$ are the set of places of $\mathcal{F}$ dividing the “tame level” of Hida families $\mathbf{f}$ and $\mathbf{g}$ respectively.

We note that there are similar recent works on such $p$ -adic $L$ -functions in various generalizations, for instance [Liu23, HY23, Dim24].

We present these theorems in Part II of this article.

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In Section 7, we recall the geometric backgrounds of modular forms, $p$ -adic modular forms and their $p$ -adic families, over unitary groups.
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In Section 8, we recall the construction of Siegel Eisenstein families and Klingen Eisenstein families over unitary groups.
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In Section 9, we $p$ -adically interpolate the GGP period integral, where Theorem C and Theorem D is proved.

1.3. Future works

Firstly, some of the assumptions in this article is technical and is expected to be removed or weakened in the future.

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For the computation of local Ichino-Ikeda integrals, Assumption 3.9 and 3.11 are expected to be removed.
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Though it is believed that the Klingen Eisenstein family over unitary groups could be constructed in full generality, it is still missing in the literature. The Klingen Eisenstein family we are using is the one in [Wan15]. That is why Assumption 6.1 and 6.2 are put in this article.

Secondly, besides the construction of $p$ -adic $L$ -functions $\mathbf{L}_{\mathbf{f},\mathbf{g}}$ in Theorem D, recall that our motivation is the $p$ -primitive property of Klingen Eisenstein series and their $p$ -adic families over unitary groups. This article acts as the first step of this project. From this perspective, we could consider next steps of this project.

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The main result in this article reduce the $p$ -primitivity problem of Klingen Eisenstein families over $\mathrm{U}(m+1,n+1)$ to certain “modulo $p$ nonvanishing” property of the special $L$ -values of Rankin-Selberg $L$ -functions and standard $L$ -functions. Nevertheless, this latter problem is not an easy one, and definitely needs new ideas. In future works, we shall first start with some lower rank cases, where more tools might be available.
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Granting the $p$ -primitivity of Klingen Eisenstein families, another key input for the machinery of Eisenstein congruences is the noncuspidal Hida theory of unitary group with general signatures. This will provide a “fundamental exact sequence of Eisenstein congruences”, which is essential in the arguments. Though it is expected to hold and shall follows from the same technique with [LR20], as far as we are aware of, it is still missing in the literature. We remark that some special cases are known. In [SU14, Chapter 6], the case $\mathrm{U}(m,m)$ is considered and in [Hsi14, Chapter 4], the case $\mathrm{U}(m,1)$ is considered, both for all positive integer $m$ .

So the project is indeed a challenging one.

1.4. Notations and conventions

In this subsection, we introduce basic notations of this article.

1.4.1. Fields and places

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Let $\mathbb{Q},\mathbb{Z}$ (resp. $\mathbb{Q}_{p},\mathbb{Z}_{p}$ ) be the field of (resp. $p$ -adic) rational numbers and the ring of (resp. $p$ -adic) integers respectively.
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Let $L/\mathbb{Q}_{p}$ be a finite extension, we write $\mathcal{O}_{L}$ be its ring of integers, and $\mathcal{O}_{L}^{{\mathrm{ur}}}$ be the completion of the maximal unramified extension of $\mathcal{O}_{L}$ .
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Let $\mathcal{M}$ be a number field, that is, a finite extension of $\mathbb{Q}$ . We put $\mathbb{A}_{\mathcal{M}},\mathbb{A}_{\mathcal{M},\mathrm{f}},\mathbb{A}_{\mathcal{M},\mathrm{f}}^{p}$ be the ring of adèles of $\mathcal{M}$ , the ring of finite adèles and the ring of finite adèles with $p$ -coordinates removed.
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Let $\mathcal{K}/\mathcal{F}$ be a CM extension, that is, $\mathcal{F}$ is a totally real number field of degree $d$ over $\mathbb{Q}$ , and $\mathcal{K}$ is an imaginary quadratic extension of $\mathcal{F}$ . We put $\mathbb{K}$ (resp. $\mathbb{A}$ ) the ring of adeles of $\mathcal{K}$ (resp. $\mathcal{F}$ ).
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Let $\mathcal{K}_{\infty}/\mathcal{K}$ be the composition of all $\mathbb{Z}_{p}$ -extensions of $\mathcal{K}$ , with Galois group $\Gamma_{\mathcal{K}}$ .
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Let $\mathrm{c}\in\operatorname{Gal}(\mathcal{K}/\mathcal{F})$ be the unique nontrivial automorphism of $\mathcal{K}$ , called the complex conjugation.
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We fix an element $\mathsf{i}\in\mathcal{K}$ such that $\mathrm{c}(\mathsf{i})=-\mathsf{i}$ and $\operatorname{Nm}^{\mathcal{K}}_{\mathcal{F}}(\mathsf{i})=1$ (where $\operatorname{Nm}^{\mathcal{K}}_{\mathcal{F}}$ is the norm map from $\mathcal{K}^{\times}$ to $\mathcal{F}^{\times}$ ), which equivalently, means that $\mathsf{i}\cdot\mathrm{c}(\mathsf{i})=1$ .

Let $\mathcal{M}$ be a number field, we denote $\mathscr{V}_{\mathcal{M}}$ or $\mathscr{S}_{\mathcal{M}}$ as the set of places of $\mathcal{M}$ . We denote by

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$\mathscr{V}_{\mathcal{F}}^{\square}$ (resp. $\mathscr{V}_{\mathcal{K}}^{\square}$ ) the set of places of $\mathcal{F}$ (resp. $\mathcal{K}$ ) above a finite set $\square$ of places of $\mathbb{Q}$ (resp. $\mathcal{F}$ ) ³³3When $\square=\{w\}$ is a singleton, we simply write $\mathscr{V}_{\mathcal{F}}^{(w)}=\mathscr{V}_{\mathcal{F}}^{\{w\}}$ .,
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$\mathscr{V}_{\mathcal{F}}^{\infty}$ (resp. $\mathscr{V}_{\mathcal{F}}^{0}$ ) the set of archimedean (resp. nonarchimedean) places of $\mathcal{F}$ .
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$\mathscr{V}_{\mathcal{F}}^{\mathrm{spl}}$ (resp. $\mathscr{V}_{\mathcal{F}}^{\mathrm{ns}}$ ) the subset of $\mathscr{V}_{\mathcal{F}}^{0}$ of those that are split (resp. nonsplit) in $\mathcal{K}$ respectively.
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$\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}$ be the set of finite places of $\mathcal{F}$ such that $\sigma_{v}$ , $\pi_{v}$ and the local extension $\mathcal{K}_{v}/\mathcal{F}_{v}$ are unramified, and $\mathscr{V}_{\mathcal{F}}^{\mathrm{bad}}$ be the complement of $\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}$ in $\mathscr{V}_{\mathcal{F}}$ .
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$\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$ be the set of finite places of $\mathcal{F}$ such that either $\sigma_{v}$ or $\pi_{v}$ or both is ramified.

We denote $\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ the set of finite places of $\mathcal{F}$ such that $\sigma_{v}$ , $\pi_{v}$ and $\chi_{v}$ , and the local extension $\mathcal{K}_{v}/\mathcal{F}_{v}$ are unramified, and $\mathscr{S}_{\mathcal{F}}^{\mathrm{bad}}$ the complement of $\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ in $\mathscr{S}_{\mathcal{F}}$ . By definition, $\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}\subseteq\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}$ . For any $?\not\in\{{\mathrm{ur}},\mathrm{bad}\}$ , we set $\mathscr{S}_{\mathcal{F}}^{?}:=\mathscr{V}_{\mathcal{F}}^{?}$ . There is also a third partition of $\mathscr{V}_{\mathcal{F}}$ , introduced in Section 6.

Let $L(\cdots)$ be a certain $L$ -function with Euler product expression $L(\cdots)=\prod_{v\in\mathscr{V}_{\mathcal{F}}}L_{v}(\cdots)$ . Then for any subset $\mathscr{V}$ of $\mathscr{V}_{\mathcal{F}}$ , we write

L_{\mathscr{V}}(\cdots):=\prod_{v\in\mathscr{V}}L_{v}(\cdots)\quad\text{ and }\quad L^{\mathscr{V}}(\cdots):=\prod_{v\in\mathscr{V}_{\mathcal{F}}\smallsetminus\mathscr{V}}L_{v}(\cdots).

1.4.2. Characters and automorphic representations

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By saying $\chi$ a Hecke character of a number field $\mathcal{M}$ , we mean automatically an algebraic Hecke character $\chi:\mathcal{M}^{\times}\backslash\mathbb{A}^{\times}_{\mathcal{M}}\rightarrow\mathbb{C}^{\times}$ .
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$\epsilon_{\mathcal{K}/\mathcal{F}}:\mathbb{A}^{\times}\rightarrow\mathbb{C}^{\times}$ is the quadratic character attached to the quadratic extension $\mathcal{K}/\mathcal{F}$ . It decomposes into local quadratic characters $\epsilon_{\mathcal{K}_{w}/\mathcal{F}_{v}}:\mathcal{F}_{v}^{\times}\rightarrow\mathbb{C}^{\times}$ .
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For any Hecke character $\chi$ of $\mathcal{K}$ , we write $\chi^{\mathcal{F}}$ for its restriction to $\mathbb{A}^{\times}$ . We write $\chi^{\mathsf{c}}$ to be the conjugation of $\chi$ on the source, defined as $x\mapsto\chi(x^{\mathsf{c}})$ . We write $\overline{\chi}$ as the conjugation of $\chi$ on the target, defined as $x\mapsto(\chi(x))^{\mathsf{c}}$ .

By a unitary automorphic representation of $\mathrm{U}(m,n)$ over $\mathcal{F}$ , we realize it as a $L^{2}$ -subspace of the space of automorphic forms over $\mathrm{U}(m,n)$ , with the inner product given by the $L^{2}$ inner product. We denote the space of automorphic forms over $\mathrm{U}(m,n)$ (resp. cuspidal automorphic forms) by $\mathcal{A}(\mathrm{U}(m,n)(\mathcal{F})\backslash\mathrm{U}(m,n)(\mathbb{A}))$ (resp. $\mathcal{A}_{\mathrm{cusp}}(\mathrm{U}(m,n)(\mathcal{F})\backslash\mathrm{U}(m,n)(\mathbb{A}))$ ).

1.4.3. General linear groups and operation on matrices

Let $m,n$ be positive integers. For any square matrix $A$ of size $n\times n$ , we put (whenever the operations makes sense)

•

$A^{\mathrm{t}}$ as the transpose of $A$ , and $A^{-\mathrm{t}}$ as the inverse of $A^{\mathrm{t}}$ ,
•

$\overline{A}$ as the conjugation of $A$ , whenever the conjugation makes sense, and
•

$A^{\star}:=\overline{A}^{\mathrm{t}}$ and $A^{-\star}:=(A^{\star})^{-1}$ .
•

We write $\mathrm{Herm}_{n}$ be the set of matrices in $M_{n}$ such that $A^{\star}=A$ .

We write $\mathbf{1}_{n}$ as the identity square matrix of size $n\times n$ . we put $M_{m,n}$ to be the set of matrices with $m$ rows and $n$ columns. We denote by

•

$\mathrm{GL}_{n}$ the general linear group of matrices of size $n\times n$ .
•

$T_{n}$ the diagonal torus of $\mathrm{GL}_{n}$ ,
•

$B_{n}$ the upper triangular matrices in $\mathrm{GL}_{n}$ , that is, the Borel subgroup of $\mathrm{GL}_{n}$ , $B_{n}^{-}$ the lower triangular matrices in $\mathrm{GL}_{n}$ ,
•

$U_{n}$ the strictly upper triangular matrices in $\mathrm{GL}_{n}$ , that is, upper triangular matrices with $1$ ’s on the diagonal. It is the unipotent subgroup of $B_{n}$ and $B_{n}=T_{n}U_{n}$ is the Levi decomposition of $B_{n}$ . In the same way we have $B_{n}^{-}=T_{n}U_{n}^{-}$ .

•

More generally, let $n=n_{1}+\cdots+n_{r}$ be a partition of $n$ . We put $P[n_{1},\ldots,n_{r}]$ as the parabolic subgroup of $\mathrm{GL}_{n}$ with respect to this partition, that is,

P[n_{1},\ldots,n_{r}]=\begin{bmatrix}\mathrm{GL}_{n_{1}}&\ast&\ast&\ast\\ &\mathrm{GL}_{n_{2}}&\ast&\ast\\ &&\ddots&\vdots\\ &&&\mathrm{GL}_{n_{r}}\end{bmatrix}.

It has a Levi decomposition $P[n_{1},\ldots,n_{r}]=M[n_{1},\ldots,n_{r}]U[n_{1},\ldots,n_{r}]$ , where

M[n_{1},\ldots,n_{r}]=\operatorname{diag}[\mathrm{GL}_{n_{1}},\mathrm{GL}_{n_{2}},\ldots,\mathrm{GL}_{n_{r}}]

is its Levi component and

U[n_{1},\ldots,n_{r}]=\begin{bmatrix}\mathbf{1}_{n_{1}}&\ast&\ast&\ast\\ &\mathbf{1}_{n_{2}}&\ast&\ast\\ &&\ddots&\vdots\\ &&&\mathbf{1}_{n_{r}}\end{bmatrix}

is its unipotent radical.

Given a square matrix of size $n\times n$ , we can group its entries into blocks by a partition of $n$ . We denote such a partition as $n=[n_{1}\mid n_{2}\mid\cdots\mid n_{r}]$ .

1.4.4. Measures and pairings

We fix a Haar measure $\,\mathrm{d}g$ on the adèle group of a reductive group $G$ over $\mathcal{F}$ . We take $\,\mathrm{d}g$ to be the Tamagawa measure for definiteness. We write $\,\mathrm{d}g=\prod_{v\in\mathscr{V}_{\mathcal{F}}}\,\mathrm{d}g_{v}$ , with $\,\mathrm{d}g_{v}$ a Haar measure on $G(\mathcal{F}_{v})$ , the $\mathcal{F}_{v}$ -points of $G$ , under the following hypothesis (following [EHLS20, Hypothesis 1.4.4]):

•

At all finite places $v$ of $\mathcal{F}$ at which the group $G$ us unramified, $\,\mathrm{d}g_{v}$ is the measure that gives volume $1$ to a hyperspecial maximal compact subgroup.
•

At all finite places $v$ at which $G(\mathcal{F}_{v})$ is isomorphic to $\prod_{i}\mathrm{GL}_{n_{i}}(\mathcal{F}_{i,w_{i}})$ , where $\mathcal{F}_{i,w_{i}}$ is a finite extension of $\mathcal{F}_{v}$ with integer $\mathcal{O}_{i}$ , the measure $\,\mathrm{d}g_{v}$ is the measure that gives volume $1$ to the group $\prod_{i}\mathrm{GL}_{n_{i}}(\mathcal{O}_{i})$ .
•

At all finite places $v$ of $\mathcal{F}$ , the values of $\,\mathrm{d}g_{v}$ on open compact subgroups are rational numbers.
•

At archimedean places $v$ of $\mathcal{F}$ , we choose measures such that $\prod_{v\in\mathscr{V}_{\mathcal{F}}^{\infty}}\,\mathrm{d}g_{v}$ is Tamagawa measure.

For the reductive group $G$ over $\mathcal{F}$ , we let $[G]$ denote the quotient $G(F)\backslash G(\mathbb{A})$ . Endowing $G(\mathbb{A})$ with the Tamagawa measure, $[G]$ is endowed with the quotient measure by the counting measure on $G(\mathcal{F})$ . Give two automorphic forms $\Phi,\Phi^{\prime}$ over $G$ , we define their Petersson inner product by

\left\langle{\Phi,\Phi^{\prime}}\right\rangle_{\mathrm{Pet}}:=\int_{[G]}\Phi(g)\Phi^{\prime}(g)\,\mathrm{d}g.

Let $\pi$ be an irreducible unitary cuspidal automorphic representation of $G$ . We denote $\pi^{\vee}$ be the contragredient representation of $\pi$ . Then the Petersson inner product is a canonically defined pairing $\pi\times\pi^{\vee}\rightarrow\mathbb{C}$ . We have factorizations

\mathrm{fac}_{\pi}:\pi\mathrel{\leavevmode\hbox to16.64pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 2.07999pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{{ {\pgfsys@beginscope \pgfsys@setdash{}{0.0pt}\pgfsys@roundcap\pgfsys@roundjoin{} {}{}{} {}{}{} \pgfsys@moveto{-2.07999pt}{2.39998pt}\pgfsys@curveto{-1.69998pt}{0.95998pt}{-0.85318pt}{0.28pt}{0.0pt}{0.0pt}\pgfsys@curveto{-0.85318pt}{-0.28pt}{-1.69998pt}{-0.95998pt}{-2.07999pt}{-2.39998pt}\pgfsys@stroke\pgfsys@endscope}} }{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\otimes^{\prime}\pi_{v},\quad\mathrm{fac}_{\pi^{\vee}}:\pi^{\vee}\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\otimes^{\prime}\pi_{v}^{\vee},

where $\pi_{v}$ is an irreducible representation of $G(\mathcal{F}_{v})$ . We have non-degenerate canonical $G(\mathcal{F}_{v})$ -pairings

\left\langle{-,-}\right\rangle_{v}:\pi_{v}\times\pi_{v}^{\vee}\rightarrow\mathbb{C}

for all $v$ . We renormalize the local pairings such that for $\mathrm{fac}_{\pi}(\Phi)=\otimes_{v}\Phi_{v}$ and $\mathrm{fac}_{\pi^{\vee}}(\Phi^{\vee})=\otimes_{v}\Phi_{v}^{\vee}$ , we have

\left\langle{\Phi,\Phi^{\vee}}\right\rangle_{\mathrm{Pet}}=\prod_{v\in\mathscr{V}_{\mathcal{F}}}\left\langle{\Phi_{v},\Phi_{v}^{\vee}}\right\rangle_{v}.

Acknowledgements

This article will be a crucial part of the the author’s doctoral thesis under the supervision of Professor Xin Wan. The author would like to express the most sincere gratitude to him for suggesting this interesting project and his patient and insightful guidance along the way. He also thanks Yangyu Fan, Zhibin Geng, Ming-Lun Hsieh, Haijun Jia, Shilin Lai, Haidong Li, Wen-Wei Li, Yifeng Liu, Loren Spice, Ye Tian and Luochen Zhao as well as many others for many insightful conversations. The author is partially supported by the grant No. E01O010201 of Professor Xin Wan.

Part I Automorphic Computations

In this part, we shall define and compute the Gan-Gross-Prasad period integral of Klingen Eisenstein series with a cuspidal automorphic form.

2. Backgrounds

In this section, we introduce backgrounds on automorphic computations, including unitary groups and Eisenstein series over them. We also recall backgrounds on doubling methods.

2.1. Unitary groups

Let $\mathcal{K}$ be a quadratic imaginary extension of a totally real number field $\mathcal{F}$ . Let $V$ be an $N$ -dimensional vector space $V$ over $\mathcal{K}$ . Let $\phi:V\times V\rightarrow\mathcal{K}$ be a non-degenerate skew-Hermitian form on $V$ . Note that we can linearly extend $\phi$ to any $\mathcal{F}$ -algebra $R$ and the $R$ -module $V\otimes_{\mathcal{F}}R$ .

Definition 2.1.

The general unitary group attached to $(V,\phi)$ is the algebraic group ${\mathrm{GU}}(V,\phi)$ over $\mathcal{F}$ , whose $R$ -points, for each $\mathcal{F}$ -algebra $R$ , are given by

{\mathrm{GU}}(V,\phi)(R):=\{g\in\operatorname{Aut}_{\mathcal{K}\otimes_{\mathcal{F}}R}(V\otimes_{\mathcal{F}}R):\phi(gv,gw)=\nu(g)\phi(v,w),\,v,w\in V\otimes_{\mathcal{F}}R,\nu(g)\in R\}.

The unitary group attached to $(V,\phi)$ is defined as the algebraic group $\mathrm{U}(V,\phi)$ over $\mathcal{F}$ , whose $R$ -points, for each $\mathcal{F}$ -algebra $R$ , are given by

\mathrm{U}(V,\phi)(R):=\{g\in\operatorname{Aut}_{\mathcal{K}\otimes_{\mathcal{F}}R}(V\otimes_{\mathcal{F}}R):\phi(gv,gw)=\phi(v,w),\,v,w\in V\otimes_{\mathcal{F}}R\}.

In Part I, we only deal with unitary groups, not general unitary groups, see Remark 8.1.

Let $\mathtt{B}$ be an ordered $\mathcal{K}$ -basis of $V$ , then $\phi$ can be expressed as a matrix $[\phi]_{\mathtt{B}}$ under this basis and the unitary group $\mathrm{U}(V,\phi)$ can be identified with the matrix group

\mathrm{U}(V,\phi,\mathtt{B}):=\{g\in\mathrm{GL}_{N}(\mathcal{K}\otimes_{\mathcal{F}}R):g[\phi]_{\mathtt{B}}g^{\star}=[\phi]_{\mathtt{B}}\}.

Let $n$ be the maximum dimension of totally $\phi$ -isotropic subspaces of $V$ and write $m=N-n$ . By [Shi97, Lemma 1.5, Lemma 1.6], we can find an ordered $\mathcal{K}$ -basis of $V$ , that is,

\mathtt{Witt}_{m,n}:\mathtt{y}^{1},\ldots,\mathtt{y}^{n},\mathtt{w}^{1},\ldots,\mathtt{w}^{m-n},\mathtt{x}^{1},\ldots,\mathtt{x}^{n}

(2.1)

called Witt basis, under which $\phi$ has a matrix representation as

J_{m,n}:=[\phi]_{\mathtt{Witt}_{m,n}}:=\begin{bmatrix}&&\mathbf{1}_{n}\\ &\vartheta&\\ -\mathbf{1}_{n}&&\end{bmatrix}.

Throughout this article, we fix such a basis of $V$ (and hence the matrix $\vartheta$ ) and denote $\mathrm{U}(V,\phi,\mathtt{Witt}_{m,n})$ as $\mathrm{U}(m,n)$ for simplicity, though it surely depends on $\vartheta$ , not only on $m$ and $n$ .

2.1.1. Unitary groups at local places

Let $v$ be any place of $\mathcal{F}$ .

•

Let $v$ be an archimedean place of $\mathcal{F}$ , which gives a real embedding $\iota_{v}:\mathcal{F}\hookrightarrow\mathbb{R}$ . We further require $\vartheta$ is a $(m-n)\times(m-n)$ diagonal matrix such that

$\iota_{v}(\mathsf{i}^{-1}\vartheta)\text{ is positive definite for all }v\in\mathscr{V}_{\mathcal{F}}^{\infty}.$ (sgn)

This implies that $\mathrm{U}(m,n)(\mathcal{F}_{v})$ is isomorphic to the real Lie group

$\mathrm{U}_{m,n}:=\left\{g\in\mathrm{GL}_{m+n}(\mathbb{C}):g^{\star}g=\begin{bmatrix}\mathbf{1}_{m}&\\ &-\mathbf{1}_{n}\end{bmatrix}\right\}$

for any archimedean place $v$ of $\mathcal{F}$ .

•

Let $v$ be a finite place of $F$ such that $v$ splits as $v=w\overline{w}$ in $\mathcal{K}$ . Then $\mathcal{K}\otimes\mathcal{F}_{v}\cong\mathcal{F}_{v}\times\mathcal{F}_{v}$ induces an isomorphism $\varrho_{v,N}:\mathrm{GL}_{N}(\mathcal{K}\otimes\mathcal{F}_{v})\cong\mathrm{GL}_{N}(\mathcal{F}_{v})\times\mathrm{GL}_{N}(\mathcal{F}_{v})$ . Moreover, if $g$ maps to $(g_{1},g_{2})$ via $\varrho_{v,N}$ , then $\overline{g}$ is sent to $(g_{2},g_{1})$ . Hence

\mathrm{U}(m,n)(\mathcal{F}_{v})\cong\{(g_{1},g_{2})\in\mathrm{GL}_{N}(\mathcal{F}_{v})\times\mathrm{GL}_{N}(\mathcal{F}_{v}):g_{2}=J_{m,n}^{\mathrm{t}}g_{1}^{-1}J_{m,n}^{-1}\}.

which is isomorphic to $\mathrm{GL}_{N}(\mathcal{F}_{v})$ by projecting to the first factor. This involves a choice of place $w$ of $\mathcal{K}$ above $v$ . We denote the isomorphism by $\varrho_{w,m,n}$ .

•

Let $v$ be a finite nonsplit place of $\mathcal{F}$ . Naturally, the $\mathcal{K}$ -vector space extends to a $\mathcal{K}_{v}$ -vector space $V_{v}$ over $\mathcal{K}_{v}$ and so does the skew-Hermitian form extended to $\phi_{v}$ . Then by [Shi97, page 76], there is a local basis $\mathtt{Witt}_{v}$ such that

$[\phi_{v}]_{\mathtt{Witt}_{v}}=\begin{bmatrix}&&\mathbf{1}_{n_{v}}\\ &\vartheta_{v}&\\ -\mathbf{1}_{n_{v}}\end{bmatrix}$

where $\vartheta_{v}$ is an anisotropic square matrix of size $r_{v}\times r_{v}$ , with $r_{v}=1$ when $N$ is odd and $r_{v}=0$ or $2$ when $N$ is even. The group $\mathrm{U}(m,n)(\mathcal{F}_{v})$ is quasi-split if and only if $r_{v}\leq 1$ . We assume

$r_{v}\leq 1\text{ for all finite nonsplit place }v\text{ of }\mathcal{F},$ (QS)

so that we will not bother dealing with the places where $\mathrm{U}(m,n)$ is not quasi-split.

2.1.2. Hermitian basis

It is sometimes easier to use another basis. One checks immediately that $\mathsf{i}\phi$ is an Hermitian form on $V$ , so there exists an ordered $\mathcal{K}$ -basis

\mathtt{Herm}_{m,n}:\mathtt{a}^{1},\ldots,\mathtt{a}^{m},\mathtt{b}^{1},\ldots,\mathtt{b}^{n}

of $V$ such that $\mathsf{i}\phi$ has a matrix representation as $[\mathsf{i}\phi]_{\mathtt{Herm}_{m,n}}:=\begin{bmatrix}\mathbf{1}_{m}&0\\ 0&-\mathbf{1}_{n}\end{bmatrix}$ . Then under this basis, our original skew-Hermitian form can be written as $[\phi]_{\mathtt{Herm}_{m,n}}:=\begin{bmatrix}-\mathsf{i}\cdot\mathbf{1}_{m}&0\\ 0&\mathsf{i}\cdot\mathbf{1}_{n}\end{bmatrix}$ . In fact, we can make explicit the transition between the Hermitian basis and Witt basis. Let $\vartheta_{0}$ be such that $\vartheta=\vartheta_{0}\vartheta_{0}^{\star}$ . Then one checks immediately that

[\mathtt{y}^{1},\ldots,\mathtt{y}^{n},\mathtt{w}^{1},\ldots,\mathtt{w}^{m-n},\mathtt{x}^{1},\ldots,\mathtt{x}^{n}]=[\mathtt{a}^{1},\ldots,\mathtt{a}^{m},\mathtt{b}^{1},\ldots,\mathtt{b}^{n}]\begin{bmatrix}\frac{1}{2}\cdot\mathbf{1}_{n}&&-\mathsf{i}\cdot\mathbf{1}_{n}\\ &\vartheta_{0}&\\ \frac{1}{2}\cdot\mathbf{1}_{n}&&\mathsf{i}\cdot\mathbf{1}_{n}\end{bmatrix}.

Denote the matrix on the right hand side by $\varrho_{m,n,\vartheta_{0}}$ , then it gives an isomorphism of unitary groups

\varrho_{m,n,\vartheta_{0}}:\mathrm{U}(V,\phi,\mathtt{Witt}_{m,n})\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathrm{U}(V,\phi,\mathtt{Herm}_{m,n}),\quad h\mapsto\varrho_{m,n,\vartheta_{0}}h\varrho_{m,n,\vartheta_{0}}^{-1}.

When $m=n$ , the matrix $\vartheta_{0}$ disappears and we omit it from the notation.

2.1.3. The $\sharp$ -space

Starting from the vector space $V$ under the basis $\mathtt{Herm}_{m,n}$ , we consider the $\mathcal{K}$ -vector space

V^{\sharp}:=\mathcal{K}\mathtt{e}\oplus\mathcal{K}\mathtt{a}^{1}\oplus\cdots\mathcal{K}\mathtt{a}^{m}\oplus\mathcal{K}\mathtt{b}^{1}\oplus\cdots\mathcal{K}\mathtt{b}^{n},

with the ordered $\mathcal{K}$ -basis $\mathtt{Herm}^{\sharp}$ listed as above. The space $V$ is equipped with a skew-Hermitian form $\phi^{\sharp}$ such that $[\mathsf{i}\phi^{\sharp}]_{\mathtt{Herm}^{\sharp}}=\operatorname{diag}[1,\mathbf{1}_{m},-\mathbf{1}_{n}]$ . Accordingly, we have the embedding of unitary groups

\jmath^{\sharp}:\mathrm{U}(V,\phi,\mathtt{Herm}_{m,n})\hookrightarrow\mathrm{U}(V^{\sharp},\phi^{\sharp},\mathtt{Herm}^{\sharp}),\quad h\mapsto\operatorname{diag}[1,h].

We go back to the Witt basis $\mathtt{Witt}^{\sharp}$ by the transition matrix $\varrho_{m+1,n,\vartheta_{0}^{\sharp}}$ where $\vartheta_{0}^{\sharp}:=\operatorname{diag}[1,\vartheta_{0}]$ . The corresponding embedding $\jmath^{\sharp}$ with respect to the Witt basis $\mathtt{Witt}^{\sharp}$ is defined such that the diagram

commutes.

2.1.4. The $\heartsuit$ -space

Next we consider adding another line to $V^{\sharp}$ . Starting from the $\mathcal{K}$ -ordered basis $\mathtt{Herm}^{\sharp}$ , we consider the $\mathcal{K}$ -vector space

V^{\heartsuit}:=\mathcal{K}\mathtt{e}\oplus\mathcal{K}\mathtt{a}^{1}\oplus\cdots\mathcal{K}\mathtt{a}^{m}\oplus\mathcal{K}\mathtt{b}^{1}\oplus\cdots\mathcal{K}\mathtt{b}^{n}\oplus\mathcal{K}\mathtt{f}.

with the ordered $\mathcal{K}$ -basis $\mathtt{Herm}^{\heartsuit}$ listed as above. The space $V^{\heartsuit}$ is equipped with a skew-Hermitian form $\phi^{\heartsuit}$ such that $[\mathsf{i}\phi^{\heartsuit}]_{\mathtt{Herm}^{\heartsuit}}=\operatorname{diag}[1,\mathbf{1}_{m},-\mathbf{1}_{n},-1]$ . Accordingly, we have the embedding of unitary groups

\jmath^{\flat}:\mathrm{U}(V^{\sharp},\phi^{\sharp},\mathtt{Herm}^{\sharp})\hookrightarrow\mathrm{U}(V^{\heartsuit},\phi^{\heartsuit},\mathrm{Herm}^{\heartsuit}),\quad g\mapsto\operatorname{diag}[g,1].

We go back to the Witt basis by the transition matrix $\varrho_{m+1,n+1,\vartheta_{0}}$ . Then corresponding $\jmath^{\flat}$ with respect to the Witt basis $\mathtt{Witt}^{\heartsuit}$ is defined such that the diagram

commutes.

To conclude, we have successive embeddings of unitary groups

\mathrm{U}(m,n)\mathrel{\leavevmode\hbox to10.36pt{\vbox to17.5pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.29448pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.49448pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.84723pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{4.89409pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\jmath^{\sharp}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathrm{U}(m+1,n)\mathrel{\leavevmode\hbox to10.36pt{\vbox to17.61pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{9.29448pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{9.49448pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{4.84723pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{4.89409pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\jmath^{\flat}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathrm{U}(m+1,n+1).

2.1.5. Doubling space

Let $V^{\blacklozenge}$ be the direct sum $V^{\heartsuit}\oplus V$ with ordered $\mathcal{K}$ -basis

\mathtt{Herm}^{\blacklozenge,\oplus}:\mathtt{e},\mathtt{a}^{1},\ldots,\mathtt{a}^{m},\mathtt{b}^{1},\ldots,\mathtt{b}^{n},\mathtt{f},\bar{\mathtt{a}}^{1},\ldots,\bar{\mathtt{a}}^{m},\bar{\mathtt{b}}^{1},\ldots,\bar{\mathtt{b}}^{n}

by directly putting the Hermitian basis $\mathtt{Herm}^{\heartsuit}$ of $V^{\heartsuit}$ and the Hermitian basis $\mathtt{Herm}_{m,n}$ of $V$ together (where we add “bars” at the basis element of $V$ to distinguish). We equip it with the skew-Hermitian form $\phi^{\blacklozenge}$ such that

[\mathsf{i}\phi^{\blacklozenge}]_{\mathtt{Herm}^{\blacklozenge,\oplus}}:=\operatorname{diag}[[\mathsf{i}\phi^{\heartsuit}]_{\mathtt{Herm}^{\heartsuit}},-[\mathsf{i}\phi]_{\mathtt{Herm}}]=\operatorname{diag}[1,\mathbf{1}_{m},-\mathbf{1}_{n},-1,-\mathbf{1}_{m},\mathbf{1}_{n}].

Accordingly, we have the embedding of unitary groups

\iota^{\heartsuit}:\mathrm{U}(V^{\heartsuit},\phi^{\heartsuit},\mathtt{Herm}^{\heartsuit})\times\mathrm{U}(V,-\phi,\mathtt{Herm}_{m,n})\hookrightarrow\mathrm{U}(V^{\blacklozenge},\phi^{\blacklozenge},\mathtt{Herm}^{\blacklozenge,\oplus}),\quad(g,h)\mapsto\operatorname{diag}[g,h].

Moreover, we have another embedding of unitary groups ⁴⁴4The two embeddings come from two viewpoints on the “direct sum” $\phi^{\blacklozenge}$ : $\operatorname{diag}[\operatorname{diag}[1,\mathbf{1}_{m},-\mathbf{1}_{n},-1],\operatorname{diag}[-\mathbf{1}_{m},\mathbf{1}_{n}]]=\operatorname{diag}[\operatorname{diag}[1,\mathbf{1}_{m},-\mathbf{1}_{n}],\operatorname{diag}[-1,-\mathbf{1}_{m},\mathbf{1}_{n}]].$ The left hand side partition gives the embedding $\iota^{\heartsuit}$ and the right hand side one gives the embedding $\iota^{\diamondsuit}$ .

\iota^{\diamondsuit}:\mathrm{U}(V^{\sharp},\phi^{\sharp},\mathtt{Herm}^{\sharp})\times\mathrm{U}(V^{\sharp},-\phi^{\sharp},\mathtt{Herm}^{\sharp})\hookrightarrow\mathrm{U}(V^{\blacklozenge},\phi^{\blacklozenge},\mathtt{Herm}^{\blacklozenge,\oplus}),\quad(g_{1},g_{2})\mapsto\operatorname{diag}[g_{1},g_{2}].

By the definition of above embedding of unitary groups, we see immediately that

\imath^{\heartsuit}(\jmath^{\flat}(g),h)=\imath^{\diamondsuit}(g,\jmath^{\sharp}(h))

(2.2)

for any $h\in\mathrm{U}(V,\phi,\mathtt{Herm}_{m,n})$ and $g\in\mathrm{U}(V^{\sharp},\phi^{\sharp},\mathtt{Herm}^{\sharp})$ .

We rearrange the order of the basis to make the corresponding metric in the standard Hermitian form, by

\mathtt{Herm}^{\blacklozenge}:\mathtt{e},\mathtt{a}^{1},\ldots,\mathtt{a}^{m},\bar{\mathtt{b}}^{1},\ldots,\bar{\mathtt{b}}^{n},\mathtt{f},\bar{\mathtt{a}}^{1},\ldots,\bar{\mathtt{a}}^{m},\mathtt{b}^{1},\ldots,\mathtt{b}^{n}.

Then under this basis,

[\phi^{\blacklozenge}]_{\mathtt{Herm}^{\blacklozenge}}:=\varsigma^{\blacklozenge}\operatorname{diag}[\psi,-\phi](\varsigma^{\blacklozenge})^{-1}=\operatorname{diag}[1,\mathbf{1}_{m},\mathbf{1}_{n},-1,-\mathbf{1}_{m},-\mathbf{1}_{n}],

where

\varsigma^{\blacklozenge}=\begin{bmatrix}\mathbf{1}_{1+m}&&&\\ &&&\mathbf{1}_{n}\\ &&\mathbf{1}_{1+m}&\\ &\mathbf{1}_{n}&&\end{bmatrix}.

This gives an isomorphism of unitary groups

\varsigma^{\blacklozenge}:\mathrm{U}(V^{\blacklozenge},\phi^{\blacklozenge},\mathtt{Herm}^{\blacklozenge})\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathrm{U}(V^{\blacklozenge},\phi^{\blacklozenge},\mathtt{Herm}^{\blacklozenge,\oplus}),\quad g\mapsto\varsigma^{\blacklozenge}g(\varsigma^{\blacklozenge})^{-1}.

And we further go back to the Witt basis by $\varrho_{m+n+1,m+n+1}$ . The embeddings $\iota^{\heartsuit}$ and $\iota^{\diamondsuit}$ are defined to make the diagrams

and

commute. Then it follows that Equation (2.2) holds for the embeddings under Witt basis. To sum up, we have two embeddings of unitary groups

	$\displaystyle\iota^{\heartsuit}$	$\displaystyle:\mathrm{U}(m+1,n+1)\times\mathrm{U}(m,n)\rightarrow\mathrm{U}(m+n+1,m+n+1),$
	$\displaystyle\iota^{\diamondsuit}$	$\displaystyle:\mathrm{U}(m+1,n)\times\mathrm{U}(m+1,n)\rightarrow\mathrm{U}(m+n+1,m+n+1).$

To ease the notation, we put

H:=\mathrm{U}(m,n),\quad G:=\mathrm{U}(m+1,n),\quad G^{\heartsuit}:=\mathrm{U}(m+1,n),\quad G^{\blacklozenge}:=\mathrm{U}(m+n+1,m+n+1)

and $\mathbf{G}=H\times G$ throughout the rest of the article.

2.2. Eisenstein series on unitary groups

In this section, we define Klingen Eisenstein series and Siegel Eisenstein series over unitary groups.

2.2.1. Klingen Eisenstein series

We first define the Klingen parabolic subgroup $P$ of $G^{\heartsuit}$ as the algebraic group $P$ over $\mathcal{F}$ such that

P(R)=\left\{g=\begin{bmatrix}a&&b&c&\ast\\ \ast&x^{-\star}&\ast&\ast&\ast\\ d&&e&f&\ast\\ h&&l&k&\ast\\ &&&&x\end{bmatrix}\in G^{\heartsuit}(R):g_{0}:=\begin{bmatrix}a&b&c\\ d&e&f\\ h&l&k\end{bmatrix}\in G(R),x\in\mathcal{K}\otimes_{\mathcal{F}}R\right\}.

for any $\mathcal{F}$ -algebra $R$ . Here the block matrix is written with respect to the partition $[n\mid 1\mid m-n\mid n\mid 1]$ . Then it has a Levi decomposition $P=MN$ , where $M$ is the Levi subgroup given by

M(R)=\left\{\mathbf{m}(g_{0},x):=\begin{bmatrix}a&&b&c&\\ &x^{-\star}&&&\\ d&&e&f&\\ h&&l&k&\\ &&&&x\end{bmatrix}\in P(R)\right\}\mathrel{\leavevmode\hbox to14.89pt{\vbox to16.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-6.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{14.22218pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}G(R)\times\operatorname{Res}_{\mathcal{K}/\mathcal{F}}\mathbb{G}_{\mathrm{m}}(R)

with the obvious isomorphism given by $\mathbf{m}(-,-)$ .

To construct an Eisenstein series on $G^{\heartsuit}$ with respect to the Klingen parabolic subgroup $P$ , we start with the following input data.

•

Let $(\sigma,V_{\sigma})$ be an irreducible tempered unitary cuspidal automorphic representation of $H$ over $\mathcal{F}$ , with its archimedean part $\sigma_{\infty}$ being a holomorphic discrete series representation. We identify $V_{\sigma}$ as a subspace of the space of cuspidal automorphic forms $\mathcal{A}_{\mathrm{cusp}}(H(\mathcal{F})\backslash H(\mathbb{A}))$
•

Let $\chi:\mathbb{K}^{\times}\rightarrow\mathbb{C}^{\times}$ be a unitary Hecke character. So clearly $\overline{\chi}=\chi^{-1}$ . We then regard it as an automorphic form over the reductive group $\operatorname{Res}_{\mathcal{F}}^{\mathcal{K}}(\mathbb{G}_{\mathrm{m}})$ .
•

Let $s$ be any complex number.

Then $\sigma\boxtimes\chi$ gives an automorphic representation of $M(\mathbb{A})$ under the identification $\mathbf{m}$ , and we extend it trivially to $P(\mathbb{A})$ by the Levi decomposition. Then we consider the induced representation ⁵⁵5Henceforth, by writing $\operatorname{Ind}_{P(\mathbb{A})}^{G^{\heartsuit}(\mathbb{A})}$ , we mean the unnormalized smooth parabolic induction. Yet here we have added $\delta_{P}^{\frac{1}{2}}$ to actually turning it into a normalized one.

I^{\mathrm{Kling}}(\sigma,\chi,s):=\operatorname{Ind}_{P(\mathbb{A})}^{G^{\heartsuit}(\mathbb{A})}(\delta_{P}^{\frac{1}{2}+s}\cdot\sigma\boxtimes\chi).

More precisely, the representation space $I^{\mathrm{Kling}}(\sigma,\chi,s)$ is the set of smooth functions $f_{\sigma,\chi,s}^{\mathrm{Kling}}:G^{\heartsuit}(\mathbb{A})\rightarrow V_{\sigma}$ such that

(1)

For any $\mathbf{m}(g_{0},x)\in M$ , $n\in N$ and $g\in G^{\heartsuit}(\mathbb{A})$ ,

f_{\sigma,\chi,s}^{\mathrm{Kling}}(\mathbf{m}(g_{0},x)ng)=\delta_{P}(\mathbf{m}(g_{0},x))^{\frac{1}{2}+s}\chi(x)\sigma(g_{0})f_{\sigma,\chi,s}^{\mathrm{Kling}}(g),

(2)

$f_{\sigma,\chi,s}^{\mathrm{Kling}}$ is right $K$ -finite, with $K$ some maximal open compact subgroup of $G^{\heartsuit}(\mathbb{A})$ (which may depends on $f_{\sigma,\chi,s}^{\mathrm{Kling}}$ ).

Elements in this representation space are called a Klingen Eisenstein sections with respect to the datum $(\sigma,\chi,s)$ .

We have a natural evaluation map

\mathrm{ev}_{g}:\mathcal{A}_{\mathrm{cusp}}(H(\mathcal{F})\backslash H(\mathbb{A}))\rightarrow\mathbb{C},\quad\Phi\mapsto\Phi(g)

for any $g\in H(\mathbb{A})$ . By our assumption, $V_{\sigma}$ lies in the space of cuspidal automorphic forms, so for every Klingen section $f_{\sigma,\chi,s}^{\mathrm{Kling}}$ , we attach it with a scalar-valued section $f_{s,\chi,\sigma}^{\mathrm{Kling},g}:=\mathrm{ev}_{g}(f_{\sigma,\chi,s}^{\mathrm{Kling}})$ . Then we define the Klingen Eisenstein series

E^{\mathrm{Kling}}(f_{\sigma,\chi,s}^{\mathrm{Kling}},g):=\sum_{\gamma\in P(\mathcal{F})\backslash G^{\heartsuit}(\mathcal{F})}f_{s,\chi,\sigma}^{\mathrm{Kling},1}(\gamma g).

It is well-known that it converges absolutely and uniformly for $(s,g)$ in compact subsets of $\{s\in\mathbb{C}:\operatorname{Re}(s)>(m+n+1)/2\}\times G^{\heartsuit}(\mathbb{A})$ .

2.2.2. Siegel Eisenstein series

As we shall see shortly, Klingen Eisenstein series are often constructed by pullbacks of certain Siegel Eisenstein series on the larger quasi-split unitary groups $G^{\blacklozenge}$ . We define the Siegel parabolic subgroup $Q$ of $G^{\blacklozenge}$ as the algebraic group $Q$ over $\mathcal{F}$ such that

Q(R)=\left\{g=\begin{bmatrix}A_{g}&B_{g}\\ 0&D_{g}\end{bmatrix}\in\mathrm{GL}_{2N+2}(R\otimes_{\mathcal{F}}\mathcal{K}):D_{g}=A_{g}^{-\star},A_{g}^{-1}B_{g}\in\mathrm{Herm}_{N+1}(R\otimes_{\mathcal{F}}\mathcal{K})\right\}

for any $\mathcal{F}$ -algebra $R$ . Here the block matrix is written with respect to the partition $[N+1\mid N+1]$ . It has a Levi decomposition $Q=M_{Q}N_{Q}$ where

	$\displaystyle M_{Q}(R)$	$\displaystyle:=\left\{\begin{bmatrix}A_{g}&0\\ 0&A_{g}^{-\star}\end{bmatrix}:A_{g}\in\mathrm{GL}_{N+1}(R\otimes_{\mathcal{F}}\mathcal{K})\right\}\simeq\mathrm{GL}_{N+1}(R\otimes_{\mathcal{F}}\mathcal{K}),$
	$\displaystyle N_{Q}(R)$	$\displaystyle:=\left\{\begin{bmatrix}\mathbf{1}_{N+1}&X_{g}\\ 0&\mathbf{1}_{N+1}\end{bmatrix}:X_{g}\in\mathrm{Herm}_{N+1}(R\otimes_{\mathcal{F}}\mathcal{K})\right\}\simeq\mathrm{Herm}_{N+1}(R\otimes_{\mathcal{F}}\mathcal{K}).$

Given any $g\in Q(R)$ , we can decompose it into

g=\begin{bmatrix}A_{g}&B_{g}\\ 0&D_{g}\end{bmatrix}=\begin{bmatrix}A_{g}&0\\ 0&A_{g}^{-\star}\end{bmatrix}\begin{bmatrix}\mathbf{1}_{N+1}&X_{g}\\ 0&\mathbf{1}_{N+1}\end{bmatrix}

with $A_{g}\in\mathrm{GL}_{N+1}(R\otimes_{\mathcal{F}}\mathcal{K})$ and $X_{g}=A_{g}^{-1}B_{g}\in\mathrm{Herm}_{N+1}(R\otimes_{\mathcal{F}}\mathcal{K})$ .

Then for any character $\chi$ of $(R\otimes_{\mathcal{F}}\mathcal{K})^{\times}$ , we regard it as a character on the Levi subgroup $M_{Q}$ via

\chi:\begin{bmatrix}A_{g}&0\\ 0&A_{g}^{-\star}\end{bmatrix}\mapsto\chi(\det A_{g}).

and extends trivially on $N_{Q}$ to get a character of $Q$ .

Let $v$ be any place of $\mathcal{F}$ . For any character $\chi_{v}:\mathcal{K}_{v}^{\times}\rightarrow\mathbb{C}^{\times}$ and $s\in\mathbb{C}$ , we define

I^{\mathrm{Sieg}}_{v}(\chi_{v},s):=\operatorname{Ind}_{Q(\mathcal{F}_{v})}^{G(\mathcal{F}_{v})}(\delta_{Q}^{\frac{1}{2}+s}\cdot\chi_{v})

as the space consisting of smooth functions $f_{s,\chi_{v}}^{\mathrm{Sieg}}:G(\mathcal{F}_{v})\rightarrow\mathbb{C}$ such that

(1)

For any $g\in Q(\mathcal{F}_{v})$ and $h\in G(\mathcal{F}_{v})$ ,

f_{s,\chi_{v}}^{\mathrm{Sieg}}\left(\begin{bmatrix}A_{g}&B_{g}\\ 0&D_{g}\end{bmatrix}h\right)=\delta_{Q}(g)^{\frac{1}{2}+s}\chi(\det A_{g})f_{s,\chi}^{\mathrm{Sieg}}(h)=\left|{\det A_{g}D_{g}^{-1}}\right|_{v}^{s+\frac{n}{2}}\chi(\det A_{g})f_{s,\chi}^{\mathrm{Sieg}}(h),

(2)

$f_{s,\chi_{v}}^{\mathrm{Sieg}}$ is right $G^{\blacklozenge}(\mathcal{O}_{\mathcal{F}_{v}})$ -finite.

Such an element is called a Siegel Eisenstein section at $v$ with respect to the datum $(\chi_{v},s)$ . In particular, when $v$ is a finite place of $F$ and $\chi_{v}$ is an unramified character, then we define the spherical Siegel Eisenstein section $f^{\mathrm{sph}}_{s,\chi_{v}}\in I^{\mathrm{Sieg}}_{v}(\chi_{v},s)$ to be the one such that $f^{\mathrm{sph}}_{s,\chi_{v}}(G^{\blacklozenge}(\mathcal{O}_{\mathcal{F}_{v}}))=1$ .

Let $\chi:\mathbb{K}^{\times}\rightarrow\mathbb{C}^{\times}$ be a unitary Hecke character with the tensor product decomposition $\chi=\otimes_{v\in\mathscr{S}_{\mathcal{F}}}^{\prime}\chi_{v}$ . Consider the restricted tensor product $I^{\mathrm{Sieg}}(\chi,s)=\otimes_{v}^{\prime}I^{\mathrm{Sieg}}_{v}(\chi_{v},s)$ with respect to the spherical Siegel sections $f^{\mathrm{sph}}_{s,\chi_{v}}$ at finite places $v$ where $\chi_{v}$ is unramified. Let $f_{\chi,s}^{\mathrm{Sieg}}\in I^{\mathrm{Sieg}}(\chi,s)$ , the Siegel Eisenstein series attached to $f_{s,\chi}^{\mathrm{Sieg}}$ is defined as

E^{\mathrm{Sieg}}(f_{s,\chi}^{\mathrm{Sieg}},g):=\sum_{\gamma\in Q(\mathcal{F})\backslash G(\mathcal{F})}f_{s,\chi}^{\mathrm{Sieg}}(\gamma g).

It is well-known that it converges absolutely and uniformly for $(s,g)$ in compact subsets of $\{s\in\mathbb{C}:\operatorname{Re}(s)>(m+n+1)/2\}\times G^{\blacklozenge}(\mathbb{A})$ .

Remark 2.2.

We note that the notion of Siegel Eisenstein sections here slightly differs from [SU14, Wan15]. As explained in [SU14, Remark on page 170], their representation $\chi$ on the Levi subgroup $M_{Q}$ is defined by

\chi:\begin{bmatrix}D_{g}^{-\star}&0\\ 0&D_{g}\end{bmatrix}\mapsto\chi(\det D_{g}).

So the induced representation $I(\chi)$ in loc.cit is actually $I^{\mathrm{Sieg}}((\chi^{\mathrm{c}})^{-1},s)$ in our setup. Their convention is convenience when dealing with the functional equations of Eisenstein series.

2.3. On multiplicity one conditions

Before we start conducting automorphic computations, we record a result on the multiplicity one theorem of automorphic representation of unitary groups. Let $\pi$ be an automorphic representation of a unitary group $\mathrm{U}(m,n)$ over $\mathcal{F}$ , and suppose that

\text{the functorial base change }\mathrm{BC}(\pi)\text{ of }\pi\text{ to }\mathrm{GL}_{m+n}(\mathcal{K})\text{ is a cuspidal automorphic representation. }

(BC)

Then $\pi$ appears in the space of cuspidal automorphic forms over $\mathrm{U}(m,n)$ of multiplicity one, as a consequence of [Mok15, KMSW14]. As a result, up to a complex scalar, there exists a unique $\mathrm{U}(m,n)(\mathbb{A})$ -invariant pairing between $\pi$ and $\pi^{\vee}$ . Throughout this article, we assume (BC) for all automorphic representations appearing in this article.

2.4. Doubling method, à la Piatetski-Shapiro and Rallis

Here we briefly recall the doubling method à la Piatetski-Shapiro and Rallis, first introduced in [GPR87].

Let $(\pi,V_{\pi})$ be an irreducible tempered unitary cuspidal automorphic representation of $G$ and $(\pi^{\vee},V_{\pi^{\vee}})$ be its contragredient representation. We thus identify $V_{\pi}$ as a subspace $\mathcal{A}_{\mathrm{cusp}}(G(\mathcal{F})\backslash G(\mathbb{A}))$ . We define the global doubling integral à la Piatetski-Shapiro and Rallis as

Z^{\diamondsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,\Psi^{\vee}):=\int_{[G^{\diamondsuit}]}E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s,\chi},\imath^{\diamondsuit}(g,h))\Psi(g)\Psi^{\vee}(h)\chi^{-1}(\det h)\,\mathrm{d}g\,\mathrm{d}h.

where $\Psi\in V_{\pi}$ and $\Psi^{\vee}\in V_{\pi^{\vee}}$ . It converges at wherever the Eisenstein series is defined, by the cuspidality of $\Phi$ and $\Phi^{\vee}$ .

2.4.1. Basic identity of Piatetski-Shapiro and Rallis

We put $G^{\diamondsuit}$ as the image of $G\times G$ via the canonical doubling embedding $\iota^{\diamondsuit}:G\times G\hookrightarrow G^{\blacklozenge}$ and $G^{\Delta}$ as the image of the diagonal embedding $G\rightarrow G\times G$ composited with $\iota^{\diamondsuit}$ .

The following fundamental result is well-known, which appears in the proof of [GPR87, “Basic Identity” on page 3], but the character $\chi$ was absent. For the readers’ convenience, we roughly sketch the proof here, reproduced from the proof of [Eis24, Theorem 4.3.4] in the setup of unitary groups.

Theorem 2.3 (Basic identity of Piatetski-Shapiro and Rallis).

Z^{\diamondsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,\Psi^{\vee})=\int_{G^{\Delta}(\mathcal{F})\backslash(G^{\diamondsuit})(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,h))\Psi(g)\Psi^{\vee}(h)\chi^{-1}(\det h)\,\mathrm{d}g\,\mathrm{d}h.

Proof.

The theorem follows from an analysis of the orbits of $G^{\diamondsuit}$ acting on $\mathcal{X}:=Q\backslash G^{\blacklozenge}$ by multiplication on the right. We write $(G^{\diamondsuit})^{\gamma}$ as the stabilizer of a point $\gamma\in\mathcal{X}$ . Then we can rewrite the Siegel Eisenstein series $E^{\mathrm{Sieg}}(f_{s,\chi}^{\mathrm{Sieg}})$ by grouping the summands by the orbits, as

E^{\mathrm{Sieg}}(f_{s,\chi}^{\mathrm{Sieg}})(h)=\sum_{[\gamma]\in Q(\mathcal{F})\backslash G^{\blacklozenge}(\mathcal{F})/G^{\diamondsuit}(\mathcal{F})}\left(\sum_{[\gamma_{0}]\in(G^{\diamondsuit})^{\gamma}\backslash G^{\diamondsuit}(\mathcal{F})}f_{s,\chi}^{\mathrm{Sieg}}(\gamma\gamma_{0}h)\right).

Here $[\gamma]$ denotes the orbit of $Q(\mathcal{F})\gamma\in\mathcal{X}(\mathcal{F})$ under the right action of $G^{\diamondsuit}(\mathcal{F})$ . Inserting this expression into the doubling integral, we have $Z^{\diamondsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,\Psi^{\vee})$ equals

	$\displaystyle\sum_{[\gamma]\in Q(\mathcal{F})\backslash G^{\blacklozenge}(\mathcal{F})/G^{\diamondsuit}(\mathcal{F})}\left(\sum_{[\gamma_{0}]\in(G^{\diamondsuit})^{\gamma}\backslash G^{\diamondsuit}(\mathcal{F})}\int_{[G^{\diamondsuit}]}f^{\mathrm{Sieg}}_{s,\chi}(\gamma\gamma_{0}\imath^{\diamondsuit}(g,h))\Psi(g)\Psi^{\vee}(h)\chi^{-1}(\det h)\,\mathrm{d}g\,\mathrm{d}h\right)$
	$\displaystyle=\sum_{[\gamma]\in Q(\mathcal{F})\backslash G^{\blacklozenge}(\mathcal{F})/G^{\diamondsuit}(\mathcal{F})}\left(\int_{(G^{\diamondsuit})^{\gamma}\backslash G^{\diamondsuit}(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\gamma\gamma_{0}\imath^{\diamondsuit}(g,h))\Psi(g)\Psi^{\vee}(h)\chi^{-1}(\det h)\,\mathrm{d}g\,\mathrm{d}h\right)$

Temporarily denote the integral in the summand as $I(\gamma)$ . Note that for each $\gamma\in G^{\blacklozenge}(\mathcal{F})$ ,

	$\displaystyle(G^{\diamondsuit})^{\gamma}(\mathcal{F})$	$\displaystyle=\{\imath^{\diamondsuit}(g,h)\in G^{\diamondsuit}(\mathcal{F}):Q(\mathcal{F})\gamma\imath^{\diamondsuit}(g,h)=Q(\mathcal{F})\gamma\}$
		$\displaystyle=\{\imath^{\diamondsuit}(g,h)\in G^{\diamondsuit}(\mathcal{F}):\gamma\imath^{\diamondsuit}(g,h)\gamma^{-1}=Q(\mathcal{F})\}.$

We first deal with the orbit $\gamma=1$ . In this case, the stabilizer is

(G^{\diamondsuit})^{1}(\mathcal{F})=Q(\mathcal{F})\cap G^{\diamondsuit}(\mathcal{K})=\{\imath^{\diamondsuit}(g,g):g\in G(\mathcal{F})\}=G^{\Delta}(\mathcal{F})

and

I(1)=\int_{(G^{\diamondsuit})^{\Delta}\backslash G^{\diamondsuit}(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\gamma\gamma_{0}\imath^{\diamondsuit}(g,h))\Psi(g)\Psi^{\vee}(h)\chi^{-1}(\det h)\,\mathrm{d}g\,\mathrm{d}h.

The remaining orbits (that is, $\gamma\neq 1$ ) are negligible in the sense of [GPR87, PartA, Chapter I], and $I(\gamma)=0$ thereof. This vanishing result essentially follows from the cuspidality of $\Psi$ and $\Psi^{\vee}$ , and that $(G^{\diamondsuit})^{\gamma}$ contains the unipotent radical of a proper parabolic subgroup of $G^{\diamondsuit}(\mathcal{F})$ as a normal subgroup. Details on these negligible orbits can be found in [GPR87, PartA, Chapter I] or the proof of [Eis24, Theorem 4.3.4]. ∎

2.4.2. Partial doubling integrals

Following [SU14, page 174], we define the partial doubling integrals

Z^{\diamondsuit,\triangleleft}(f^{\mathrm{Sieg}}_{s,\chi},\Psi^{\vee};g):=\int_{[G]}E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s,\chi},\imath^{\diamondsuit}(g,h))\Psi^{\vee}(h)\chi^{-1}(\det h)\,\mathrm{d}h,

and

Z^{\diamondsuit,\triangleright}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,h):=\int_{[G]}E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s,\chi},\imath^{\diamondsuit}(g,h))\Psi(g)\,\mathrm{d}g.

They converges by the cuspidality of $\Psi$ and $\Psi^{\vee}$ . Then we see that

Z^{\diamondsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,\Psi^{\vee})=\left\langle{\Psi,Z^{\diamondsuit,\triangleleft}(f^{\mathrm{Sieg}}_{s,\chi},\Psi^{\vee};-)}\right\rangle_{\mathrm{Pet}}=\left\langle{Z^{\diamondsuit,\triangleright}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,-),\Psi^{\vee}\cdot\chi^{-1}(\det-)}\right\rangle_{\mathrm{Pet}}.

(2.3)

We shall use the following corollary of the basic identity below.

Corollary 2.4.

Notations being as above.

(1)

Let $g_{0}\in G(\mathbb{A})$ , then

	$\displaystyle Z^{\diamondsuit,\triangleleft}(f^{\mathrm{Sieg}}_{s,\chi},\Psi^{\vee};g)$	$\displaystyle=\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,h))\overline{\chi}(\det h)\Psi^{\vee}(h)\,\mathrm{d}h$
		$\displaystyle=\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{0},h))\overline{\chi}(\det h)\Psi^{\vee}(gg_{0}^{-1}h)\,\mathrm{d}h$

(2)

Let $h_{0}\in G(\mathbb{A})$ , then

	$\displaystyle Z^{\diamondsuit,\triangleright}(f^{\mathrm{Sieg}}_{s,\chi},\Psi;h)$	$\displaystyle=\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,h))\Psi(g)\,\mathrm{d}g$
		$\displaystyle=\chi(\det hh_{0}^{-1})\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,h_{0}))\Psi(hh_{0}^{-1}g)\,\mathrm{d}g.$

(3)

As a result,

	$\displaystyle Z^{\diamondsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,\Psi^{\vee})$	$\displaystyle=\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{0},h))\left\langle{\Psi,\pi^{\vee}(g_{0}^{-1}h)\Psi^{\vee}}\right\rangle_{\mathrm{Pet}}\,\mathrm{d}h$
		$\displaystyle=\overline{\chi}(\det h_{0})\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,h_{0}))\left\langle{\pi(h_{0}^{-1}g)\Psi,\Psi^{\vee}}\right\rangle_{\mathrm{Pet}}\,\mathrm{d}g$

Proof.

We note that there are two ways of identifying $G^{\Delta}(\mathcal{F})\backslash(G\times G)(\mathbb{A})$ and $G(\mathbb{A})\times(G(\mathcal{F})\backslash G(\mathbb{A}))$ , by sending $(g,h)$ to either $(g,h)$ or $(h,g)$ . Then Theorem 2.3, together with (2.3), gives the first equality of (1) and (2) accordingly, by noting that the Petersson pairing $\left\langle{-,-}\right\rangle_{\mathrm{Pet}}$ is perfect.

To deduce the second equality of (1), we note that

f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,h))=f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(gg_{0}^{-1},g)\imath^{\diamondsuit}(g_{0},g_{0}g^{-1}h))=\chi(\det gg_{0}^{-1})f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{0},g_{0}g^{-1}h)).

Putting into the integral on the right hand side of the first equality of (1), we see that

	$\displaystyle Z^{\diamondsuit,\triangleleft}(f^{\mathrm{Sieg}}_{s,\chi},\Psi^{\vee};g)$	$\displaystyle=\chi(\det gg_{0}^{-1})\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{0},g_{0}g^{-1}h))\overline{\chi}(\det h)\Psi^{\vee}(h)\,\mathrm{d}h$
		$\displaystyle=\chi(\det gg_{0}^{-1})\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{0},h^{\prime}))\overline{\chi}(\det gg_{0}^{-1}h^{\prime})\Psi^{\vee}(gg_{0}^{-1}h^{\prime})\,\mathrm{d}h^{\prime}$
		$\displaystyle=\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{0},h^{\prime}))\overline{\chi}(\det h^{\prime})\Psi^{\vee}(gg_{0}^{-1}h^{\prime})\,\mathrm{d}h^{\prime}$

by a change of variable $h^{\prime}=g_{0}g^{-1}h$ . This shows the second equality in (1).

The second equality of (2) is deduced in the same way. We note that

f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,h))=f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(hh_{0}^{-1},hh_{0}^{-1})\imath^{\diamondsuit}(h_{0}h^{-1}g,h_{0}))=\chi(\det hh_{0}^{-1})f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(h_{0}h^{-1}g,h_{0})).

Putting into the integral on the right hand side of the first equality of (2), we see that

	$\displaystyle Z^{\diamondsuit,\triangleright}(f^{\mathrm{Sieg}}_{s,\chi},\Psi;h)$	$\displaystyle=\chi(\det hh_{0}^{-1})\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(h_{0}h^{-1}g,h_{0}))\Psi(g)\,\mathrm{d}g$
		$\displaystyle=\chi(\det hh_{0}^{-1})\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g^{\prime},h_{0}))\Psi(hh_{0}^{-1}g^{\prime})\,\mathrm{d}g^{\prime}$

by a change of variable $g^{\prime}=h_{0}h^{-1}g$ . This shows the second equality in (2). The equalities in (3) follows from the second equalities of (1) and (2) by (2.3). ∎

2.4.3. Local doubling integrals

We also consider the local counterparts of the doubling integrals. Let $g_{0},h_{0}\in G(\mathcal{F}_{v})$ . In the spirit of Corollary 2.4 (where $g$ and $h$ are set to $\mathbf{1}_{m+n+1}$ ), define

Z_{v}^{\diamondsuit,\triangleleft}(f^{\mathrm{Sieg}}_{s,\chi,v},\Psi^{\vee}_{v};g_{v}):=\int_{G(\mathcal{F}_{v})}f^{\mathrm{Sieg}}_{s,\chi,v}(\imath^{\diamondsuit}(g_{0},h_{v}))\overline{\chi}(\det h_{v})\pi^{\vee}(g_{v}g_{0}^{-1}h_{v})\Psi_{v}^{\vee}\,\,\mathrm{d}h_{v}

and

Z_{v}^{\diamondsuit,\triangleright}(f^{\mathrm{Sieg}}_{s,\chi,v},\Psi_{v};h_{v}):=\overline{\chi}_{v}(\det h_{0})\int_{G(\mathcal{F}_{v})}f^{\mathrm{Sieg}}_{s,\chi,v}(\imath^{\diamondsuit}(g_{v},h_{0}))\pi(h_{v}h_{0}^{-1}g_{v})\Psi\,\,\mathrm{d}g_{v}.

Here, the integrals $Z_{v}^{\diamondsuit,\triangleleft}$ and $Z_{v}^{\diamondsuit,\triangleright}$ are understood as vector-valued integrals (see, for example, [Gar18, Chapter 14] for a precise definition).

By the uniqueness of $G$ -invariant pairings between $\pi$ and $\pi^{\vee}$ , as provided by (BC), and Item (3) of Corollary 2.4, we obtain the following result.

Corollary 2.5.

Under the assumptions above, We have

Z^{\diamondsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,\Psi^{\vee})=\left\langle{\Psi,\Psi^{\vee}}\right\rangle\prod_{v\in\mathscr{V}_{\mathcal{F}}}\frac{Z^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{s,\chi,v},\Psi_{v},\Psi_{v}^{\vee})}{\left\langle{\Psi_{v},\Psi^{\vee}_{v}}\right\rangle},

with

	$\displaystyle Z^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\Psi,\Psi^{\vee})$	$\displaystyle=\int_{G(\mathcal{F}_{v})}f^{\mathrm{Sieg}}_{v,s,\chi}(\imath^{\diamondsuit}(g_{0},h_{v}))\left\langle{\Psi,\pi^{\vee}(g_{0}^{-1}h_{v})\Psi^{\vee}}\right\rangle_{\mathrm{Pet}}\,\,\mathrm{d}h_{v}$
		$\displaystyle=\overline{\chi}(\det h_{0})\int_{G(\mathcal{F}_{v})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{v},h_{0}))\left\langle{\pi(h_{0}^{-1}g_{v})\Psi,\Psi^{\vee}}\right\rangle_{\mathrm{Pet}}\,\,\mathrm{d}g_{v};$

We remark that $g_{0}$ and $h_{0}$ can certainly be $1\in G(\mathcal{F}_{v})$ , and actually this is the only case we shall use. In other applications, a flexibility to choose $g_{0},h_{0}\in G(\mathcal{F}_{v})$ may be convenient for computations. By the uniqueness of $G$ -invariant pairings between $\pi$ and $\pi^{\vee}$ granted by (BC), we see the local quotients is independent of the choice of local vectors $\Psi_{v}$ and $\Psi_{v}^{\vee}$ , but only depend on the local Siegel Eisenstein section $f^{\mathrm{Sieg}}_{v,s,\chi}$ and the local representation $\pi_{v}$ . We therefore denote

\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v}):=\frac{Z^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{s,\chi,v},\Psi_{v},\Psi_{v}^{\vee})}{\left\langle{\Psi_{v},\Psi^{\vee}_{v}}\right\rangle}

for simplicity.

2.5. Doubling method, à la Garrett

The primary method for explicitly constructing Klingen Eisenstein series is via pullbacks of Siegel Eisenstein series. This approach generalizes the doubling method à la Piatetski-Shapiro and Rallis, initiated by Garrett in [Gar84, Gar89] and later further developed by Shimura [Shi97].

We define the pullback integral formally as the vector-valued integral

F^{\heartsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;g_{0}):=\int_{H(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\heartsuit}(g_{0},h))\chi^{-1}(\det h)\pi(h)\Phi\,\,\mathrm{d}h,

(2.4)

with values in $V_{\pi}$ , for any Siegel section $f^{\mathrm{Sieg}}_{s,\chi}\in I^{\mathrm{Sieg}}(\chi,s)$ and any $g_{0}\in G^{\heartsuit}(\mathbb{A})$ . Composing this with $\mathrm{ev}_{1}$ (the evaluation map at $\mathbf{1}_{m+n+1}$ from $V_{\pi}$ to $\mathbb{C}$ ), we formally have

F^{\heartsuit,1}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;g_{0}):=\int_{H(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\heartsuit}(g_{0},h))\chi^{-1}(\det h)\Phi(h)\,\,\mathrm{d}h.

Proposition 2.6.

With the above notations:

(1)

The vector $F^{\heartsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;-)$ is a Klingen Eisenstein section whenever it exists. The integral $F^{\heartsuit,1}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;-)$ converges for $(s,g)$ in compact subsets of $\{\operatorname{Re}(s)>(N+1)/2\}\times G^{\heartsuit}(\mathbb{A})$ . ⁶⁶6There appears to be a typographical error in the convergence range stated in [Wan15, Proposition 3.5 (ii)]. More precisely, $\{\operatorname{Re}(z)>r+s+1/2\}$ should be $\{\operatorname{Re}(z)>(r+s+1)/2\}$ with the notations in loc.cit..

(2)

Moreover,

\int_{H(F)\backslash H(\mathbb{A})}E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s,\chi};\imath^{\heartsuit}(g_{0},h))\chi^{-1}(\det h)\Phi(h)\,\,\mathrm{d}h=E^{\mathrm{Kling}}(F^{\heartsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;-);g_{0}).

Proof.

This is [Wan15, Proposition 3.5]. The convergence issue is discussed in the proof there. The fact that $F^{\heartsuit,1}$ is a Klingen Eisenstein section is well-known and can be directly verified. The reader may refer to the proof of [Zha13, Theorem 2.6] for the case when $H=\mathrm{U}(2,0)$ . The computation there can be generalized to the broader case. ∎

3. The Gan-Gross-Prasad period integral

We inherit all notations and conventions from previous sections. Let

•

$(\pi,V_{\pi})$ be an irreducible unitary cuspidal automorphic representation of $G$ over $\mathcal{F}$ and identify $V_{\pi}$ as a subspace of the $\mathcal{A}_{\mathrm{cusp}}(G(\mathcal{F})\backslash G(\mathbb{A}))$ , and
•

$E^{\mathrm{Kling}}(f^{\mathrm{Kling}}_{\sigma,\chi,s},-)$ be a Klingen Eisenstein series on $G^{\heartsuit}$ , defined in Section 2.2.1.

Let $\Psi\in V_{\pi}$ , we define the Gan-Gross-Prasad period integral (GGP period integral) of $E^{\mathrm{Kling}}$ with the cusp form $\Psi$ as the integral

\mathcal{P}(\Psi,E^{\mathrm{Kling}}(f^{\mathrm{Kling}}_{\sigma,\chi,s},-)):=\int_{G(\mathcal{F})\backslash G(\mathbb{A})}E^{\mathrm{Kling}}(f^{\mathrm{Kling}}_{\sigma,\chi,s},\jmath^{\flat}(g))\Psi(g)\,\mathrm{d}g.

(3.5)

By the cuspidality of $\Psi$ , this converges absolutely for those values of $s$ at which $E^{\mathrm{Kling}}(f^{\mathrm{Kling}}_{\sigma,\chi,s},-)$ is defined. ⁷⁷7We thank Wen-Wei Li for his hint on this issue.

In this section, we consider in particular the Klingen Eisenstein series constructed by pulling back from Siegel Eisenstein series (see Section 2.5). Our plan is as follows.

(1)

We first unfold the GGP period integral (3.5) to reduce it to certain “cuspidal GGP period integral”. See Proposition 3.1.
(2)

Then we square the integral and use Ichino-Ikeda formula (Theorem 3.2) to decompose it into products of local integrals, involving local Ichino-Ikeda integrals and local doubling integrals. See Theorem 3.4.
(3)

We further invoke the unramified computations of such local integrals to see that the Rankin-Selberg local $L$ -factors and standard local $L$ -factors arise at “good” places (which covers all but finitely many places) of $\mathcal{F}$ . See Theorem 3.7.

There are finitely many “bad” places remained, which we shall deal with in coming up sections.

3.1. Reduce to the cuspidal GGP period integral

We write

\mathcal{P}^{\mathrm{Kling}}(\Phi,\Psi,\chi,s):=\mathcal{P}(\Psi,E^{\mathrm{Kling}}(F^{\heartsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;-)))

(3.6)

We write $\mathcal{P}^{\mathrm{Kling}}(\Phi,\Psi,\chi,s)$ as $\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}$ in this part for simplicity. By Proposition 2.6,

\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}=\int_{[G]}\int_{[H]}\Psi(g)E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s,\chi};\imath^{\heartsuit}(\jmath^{\flat}(g),h))\chi^{-1}(\det h)\Phi(h)\,\mathrm{d}h\,\mathrm{d}g.

Interchanging the two integrals, we isolate

\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}=\int_{[H]}\left(\int_{[G]}\Psi(g)E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s,\chi};\imath^{\heartsuit}(\jmath^{\flat}(g),h))\,\mathrm{d}g\right)\chi^{-1}(\det h)\Phi(h)\,\mathrm{d}h.

By the observation (2.2),

\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}=\int_{[H]}\left(\int_{[G]}\Psi(g)E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s,\chi};\imath^{\diamondsuit}(g,\jmath^{\sharp}(h))\,\mathrm{d}g\right)\chi^{-1}(\det h)\Phi(h)\,\mathrm{d}h.

The inner integral is nothing but $Z^{\diamondsuit,\triangleright}(f^{\mathrm{Sieg}}_{s,\chi},\Psi,\jmath^{\sharp}(h))$ . By Corollary 2.4 (2), putting $h_{0}=1$ there, we obtain

	$\displaystyle\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}$	$\displaystyle=\int_{[H]}\left(\chi(\det\jmath^{\sharp}(h))\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,1))\Psi(\jmath^{\sharp}(h)g)\,\mathrm{d}g\right)\chi^{-1}(\det h)\Phi(h)\,\mathrm{d}h$
		$\displaystyle=\int_{[H]}\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,1))\Psi(\jmath^{\sharp}(h)g)\Phi(h)\,\mathrm{d}g\,\mathrm{d}h$

Then we interchange the two integrals back, it yields

\displaystyle\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}=\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,1))\left(\int_{[H]}\Phi(h)\cdot(\pi(g)\Psi)(\jmath^{\sharp}(h))\,\mathrm{d}h\right)\,\mathrm{d}g.

The inner integral is a cuspidal GGP period integral. In general, for $\Phi\in\sigma$ and $\Psi\in\pi$ , we define the cuspidal GGP period integral of $\Psi$ with $\Phi$ as

\mathcal{P}^{\sharp}(\Phi,\Psi):=\int_{[H]}\Phi(h)\Psi(\jmath^{\sharp}(h))\,\mathrm{d}h,

which converges by the cuspidality of $\Phi$ and $\Psi$ . To sum up, we have proved the following result.

Proposition 3.1.

Let $\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}$ be the GGP period integral defined in (3.6), then

\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}=\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g,1))\mathcal{P}^{\sharp}(\Phi,\pi(g)\Psi)\,\mathrm{d}g.

3.2. Break into local integrals

Before conducting concrete computations, we make some preparations.

3.2.1. Contragredient, conjugation and MVW involutions

Given an irreducible cuspidal automorphic representation $\pi\subseteq\mathcal{A}_{0}(G(\mathcal{F})\backslash G(\mathbb{A}))$ and its complex conjugation $\overline{\pi}\subseteq\mathcal{A}_{0}(G(\mathcal{F})\backslash G(\mathbb{A}))$ , which is isomorphic to the contragredient of $\pi$ ⁸⁸8See, for example, [GH11a, Proposition 8.9.6] for the case of $\mathrm{GL}_{2}(\mathbb{A}_{\mathbb{Q}})$ . The proof goes the same for any reductive group., we fix factorizations

\mathrm{fac}_{\pi}:\pi\cong\otimes^{\prime}_{v\in\mathscr{V}_{\mathcal{F}}}\pi_{v},\quad\mathrm{fac}_{\overline{\pi}}:\overline{\pi}\cong\otimes^{\prime}_{v\in\mathscr{V}_{\mathcal{F}}}\pi^{\vee}_{v}

with the restricted tensor product is taken with respect to spherical elements $\Psi_{v}^{\circ}$ and $\Psi_{v}^{\vee,\circ}$ at the places $v$ of $\mathcal{F}$ where $\pi_{v}$ is unramified.

Besides the complex conjugation, another model for the contragredient representation $\pi^{\vee}$ is established by Moeglin, Vignéras and Waldspurger in [MVW87]. Consider in general a unitary group $\mathrm{U}(V,\phi)$ . By [MVW87, page 74], there exists a (unique) element $\delta\in\operatorname{Aut}_{\mathcal{K}}(V)$ such that $\phi(\delta v,\delta w)=\phi(w,v)$ for any $v,w\in V$ . Conjugation by $\delta$ gives an automorphism of the group $\mathrm{U}(V,\phi)$ :

(-)^{\ast}:g\mapsto\delta g\delta^{-1}.

This is called the MVW involution on $\mathrm{U}(V,\phi)$ .

We fix an MVW involution $\ast$ on $\mathbf{G}$ that stabilizes $H$ . For every $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{spl}}$ , we fix a standard isomorphism

\mathbf{G}(\mathcal{F}_{v})\cong\mathrm{GL}_{n}(\mathcal{F}_{v})\times\mathrm{GL}_{n+1}(\mathcal{F}_{v}),

under which the MVW involution $\ast$ coincide with the transpose-inverse. For every element $\Psi=\otimes_{v}^{\prime}\Psi_{v}\in\pi$ written through $\mathrm{fac}_{\pi}$ , we have the function $\Psi^{\ast}$ defined by the formula $\Psi^{\ast}(g):=\Psi(g^{\ast})$ . Then $\Psi^{\ast}=\otimes^{\prime}_{v}\Psi^{\ast}_{v}$ is again decomposable and belongs to $\pi^{\vee}$ , where each $\Psi^{\ast}_{v}\in\pi_{v}^{\vee}$ . By a change of variables, we see immediately that

\mathcal{P}^{\sharp}(\Phi^{\ast},\Psi^{\ast})=\mathcal{P}^{\sharp}(\Phi,\Psi),

(3.7)

which turns out to be an advantage for the MVW involution in the computation.

3.2.2. Ichino-Ikeda formula

Recall $\mathbf{G}=H\times G$ . Let $\Pi=\sigma\boxtimes\pi$ be the cuspidal representation of $\mathbf{G}(\mathbb{A})$ .

The product $L$ -series associated to $\sigma$ and $\pi$ is defined as

L(s,\sigma\times\pi):=L^{\mathrm{JPSS}}(s,\mathrm{BC}(\sigma)\times\mathrm{BC}(\pi)),

where $\mathrm{BC}(\sigma)$ (resp. $\mathrm{BC}(\pi)$ ) is the functorial lift of $\sigma$ (resp. $\pi$ ) to an automorphic representation of $\mathrm{GL}_{m+n}(\mathbb{K})$ (resp. $\mathrm{GL}_{m+n+1}(\mathbb{K})$ ). The right hand side is the $L$ -factor defined by Jacquet, Piateski-Shapiro and Shalika in [JPS83]. Let $L(s,\sigma,\operatorname{Ad})$ denote the adjoint $L$ -series for $\sigma$ .

Assume that both $\sigma$ and $\pi$ is tempered, then we put

\mathscr{L}(\sigma\times\pi)=\dfrac{L(\frac{1}{2},\sigma\times\pi)}{L(1,\sigma,\operatorname{Ad})L(1,\pi,\operatorname{Ad})}\prod_{i=1}^{m+n+1}L(i,\epsilon_{\mathcal{K}/\mathcal{F}}^{i})

and

\mathscr{L}(\sigma_{v}\times\pi_{v})=\dfrac{L(\frac{1}{2},\sigma_{v}\times\pi_{v})}{L(1,\sigma_{v},\operatorname{Ad})L(1,\pi_{v},\operatorname{Ad})}\prod_{i=1}^{m+n+1}L(i,\epsilon_{\mathcal{K}_{v}/\mathcal{F}_{v}}^{i})

for any $v\in\mathscr{V}_{\mathcal{F}}$ .

Let $\Xi$ be a cusp form on $\mathbf{G}$ , we define the integral

\mathcal{P}(\Xi):=\int_{[H]}\Xi(\Delta^{\sharp}(h))\,\mathrm{d}h.

where $\Delta^{\sharp}:H\rightarrow\mathbf{G}$ is defined by $h\mapsto(\jmath^{\sharp}(h),h)$ . If $\Xi=\otimes^{\prime}_{v}\Xi_{v}\in\Pi$ and $\Xi^{\prime}=\otimes^{\prime}_{v}\Xi^{\prime}_{v}\in\Pi^{\vee}$ are factorizable, we define the local Ichino-Ikeda integral

I(\Xi_{v},\Xi_{v}^{\prime}):=\int_{H(F_{v})}\left\langle{\Pi_{v}(\Delta^{\sharp}(h_{v}))\Xi_{v},\Xi^{\prime}_{v}}\right\rangle_{\Pi_{v}}\,\mathrm{d}h_{v}.

It is convergent if $\Pi_{v}$ is tempered. Actually $\mathcal{P}(\Psi\boxtimes\Phi)=\mathcal{P}^{\sharp}(\Phi,\Psi)$ .

Recently, there has been great progress on the global Gan-Gross-Prasad conjecture for unitary groups, for example, [II10, Har14, BLZZ21, BCZ22]. We take a version reinterpreted in [HY23, Theorem 4.2] ⁹⁹9Following our convention in Section 1.4.4, the constant $C_{H}$ in [HY23, Theorem 4.2] is $1$ in our article..

Theorem 3.2 (Ichino-Ikeda formula).

Let $\Pi$ be an irreducible tempered cuspidal automorphic representation of $\mathbf{G}(\mathbb{A})$ . If $\Xi=\otimes^{\prime}_{v}\Xi_{v}\in\Pi$ and $\Xi^{\prime}=\otimes^{\prime}_{v}\Xi^{\prime}_{v}\in\Pi^{\vee}$ are factorizable, then

\dfrac{\mathcal{P}(\Xi)\mathcal{P}(\Xi^{\prime})}{(\Xi,\Xi^{\prime})_{\mathrm{Pet}}}=\dfrac{\mathscr{L}(\sigma\times\pi)}{2^{\varkappa_{\sigma}+\varkappa_{\pi}}}\prod_{v\in\mathscr{V}_{\mathcal{F}}}\dfrac{I(\Xi_{v},\Xi_{v}^{\prime})}{\mathscr{L}(\sigma_{v}\times\pi_{v})\left\langle{\Xi_{v},\Xi_{v}^{\prime}}\right\rangle_{\Pi_{v}}},

where $2^{\varkappa_{\pi}}$ (resp. $2^{\varkappa_{\sigma}}$ ) is the order of the component group associated to the $L$ -parameter of $\pi$ (resp. $\sigma$ ).

3.2.3. Reduce to local Ichino-Ikeda integrals

We square the GGP period integral, using Proposition 3.1, to get

	$\displaystyle(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}$	$\displaystyle=\int_{G(\mathbb{A})\times G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{1},1))f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\mathcal{P}^{\sharp}(\Phi,\pi(g_{2})\Psi)\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$
		$\displaystyle=\int_{G(\mathbb{A})\times G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{1},1))f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\mathcal{P}^{\sharp}(\Phi^{\ast},(\pi(g_{2})\Psi)^{\ast})\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$
		$\displaystyle=\int_{G(\mathbb{A})\times G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{1},1))f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\mathcal{P}^{\sharp}(\Phi^{\ast},\pi^{\vee}(g_{2})\Psi^{\ast})\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$

Here in the second equality, we use (3.7) and the third equality follows from the definition of the MVW involution on $\Psi$ .

Remark 3.3 (On complex conjugation).

We also have the naive approach by taking complex conjugation. Then we see

	$\displaystyle\left\|{\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}}\right\|^{2}$	$\displaystyle=\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi}\cdot(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{\mathrm{c}}$
		$\displaystyle=\int_{G(\mathbb{A})\times G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{1},1))\overline{f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))}\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\overline{\mathcal{P}^{\sharp}(\Phi,\pi(g_{2})\Psi)}\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$
		$\displaystyle=\int_{G(\mathbb{A})\times G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{1},1))\overline{f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))}\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\mathcal{P}^{\sharp}(\overline{\Phi},\pi^{\vee}(g_{2})\overline{\Psi})\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$

This is acceptable by purely automorphic computation, but seems useless for $p$ -adic interpolations and further applications.

In this subsection, our main focus is on the part $\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\mathcal{P}^{\sharp}(\Phi^{\ast},\pi^{\vee}(g_{2})\Psi^{\ast})$ . Such a product can be handled by Theorem 3.2: putting

\Xi:=\Phi\boxtimes\pi(g_{1})\Psi,\quad\Xi^{\prime}:=\Phi^{\ast}\boxtimes\pi^{\vee}(g_{2})\Psi^{\ast},

we see that

\dfrac{\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\mathcal{P}^{\sharp}(\Phi^{\ast},\pi^{\vee}(g_{2})\Psi^{\ast})}{\left|\left|{\Phi}\right|\right|_{\mathrm{Pet}}^{2}(\pi(g_{1})\Psi,\pi^{\vee}(g_{2})\Psi^{\ast})_{\pi,\mathrm{Pet}}}\\ =\dfrac{\mathscr{L}(\sigma\times\pi)}{2^{\aleph_{\sigma}+\aleph_{\pi}}}\prod_{v\in\mathscr{V}_{\mathcal{F}}}\dfrac{I(\Phi_{v}\boxtimes\pi(g_{1})\Psi_{v},\Phi_{v}^{\ast}\boxtimes\pi^{\vee}(g_{2})\Psi^{\ast}_{v})}{\mathscr{L}(\sigma_{v}\times\pi_{v})\left\langle{\Phi_{v}\boxtimes\pi(g_{1})\Psi_{v},\Phi^{\ast}_{v}\boxtimes\pi^{\vee}(g_{2})\Psi^{\ast}_{v}}\right\rangle_{\Pi_{v}}}.

(3.8)

Here and afterwards, we denote $\left|\left|{\Phi}\right|\right|_{\mathrm{Pet}}^{2}:=\left\langle{\Phi,\Phi^{\ast}}\right\rangle_{\mathrm{Pet}}$ and similarly for $\Psi$ .

By the uniqueness (up to scalar) of the $\mathbf{G}$ -invariant pairings $\Pi\times\Pi^{\vee}\rightarrow\mathbb{C}$ , granted by (BC), we see that

\mathscr{I}_{v}(\sigma_{v},\pi_{v}):=\dfrac{I(\Phi_{v}\boxtimes\pi(g_{1})\Psi_{v},\Phi_{v}^{\ast}\boxtimes\pi^{\vee}(g_{2})\Psi^{\ast}_{v})}{\left\langle{\Phi_{v}\boxtimes\pi(g_{1})\Psi_{v},\Phi^{\ast}_{v}\boxtimes\pi^{\vee}(g_{2})\Psi^{\ast}_{v}}\right\rangle_{\Pi_{v}}}=\dfrac{I(\Phi_{v,0}\boxtimes\Psi_{v,0},\Phi_{v,0}^{\vee}\boxtimes\Psi^{\vee}_{v,0})}{\left\langle{\Phi_{v,0}\boxtimes\Psi_{v,0},\Phi^{\vee}_{v,0}\boxtimes\Psi^{\vee}_{v,0}}\right\rangle_{\Pi_{v}}}

(3.9)

for any $\Phi_{v,0},\Psi_{v,0},\Phi_{v,0}^{\vee},\Psi_{v,0}^{\vee}$ in corresponding spaces, hence (3.9) depends only on the local representation $\Pi_{v}$ , not on the particular vectors in $\Pi_{v}$ . Rewriting (3.8), we have

\dfrac{\mathcal{P}^{\sharp}(\Phi,\pi(g_{1})\Psi)\mathcal{P}^{\sharp}(\Phi^{\ast},\pi^{\vee}(g_{2})\Psi^{\ast})}{\left|\left|{\Phi}\right|\right|_{\mathrm{Pet}}^{2}(\pi(g_{1})\Psi,\pi^{\vee}(g_{2})\Psi^{\ast})_{\pi,\mathrm{Pet}}}=C_{H}\dfrac{\mathscr{L}(\sigma\times\pi)}{2^{\aleph_{\sigma}+\aleph_{\pi}}}\prod_{v\in\mathscr{V}_{\mathcal{F}}}\mathscr{L}(\sigma_{v}\times\pi_{v})^{-1}\mathscr{I}_{v}(\sigma_{v},\pi_{v}).

(3.10)

3.2.4. Reduce to the doubling integrals

To ease the notation in this subsection, we denote the right hand side of (3.10) simply by $(\star)$ . We have seen that it is independent of $g_{1}$ and $g_{2}$ . So in $(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}$ , we drag it out of the integrals and obtain

	$\displaystyle(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}$	$\displaystyle=(\star)\cdot\left\|\left\|{\Phi}\right\|\right\|_{\sigma,\mathrm{Pet}}^{2}\int_{G(\mathbb{A})\times G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{1},1))f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))\left\langle{\pi(g_{1})\Psi,\pi^{\vee}(g_{2})\Psi^{\ast}}\right\rangle_{\pi,\mathrm{Pet}}\,\mathrm{d}g_{1}\,\mathrm{d}g_{2}$
		$\displaystyle=(\star)\cdot\left\|\left\|{\Phi}\right\|\right\|_{\sigma,\mathrm{Pet}}^{2}\int_{G(\mathbb{A})}\left(\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{1},1))\left\langle{\pi(g_{1})\Psi,\pi^{\vee}(g_{2})\Psi^{\ast}}\right\rangle_{\pi,\mathrm{Pet}}\,\mathrm{d}g_{1}\right)f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))\,\mathrm{d}g_{2}.$

The main focus of this subsection is to deal with such integrals. One notes that the integral in the parentheses is just $Z^{\diamondsuit}(\Psi,\pi^{\vee}(g_{2})\Psi^{\ast},f^{\mathrm{Sieg}}_{s,\chi})$ by Corollary 2.4 (3) (putting $h_{0}=1$ ). Breaking it into local doubling integrals by Corollary 2.5 (1), we have

(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}=(\star)\cdot\left|\left|{\Phi}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\cdot\prod_{v\in\mathscr{V}_{\mathcal{F}}}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v})\cdot\int_{G(\mathbb{A})}f^{\mathrm{Sieg}}_{s,\chi}(\imath^{\diamondsuit}(g_{2},1))\left\langle{\Psi,\pi^{\vee}(g_{2})\Psi^{\ast}}\right\rangle_{\pi,\mathrm{Pet}}\,\mathrm{d}g_{2}.

Observe that $\left\langle{\Psi,\pi^{\vee}(g_{2})\Psi^{\ast}}\right\rangle_{\pi,\mathrm{Pet}}=\left\langle{\pi^{\vee}(g_{2})\Psi^{\ast},\Psi}\right\rangle_{\pi^{\vee},\mathrm{Pet}}$ by identifying $(\pi^{\vee})^{\vee}$ with $\pi$ , the remaining integral is just $Z^{\diamondsuit}(\Psi^{\ast},\Psi,f^{\mathrm{Sieg}}_{s,\chi})$ by Corollary 2.4 (3) (putting $h_{0}=1$ ). Again break it into local doubling integrals by Corollary 2.5 (1), we obtain

(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}=(\star)\cdot\left|\left|{\Phi}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\Psi}\right|\right|_{\pi,\mathrm{Pet}}^{2}\cdot\prod_{v\in\mathscr{V}_{\mathcal{F}}}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v}^{\vee}).

To conclude, we have finally obtain the following result.

Theorem 3.4.

Notations being as above, we have

\dfrac{(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}}{\left|\left|{\Phi}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\Psi}\right|\right|_{\pi,\mathrm{Pet}}^{2}}=\dfrac{\mathscr{L}(\sigma\times\pi)}{2^{\varkappa_{\sigma}+\varkappa_{\pi}}}\prod_{v\in\mathscr{V}_{\mathcal{F}}}\dfrac{\mathscr{I}_{v}(\sigma_{v},\pi_{v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v}^{\vee})}{\mathscr{L}(\sigma_{v}\times\pi_{v})}.

(3.11)

3.3. Unramified Computations

We have the following standard results relating local integrals to local $L$ -factors at unramified places.

Theorem 3.5 ([Har14, Theorem 2.12]).

Let $v\in\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}$ , then $\mathscr{I}_{v}(\sigma_{v},\pi_{v})=\mathscr{L}_{v}(\sigma_{v}\times\pi_{v})$ .

Theorem 3.6 ([EHLS20, Section 4.2.1]).

Let $v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ , then

\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg},\mathrm{sph}}_{v,s,\chi},\pi_{v})=d_{N+1,v}(s,\chi_{v})^{-1}\cdot L_{v}\left(s+\dfrac{1}{2},\pi_{v},\chi_{v}\right),

where

•

$d_{n,v}(s,\chi_{v}):=\prod_{j=1}^{n}L_{v}(2s+j,\chi^{\mathcal{F}}\cdot\epsilon_{\mathcal{K}/\mathcal{F}}^{n-j})$ is the product of loca $L$ -factors for Hecke characters over $\mathcal{F}$ ,
•

$L_{v}(s+\dfrac{1}{2},\pi_{v},\chi_{v})=L_{v}(s,\mathrm{BC}(\pi_{v})\otimes\chi_{v}\circ\det)$ , where the right hand side is the standard local Godement-Jacquet $L$ -factor and $\mathrm{BC}(\pi_{v})$ is the local base change from $G(\mathcal{F}_{v})$ to $\mathrm{GL}_{N+1}(\mathcal{K}_{v})$ .

Combining Theorem 3.4, 3.5 and 3.6, we have the following result.

Theorem 3.7.

We assume Assumption (BC). We choose Siegel Eisenstein sections at $v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ as spherical sections $f^{\mathrm{Sieg},\mathrm{sph}}_{v,s,\chi}$ in the construction of the Klingen Eisenstein series. Then

\dfrac{(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}}{\left|\left|{\Phi}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\Psi}\right|\right|_{\pi,\mathrm{Pet}}^{2}}=\dfrac{1}{2^{\varkappa_{\sigma}+\varkappa_{\pi}}}\cdot\mathscr{L}_{\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}}(\sigma\times\pi)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v},\chi_{v}\right)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v}^{\vee},\chi_{v}\right)\\ \times\prod_{v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{bad}}}\mathscr{I}_{v}(\sigma_{v},\pi_{v})\prod_{v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}d_{N+1,v}(s,\chi_{v})^{-1}\prod_{v\in\mathscr{S}_{\mathcal{F}}^{\mathrm{bad}}}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v}^{\vee}),

where $\mathscr{L}_{\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}}(\cdots)$ and $L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}(\cdots)$ is the product of local $L$ -factors $\mathscr{L}_{v}(\cdots)$ and $L_{v}(\cdots)$ for $v$ running through $\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}$ or $\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ respectively.

3.4. Further assumptions

In what follows, we shall

(1)

compute the local Ichino-Ikeda integrals at bad places $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{bad}}$ , from Section 4 to 5.
(2)

choose Siegel Eisenstein sections carefully at bad places $v\in\mathscr{S}_{\mathcal{F}}^{\mathrm{bad}}$ such that the resulting Siegel Eisenstein series has $p$ -adic interpolatable $q$ -expansions, and therefore Siegel Eisenstein series, together with the Klingen Eisenstein series via pullback, can be interpolated into $p$ -adic families (see Section 8).

For the convenience in step (1), we put more assumptions on the cuspidal representations $\sigma$ and $\pi$ and their “relative position”.

Assumption 3.8 (Weight interlacing assumption).

We assume that for any $v\in\mathscr{V}_{\mathcal{F}}^{\infty}$ , representations $\sigma_{v}^{\vee}$ and $\pi_{v}$ of real Lie groups $\mathrm{U}_{m,n}$ and $\mathrm{U}_{m,n+1}$ are irreducible discrete series representation and satisfy the “Gan-Gross-Prasad weight interlacing property”. We shall recall it in Section 4.

Assumption 3.9 (Unramified assumption).

We assume that $\mathcal{K}/\mathcal{F}$ is unramified at any finite places of $\mathcal{F}$ .

Assumption 3.10 (Splitting assumption).

We assume that every $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$ splits in $\mathcal{K}$ .

Assumption 3.11 (Disjointly ramified assumption).

We assume that every $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$ , one of $\pi_{v}$ and $\sigma_{v}$ is unramified, i.e. they cannot be both ramified.

Very roughly speaking, Assumption 3.11 is some kind of “Heegner hypothesis”, and it is known that there are infinitely many imaginary quadratic extensions $\mathcal{K}$ of $\mathcal{F}$ such that this assumption is satisfied. When understanding $\sigma$ and $\pi$ as automorphic representations “generated” by “modular forms” of “tame levels” $N_{\sigma}$ and $N_{\pi}$ , Assumption 3.11 is requiring $N_{\sigma}$ and $N_{\pi}$ be coprime.

Assumption 3.9 may be awkward. This is because the computation of local Ichino-Ikeda integrals at places where $\mathcal{K}/\mathcal{F}$ is ramified with both $\pi_{v}$ and $\sigma_{v}$ unramified is still out of reach for us, for general signatures. It is proved that

\mathscr{I}_{v}(\sigma_{v},\pi_{v})=\mathscr{L}_{v}(\sigma_{v}\times\pi_{v}).

in the case $H=\mathrm{U}(2,0)$ and $G=\mathrm{U}(3,0)$ in [HY23, Appendix B]. It is believed that their method can be generalized to arbitrary signature case, by combinatorial brute force. We shall pursue this in future works. In particular, we note that this assumption exclude the case where $\mathcal{K}/\mathcal{F}$ is an imaginary quadratic extension of $\mathbb{Q}$ .

For simplicity, for any place $v$ of $\mathcal{F}$ , when it is clear from the context, we shall write

•

$H:=H(\mathcal{F}_{v})$ , $G:=G(\mathcal{F}_{v})$ and similarly for other groups,
•

$\sigma:=\sigma_{v}$ , $\pi:=\pi_{v}$ and similarly for other local representations, and
•

extension of local fields $\mathcal{K}_{v}/\mathcal{F}_{v}$ will be written as $K/F$ if this will not cause any confusion. In this case, we write $\mathfrak{o},\mathfrak{p},\varpi$ for the ring of integers, the maximal ideal, and a fixed choice of uniformizer of $F$ respectively, and let $q$ be the cardinality of the residue field $\mathfrak{o}/\mathfrak{p}$ of $F$ . The absolute value $\left|{-}\right|$ on $F$ is normalized via $\left|{\varpi}\right|=q^{-1}$ and the corresponding valuation on $F$ is denoted by $v$ .

4. Local Ichino-Ikeda integrals at archimedean places

In this section, we consider $v\in\mathscr{V}_{\mathcal{F}}^{\infty}$ , an archimedean place of $\mathcal{F}$ .

4.1. Weight interlacing property

As promised, we shall first introduce the weight interlacing assumption (Assumption 3.8). The primary reference is [He17].

Let $\sigma$ be a discrete series representation of $\mathrm{U}_{m,n}$ . It is known that discrete series of $\mathrm{U}_{m,n}$ are parameterized by Harish-Chandra parameters $(\xi,z)$ , where $\xi\in\mathbb{R}^{m+n}$ is a sequence of distinct integers or half-integers and $z\in\{+,-\}^{m+n}$ is a sequence of $+$ and $-$ corresponding to each entry in $\xi$ , such that the total number of $+$ ’s must be $m$ and the total number of $-$ ’s must be $n$ . We denote such a discrete series as $\mathcal{D}(\xi,z)$ .

Let $\sigma=\mathcal{D}(\chi,z)$ be a discrete series of $\mathrm{U}_{m,n}$ and $\pi=\mathcal{D}(\eta,t)$ be a discrete series of $\mathrm{U}_{m+1,n}$ . Here we represent the $+,-$ in $t$ by $\oplus,\ominus$ . We say $\sigma$ and $\pi$ satisfies the Gan-Gross-Prasad weight interlacing relation if one can line up $\xi$ and $\eta$ in the descending ordering such that the corresponding sequence of signs from $z$ and $t$ only has the following eight adjacent pairs

(\oplus,+),(+,\oplus),(-,\ominus),(\ominus,-),(+,-),(-,+),(\oplus,\ominus),(\ominus,\oplus).

The local Gan-Gross-Prasad conjecture at archimedean places for discrete series, now being a theorem of He [He17, Theorem 1.1], is the following result.

Theorem 4.1.

The discrete series $\mathcal{D}(\eta,t)$ of $\mathrm{U}_{m,n}$ appears as a subrepresentation of the restriction $\mathcal{D}(\xi,z)|_{\mathrm{U}_{m,n}}$ if and only if $\mathcal{D}(\eta,t)$ and $\mathcal{D}(\xi,z)$ satisfy the Gan-Gross-Prasad weight interlacing property.

4.2. Local Ichino-Ikeda integral

By adjointness of corresponding functors, we have

\operatorname{Hom}_{H}(\Pi,\mathbb{C})=\operatorname{Hom}_{H}\left(\pi|_{H}\boxtimes\sigma,\mathbb{C}\right)=\operatorname{Hom}_{H}(\pi|_{H},\operatorname{Hom}_{H}(\sigma,\mathbb{C}))=\operatorname{Hom}_{H}(\pi|_{H},\sigma^{\vee})

Note that since $\sigma$ is an irreducible admissible representation of $H$ , so does its contragredient $\sigma^{\vee}$ . By Schur’s lemma, we see that $\operatorname{Hom}_{H}(\Pi,\mathbb{C})$ is nonzero if and only if $\sigma^{\vee}$ is a subrepresentation of $\pi|_{H}$ , hence if and only if $\sigma^{\vee}$ and $\pi$ satisfy the GGP weight interlacing property by Theorem 4.1.

One further notes that the condition “ $\operatorname{Hom}_{H}(\Pi,\mathbb{C})\neq\{0\}$ ” says that there are nontrivial $H$ -invariant vectors $\Xi^{\circ}\in\Pi$ . Therefore, it follows directly from the definition that

I(\Xi^{\circ},\Xi^{\vee}):=\int_{H}\left\langle{\Pi(\Delta^{\sharp}(h))\Xi^{\circ},\Xi^{\vee}}\right\rangle_{\Pi}\,\mathrm{d}h=\left\langle{\Xi^{\circ},\Xi^{\vee}}\right\rangle

for any $\Xi^{\vee}\in\Pi^{\vee}$ . Therefore, we have the following proposition.

Proposition 4.2.

Under Assumption 3.8, for any $v\in\mathscr{V}_{\mathcal{F}}^{\infty}$ , $\mathscr{I}_{v}(\sigma_{v},\pi_{v})=1$ .

We remark that the weight interlacing assumption (i.e. Assumption 3.8) guarantees that the local Ichino-Ikeda integral is nonzero.

5. Local Ichino-Ikeda integrals at split primes

In this section, we deal with the local Ichino-Ikeda integrals at places $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{spl}}$ , i.e. finite places of $\mathcal{F}$ that splits in $\mathcal{K}$ . It includes places of $\mathcal{F}$ above $p$ .

In Section 2.1.1, we have seen that there are isomorphisms

\varrho_{w,m,n}:H\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathrm{GL}_{m+n}(F),\quad\varrho_{w,m,n+1}:G\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathrm{GL}_{m+n+1}(F)

To make it more convenience when dealing with local Ichino-Ikeda integrals, we adjust the isomorphisms $\varrho_{w,n}$ and $\varrho_{w,n+1}$ such that the diagram

commutes, with

\jmath^{\mathrm{GL}}:\mathrm{GL}_{m+n}(\mathcal{F}_{v})\hookrightarrow\mathrm{GL}_{m+n+1}(\mathcal{F}_{v}),\quad g\mapsto\operatorname{diag}[g,1]

Under isomorphisms $\varrho_{w,m+n}$ and $\varrho_{w,m+n+1}$ , we regard $\sigma$ and $\pi$ as representations of $\mathrm{GL}_{m+n}(F)$ and $\mathrm{GL}_{m+n+1}(F)$ respectively.

5.1. Basic representation theory of $\mathrm{GL}_{n}(F)$

In this subsection, we briefly review some basic representation theory of $\mathrm{GL}_{n}(F)$ . Solely in this subsection, we let $m,n,l$ be a general positive integers (that are not related to the unitary groups before).

We fix an additive character $\boldsymbol{\psi}:F\rightarrow\mathbb{C}^{\times}$ which is trivial on $\mathfrak{o}$ and nontrivial on $\mathfrak{p}^{-1}$ . We write $\mathscr{S}(F)$ for the space of locally constant compactly supported functions on $F$ . The Fourier transform of $\phi\in\mathscr{S}(F)$ is defined by

\widehat{\phi}(y)=\int_{F}\phi(x)\boldsymbol{\psi}(-xy)\,\mathrm{d}x.

The measure $\,\mathrm{d}x$ is chosen so that $\widehat{\widehat{\phi}}(x)=\phi(-x)$ .

Put $\tau:F^{\times}\rightarrow\mathbb{C}^{\times}$ an arbitrary multiplicative character of $F^{\times}$ , we define $\mathfrak{g}(\tau,\boldsymbol{\psi},y)$ denotes the Gauss sum

\mathfrak{g}(\tau,\boldsymbol{\psi},y):=\int_{\mathfrak{o}^{\times}}\tau(z)\boldsymbol{\psi}(yz)\,\mathrm{d}^{\times}z

for $y\in F$ . When $y=1$ and $\boldsymbol{\psi}$ is clear from contexts, we shall simply write $\mathfrak{g}(\tau)$ for short.

5.1.1. Whittaker models

Extend $\boldsymbol{\psi}$ to $U_{n}(F)$ by the rule

\boldsymbol{\psi}(u):=\prod_{i=1}^{n-1}\boldsymbol{\psi}(u_{i,i+1}),\,u=(u_{i,j})\in U_{n}(F).

This is a generic character. Note that it is trivial on $U_{n}(\mathfrak{o})$ .

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_{n}(F)$ . It is called generic if

\operatorname{Hom}_{\mathrm{GL}_{n}(F)}(\pi,\operatorname{Ind}_{U_{n}(F)}^{\mathrm{GL}_{n}(F)}\boldsymbol{\psi})\neq 0.

It is known that tempered representations are always generic. By Frobenius reciprocity, this means that there exists a nonzero linear form $\lambda:V_{\pi}\rightarrow\mathbb{C}$ such that

\lambda(\pi(u)v)=\boldsymbol{\psi}(u)\lambda(v),\quad v\in V_{\pi},u\in U_{n}(F).

It is known that for a generic $\pi$ , $\operatorname{Hom}_{\mathrm{GL}_{n}(F)}(\pi,\operatorname{Ind}_{U_{n}(F)}^{\mathrm{GL}_{n}(F)}\boldsymbol{\psi})$ is of dimension one. The Whittaker model $\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ of $\pi$ with respect to $\boldsymbol{\psi}$ is defined as

\mathcal{W}_{\boldsymbol{\psi}}(\pi):=\{W_{v}:\mathrm{GL}_{n}(F)\rightarrow\mathbb{C}:W_{v}(g)=\lambda(\pi(g)v),v\in V_{\pi}\}.

Then $\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ is independent of the choice of $\lambda$ , and for $u\in U_{n}(F)$ , $g\in\mathrm{GL}_{n}(F)$ ,

W_{v}(ug)=\boldsymbol{\psi}(u)W_{v}(g),\,W_{v}(g)=W_{\pi(g)v}(\mathbf{1}_{n}).

Then the map $v\mapsto W_{v}$ gives an isomorphism $\mathsf{W}_{\pi}:V_{\pi}\rightarrow\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ . One can define an invariant perfect pairing

\left\langle{-,-}\right\rangle:\mathcal{W}_{\boldsymbol{\psi}}(\pi)\otimes\mathcal{W}_{\boldsymbol{\psi}^{-1}}(\pi^{\vee})\rightarrow\mathbb{C}

such that $\left\langle{\phi,\phi^{\vee}}\right\rangle=\left\langle{\mathsf{W}_{\pi}(\phi),\mathsf{W}_{\pi^{\vee}}(\phi^{\vee})}\right\rangle$ , where the pairing $\left\langle{-,-}\right\rangle$ on the left is the canonical pairing between $\pi$ and $\pi^{\vee}$ .

Given $W\in\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ , we define $\widetilde{W}\in\mathcal{W}_{\boldsymbol{\psi}^{-1}}(\pi^{\vee})$ by $\widetilde{W}(g)=W(w_{n}g^{-\mathrm{t}})$ , where $w_{n}:=\begin{bmatrix}&&&1\\ &&1&\\ &\iddots&&\\ 1&&&\end{bmatrix}$ is the longest Weyl element.

5.1.2. The JPSS integrals and local JPSS $L$ -factors

Now let $\pi$ be an irreducible admissible generic representation of $\mathrm{GL}_{m+1}(F)$ . Let $n$ be a positive integer which is equal or less than $m$ . Put $l=m-n$ . Let $\sigma$ be an irreducible admissible generic representation of $\mathrm{GL}_{n}(F)$ whose central character is $\omega_{\sigma}$ . We associate to Whittaker functions $W\in\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ and $W^{\prime}\in\mathcal{W}_{\boldsymbol{\psi}^{-1}}(\sigma)$ the local zeta integrals ¹⁰¹⁰10There appears to be a typographical error on [HY23, page 37] where it was written that $W^{\prime}\in\mathcal{W}_{\boldsymbol{\psi}}(\sigma)$ .

	$\displaystyle Z(s,W,W^{\prime})$	$\displaystyle=\int_{U_{n}(F)\backslash\mathrm{GL}_{n}(F)}W\left(\begin{bmatrix}h&\\ &\mathbf{1}_{l+1}\end{bmatrix}\right)W^{\prime}(h)\left\|{\det h}\right\|^{s-\frac{l+1}{2}}\,\mathrm{d}h,$
	$\displaystyle\widetilde{Z}(s,\widetilde{W},\widetilde{W^{\prime}})$	$\displaystyle=\int_{U_{n}(F)\backslash\mathrm{GL}_{n}(F)}\int_{\mathrm{M}_{l\times n}(F)}\widetilde{W}\left(\begin{bmatrix}h&&\\ x&\mathbf{1}_{l}&\\ &&1\end{bmatrix}\right)\widetilde{W^{\prime}}(h)\left\|{\det h}\right\|^{s-\frac{l+1}{2}}\,\mathrm{d}x\,\mathrm{d}h,$

which converge absolutely for $\operatorname{Re}s\gg 0$ , where $\,\mathrm{d}h$ is the Haar measure on $\mathrm{GL}_{n}(F)$ giving $\mathrm{GL}_{n}(\mathfrak{o})$ volume $1$ .

We write $L^{\mathrm{JPSS}}(s,\pi\times\sigma)$ , $\varepsilon^{\mathrm{JPSS}}(s,\pi\times\sigma,\boldsymbol{\psi})$ and $\gamma^{\mathrm{JPSS}}(s,\pi\times\sigma,\boldsymbol{\psi})$ for the $L$ , epsilon and gamma factors associated to $\pi$ and $\sigma$ . These local factors are studied extensively in [JPS83]. The gamma factor is defined as the proportionality constant of the functional equation

Z(1-s,\pi^{\vee}(w_{m+1,n})\widetilde{W},\widetilde{W^{\prime}})=\omega_{\sigma}(-1)^{m}\gamma^{\mathrm{JPSS}}(s,\pi\times\sigma,\boldsymbol{\psi})\widetilde{Z}(s,W,W^{\prime}),

(5.12)

where

w_{m+1,n}=\begin{bmatrix}\mathbf{1}_{n}&\\ &w_{m-n+1}\end{bmatrix}.

Remark 5.1.

We are only interested in the case $n=m$ (so $l=0$ ). By a change of variables, we see that

Z(s,W,\sigma(h_{0})W^{\prime})=\left|{\det h_{0}}\right|^{\frac{1}{2}-s}Z(s,\pi(\jmath^{\mathrm{GL}}(h_{0}^{-1}))W,W^{\prime})

(5.13)

for any $h_{0}\in\mathrm{GL}_{n}(F)$ . This property is called the invariance of JPSS integrals.

When we view $\pi$ and $\sigma$ are representations of unitary groups over the split quadratic algebra $K=F\oplus F$ ,

L(s,\pi\times\sigma)=L^{\mathrm{JPSS}}(s,\pi\times\sigma)L^{\mathrm{JPSS}}(s,\pi^{\vee}\times\sigma^{\vee}).

(5.14)

When $n=1$ and $\chi$ is a character of $F^{\times}$ , we have

	$L^{\mathrm{GJ}}(s,\pi\otimes\chi)=L^{\mathrm{JPSS}}(s,\pi\times\chi),$		(5.15a)
	$\varepsilon^{\mathrm{GJ}}(s,\pi\otimes\chi,\boldsymbol{\psi})=\varepsilon^{\mathrm{JPSS}}(s,\pi\times\chi,\boldsymbol{\psi}),$		(5.15b)
	$\gamma^{\mathrm{GJ}}(s,\pi\otimes\chi,\boldsymbol{\psi})=\gamma^{\mathrm{JPSS}}(s,\pi\times\chi,\boldsymbol{\psi}),$		(5.15c)

where the local $L$ -factors are the Godement-Jacquet $L$ -factors studied extensively in [GJ72]. Moreover, recall that

L(s,\pi,\operatorname{Ad})=L^{\mathrm{JPSS}}(s,\pi\times\pi^{\vee}),\quad L(s,\sigma,\operatorname{Ad})=L^{\mathrm{JPSS}}(s,\sigma\times\sigma^{\vee}),

(5.16)

where we regard $\pi$ and $\sigma$ as representation of unitary groups on the left hand side and representations of general linear groups on the right hand side. Combining (5.14) and (5.16), we obtain

\mathscr{L}(\pi\times\sigma)=\dfrac{L^{\mathrm{JPSS}}\left(\dfrac{1}{2},\pi\times\sigma\right)L^{\mathrm{JPSS}}\left(\dfrac{1}{2},\pi^{\vee}\times\sigma^{\vee}\right)}{L^{\mathrm{JPSS}}\left(\dfrac{1}{2},\pi\times\pi^{\vee}\right)L^{\mathrm{JPSS}}\left(\dfrac{1}{2},\sigma\times\sigma^{\vee}\right)}.

(5.17)

5.1.3. The naive local Rankin-Selberg $L$ -factor

We keep the notations and conventions of previous section, and consider the special case $n=m$ (so $l=0$ ). By definition, the local Godement-Jacquet $L$ -function $L^{\mathrm{GJ}}(s,\sigma)$ is of the form $P_{\sigma}(q^{-s})^{-1}$ , where $P_{\sigma}\in\mathbb{C}[X]$ has degree at most $n$ and satisfies $P_{\sigma}(0)=1$ , We may then find $n$ complex numbers $\{\alpha_{i}\}_{i=1}^{n}$ (some of them may be zero) such that

L^{\mathrm{GJ}}(s,\sigma)=\prod_{i=1}^{n}(1-\alpha_{i}q^{-s})^{-1}.

We call the set $\{\alpha_{i}\}$ the Langlands parameter of $\sigma$ . Let $\{\gamma_{i}\}_{j=1}^{n+1}$ be the Langlands parameter of $\pi$ . Then we define

L^{\mathrm{RS}}(s,\pi\times\sigma):=\prod_{i=1}^{n}\prod_{j=1}^{n+1}(1-\alpha_{i}\gamma_{j}q^{-s})^{-1},

to be the naive local Rankin-Selberg $L$ -factor. We shall compare it with the local JPSS $L$ -factor. Morally speaking, $L^{\mathrm{RS}}(s,\pi\times\sigma)$ sees the “unramified part” of $L^{\mathrm{JPSS}}(s,\pi\times\sigma)$ .

We need Bernstein-Zelevinsky’s classification theorm of irreducible admissible representations of $\mathrm{GL}_{n}(F)$ . For our purpose, we restrict us to the classification of tempered ones ¹¹¹¹11See, for example, [GH11b, Theorem 14.6.4-14.6.5] for the case $\mathrm{GL}_{n}(\mathbb{Q}_{p})$ . There are two phrases.

•

Discrete series. Let $r,d$ be two positive integers such that $rd=n$ and let $\eta$ be an irreducible supercuspidal representation of $\mathrm{GL}_{r}(F)$ with unitary central character. Then

\operatorname{Ind}_{P[r,\ldots,r]}^{\mathrm{GL}_{n}(F)}(\eta,\boldsymbol{\alpha}_{F}^{(1-d)/2}\eta,\ldots,\boldsymbol{\alpha}_{F}^{(d-1)/2}\eta)

has a unique irreducible quotient, denoted by $\mathrm{BZ}(\eta,d)$ . Every discrete series representation of $\mathrm{GL}_{n}(F)$ is isomorphic to some $\mathrm{BZ}(\eta,d)$ .

•

Tempered representations. Let $n=r_{1}+\cdots+r_{s}$ be a partition of $n$ and $\tau_{i}$ be discrete series of $\mathrm{GL}_{r_{i}}(F)$ for $i=1,\ldots,s$ . Then

$\tau_{1}\boxplus\cdots\boxplus\tau_{s}:=\operatorname{Ind}_{P[r_{1},\ldots,r_{s}]}^{\mathrm{GL}_{n}(F)}(\tau_{1},\ldots,\tau_{s})$

is irreducible and tempered. Every tempered representation is isomorphic to this form. The operation “ $\boxplus$ ” is called the isobaric sum.

Let $\sigma=\tau_{1}\boxplus\cdots\boxplus\tau_{s}$ be a tempered representation of $\mathrm{GL}_{n}(F)$ . Let $(\tau_{j_{1}},\ldots,\tau_{j_{r}})$ be the tuple of unramified isobaric summands of $\sigma$ with increasing $j_{k}$ ’s. We define the unramified socle of $\sigma$ as the isobaric sum $\sigma_{{\mathrm{ur}}}:=\boxplus_{k=1}^{r}\tau_{j_{k}}$ as an unramified tempered representation of $\mathrm{GL}_{r_{j_{1}}+\cdots r_{j_{r}}}(F)$ . Obviously if $\sigma$ itself is unramified, then $\sigma_{{\mathrm{ur}}}=\sigma$ .

Proposition 5.2.

With conventions and notations above, we have:

(1)

The local $L$ -factors $L^{\mathrm{RS}}(s,-\times-)$ and $L^{\mathrm{JPSS}}(s,-\times-)$ are bi-additive under the isobaric sum, i.e.

	$\displaystyle L^{\bullet}(s,\pi\times(\sigma\boxplus\sigma^{\prime}))$	$\displaystyle=L^{\bullet}(s,\pi\times\sigma)L^{\bullet}(s,\pi\times\sigma^{\prime}),$
	$\displaystyle L^{\bullet}(s,(\pi\boxplus\pi^{\prime}),\sigma)$	$\displaystyle=L^{\bullet}(s,\pi\times\sigma)L^{\bullet}(s,\pi^{\prime}\times\sigma),$

for $\bullet\in\{\mathrm{RS},\mathrm{JPSS}\}$ and all irreducible admissible representations $\pi,\pi^{\prime},\sigma,\sigma^{\prime}$ of general linear groups.

(2)

If both $\sigma$ and $\pi$ are unramified irreducible admissible representations of $\mathrm{GL}_{n}(F)$ and $\mathrm{GL}_{n+1}(F)$ respectively, then $L^{\mathrm{RS}}(s,\pi\times\sigma)=L^{\mathrm{JPSS}}(s,\pi\times\sigma)$ .
(3)

If either $\sigma$ or $\pi$ is ramified discrete series representations of $\mathrm{GL}_{n}(F)$ and $\mathrm{GL}_{n+1}(F)$ respectively, then $L^{\mathrm{RS}}(s,\pi\times\sigma)=1$ .

Therefore, we have $L^{\mathrm{RS}}(s,\pi\times\sigma)=L^{\mathrm{JPSS}}(s,\pi\times\sigma_{{\mathrm{ur}}})=L^{\mathrm{JPSS}}(s,\pi_{{\mathrm{ur}}}\times\sigma)$ .

Proof.

The bi-additive property of $L^{\mathrm{RS}}$ follows easily from the definition, and that of $L^{\mathrm{JPSS}}$ is [JPS83, Section 9.5, Theorem]. The result in (2) is stated in [JPS83, Equation (14) on page 371]. For (3), it follows from the definition of $L^{\mathrm{RS}}$ that it suffices to show the following claim: let $\tau$ be a ramified discrete series of $\mathrm{GL}_{n}(F)$ , then $L^{\mathrm{GJ}}(s,\tau)=1$ . (Clearly, this is equivalent to that all Langlands parameters of $\tau$ are zero.) To show this claim, by Bernstein-Zelevinsky’s classification, $\tau=\mathrm{BZ}(\eta,d)$ . Then we know

L^{\mathrm{GJ}}(s,\tau)=\begin{cases}L(s,\boldsymbol{\alpha}_{F}^{1-n}\eta),&\quad d=n,\\ 1,&\quad d\neq n,\end{cases}

where the $L$ -factor on the first line is the local Hecke $L$ -factor for Hecke $L$ -functions. When $d\neq n$ , the claim follows from this fact. When $d=n$ , if $\tau$ is ramified, $\eta$ is then ramified and hence $L(s,\boldsymbol{\alpha}_{F}^{1-n}\eta)=1$ as well. So the claim is proved. The final equalities in the proposition follows directly from (1) to (3), and the definition of unramified socles. ∎

5.1.4. Essential Whittaker vectors

Now we review the theory of the essential Whittaker vector associated to an irreducible admissible generic representation $\pi$ of $\mathrm{GL}_{n}(F)$ . Given an open compact subgroup $\Gamma$ of $\mathrm{GL}_{n}(F)$ and its character $\mathcal{X}:\Gamma\rightarrow\mathbb{C}^{\times}$ , we put

\mathcal{W}_{\boldsymbol{\psi}}(\pi,\Gamma,\mathcal{X})=\{W\in\mathcal{W}_{\psi}(\pi):\pi(\gamma)W=\mathcal{X}(\gamma)W\text{ for }\gamma\in\Gamma\}.

For any integer $f\geq 0$ , we consider the following two compact open subgroups of $\mathrm{GL}_{n}(F)$ :

	$\displaystyle K_{1}(\mathfrak{p}^{f})$	$\displaystyle:=\left\{g\in\mathrm{GL}_{n}(\mathfrak{o}):g\equiv\begin{bmatrix}\ast&\ast\\ 0_{1\times(n-1)}&\mathbf{1}_{1}\end{bmatrix}\pmod{\mathfrak{p}^{f}}\right\},$
	$\displaystyle K_{0}(\mathfrak{p}^{f})$	$\displaystyle:=\left\{g\in\mathrm{GL}_{n}(\mathfrak{o}):g\equiv\begin{bmatrix}\ast&\ast\\ 0_{1\times(n-1)}&\ast\end{bmatrix}\pmod{\mathfrak{p}^{f}}\right\},$

so that $K_{1}(\mathfrak{p}^{f})$ is a normal subgroup of $K_{0}(\mathfrak{p}^{f})$ , with quotient $K_{0}(\mathfrak{p}^{f})/K_{1}(\mathfrak{p}^{f})\cong(\mathfrak{o}/\mathfrak{p}^{f})^{\times}$ .

Let $c(\pi)$ be the exponent of the conductor of $\pi$ , i.e. the epsilon factor of $\pi$ satisfies

\varepsilon\left(s+\dfrac{1}{2},\pi,\boldsymbol{\psi}\right)=q^{-c(\pi)s}\varepsilon\left(\dfrac{1}{2},\pi,\boldsymbol{\psi}\right).

According to [JPS81, Jac12],

\dim_{\mathbb{C}}\mathcal{W}_{\boldsymbol{\psi}}(\pi,K_{1}(\mathfrak{p}^{c(\pi)}),\mathbf{1})=1.

This subspace of $\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ is called the essential line of $\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ , and we define the normalized essential Whittaker vector of $\pi$ with respect to $\boldsymbol{\psi}$ to be $W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}}\in\mathcal{W}_{\boldsymbol{\psi}}(\pi,K_{1}(\mathfrak{p}^{c(\pi)}),\mathbf{1})$ satisfying

W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}}\left(\begin{bmatrix}gh&\\ &1\end{bmatrix}\right)=W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}}\left(\begin{bmatrix}g&\\ &1\end{bmatrix}\right)\,\text{ for all }h\in\mathrm{GL}_{n-1}(\mathfrak{o}).

Remark 5.3.

(1)

If $\pi$ is unramified, then $c(\pi)=0$ and by the uniqueness of the essential vectors, $W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}}$ is nothing but the normalized spherical function.

(2)

The larger compact group $K_{0}(\mathfrak{p}^{c(\pi)})$ acts on the essential line via the central character $\omega_{\pi}$ of $\pi$ . Precisely, for $g=(g_{i,j})\in K_{0}(\mathfrak{p}^{c(\pi)})$ , define

\omega_{\pi}^{\downarrow}(g)=\begin{cases}1,&\quad\text{ if }c(\pi)=0,\\ \omega_{\pi}(g_{n,n}),&\quad\text{ if }c(\pi)>0.\end{cases}

Then clearly $\omega_{\pi}^{\downarrow}$ is a character of $K_{0}(\mathfrak{p}^{c(\pi)})$ trivial on $K_{1}(\mathfrak{p}^{c(\pi)})$ and

\pi(g)W_{\pi}^{\mathrm{ess}}=\omega_{\pi}^{\downarrow}(g)W_{\pi}^{\mathrm{ess}}\,\text{ for all }g\in K_{0}(\mathfrak{p}^{c(\pi)}).

(3)

We recall the definition of conductor of a multiplicative character $\chi$ of $F^{\times}$ , denoted by $c(\chi)$ . If $\chi$ is trivial, then the conductor of $\chi$ is $\mathfrak{o}$ , otherwise $c(\chi)=\mathfrak{p}^{n}$ , where $n\geq 1$ is the least integer such that $\chi$ is trivial on $1+\mathfrak{p}^{n}$ .

5.1.5. Test vector problem

Let $\sigma$ and $\pi$ be irreducible admissible tempered representation of $\mathrm{GL}_{n}(F)$ and $\mathrm{GL}_{n+1}(F)$ respectively. Previously we have defined the JPSS integrals $Z(s,W,W^{\prime})$ for Whittaker functions $W\in\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ and $W^{\prime}\in\mathcal{W}_{\boldsymbol{\psi}}(\sigma)$ . The test vector problem is to find Whittaker functions $W^{\circ}\in\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ and $W^{\prime,\circ}\in\mathcal{W}_{\boldsymbol{\psi}}(\sigma)$ such that

Z(s,W^{\circ},W^{\prime,\circ})=L^{\mathrm{JPSS}}(s,\pi\times\sigma).

In this section we introduce two partial results on this problem, which are sufficient to compute the local Ichino-Ikeda integral under Assumption 3.11.

Theorem 5.4 ([JPS81, Théorème on page 208]).

Suppose $\sigma$ is unramified, then

Z\left(\dfrac{1}{2},W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}},W_{\sigma,\boldsymbol{\psi}^{-1}}^{\mathrm{ess}}\right)=L^{\mathrm{JPSS}}\left(\dfrac{1}{2},\pi\times\sigma\right).

The case when $\sigma$ is ramified is more complicated. In this case, Booker, Krishmanrthy and Lee [BKL20] modified $W_{\pi,\psi}^{\mathrm{ess}}$ through a process of unipotent averaging. This method dates back to [Sch93], etc..

Let $\mathfrak{n}$ , $\mathfrak{q}$ , $\mathfrak{c}$ denote the conductor of $\pi$ , $\sigma$ and $\omega_{\sigma}$ (the central character of $\sigma$ ), respectively. Consider $\beta=(\beta_{1},\ldots,\beta_{n})\in F^{n}$ with $\beta_{i}\in\mathfrak{q}^{-1}$ for $i=1,\ldots,n$ . Let $\mathbf{u}(\beta)=(u_{i,j})$ denote the matrix

\mathbf{u}(\beta)=\begin{bmatrix}1&&&&\beta_{1}\\ &1&&&\beta_{2}\\ &&\ddots&&\vdots\\ &&&1&\beta_{n}\\ &&&&1\end{bmatrix}\in U_{n+1}(F).

We define for any $W\in\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ the unipotent averaging operator

\Theta W:=\dfrac{1}{[\mathfrak{o}:\mathfrak{q}]^{n-1}}\sum_{(\beta_{1},\ldots,\beta_{n-1})\in(\mathfrak{q}^{-1}/\mathfrak{o})^{n-1}}\pi(\mathbf{u}(\beta_{1},\ldots,\beta_{n-1},\varpi^{-c(\omega_{\sigma})}))W\in\mathcal{W}_{\boldsymbol{\psi}}(\pi).

(When $n=1$ , we understand there to be one summand, so that $\Theta W=\pi(\varpi^{-c(\omega_{\sigma})})W$ ).

Then we have the following theorem.

Theorem 5.5.

Notations and conventions as above. Under Assumption 3.11, when $\sigma_{v}$ is ramified, we have

Z\left(\dfrac{1}{2},\Theta W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}},W_{\sigma,\boldsymbol{\psi}^{-1}}^{\mathrm{ess}}\right)=C_{\sigma,\boldsymbol{\psi}}\cdot L^{\mathrm{JPSS}}\left(\dfrac{1}{2},\pi\times\sigma\right),

where $C_{\sigma,\boldsymbol{\psi}}$ is an explicit nonzero number

C_{\sigma,\boldsymbol{\psi}}:=\dfrac{\mathfrak{g}(\omega_{\sigma},\boldsymbol{\psi},\varpi^{-v(\mathfrak{c})})}{[\mathrm{GL}_{n}(\mathfrak{o}):K_{0}(\mathfrak{q})]}\neq 0.

Proof.

The main theorem of [BKL20] is that

Z\left(s,\Theta W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}},W_{\sigma,\boldsymbol{\psi}^{-1}}^{\mathrm{ess}}\right)=C_{\sigma,\boldsymbol{\psi}}\cdot L^{\mathrm{RS}}\left(s,\pi\times\sigma\right),

with the constant $C_{\sigma,\boldsymbol{\psi}}$ hidden on [BKL20, page 46]. Under Assumption 3.11, when $\sigma_{v}$ is ramified, $\pi_{v}$ is then unramified. Therefore by Proposition 5.2, we see $L^{\mathrm{RS}}(s,\pi\times\sigma)=L^{\mathrm{JPSS}}(s,\pi_{{\mathrm{ur}}}\times\sigma)=L^{\mathrm{JPSS}}(s,\pi\times\sigma)$ . Taking $s=1/2$ gives the result. ∎

5.2. The splitting lemma

Finally we finshed the preparations on local representation theory of general linear groups. We go back to the computation of local Ichino-Ikeda integrals.

Let $\pi$ be an irreducible admissible tempered representation of $\mathrm{GL}_{m+n+1}(F)$ and $\sigma$ that of $\mathrm{GL}_{m+n}(F)$ . We consider the integral

J(W_{1},W_{2},W_{1}^{\prime},W_{2}^{\prime}):=\int_{\mathrm{GL}_{m+n}(F)}\left\langle{\pi(\jmath^{\mathrm{GL}}(h))W_{1},W_{2}}\right\rangle_{\pi}\left\langle{\sigma(h)W_{1}^{\prime},W_{2}^{\prime}}\right\rangle_{\sigma}\,\mathrm{d}h.

where

W_{1}\in\mathcal{W}_{\boldsymbol{\psi}}(\pi),\,W_{2}\in\mathcal{W}_{\boldsymbol{\psi}^{-1}}(\pi^{\vee}),\,W_{1}^{\prime}\in\mathcal{W}_{\boldsymbol{\psi}^{-1}}(\sigma),\,W_{2}^{\prime}\in\mathcal{W}_{\boldsymbol{\psi}}(\sigma^{\vee}).

This integral converges.

An essential tool is the splitting lemma of Wei Zhang ¹²¹²12Note that [Zha14], Zhang used unnormalized local Haar measures while here we are using normalized ones, reformulated as [HY23, Lemma 5.2]. Interested readers can turn to [HY23, Remark 5.3] for some familiar special cases of this lemma, as follows.

Theorem 5.6 (Splitting lemma, [Zha14, Proposition 4.10]).

Notations being as above, we have

J(W_{1},W_{2},W_{1}^{\prime},W_{2}^{\prime})=Z\left(\dfrac{1}{2},W_{1},W_{1}^{\prime}\right)Z\left(\dfrac{1}{2},W_{2},W_{2}^{\prime}\right)\prod_{i=1}^{m+n-1}\zeta_{F}(i).

Therefore, the computation of local Ichino-Ikeda integrals reduces to that of local JPSS integrals. Indeed, under the identifications of unitary groups and general linear groups via $\varrho_{w,m,n}$ and $\varrho_{w,m,n+1}$ , and local Whittaker models, we have the following corollary.

Corollary 5.7.

Notations being as above, we have

\mathscr{I}(\sigma,\pi)=\dfrac{J(W_{1},W_{2},W_{1}^{\prime},W_{2}^{\prime})}{\left\langle{W_{1},W_{2}}\right\rangle\left\langle{W_{1}^{\prime},W_{2}^{\prime}}\right\rangle}=\dfrac{Z\left(1/2,W_{1},W_{1}^{\prime}\right)Z\left(1/2,W_{2},W_{2}^{\prime}\right)}{\left\langle{W_{1},W_{2}}\right\rangle\left\langle{W_{1}^{\prime},W_{2}^{\prime}}\right\rangle}\prod_{i=1}^{m+n-1}\zeta_{F}(i).

5.3. Local Ichino-Ikeda integrals at $\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$

Taking

\mathcal{B}_{\pi}(W_{1},W_{2}):=\dfrac{\left\langle{W_{1},W_{2}}\right\rangle}{L^{\mathrm{JPSS}}(1/2,\pi\times\pi^{\vee})},\quad\mathcal{B}_{\sigma}(W_{1}^{\prime},W_{2}^{\prime}):=\dfrac{\left\langle{W_{1}^{\prime},W_{2}^{\prime}}\right\rangle}{L^{\mathrm{JPSS}}(1/2,\sigma\times\sigma^{\vee})}

and further write ¹³¹³13We note that this is a little bit different from the definition of $\mathcal{B}_{\pi_{l}}$ and $\mathcal{B}_{\sigma_{l}}$ in [HY23, Sect. 4.7].

\mathcal{B}_{\pi}^{\mathrm{ess}}:=\mathcal{B}_{\pi}(W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}},W_{\pi^{\vee},\boldsymbol{\psi}^{-1}}^{\mathrm{ess}}),\quad\mathcal{B}_{\sigma}^{\mathrm{ess}}:=\mathcal{B}_{\sigma}(W_{\sigma,\boldsymbol{\psi}^{-1}}^{\mathrm{ess}},W_{\sigma,\boldsymbol{\psi}}^{\mathrm{ess}}).

They can be regarded as certain invariant of the local representations $\pi$ and $\sigma$ .

5.3.1. The case when $\sigma$ is unramified

We first consider the case when $\sigma$ is unramified. We take

W_{1}=W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}},\quad W_{1}^{\prime}=W_{\sigma,\boldsymbol{\psi}^{-1}}^{\mathrm{ess}},\quad W_{2}=W_{\pi^{\vee},\boldsymbol{\psi}^{-1}}^{\mathrm{ess}},\quad W_{2}^{\prime}=W_{\sigma^{\vee},\boldsymbol{\psi}}^{\mathrm{ess}}

in Corollary 5.7 and apply Theorem 5.4, together with (5.17), to obtain the following proposition.

Proposition 5.8.

Notations being as above, let $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$ such that $\sigma_{v}$ is unramified, then

\mathscr{I}(\sigma,\pi)=\mathscr{L}(\pi\times\sigma)\cdot\mathcal{B}_{\pi}^{\mathrm{ess}}\mathcal{B}_{\sigma}^{\mathrm{ess}}\cdot\prod_{i=1}^{m+n-1}\zeta_{F}(i).

5.3.2. The case when $\sigma$ is ramified

We take

W_{1}=\Theta W_{\pi,\boldsymbol{\psi}}^{\mathrm{ess}},\quad W_{1}^{\prime}=W_{\sigma,\boldsymbol{\psi}^{-1}}^{\mathrm{ess}},\quad W_{2}=\Theta W_{\pi^{\vee},\boldsymbol{\psi}^{-1}}^{\mathrm{ess}},\quad W_{2}^{\prime}=W_{\sigma^{\vee},\boldsymbol{\psi}}^{\mathrm{ess}}

in Corollary 5.7 and apply Theorem 5.5, together with (5.17), to obtain the following proposition.

Proposition 5.9.

Notations being as above, let $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$ such that $\sigma_{v}$ is ramified, then under Assumption 3.11, we have

\mathscr{I}(\sigma,\pi)=\mathscr{L}(\pi\times\sigma)\cdot C_{\sigma,\boldsymbol{\psi}}C_{\sigma^{\vee},\boldsymbol{\psi}^{-1}}\cdot\mathcal{B}_{\pi}^{\mathrm{ess}}\mathcal{B}_{\sigma}^{\mathrm{ess}}\cdot\prod_{i=1}^{m+n-1}\zeta_{F}(i).

5.4. Further representation theory of $\mathrm{GL}_{n}(F)$ : ordinary condition

To handle the local Ichino-Ikeda integrals at places $v$ of $F$ above $p$ , we introduce the background on the ordinary condition on irreducible admissible representations of $\mathrm{GL}_{n}(F)$ . Again solely in this subsection, we let $m,n$ be a general positive integers (that are not related to the unitary groups before).

Recall that at the very beginning of this article, we have fixed an identification $\iota_{p}:\mathbb{C}\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\overline{\mathbb{Q}}_{p}^{\times}$ . Through $\iota_{p}$ , we regard characters of $F$ as valued in $\overline{\mathbb{Q}}_{p}$ and every irreducible admissible representations $\pi$ of $\mathrm{GL}_{n}(F)$ has coefficient $\overline{\mathbb{Q}}_{p}$ .

5.4.1. Ordinary line

Let $\underline{\mu}=(\mu_{1},\ldots,\mu_{n})$ be a tuple of characters $F^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ , which naturally gives a character $\mu$ of $T_{n}(F)$ . We have an induced character $\mu^{\natural}$ of $T_{n}(F)$ given by

\mu^{\natural}:\operatorname{diag}[x_{1},\ldots,x_{n}]\mapsto\prod_{i=1}^{n}\left|{x_{i}}\right|^{n-i}\mu_{i}(x_{i}).

hence a character of $B_{n}(F)$ by inflation. Then define an algebraically induced principal series representation ¹⁴¹⁴14The representation $\left|{\det(-)}\right|^{\frac{1-n}{2}}\otimes I_{B_{n}}^{\mathrm{GL}_{n}}(\mu)$ agrees with the normalized induction of $\mu$ from $B_{n}(F)$ to $\mathrm{GL}_{n}(F)$ .

I_{B_{n}}^{\mathrm{GL}_{n}}(\mu):=\{f:\mathrm{GL}_{n}(F)\rightarrow\overline{\mathbb{Q}}_{p}^{\times}\text{ locally constant}:f(bg)=\mu^{\natural}(b)f(g),\text{ for all }b\in B_{n}(F),g\in\mathrm{GL}_{n}(F)\}

as an admissible representation of $\mathrm{GL}_{n}(F)$ via the right translation.

Definition 5.10.

Let $\pi$ be an irreducible admissible representation of $\mathrm{GL}_{n}(F)$ .

(1)

We say $\pi$ is ordinary if there exists a (unique) tuple $\underline{\mu}$ of admissible characters satisfying $\left|{x}\right|^{i-1}\mu_{n+1-i}(x)$ are $p$ -adic units for $1\leq i\leq n$ and every $x\in F^{\times}$ , such that $\pi$ is isomorphic to $I_{B_{n}}^{\mathrm{GL}_{n}}(\mu)$ .
(2)

We say $\pi$ is semi-stably ordinary if furthermore $\mu_{i}$ are all unramified.
(3)

We say $\pi$ is regularly ordinary if $\pi$ is ordinary and the $p$ -adic valuations of $\mu_{i}(\varpi)$ are all distinct. We say $\pi$ is regularly semi-stably ordinary if it is regularly ordinary and semi-stably ordinary.

Remark 5.11.

Note that if $\pi$ satisfies the properties in Definition above, then so does $\pi^{\vee}$ with respect to the tuple $\check{\underline{\mu}}:=(\boldsymbol{\alpha}_{F}^{1-n}\mu_{n}^{-1},\ldots,\boldsymbol{\alpha}_{F}^{1-n}\mu_{1}^{-1})$ .

Definition 5.12.

Let $x\in\mathfrak{o}\cap F^{\times}$ and put $\mathbf{d}(x):=\operatorname{diag}[x^{n-1},\ldots,x,1]\in\mathrm{GL}_{n}(F)$ , we define an operator $\mathbb{V}_{n}^{x}$ on $\pi^{U_{n}(\mathfrak{o})}$ as

\mathbb{V}_{n}^{x}:=\sum_{u\in U_{n}(\mathfrak{o})/(U_{n}(\mathfrak{o})\,\cap\,\mathbf{d}(x)U_{n}(\mathfrak{o})\mathbf{d}(x)^{-1})}\pi(u\mathbf{d}(x)).

We remark that when taking $x=\varpi$ , the operator $\mathbb{V}_{n}^{\varpi}$ is the $\mathbb{U}_{\mathfrak{p}}$ -operator (see, for example, [Jan24, Section 1.4]).

Then we have the following proposition, proved in [Liu23, Lemma 4.4]¹⁵¹⁵15We remark that in the statement of [Liu23, Lemma 4.4], the eigenvalue is $\left(\prod_{m=1}^{n-1}\prod_{i=1}^{m}\left|{x}\right|^{i-1}\mu_{i}(x)\right),$ reordering $(\mu_{1},\ldots,\mu_{n})$ into $(\mu_{n},\mu_{n-1},\ldots,\mu_{1})$ . The ordinary condition we defined in Definition 5.10 is adjusted accordingly, which is different from loc.cit. Since we have required $\pi$ to be irreducible, the ordering of $\mu_{i}$ ’s does not harm. The case where $\operatorname{Ind}^{\mathrm{GL}_{n}}_{B_{n}}(\mu)$ being reducible is more subtle. For example when $n=2$ , that means $\mu_{1}/\mu_{2}=\left|{-}\right|^{\pm 1}$ . Whether the Steinberg representation is a subrepresentation or a quotient representation of $\operatorname{Ind}^{\mathrm{GL}_{n}}_{B_{n}}(\mu)$ does depend on the ordering of $\mu_{1}$ and $\mu_{2}$ . Note that the definition in [Liu23] insists on requiring $\pi$ to be a subrepresentation of $\operatorname{Ind}^{\mathrm{GL}_{n}}_{B_{n}}(\mu)$ , our adjustment in Definition 5.10 is just replacing “subrepresentation” by “quotient representation”. We thank Yifeng Liu for his guidance on this issue..

Proposition 5.13.

Suppose that $\pi$ is regularly ordinary, then there exists a unique up to scalar nonzero element $f^{\operatorname{ord}}_{\pi}\in\pi^{U_{n}(\mathfrak{o})}$ satisfying that

\mathbb{V}_{n}^{x}\cdot f^{\operatorname{ord}}_{\pi}=\left(\prod_{m=1}^{n-1}\prod_{i=1}^{m}\left|{x}\right|^{i-1}\mu_{n+1-i}(x)\right)f^{\operatorname{ord}}_{\pi}(g)

holds for every $x\in\mathfrak{o}\cap F^{\times}$ . In particular, the $\mathbb{V}_{n}^{x}$ -eigenvalue is a $p$ -adic unit.

We call the one-dimensional $\overline{\mathbb{Q}}_{p}$ -subspace of $\pi^{U_{n}(\mathfrak{o})}$ generated by $f^{\operatorname{ord}}_{\pi}$ the ordinary line of $\pi$ , denoted by $\pi^{\operatorname{ord}}$ and a nonzero element of it an ordinary vector. Let $\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ be the Whittaker model of $\pi$ with respect to $\boldsymbol{\psi}$ . For every $f\in V_{\pi}$ supported on $B_{n}(F)w_{n}B_{n}(F)$ and $g\in B_{n}(F)w_{n}B_{n}(F)$ , consider for the normalized Haar measure $\,\mathrm{d}u$ on $U_{n}(F)$ the integral

W_{f}(g):=\int_{U_{n}(F)}f(w_{n}ug)\overline{\boldsymbol{\psi}}(u)\,\mathrm{d}u.

By a well-known result of Rodier, this integral converges and extends uniquely to an intertwining operator

\mathsf{W}:I_{B_{n}}^{\mathrm{GL}_{n}}(\mu)\rightarrow\operatorname{Ind}_{U_{n}(F)}^{\mathrm{GL}_{n}(F)}\boldsymbol{\psi}.

See [CS80, Corollary 1.8] for details. By the uniqueness of Whittaker models for $I_{B_{n}}^{\mathrm{GL}_{n}}(\mu)$ , we have that $W$ gives the Whittaker model of $\pi$ , i.e.

\mathsf{W}_{\pi}:I_{B_{n}}^{\mathrm{GL}_{n}}(\mu)\rightarrow\mathcal{W}_{\boldsymbol{\psi}}(\pi)\hookrightarrow\operatorname{Ind}_{U_{n}(F)}^{\mathrm{GL}_{n}(F)}\boldsymbol{\psi}.

The image of the ordinary line $\pi^{\operatorname{ord}}$ in $\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ is denoted by $\mathcal{W}^{\operatorname{ord}}_{\boldsymbol{\psi}}(\pi)$ .

Actually we can explicitly construct a canonical ordinary vector. We define a big-cell section ¹⁶¹⁶16It is called the big cell section since it is supported on the big cell of the Bruhat decomposition, i.e. the cell of longest Weyl element. We use the notation “ $\dagger$ ” to denote such sections. $f^{\dagger}_{\pi}:\mathrm{GL}_{n}(F)\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ as

f^{\dagger}_{\pi}:g\mapsto\begin{cases}\mu^{\natural}(b),&\quad\text{ if }g=bw_{n}r,\text{ with }b\in B_{n}(F),\,r\in U_{n}(\mathfrak{o}),\\ 0,&\quad\text{ otherwise}.\end{cases}

Let $W_{\pi}^{\dagger}$ be the corresponding Whittaker vector of $f^{\dagger}_{\pi}$ . Here are some basic properties of $f^{\dagger}_{\pi}$ .

Proposition 5.14.

Notations being as above, we have

(1)

$f^{\dagger}_{\pi}$ is an ordinary vector.
(2)

$W^{\dagger}_{\pi}(\mathbf{1}_{n})=\operatorname{vol}(U_{n}(\mathfrak{o}))\neq 0$ .

Proof.

It is immediate to see $f_{\pi}^{\dagger}\in\pi^{U_{n}(\mathfrak{o})}$ . We fix $x\in\mathfrak{o}\cap F^{\times}$ . Let $g\in\mathrm{GL}_{n}(F)$ such that $\mathbb{V}_{n}^{x}f_{\pi}^{\dagger}(g)\neq 0$ . Then by the definition of $\mathbb{V}_{n}^{x}$ , there exists $u\in U_{n}(\mathfrak{o})$ such that $f_{\pi}^{\dagger}(gu\mathbf{d}(x))\neq 0$ . We have $gu\mathbf{d}(x)\in B_{n}(F)w_{n}U_{n}(\mathfrak{o})$ . One verifies for any $u=(u_{i,j})\in U_{n}(\mathfrak{o})$ ,

	$\mathbf{d}(x)u\mathbf{d}(x)^{-1}=\begin{bmatrix}1&xu_{1,2}&x^{2}u_{2,3}&\cdots&x^{n-1}u_{1,n}\\ &1&xu_{2,3}&\cdots&x^{n-2}u_{2,n}\\ &&\ddots&\vdots&\vdots\\ &&&1&xu_{n-1,n}\\ &&&&1\end{bmatrix}\in U_{n}(\mathfrak{o})$	(5.18a)
and
	$\mathbf{d}(x)^{-1}u\mathbf{d}(x)=\begin{bmatrix}1&x^{-1}u_{1,2}&x^{-2}u_{2,3}&\cdots&x^{1-n}u_{1,n}\\ &1&x^{-1}u_{2,3}&\cdots&x^{2-n}u_{2n}\\ &&\ddots&\vdots&\vdots\\ &&&1&x^{-1}u_{n-1,n}\\ &&&&1\end{bmatrix}\in U_{n}\left(\mathfrak{p}^{-nv(x)}\mathfrak{o}\right).$	(5.18b)

Then we get $g\in B_{n}(F)w_{n}U_{n}(\mathfrak{o})$ . Therefore, it reduces to compute $\mathbb{V}_{n}^{x}f^{\dagger}_{\pi}(g)$ for $g\in B_{n}(F)w_{n}U_{n}(\mathfrak{o})$ . Moreover, since $\mathbb{V}_{n}^{x}f^{\dagger}_{\pi}$ is $U_{n}(\mathfrak{o})$ -invariant, we may assume $g\in B_{n}(F)w_{n}$ .

By (5.18a), we see that the quotient $U_{n}(\mathfrak{o})/(\mathbf{d}(x)u\mathbf{d}(x)^{-1})$ has a finite set of complete representative elements

U_{n}(\mathfrak{o})/(\mathbf{d}(x)u\mathbf{d}(x)^{-1})\mathrel{\leavevmode\hbox to21.17pt{\vbox to18.24pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-6.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{}{}{{{}{}}}{}{}{}{}{{}}{}{}{}{{}}\pgfsys@moveto{0.4pt}{0.0pt}\pgfsys@lineto{20.09987pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{-1.0}{0.0}{0.0}{-1.0}{0.2pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{20.29987pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.24992pt}{-3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle 1:1$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\begin{bmatrix}1&\mathfrak{o}/\mathfrak{p}^{v(x)}&\mathfrak{o}/\mathfrak{p}^{2v(x)}&\cdots&\mathfrak{o}/\mathfrak{p}^{(n-1)v(x)}\\ &1&\mathfrak{o}/\mathfrak{p}^{v(x)}&\cdots&\mathfrak{o}/\mathfrak{p}^{(n-2)v(x)}\\ &&\ddots&\vdots&\vdots\\ &&&1&\mathfrak{o}/\mathfrak{p}^{v(x)}\\ &&&&1\end{bmatrix}.

(5.19)

Here we denote for any positive integer $t$ ,

\mathfrak{o}/\mathfrak{p}^{t}:=\left\{\sum_{k=0}^{t-1}s_{k}\varpi^{k}:s_{k}=0,\ldots,q-1\right\}.

So it suffices to compute $\pi(u\mathbf{d}(x))f^{\dagger}_{\pi}(g)$ for $g=bw_{n}\in B_{n}(F)w_{n}$ and $u$ in the right hand side of (5.19). We have

	$\displaystyle\pi(u\mathbf{d}(x))f^{\dagger}_{\pi}(g)$	$\displaystyle=f^{\dagger}_{\pi}(bw_{n}u\mathbf{d}(x))$
		$\displaystyle=f^{\dagger}_{\pi}(b(w_{n}\mathbf{d}(x)w_{n}^{-1})w_{n}(\mathbf{d}(x)^{-1}u\mathbf{d}(x)))$

By (5.18b) and the definition of $f_{\pi}^{\dagger}$ , we see that $\mathbf{d}(x)^{-1}u\mathbf{d}(x)\in U_{n}(\mathfrak{o})$ if and only if $u=1$ . Moreover,

w_{n}\mathbf{d}(x)w_{n}^{-1}=\operatorname{diag}[1,x,x^{2},\ldots,x^{n-1}].

With these observations, we obtain ¹⁷¹⁷17The trick of getting the third equality is to write the factors in the following table, then take products of nonzero entries column by column: $\begin{pmatrix}1&\mu_{n}(x)&\mu_{n}(x)&\cdots&\cdots&\mu_{n}(x)\\ &1&\left|{x}\right|\mu_{n-1}(x)&\cdots&\cdots&\left|{x}\right|\mu_{n-1}(x)\\ &&\ddots&\vdots&\vdots&\vdots\\ &&&1&\left|{x}\right|^{n-3}\mu_{3}(x)&\left|{x}\right|^{n-3}\mu_{3}(x)\\ &&&&1&\left|{x}\right|^{n-2}\mu_{2}(x)\\ &&&&&1\end{pmatrix},$ with rows indexed by $i=1,\ldots,n$ and columns indexed by $m=0,1,\ldots,n-1$ .

	$\displaystyle\mathbb{V}_{n}^{x}f^{\dagger}_{\pi}(g)$	$\displaystyle=\mu^{\natural}(\operatorname{diag}[1,x,x^{2},\ldots,x^{n-1}])f^{\dagger}_{\pi}(g)$
		$\displaystyle=\prod_{i=1}^{n}\left(\left\|{x}\right\|^{n-i}\mu_{i}(x)\right)^{i-1}f^{\dagger}_{\pi}(g)=\left(\prod_{m=1}^{n-1}\prod_{i=1}^{m}\left\|{x}\right\|^{i-1}\mu_{n+1-i}(x)\right)f^{\dagger}_{\pi}(g).$

This verifies (1). For (2),

W_{\pi}^{\dagger}=\mathsf{W}_{\pi}(f^{\dagger}_{\pi}):\,g\mapsto\int_{U_{n}(F)}f_{\pi}^{\dagger}(w_{n}ug)\overline{\boldsymbol{\psi}}(u)\,\mathrm{d}u.

It evaluates at $\mathbf{1}_{n}$ to

W_{\pi}^{\dagger}(\mathbf{1}_{n})=\int_{U_{n}(F)}f_{\pi}^{\dagger}(w_{n}u)\overline{\boldsymbol{\psi}}(u)\,\mathrm{d}u=\int_{U_{n}(\mathfrak{o})}\,\mathrm{d}u=\operatorname{vol}(U_{n}(\mathfrak{o}))\neq 0

because the integrand vanishes for $u\not\in U_{n}(\mathfrak{o})$ and $f_{\pi}^{\dagger}(w_{n}u)=1$ for $u\in U_{n}(\mathfrak{o})$ . ∎

5.4.2. Iwahori type

For any integer $\alpha\geq 0$ , we define the Iwahori subgroup of level $\alpha$ , denoted by $\mathrm{Iw}_{\alpha}$ , to be the subgroup of matrices in $\mathrm{GL}_{n}(\mathfrak{o})$ that become upper triangular modulo $\varpi^{\alpha}$ .

One checks that $\mathrm{Iw}_{\alpha}$ acts on $\mathcal{W}_{\boldsymbol{\psi}}(\pi)^{U_{n}(\mathfrak{o})}$ naturally. Since $\mathcal{W}^{\operatorname{ord}}_{\boldsymbol{\psi}}(\pi)$ is one- dimensional, there exists a character $\vartheta:\mathrm{Iw}_{\alpha}\rightarrow\mathbb{C}^{\times}$ such that

W(gr)=\vartheta(r)W(g),\quad\forall g\in\mathrm{GL}_{n}(F),r\in\mathrm{Iw}_{\alpha}.

It is called the Iwahori type of $\pi$ . Moreover, we have the decomposition

\mathrm{Iw}_{\alpha}=[\mathrm{Iw}_{\alpha},\mathrm{Iw}_{\alpha}]\cdot T_{n}(\mathfrak{o}),

where $[\mathrm{Iw}_{\alpha},\mathrm{Iw}_{\alpha}]$ is the derived subgroup of $\mathrm{Iw}_{\alpha}$ , consisting of matrices in $\mathrm{GL}_{n}(\mathfrak{o})$ that becomes strictly upper triangular when reduced modulo $\varpi^{\alpha}$ . Hence $\vartheta$ arise from the subtorus of $\mathrm{Iw}_{\alpha}$ , i.e. there exist characters $\theta_{1},\ldots,\theta_{n}:F^{\times}\rightarrow\mathbb{C}^{\times}$ such that

\vartheta((r_{ij})_{i,j})=\prod_{i=1}^{n}\theta_{i}(r_{ii}).

The following lemma characterizes the Iwahori type of $\pi$ . ¹⁸¹⁸18We appreciate the helpful guidance of Loren Spice on understanding this part.

Proposition 5.15.

Let $\pi=I_{B_{n}}^{\mathrm{GL}_{n}}(\mu)$ be a regularly ordinary representation of $\mathrm{GL}_{n}(F)$ , then the Iwahori type of $\pi$ is $(\breve{\mu})^{\natural}$ , where for $\underline{\mu}=(\mu_{1},\ldots,\mu_{n})$ , we denote $\breve{\underline{\mu}}:=(\mu_{n},\mu_{n-1},\ldots,\mu_{1})$ .

Proof.

Since $W_{\pi}^{\dagger}(\mathbf{1}_{n})$ equals $\operatorname{vol}(U_{n}(\mathfrak{o}))$ , we have that for all $r\in\mathrm{Iw}_{\alpha}\cap T_{n}(F)=T_{n}(\mathfrak{o})$ , $W_{\pi}^{\dagger}(r)$ equals

	$\displaystyle\int_{U_{n}(F)}f_{\pi}^{\dagger}(w_{n}ur)\overline{\boldsymbol{\psi}}(u)\,\mathrm{d}u$	$\displaystyle=\int_{U_{n}(F)}f_{\pi}^{\dagger}((w_{n}rw_{n}^{-1})w_{n}(r^{-1}ur))\overline{\boldsymbol{\psi}}(u)\,\mathrm{d}u$
		$\displaystyle=\mu^{\natural}(w_{n}rw_{n}^{-1})\int_{U_{n}(F)}f_{\pi}^{\dagger}(w_{n}u^{\prime})\overline{\boldsymbol{\psi}}(u^{\prime})\,\mathrm{d}u^{\prime}$
		$\displaystyle=(\breve{\mu})^{\natural}(r)\operatorname{vol}(U_{n}(\mathfrak{o}))$
		$\displaystyle=(\breve{\mu})^{\natural}(r)W^{\dagger}_{\pi}(\mathbf{1}_{n})$

Here the second equality follows from that $r$ normalises $U_{n}(\mathfrak{o})$ . The third equality follows from the calculation

w_{n}\operatorname{diag}[r_{1},\ldots,r_{n}]w_{n}^{-1}=\operatorname{diag}[r_{n},r_{n-1},\ldots,r_{1}].

The last equality is Proposition 5.14 (2). Hence $r$ acts by multiplication by $(\breve{\mu})^{\natural}(r)$ . ∎

Remark 5.16.

In particular, when $\pi$ is furthermore regularly semi-stably ordinary, that is, all $\mu_{i}$ ’s are unramified characters, we see that $\mu(\mathfrak{o}^{\times})=1$ , hence the Iwahori subgroup $\mathrm{Iw}_{\alpha}$ acts trivially on the ordinary line by Proposition 5.15. This recovers [Liu23, Lemma 4.8]. Another explanation using $L$ -functions can be found in [Jan24, Remark 1.4].

5.4.3. Test vector problem: Januszewski’s generalization of local Birch lemma

The main input is Januszewski’s generalization of local Birch lemma, which we shall first introduce.

Solely in this subsection, let $\sigma=I_{B_{n}}^{\mathrm{GL}_{n}}(\underline{\mu}^{\prime})$ and $\pi=I_{B_{n+1}}^{\mathrm{GL}_{n+1}}(\underline{\mu})$ be regularly ordinary representations of $\mathrm{GL}_{n}(F)$ and $\mathrm{GL}_{n+1}(F)$ respectively.

We choose an auxiliary local character $\kappa:F^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ of conductor $\varpi^{\alpha}$ , and require it to satisfy the constant conductor condition: for all $1\leq i\leq n+1$ and all $1\leq j\leq n$ , the conductors of $\kappa\mu_{i}\mu_{j}^{\prime}$ are all nontrivial, all agree and are generated by $\varpi^{\beta}$ . Moreover, we require $\kappa(\varpi)=1$ . Such an character $\kappa$ exists since any character of sufficiently large conductor could be made to satisfy this condition.

Then what follows is the local Birch lemma, generalized by F. Januszewski [Jan24, Theorem 2.8]. Here we state a simplified version. Let

h_{n}:=\begin{bmatrix}w_{n}&1\\ &1\end{bmatrix},\quad\mathbf{t}(x):=\operatorname{diag}[x^{n},x^{n-1},\ldots,x]\in\mathrm{GL}_{n}(F)

for $x\in F^{\times}$ . We write $h_{n}^{(x)}:=(\jmath^{\mathrm{GL}}(\mathbf{t}(x)))^{-1}h_{n}\jmath^{\mathrm{GL}}(\mathbf{t}(x))$ .

Theorem 5.17 (Local Birch lemma).

Notations being as above. Let $W_{\pi}\in\mathcal{W}_{\boldsymbol{\psi}}(\pi)$ and $W_{\sigma}\in\mathcal{W}_{\boldsymbol{\psi}^{-1}}(\sigma)$ be Whittaker vectors of Iwahori type $\mu$ and $\lambda$ respectively, then,

Z\left(\frac{1}{2},\pi(h_{n}^{\left(-\varpi^{\alpha}\right)})W_{\pi},W_{\sigma}\right)=\Delta_{\beta}\mathfrak{G}(\kappa,\underline{\mu},\underline{\lambda})W_{\pi}(\jmath^{\mathrm{GL}}(\mathbf{t}(\varpi^{\alpha-\beta})))W_{\sigma}(\mathbf{t}(\varpi^{\alpha-\beta})),

where

\Delta_{\beta}:=\prod_{j=1}^{n}(1-q^{-j})q^{-\frac{(n+2)(n+1)n\beta}{6}},\quad\mathfrak{G}(\kappa,\underline{\mu},\underline{\lambda}):=\prod_{j=1}^{n}\prod_{i=1}^{j}\mu_{j+1-i}\mu_{i}^{\prime}(\varpi^{\beta})\mathfrak{g}(\kappa\mu_{j+1-i}\lambda_{i}).

Proof.

This is a modification of Januszewski’s generalization of local Birch lemma, i.e. [Jan24, Theorem 2.8], which gives

\int_{U_{n}(F)\backslash\mathrm{GL}_{n}(F)}\pi(h_{n}\jmath^{\mathrm{GL}}(\mathbf{t}(-\varpi^{\alpha})))W(\jmath^{\mathrm{GL}}(g))\cdot\sigma(\mathbf{t}(-\varpi^{\alpha}))W_{\sigma}(g)\cdot\kappa(\det g)\,\mathrm{d}g\\ =\Delta_{\beta}\mathfrak{G}(\kappa,\underline{\mu},\underline{\lambda})W_{\pi}(\jmath^{\mathrm{GL}}(\mathbf{t}(\varpi^{\alpha-\beta})))W_{\sigma}(\mathbf{t}(\varpi^{\alpha-\beta})).

We put for $l\geq 0$ , $\mathbf{s}(l):=\operatorname{diag}[1+\varpi^{l},1,\ldots,1]\in\mathrm{GL}_{n}(F)$ as an adjusting matrix. It is defined to satisfy that for any $g\in\mathrm{GL}_{n}(F)$ and $l\geq\alpha$ , $\kappa(\det g\mathbf{s}(l))=1$ . Taking the change of variable $g=g^{\prime}\mathbf{s}(l)$ for $l\geq\alpha$ , then we see that

\int_{U_{n}(F)\backslash\mathrm{GL}_{n}(F)}\pi(h_{n}\jmath^{\mathrm{GL}}(\mathbf{t}(-\varpi^{\alpha})))W_{\pi}(\jmath^{\mathrm{GL}}(g^{\prime}\mathbf{s}(l)))\cdot\sigma(\mathbf{t}(-\varpi^{\alpha}))W_{\sigma}(g^{\prime}\mathbf{s}_{l})\,\mathrm{d}g^{\prime}\\ =\Delta_{\beta}\mathfrak{G}(\kappa,\underline{\mu},\underline{\lambda})W_{\pi}(\jmath^{\mathrm{GL}}(\mathbf{t}(\varpi^{\alpha-\beta})))W_{\sigma}(\mathbf{t}(\varpi^{\alpha-\beta}))

The left hand side integral is then nothing but

	$\displaystyle Z\left(\frac{1}{2},\pi(\jmath^{\mathrm{GL}}(\mathbf{s}(l))h_{n}\jmath^{\mathrm{GL}}(\mathbf{t}(-\varpi^{\alpha})))W_{\pi},\sigma(\mathbf{s}(l)\mathbf{t}(-\varpi^{\alpha}))W_{\sigma}\right)$
	$\displaystyle=Z\left(\frac{1}{2},\pi(\jmath^{\mathrm{GL}}(\mathbf{s}(l)\mathbf{t}(-\varpi^{\alpha}))^{-1})\pi(\jmath^{\mathrm{GL}}(\mathbf{s}_{l})h_{n}\jmath^{\mathrm{GL}}(\mathbf{t}_{-\varpi^{\alpha}}))W_{\pi},W_{\sigma}\right)$
	$\displaystyle=Z\left(\frac{1}{2},\pi(\jmath^{\mathrm{GL}}(\mathbf{t}(-\varpi^{\alpha}))^{-1}h_{n}\jmath^{\mathrm{GL}}(\mathbf{t}(-\varpi^{\alpha})))W_{\pi},W_{\sigma}\right)$
	$\displaystyle=Z\left(\frac{1}{2},\pi(h_{n}^{\left(-\varpi^{\alpha}\right)})W_{\pi},W_{\sigma}\right)$

Here in the first equality, we used the invariance of JPSS integrals (5.13). ∎

5.5. Local Ichino-Ikeda integrals at $\mathscr{V}_{F}^{(p)}$

We go back to the context of local Ichino-Ikeda integrals. Let $\sigma=I_{B_{m+n}}^{\mathrm{GL}_{m+n}}(\underline{\lambda})$ and $\pi=I_{B_{m+n+1}}^{\mathrm{GL}_{m+n+1}}(\underline{\mu})$ be regularly ordinary representations of $\mathrm{GL}_{m+n}(F)$ and $\mathrm{GL}_{m+n+1}(F)$ respectively.

We take

W_{1}=\pi(h_{m+n}^{(-\varpi^{\alpha})})W_{\pi}^{\dagger},\quad W_{1}^{\prime}=W_{\sigma}^{\dagger},\quad W_{2}=\pi^{\vee}(h_{m+n}^{(-\varpi^{\alpha})})W_{\pi^{\vee}}^{\dagger},\quad W_{2}^{\prime}=W_{\sigma^{\vee}}^{\dagger}

in Corollary 5.7. Then by Theorem 5.17, with Remark 5.11 and Proposition 5.15, we obtain the following result.

Proposition 5.18.

Notations and conventions being as above, with $\kappa:F^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ a sufficiently ramified character with $\kappa(\varpi)=1$ , then

\mathscr{I}(\sigma,\pi)=\Delta_{\beta}^{2}\cdot\mathfrak{G}(\kappa,\breve{\underline{\mu}},\breve{\underline{\lambda}})\mathfrak{G}(\kappa,\breve{\check{\underline{\mu}}},\breve{\check{\underline{\lambda}}})\cdot\mathcal{C}_{\sigma}^{\operatorname{ord}}\mathcal{C}_{\pi}^{\operatorname{ord}}\cdot\prod_{i=1}^{m+n-1}\zeta_{F}(i),

where

\mathcal{C}_{\pi}^{\operatorname{ord}}:=\dfrac{W_{\pi}^{\dagger}(\jmath^{\mathrm{GL}}(\mathbf{t}(\varpi^{\alpha-\beta})))W_{\pi}^{\dagger}(\jmath^{\mathrm{GL}}(\mathbf{t}(\varpi^{\alpha-\beta})))}{\left\langle{W_{\pi}^{\dagger},W_{\pi^{\vee}}^{\dagger}}\right\rangle}\text{ and }\mathcal{C}_{\sigma}^{\operatorname{ord}}:=\dfrac{W_{\sigma}^{\dagger}(\mathbf{t}(\varpi^{\alpha-\beta}))W_{\sigma}^{\dagger}(\mathbf{t}(\varpi^{\alpha-\beta}))}{\left\langle{W_{\sigma}^{\dagger},W_{\sigma}^{\dagger}}\right\rangle}.

6. A summary of automorphic computations

In this section, we introduce the choice of appropriate local Siegel sections $f_{s,\chi,v}^{\mathrm{Sieg}}$ , and sum up the results we have obtained so far.

6.1. The choice of Siegel Eisenstein sections

Though it has been expected by experts that the Klingen Eisenstein series (via pullback from Siegel Eisenstein series) can be defined in full generalities, due to the lack of references, we restrict us to the Klingen Eisenstein series (and their $p$ -adic family) constructed in [Wan15], where additional assumptions are needed in the automorphic computations.

Assumption 6.1 (Scalar weight assumption).

We assume that $\sigma_{v}$ is a holomorphic discrete series representation associated to the scalar weight $(0,0,\ldots,\kappa,\ldots,\kappa)$ with $m$ zeroes and $s$ kappas, for any $v\in\mathscr{S}_{\mathcal{F}}^{\infty}$ .

Assumption 6.2 (Sufficiently ramified assumption).

We assume that $(\sigma_{v},\chi_{v})$ is “generic” in the sense of [Wan15, Definition 4.42] for any $v\in\mathscr{S}_{\mathcal{F}}^{(p)}$ . Basically this puts restrictions on the ramification of the $\pi$ at primes dividing $p$ , requiring it to be sufficiently ramified. ¹⁹¹⁹19This is the reason why we are not satisfied with restricting ourselves in the “semi-stably ordinary” case as in [Liu23].

Here we briefly recall the choice of Siegel Eisenstein sections $f^{\mathrm{Sieg}}_{v,s,\chi}$ for places $v$ of $\mathcal{F}$ in [Wan15, Chapter 4]. We bring in a new partition of $\mathscr{V}_{\mathcal{F}}$ as follows.

•

Let $\mathscr{T}_{\mathcal{F}}^{{\mathrm{ur}}}$ be the set of finite places of $\mathcal{F}$ away from $\mathscr{V}_{\mathcal{F}}^{(p)}$ such that $\sigma_{v}$ , $\chi_{v}$ and $\mathcal{K}_{v}/\mathcal{F}_{v}$ are all unramified. Note that $\mathscr{T}_{\mathcal{F}}^{{\mathrm{ur}}}\cap\mathscr{V}_{\mathcal{F}}^{{\mathrm{ur}}}=\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}$ .
•

Let $\mathscr{T}_{\mathcal{F}}^{\mathrm{bad}}$ be complement of $\mathscr{T}_{\mathcal{F}}^{{\mathrm{ur}}}$ in $\mathscr{V}_{\mathcal{F}}$ , with elements called “bad places” for Siegel Eisenstein sections.
•

Let $\mathscr{T}_{\mathcal{F}}^{\mathrm{ram}}$ be the subset of $\mathscr{T}_{\mathcal{F}}^{\mathrm{bad}}$ removing all places of $F$ above $p$ and archimedean places.

6.1.1. Archimedean places

Let $v\in\mathscr{S}_{\mathcal{F}}^{\infty}$ be an archimedean place of $\mathcal{F}$ . Following [Wan15, Section 4A2], let

\mathbf{i}:=\operatorname{diag}\left[\dfrac{1}{2}\mathrm{i}\mathbf{1}_{n},\mathrm{i},\dfrac{1}{2}\vartheta,\dfrac{1}{2}\mathrm{i}\mathbf{1}_{n}\right]

be the distinguished point in the (unbounded realization of the) symmetric domain for ${\mathrm{GU}}(m+n+1,m+n+1)$ . We define the Siegel Eisenstein section as

f^{\mathrm{Sieg}}_{v,s,\chi}\left(g=\begin{bmatrix}A_{g}&B_{g}\\ C_{g}&D_{g}\end{bmatrix}\right)=\det(C_{g}\mathbf{i}+D_{g})^{-\kappa}\left|{\det(C_{g}\mathbf{i}+D_{g})}\right|^{\kappa-2s-(m+n)},\,g\in G^{\blacklozenge}(\mathcal{F}_{v}).

which depends on $\kappa$ .

6.1.2. Unramified places

Let $v\in\mathscr{T}_{\mathcal{F}}^{{\mathrm{ur}}}$ , we choose $f_{v,s,\chi}^{\mathrm{Sieg}}$ to be the spherical section $f_{v,s,\chi}^{\mathrm{Sieg},\mathrm{sph}}$ .

6.1.3. Ramified places away from places above $p$

At the ramified places $v\in\mathscr{T}_{\mathcal{F}}^{\mathrm{ram}}$ , we let $f_{s,\chi,v}^{\dagger}$ be the big-cell section defined as the Siegel section supported on the big-cell $Q(\mathcal{F}_{v})w_{N+1}N_{Q}(\mathcal{O}_{\mathcal{F}_{v}})$ and that $f_{s,\chi,v}^{\dagger}(w_{N+1}N_{Q}(\mathcal{O}_{\mathcal{F}_{v}}))=1$ . We put

\gamma_{v}:=\begin{bmatrix}\mathbf{1}_{N+1}&\gamma_{v}^{\mathrm{upper-right}}\\ 0&\mathbf{1}_{N+1},\end{bmatrix},\quad\gamma_{v}^{\mathrm{upper-right}}:=\begin{bmatrix}0&0&0&x^{-1}\mathbf{1}_{n}\\ 0&0&0&0\\ 0&0&(y\overline{y})^{-1}\mathbf{1}_{m-n}&0\\ \overline{x}^{-1}\mathbf{1}_{n}&0&0&0\end{bmatrix},

where $x$ and $y$ are fixed constants in $\mathcal{K}$ which are divisible by some high power of $\varpi_{\mathcal{K}_{v}}$ . Then we define the Siegel section to be $f_{s,\chi,v}^{\mathrm{Sieg}}(-)=f_{v,s,\chi}^{\dagger}(-\cdot\gamma_{v})$ .

6.1.4. Places above $p$

Let $v\in\mathscr{S}_{\mathcal{F}}^{(p)}$ . We remark that in [Wan15, Section 4D], $p$ is assumed to be split completely in $\mathcal{K}$ , while here we only assume that $p$ is unramified in $\mathcal{F}$ and every places of $\mathcal{F}$ above $p$ splits in $\mathcal{K}$ . The stronger assumption of Wan is just in the purpose of easing the notational issues, and can be generalized to our setup by bookkeeping.

Recall that we have started with a Hecke character $\chi$ of $\mathcal{K}$ , which decomposes as $\chi=\otimes_{w\in\mathscr{S}_{\mathcal{K}}}\chi_{w}$ . For $v\in\mathscr{S}_{\mathcal{F}}^{(p)}$ that splits as $v=w\overline{w}$ in $\mathcal{K}$ , we put

\chi_{1}:=\chi_{w},\quad\chi_{2}:=\chi_{\overline{w}}^{-1},

where we identify $\mathcal{K}_{w}=\mathcal{K}_{\overline{w}}=\mathcal{F}_{v}$ . Then the two local characters determines a character

\chi_{w,s}:B_{2(N+1)}(\mathcal{K}_{w})\rightarrow\mathbb{C}^{\times},\quad\begin{bmatrix}A&B\\ 0&D\end{bmatrix}\mapsto\chi_{1}(\det D)\chi_{2}(\det A)\left|{\det AD^{-1}}\right|_{w}^{s}.

Here the matrix is participated as $[N+1\mid N+1]$ . Here one checks that for each

f_{w,s}\in\operatorname{Ind}_{B_{2(N+1)}(\mathcal{K}_{w})}^{\mathrm{GL}_{2(N+1)(\mathcal{K}_{w})}}(\chi_{w,s}),

(6.20)

we have $f_{v,s}(g_{v}):=f_{w,s}(g_{w})\in I_{v}^{\mathrm{Sieg}}((\chi_{1},\chi_{2}),s)$ , where $g_{w}$ is the projection of $g_{v}$ under $\varrho_{w,N+1,N+1}$ . This process reduces us to the construction of sections in (6.20), over $\mathrm{GL}_{2(N+1)}(\mathcal{K}_{w})$ , which could be simpler. Additionally, we have the local representation $\sigma_{v}=I_{B_{N}}^{\mathrm{GL}_{N}}(\underline{\lambda})$ , with the conductors of $\underline{\lambda}$ being $\varpi_{v}^{t_{1}},\ldots,\varpi_{v}^{t_{m+n}}$ . We write

\xi_{i}:=\begin{cases}\lambda_{i}\chi_{1}^{-1},&\quad 1\leq i\leq m,\\ 1&\quad i=m+1,\\ \lambda_{i-1}^{-1}\chi_{2},&\quad m+2\leq i\leq m+n+1.\end{cases}

We define the big-cell section $\tilde{f}_{v,s,\chi}^{\dagger}$ to be the Siegel Eisenstein section such that

•

it is supported on the big-cell $Q(\mathcal{F}_{v})w_{N+1}K_{Q}(p^{t})$ , with $K_{Q}(p^{t})$ be the subgroup of $\mathrm{GL}_{2(N+1)}(\mathcal{O}_{\mathcal{F}_{v}})$ consisting of matrices which are blockwise upper triangular under the partition $[N+1\mid N+1]$ modulo $p^{t}$ , and $t$ is such that the conductor of $(\chi^{\mathcal{F}})_{v}$ is $\varpi_{v}^{t}$ , and
•

$\tilde{f}_{v,s,\chi}^{\dagger}\left(w_{N+1}\begin{bmatrix}A&B\\ C&D\end{bmatrix}\right)=\chi_{v}(\det D)$ for $\begin{bmatrix}A&B\\ C&D\end{bmatrix}\in K_{Q}(p^{t})$ .

We define, following [Wan15, Section 4D4], the Siegel Eisenstein section ²⁰²⁰20There are subtle differences between the Siegel Eisenstein series we write here and the one in [Wan15, Section 4D4]. Following [Wan19, Section 4.7], we decide to add the factor $\left(\mathfrak{g}(\overline{\chi}_{\mathsf{P},v})^{m+n+1}c_{m+n+1}(\chi_{\mathsf{P},v},-s_{\mathsf{P}})\right)^{-1}$ here in the Siegel Eisenstein section, instead of in the normalization factor $B_{\mathscr{D}}$ at the beginning of [Wan15, Section 5C1].

f_{v,s,\chi}^{\dagger}:=p^{-\sum_{i=1}^{m}it_{i}-\sum_{i=1}^{n}it_{m+i}}\prod_{i=1}^{m}\mathfrak{g}(\xi_{i})\xi_{i}(-1)\prod_{i=1}^{n}\mathfrak{g}(\xi_{m+1+i})\xi_{m+1+i}(-1)\\ \times\sum_{A,B,C,D,E}\prod_{i=1}^{m-n}\overline{\xi}_{i}\left(\dfrac{\det A_{i}}{\det A_{i-1}}p^{t_{i}}\right)\prod_{i=1}^{n}\overline{\xi}_{m-n+i}\left(\dfrac{\det D_{i}}{\det D_{i-1}}p^{t_{m-n+i}}\right)\prod_{i=1}^{n}\overline{\xi}_{m+1+i}\left(\dfrac{\det E_{i}}{\det E_{i-1}}p^{t_{m+1+i}}\right)\\ \times\left(\mathfrak{g}((\overline{\chi}^{\mathcal{F}})_{v})^{m+n+1}c_{m+n+1}((\chi^{\mathcal{F}})_{v},-s)\right)^{-1}\tilde{f}^{\dagger}_{v,s,\chi}\left(gw_{\mathrm{Borel}}^{-1}\begin{bmatrix}\mathbf{1}_{N+1}&\circledast\\ 0&\mathbf{1}_{N+1}\end{bmatrix}w_{\mathrm{Borel}}\right),

where

\circledast:=\begin{bmatrix}&&C&D\\ &&&\\ &&A&B\\ E&&&\end{bmatrix},

under the partition $[n\mid 1\mid m-n\mid n]$ with $A,B,C,D,E$ run over the set defined in [Wan15, Lemma 4.29], with $A_{i}$ being the $i$ -th upper-left minor of $A$ , $D_{i}$ being the $(m-n)$ -th upper-left minor of $\begin{bmatrix}A&B\\ C&D\end{bmatrix}$ and $E_{i}$ is the $i$ -th upper-left minor of $E$ . The $w_{\mathrm{Borel}}$ is the element in $G^{\blacklozenge}(\mathcal{F}_{v})$ such that its projection via $\varrho_{w,N+1,N+1}$ is the Weyl element $\operatorname{diag}[w_{m},1]\in\mathrm{GL}_{2(N+1)}(\mathcal{K}_{w})$ . The factor $c_{m+n+1}(-)$ is defined in [Wan15, (13)].

Finally we define our Siegel Eisenstein section as

f_{v,s,\chi}^{\mathrm{Sieg}}(g):=\mathsf{M}_{v}(-s,f^{\dagger}_{v,s,\overline{\chi}^{\mathrm{c}}})(g),

with the intertwining operator defined as

\mathsf{M}_{v}(s,f)\in I_{v}^{\mathrm{Sieg}}(\overline{\chi}_{v}^{\mathrm{c}},-s):\quad g\mapsto\int_{N_{Q}(\mathcal{F}_{v})}f(w_{n}rg)\,\mathrm{d}r

for $f\in I_{v}^{\mathrm{Sieg}}(\chi_{v},s)$ .

6.2. Summary of automorphic computations

Put all the calculations in this part together, we have the following main theorem.

Theorem 6.3.

Notations being as above and assume (BC), Assumption 3.8, 3.10, 3.11 and Assumption 6.1, 6.2. Suppose $\sigma$ and $\pi$ are both regularly ordinary at any places $v$ above $p$ . Then

\dfrac{(\mathcal{P}^{\mathrm{Kling}}_{\Phi,\Psi})^{2}}{\left|\left|{\Phi}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\Psi}\right|\right|_{\pi,\mathrm{Pet}}^{2}}=\dfrac{1}{2^{\varkappa_{\sigma}+\varkappa_{\pi}}}\cdot\mathscr{L}^{\mathscr{V}_{\mathcal{F}}^{(p)}}(\sigma\times\pi)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v},\chi_{v}\right)L_{\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}\left(s+\dfrac{1}{2},\pi_{v}^{\vee},\chi_{v}\right)\\ \times\prod_{v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}}\zeta_{v}C_{\sigma_{v},\boldsymbol{\psi}_{v}}C_{\sigma_{v}^{\vee},\boldsymbol{\psi}_{v}^{-1}}\mathcal{B}_{\pi_{v}}^{\mathrm{ess}}\mathcal{B}_{\sigma_{v}}^{\mathrm{ess}}\prod_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}\zeta_{v}\Delta_{\beta}^{2}\cdot\mathfrak{G}(\kappa,\breve{\underline{\mu}},\breve{\underline{\lambda}})\mathfrak{G}(\kappa,\breve{\check{\underline{\mu}}},\breve{\check{\underline{\lambda}}})\mathcal{C}_{\sigma}^{\operatorname{ord}}\mathcal{C}_{\pi}^{\operatorname{ord}}\\ \times\prod_{v\in\mathscr{S}_{\mathcal{F}}^{{\mathrm{ur}}}}d_{N+1,v}(s,\chi_{v})^{-1}\prod_{v\in\mathscr{S}_{\mathcal{F}}^{\mathrm{bad}}}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{v}^{\vee}).

Here for $v\in\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$ such that $\sigma_{v}$ is unramified, we set $C_{\sigma_{v},\boldsymbol{\psi}_{v}}=C_{\sigma_{v}^{\vee},\boldsymbol{\psi}_{v}^{-1}}=1$ , and write $\zeta_{v}:=\prod_{i=1}^{m+n+1}\zeta_{\mathcal{F}_{v}}(i)$ .

Part II $p$ -adic Interpolation

In this part, we $p$ -adically interpolate the (square of) the Gan-Gross-Prasad period integral of the Klingen Eisenstein series with a cusp form.

7. Modular forms over unitary groups

In this section, we recall the geometric backgrounds of modular forms over unitary groups, their $p$ -adic analogues and $p$ -adic families. We try to put ourselves in the most general setup. The materials are largely taken from [EHLS20, Hsi14].

7.1. Generalities on PEL datums and unitary Shimura datums

7.1.1. PEL-type Shimura datums

Definition 7.1.

By a PEL-type Shimura datum, we mean a tuple $\mathscr{P}=(B,\ast,V,\psi_{V},h)$ , where

•

$B$ is a finite semisimple $\mathbb{Q}$ -algebra with a positive involution $\ast$ ,
•

$V$ is a symplectic $(B,\ast)$ -module, that is, a $B$ -module $V$ with a skew-symmetric nondegenerate $\mathbb{Q}$ -bilinear form $\psi_{V}:V\times V\rightarrow\mathbb{Q}$ such that $\psi_{V}(b^{\ast}u,v)=\psi_{V}(u,bv)$ for any $u,v\in V$ and $b\in B$ .
•
Let $\mathcal{C}:=\operatorname{End}_{B}(V)$ . It carries an adjoint involution defined by, for $\alpha\in\operatorname{End}_{B}(V)$ , the $\alpha^{\ast}\in\mathcal{C}$ such that

$\psi_{V}(\alpha^{\ast}v,w)=\psi_{V}(v,\alpha w),\quad\text{ for all }v,w\in V.$

Then $h$ is defined to be an $\mathbb{R}$ -algebra map $h:\mathbb{C}\rightarrow\mathcal{C}_{\mathbb{R}}$ such that
- –
  
  $h(\overline{z})=h(z)^{\ast}$ , where the $\ast$ on the right-hand-side is the adjoint involution on $\operatorname{End}_{B}(V)_{\mathbb{R}}$ ,
- –
  
  $(u,v)\mapsto\psi_{V}(u,h(\mathrm{i})v)$ is positive-definite and symmetric.

We say $\mathscr{P}$ is a simple PEL-type Shimura datum if $B$ is a simple $\mathbb{Q}$ -algebra. Given a PEL-type Shimura datum $\mathscr{P}$ , we associate it with

•

the field $F$ , defined to be the center of $B$ ,
•

the field $F_{0}:=\{b\in F:b^{\ast}=b\}$ , i.e. the subalgebra of $\ast$ -invariants in $F$ , and

•

two algebraic groups over $\mathbb{Q}$ , defined as

G(R):=\{g\in\mathrm{GL}_{B\otimes_{\mathbb{Q}}R}(V\otimes_{\mathbb{Q}}R):\psi_{V}(gu,gv)=\nu(g)\psi_{V}(u,v),\,u,v\in V\otimes_{\mathbb{Q}}R,\,\nu(g)\in R^{\times}\}

and

G_{1}(R):=\{g\in\mathrm{GL}_{B\otimes_{\mathbb{Q}}R}(V\otimes_{\mathbb{Q}}R):\psi_{V}(gu,gv)=\nu(g)\psi_{V}(u,v),\,u,v\in V\otimes_{\mathbb{Q}}R,\,\nu(g)\in(F_{0}\otimes_{\mathbb{Q}}R)^{\times}\}

for any $\mathbb{Q}$ -algebra $R$ . Clearly, $G$ is a subgroup of $G_{1}$ .

Let $X$ be the $G(\mathbb{R})$ -conjugacy class of $h^{-1}:\mathbb{C}^{\times}\rightarrow G_{\mathbb{R}}$ , then $(G,X)$ is a Shimura datum à la Deligne. For each neat compact open subgroup $K$ of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}})$ , by the result of Deligne, there is an algebraic variety $\mathrm{Sh}_{K}(G,X)$ over the reflex field $E$ of the PEL Shimura datum $\mathscr{P}$ such that

\mathrm{Sh}_{K}(G,X)(\mathbb{C})=G(\mathbb{Q})\backslash X\times G(\mathbb{A}_{\mathbb{Q},\mathrm{f}})/K.

Under certain conditions, this model is actually unique. We call it the Shimura variety of the Shimura datum $(G,X)$ , or the Shimura variety of the PEL Shimura datum $\mathscr{P}$ .

Let $p$ be a prime number. We hope to construct an integral model of the Shimura variety $\mathrm{Sh}_{K}(G,X)$ at $p$ , that is, a smooth model over the ring $\mathcal{O}_{E}\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$ . For this purpose, we need some extra data and assumptions.

Definition 7.2.

By an integral PEL-type Shimura datum, we mean a tuple $(\mathscr{P},\mathcal{O}_{B},L)$ , where

•

$\mathscr{P}=(B,\ast,V,\psi_{V},h)$ is a PEL-type Shimura datum,
•

$\mathcal{O}_{B}$ is a $\mathbb{Z}_{(p)}$ -order in $B$ which is stable under the involution $\ast$ on $B$ , and $\mathcal{O}_{B}\otimes_{\mathbb{Z}_{(p)}}\mathbb{Z}_{p}$ is a maximal order in $B$ . We require $B$ is unramified at $p$ , which means that $B_{\mathbb{Q}_{p}}$ is isomorphic to a product of matrix algebras over unramified extensions of $\mathbb{Q}_{p}$ .
•

$L$ is a $\mathbb{Z}_{p}$ -lattice in $V_{\mathbb{Q}_{p}}$ such that $L$ is stable under $\mathcal{O}_{B}$ and $L$ is self-dual with respect to the pairing $\psi_{V}$ .

Given an integral PEL-type Shimura datum with the extra unramified condition of $B$ at $p$ , $G_{\mathbb{Q}_{p}}$ is unramified. Indeed, let

K_{p}^{0}:=\{g\in G(\mathbb{Q}_{p}):g\cdot L\subseteq L\},

i.e. the subgroup of $G(\mathbb{Q}_{p})$ that stablizes the lattice $L$ , then $K_{p}^{0}$ is the hyperspecial subgroup of the $\mathbb{Z}_{p}$ -point of the smooth reductive model $\mathcal{G}$ of $G_{\mathbb{Q}_{p}}$ over $\mathbb{Z}_{p}$ .

7.1.2. Unitary Shimura datums

Definition 7.3.

We consider a special integral PEL-type Shimura datum $(\mathscr{P}=(B,\ast,V,\psi_{V},h),\mathcal{O}_{B},L)$ , where

•

$B=\mathcal{K}^{m}$ , the product of $m$ copies of $\mathcal{K}$ ,
•

$\ast$ is the complex conjugation on each factor $\mathcal{K}$ of $B$ ,
•

For $1\leq i\leq m$ , let $V_{i}$ be a finite dimensional $\mathcal{K}$ -vector space of dimension $n_{i}$ , equipped with an Hermitian form $\left\langle{-,-}\right\rangle_{V_{i}}:V\times V\rightarrow\mathcal{K}$ relative to $\mathcal{K}/\mathcal{F}$ . Let $\delta\in\mathcal{O}_{\mathcal{K}}$ be a totally imaginary element that is prime to $p$ . We then put $\psi_{V_{i}}:=\operatorname{Tr}_{\mathcal{K}/\mathbb{Q}}\delta\left\langle{-,-}\right\rangle_{V_{i}}$ . Then $V$ is taken to be $V_{1}\times\cdots\times V_{m}$ , and $\psi_{V}:=\psi_{V_{1}}\times\cdots\psi_{V_{m}}$ .
•

For each $\sigma\in\mathscr{V}_{\mathcal{K}}^{\infty}$ , $V_{i,\sigma}:=V_{i}\otimes_{\mathcal{K},\sigma}\mathbb{C}$ has a $\mathbb{C}$ -basis with respect to which $\psi_{V_{i}}$ is given by a matrix of the form $\operatorname{diag}[\mathbf{1}_{r_{i},\sigma},-\mathbf{1}_{{s_{i}},\sigma}]$ . Fixing such a basis, let $h_{i,\sigma}:\mathbb{C}\rightarrow\operatorname{End}_{\mathbb{R}}(V_{i,\sigma})$ be $z\mapsto\operatorname{diag}[z\mathbf{1}_{r_{i},\sigma},\overline{z}\mathbf{1}_{{s_{i}},\sigma}]$ . Let $h_{i}:=\prod_{\sigma\in\Sigma_{\mathcal{K}}}h_{i,\sigma}$ and $h:=\prod_{i=1}^{m}h_{i}$ .
•

$\mathcal{O}_{B}=\mathcal{O}_{K}^{m}$ .
•

Let $L_{i}\subset V_{i}$ be a free $\mathcal{O}_{\mathcal{K}}$ -module of rank $\dim_{\mathcal{K}}V$ , such that $\left\langle{L,L}\right\rangle_{\mathbb{Q}}\subset\mathbb{Z}$ and $\psi_{V_{i}}$ is a perfect pairing on $L\otimes\mathbb{Z}_{p}$ .

We say the PEL-type Shimura datum as a unitary Shimura datum. It is called a simple unitary Shimura datum if $m=1$ .

Given such a unitary Shimura datum, the corresponding objects are as follows.

•

the field $F=\mathcal{K}^{m}$ ,
•

the field $F_{0}=\mathcal{F}^{m}$ ,
•

the group $G_{1}=\operatorname{Res}^{\mathcal{F}}_{\mathbb{Q}}\left(\prod_{i=1}^{m}{\mathrm{GU}}(V_{i},\left\langle{-}\right\rangle_{V_{i}})\right)$ , where for $1\leq i\leq m$ , ${\mathrm{GU}}(V_{i})$ is the general unitary group over $\mathcal{F}$ attached to the Hermitian space $V_{i}$ , defined in Part 1.
•

the group $G$ is the subgroup of $G_{1}$ with rational silimitudes.

So in particular when $\mathcal{F}=\mathbb{Q}$ and $m=1$ , then $G_{1}=G$ is the general unitary group defined in Definition 2.1.

7.1.3. Hodge structures, lattices and level subgroups at $p$

Now we concentrate on the case of unitary Shimura datum. For each $1\leq i\leq m$ and $\sigma\in\mathscr{V}_{\mathcal{K}}^{\infty}$ , $h_{i,\sigma}$ determines a pure Hodge structure of weight $-1$ on $V_{i,\sigma}:=L_{i}\otimes_{\mathcal{O}_{\mathcal{K}},\sigma}\mathbb{C}$ . Let $V_{i,\sigma}^{0}$ be the degree zero piece of the Hodge filtration. This is an $\mathcal{O}_{\mathcal{K}}\otimes_{\sigma}\mathbb{C}$ -submodule of $V$ . For each $\sigma\in\mathscr{V}_{\mathcal{K}}^{\infty}$ , let $a_{\sigma,i}=\dim_{\mathbb{C}}(V_{i,\sigma}^{0}\otimes_{\mathcal{O}_{\mathcal{K}}\otimes_{\sigma}\mathbb{C}}\mathbb{C})$ and $b_{i,\sigma}:=n_{i}-a_{\sigma,i}$ . We note that for $\sigma\in\mathscr{V}_{\mathcal{K}}^{\infty}$ ,

(a_{\sigma,i},b_{\sigma,i})=(b_{\mathsf{c}\sigma,i},a_{\mathsf{c}\sigma,i})=(r_{i,\sigma},s_{i,\sigma}).

We assume throughout the following fundamental hypothesis, called the ordinary hypothesis

v_{\sigma}=v_{\sigma^{\prime}}\Rightarrow r_{\sigma,i}=r_{\sigma^{\prime},i},\quad i=1,\ldots,m

(ord)

for any $\sigma,\sigma^{\prime}\in\mathscr{V}_{\mathcal{K}}^{\infty}$ with $v_{\sigma}$ and $v_{\sigma^{\prime}}$ the corresponding $p$ -adic places of them given by the embedding $\iota_{p}$ .

Then for each place $w$ of $\mathcal{K}$ above $p$ , we can then define $(a_{w,i},b_{w,i})=(a_{\sigma,i},b_{\sigma,i})$ for any $\sigma\in\Sigma_{\mathcal{K}}$ corresponding to $w$ . Let $L_{i,w}:=L\otimes_{\mathcal{O}_{\mathcal{K}}}\mathcal{O}_{\mathcal{K},w}$ . For each $1\leq i\leq m$ , we fix an $\mathcal{O}_{\mathcal{K}}\otimes\mathbb{Z}_{p}$ -decomposition $L_{i}\otimes\mathbb{Z}_{p}=L_{i}^{+}\oplus L_{i}^{-}$ such that

•

$L_{i}^{+}$ is an $\mathcal{O}_{K,w}$ -module with $\operatorname{rank}_{\mathcal{O}_{\mathcal{K},w}}L_{i}^{+}=a_{w,i}$ . So $L_{i}^{+}$ is an $\mathcal{O}_{K,w}$ -module with $\operatorname{rank}_{\mathcal{O}_{\mathcal{K},w}}L_{i}^{-}=b_{w,i}$ and $L_{i,w}=L_{i,w}^{+}\oplus L_{i,w}^{-}$ .
•

$L_{i,w}^{\pm}$ is the annihilator of $L_{i,\overline{w}}^{\pm}$ for the perfect pairing $\psi_{V}:L_{i,w}\times L_{i,\overline{w}}\rightarrow\mathbb{Z}_{p}(1)$ .

For each $i$ and $w$ , we fix a basis of $L_{i,w}^{+}$ to regard it as a direct sum of copies of $\mathcal{O}_{\mathcal{K},w}$ . ²¹²¹21In [Hsi14, Section 1.8], Hsieh very carefully chose the precise basis of $L_{i,w}^{+}$ . Here we only roughly speak of such a basis without explicitly specifying it. Taking $\mathbb{Z}_{p}$ -duals via $\psi_{V}$ yields a decomposition of $L_{i,w}^{-}$ as a direct sum of copies of $\mathcal{O}_{\mathcal{K},\overline{w}}$ . The choices of these decompositions determines isomorphisms

\mathrm{GL}_{\mathcal{O}_{\mathcal{K},w}}(L_{i,w}^{+})\simeq\mathrm{GL}_{a_{w,i}}(\mathcal{O}_{\mathcal{K},w})

\mathrm{GL}_{\mathcal{O}_{\mathcal{K},w}}(L_{i,w}^{-})\simeq\mathrm{GL}_{b_{w,i}}(\mathcal{O}_{\mathcal{K},w})

\mathrm{GL}_{\mathcal{O}_{\mathcal{K},w}}(L_{i,w})\simeq\mathrm{GL}_{n_{i}}(\mathcal{O}_{\mathcal{K},w})

and the embedding

\mathrm{GL}_{\mathcal{O}_{\mathcal{K},w}}(L_{i,w}^{+})\times\mathrm{GL}_{\mathcal{O}_{\mathcal{K},w}}(L_{i,w}^{-})\hookrightarrow\mathrm{GL}_{\mathcal{O}_{\mathcal{K},w}}(L_{i,w})

is given by $(A,B)\mapsto\operatorname{diag}[A,B]$ .

To define appropriate level subgroups at $p$ , we start with defining

H_{\mathscr{P}}=\mathrm{GL}_{\mathcal{O}_{B}\otimes\mathbb{Z}_{p}}(L^{+})\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\prod_{w\mid p}\prod_{i=1}^{m}\mathrm{GL}_{a_{w,i}}(\mathcal{O}_{\mathcal{K},w}).

Let $B_{\mathscr{P}}\subset H_{\mathscr{P}}$ be the Borel subgroup corresponds via this isomorphism with the product of the upper-triangular Borel subgroups of general linear groups. Let $N_{\mathscr{P}}$ be its unipotent radical. Let $T_{\mathscr{P}}:=B_{\mathscr{P}}/N_{\mathscr{P}}$ , this is identified by the isomorphism with diagonal matrices. Let $B^{+}_{\mathscr{P}}\subset G_{\mathbb{Z}_{p}}$ be the Borel subgroup that stablizes $L^{+}$ and such that

B^{+}_{\mathscr{P}}\twoheadrightarrow\mathbb{G}_{\mathrm{m}}\times B_{\mathscr{P}}\subset\mathbb{G}_{\mathrm{m}}\times H_{\mathscr{P}},

(7.21)

where the map to the first factor is the silimitude character $\nu$ and the second projection is the projection to $H_{\mathscr{P}}$ . Let $N_{\mathscr{P}}^{+}$ be its unipotent radical. Under the identification,

B^{+}_{\mathscr{P}}\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathbb{G}_{\mathrm{m}}\times\prod_{w\in\mathscr{V}_{\mathcal{K}}^{(p)}}\prod_{i=1}^{m}\left\{\begin{bmatrix}A&B\\ 0&D\end{bmatrix}\in\mathrm{GL}_{n_{i}}(\mathcal{O}_{\mathcal{K},w}):A\in B_{a_{w,i}}(\mathcal{O}_{\mathcal{K},w}),D\in B^{-}_{b_{w,i}}(\mathcal{O}_{\mathcal{K},w})\right\}.

(7.22)

Then we define the following level subgroups at $p$ . They are all defined under the chosen basis of $L^{\bullet}$ above.

•

We have the hyperspecial subgroup $K_{p}^{0}=\mathbb{G}_{\mathrm{m}}\times H_{\mathscr{P}}$ , where $\mathbb{G}_{\mathrm{m}}$ is the similitude factor part.
•

We have the congruence level subgroup $K_{p,r}^{0}\subset G(\mathbb{Z}_{p})$ consists of those $g$ such that $g\mod p^{r}\in B_{\mathscr{P}}^{+}(\mathbb{Z}/p^{r})$ .
•

We define the principal level subgroup $K_{p,r}^{1}$ as the subgroup of $K_{p,r}^{0}$ consists of those $g$ projecting under the surjection (7.21) to an element in $(\mathbb{Z}/p^{r})^{\times}\times N_{\mathscr{P}}^{+}(\mathbb{Z}/p^{r})$ .

Then it follows that $K_{p,r}^{0}/K_{p,r}^{1}\simeq T_{\mathscr{P}}(\mathbb{Z}/p^{r})$ .

7.2. Moduli problems

In this subsection, we start with an integral PEL-type Shimura datum $(\mathscr{P}=(B,\ast,V,\psi_{V},h),\mathcal{O}_{B},L)$ , and omit it from the following-up notations.

Let $\square$ be a finite set of primes, which is usually taken to be the empty set or $\{p\}$ . Let $S$ be a locally noetherian connected $\mathcal{O}_{E}\times\mathbb{Z}_{(\square)}$ -scheme.

Definition 7.4.

We say that a tuple $\underline{A}=(A,\lambda,\iota)$ is a $\mathbb{Z}_{(\square)}$ -polarized abelian scheme with an action of $\mathcal{O}_{B}$ if

•

$A$ is an abelian scheme over $S$ ,
•

$\lambda$ is a prime-to- $\square$ polarization of $A$ over $S$ .
•

$\iota:\mathcal{O}_{B}\hookrightarrow\operatorname{End}_{S}A\otimes_{\mathbb{Z}}\mathbb{Z}_{(\square)}$ which respects involutions on both sides: the involution $\ast$ on the left and the Rosati involution coming from $\lambda$ on the right.

Let $K$ be a compact open subgroup of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}})$ , we define a $K^{\square}$ -level structure $\overline{\eta}^{(\square)}=\eta^{(\square)}K$ of $\underline{A}$ to be a $\pi_{1}(S,\overline{s})$ -invariant $K$ -orbit of the isomorphism of $\mathcal{O}_{B}\otimes\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{\square}$ -modules

\eta^{(\square)}:L\otimes\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{\square}\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathrm{H}_{1}(A_{t},\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{\square}),

which identify $\psi_{V}$ with a $\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{\square,\times}$ -multiple of the symplectic pairing on the Tate module $\mathrm{H}_{1}(A_{t},\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{\square})$ defined by $\lambda$ and the Weil pairing.

Let $K$ be a compact open subgroup of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}})$ .

Definition 7.5.

Let $\mathfrak{M}_{K}^{\square}$ be the following category fibered in groupoids over the category of $\mathcal{O}_{E}\otimes_{\mathbb{Z}}\mathbb{Z}_{(\square)}$ :

•

The objects over a scheme $S$ are quadruples $(\underline{A},\overline{\eta}^{\square})$ , where $\underline{A}$ is a $\mathbb{Z}_{(\square)}$ -polarized abelian scheme with an action of $\mathcal{O}_{B}$ , and $\overline{\eta}^{(\square)}$ is a $K^{\square}$ -level structure, such that $\operatorname{Lie}_{S}A$ satisfies the Kotwitz determinant condition defined by $(L\otimes\mathbb{R},\psi_{V},h)$ ²²²²22The requirements on the dimension of the abelian scheme $A$ and the information on the signature of the Hermitian spaces $V_{i}$ in the case of unitary Shimura datums, are encoded in this Kotwitz determinant condition. So here the definition coincide with our usual definition, for example, in [Hsi14, Section 2.1].
•

The morphisms from $(\underline{A},\overline{\eta}^{\square})$ to $(\underline{A}^{\prime},\overline{\eta}^{\square,\prime})$ are given by a $\mathbb{Z}_{(\square)}$ -isogeny $f:A\rightarrow A^{\prime}$ that is compatible with the action of $\mathcal{O}_{B}$ and the level structures.

We have the following well-known representability theorems in the case when $\square$ is the empty set or $\{p\}$ .

Theorem 7.6 (Degline-Kotwitz).

When $\square$ is the empty set and $K$ is neat, $\mathfrak{M}_{K}^{\emptyset}$ is representable by a scheme $\mathcal{S}_{G}(K)$ . Moreover, in the case when $\mathscr{P}$ is a unitary Shimura datum, we have

\mathcal{S}_{G}(K)=\sqcup_{G^{\prime}\in\ker^{1}(\mathbb{Q},G)}\mathrm{Sh}_{K}(G,X),

where $\ker^{1}(\mathbb{Q},G)$ the set of locally trivial elements of $\mathrm{H}^{1}(\mathbb{Q},G)$ .

More precisely, the elements of $\ker^{1}(\mathbb{Q},G)$ classify isomorphism classes of Hermitian tuples $(V_{j}^{\prime},\left\langle{-,-}\right\rangle_{V_{j}^{\prime}})_{1\leq j\leq m}$ that are locally isomorphic to $(V_{j},\left\langle{-,-}\right\rangle_{V_{j}})_{1\leq j\leq m}$ . Then $\mathcal{S}_{G}(K)$ is a disjoint union of copies of $\mathrm{Sh}_{K}(G,X)$ . This will not cause too much trouble since for applications to automorphic forms, and we only need one copy of them. See [EHLS20, Section 2.3] for details.

Theorem 7.7.

When $\square=\{p\}$ , the category $\mathfrak{M}_{K}^{p}$ is a smooth Deligne-Mumford stack. When $K^{p}$ is neat, it is representable by a quasi-projective scheme $\mathcal{S}_{G}(K^{p})$ . Moreover, in the case when $\mathscr{P}$ is a unitary Shimura datum, we have

\mathcal{S}_{G}(K^{p})\times_{\mathcal{O}_{E}\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}}\operatorname{Spec}E\mathrel{\leavevmode\hbox to14.89pt{\vbox to14.3pt{\pgfpicture\makeatletter\hbox{\hskip 0.33301pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{13.82219pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.02219pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{7.11108pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.0pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\sim$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\mathcal{S}_{G}(K^{p}K_{p}^{0}).

In the case when $\mathscr{P}$ is a simple unitary Shimura datum given by an Hermitian $\mathcal{K}$ -vector space $V$ , by abuse of notation, we call $\mathcal{S}_{G}(K)$ (resp. $\mathcal{S}_{G}(K^{p})$ ) the (unitary) Shimura variety attached to ${\mathrm{GU}}(V)$ . The geometric theory of modular forms and their $p$ -adic theory over ${\mathrm{GU}}(V)$ are build up over $\mathcal{S}_{G}(K)$ and $\mathcal{S}_{G}(K^{p})$ .

In the following-up sections, we concentrate on the case of simple unitary PEL datums.

7.3. Compactifications

The theory of toroidal compactification of unitary Shimura varieties are done in [Lan13]. Fixing certain smooth projective polyhedral cone decomposition (which we do not make precise here), one can attach the toroidal compactification $\mathcal{S}_{K^{\square}}^{\mathrm{tor}}$ of $\mathcal{S}_{K^{\square}}$ . We focus on the case $\square=\{p\}$ . Then we know:

•

The toroidal compactification $\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})$ contains $\mathcal{S}_{G}(K^{p})$ as an open dense subscheme. The complement of $\mathcal{S}_{G}(K^{p})$ is a relative Cartier divisor with normal crossings. We denote by $\mathscr{I}_{\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})}$ the ideal sheaf of the boundary of $\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})$ .
•

There is a quadruple $\underline{\mathscr{G}}:=(\mathscr{G},\lambda,\iota,\eta)$ over $\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})$ , where $\mathscr{G}$ is a semi-abelian scheme with an $\mathcal{O}_{\mathcal{K}}$ -action by $\iota$ and a homomorphism $\lambda:\mathscr{G}\rightarrow\mathscr{G}^{\vee}$ , such that $\underline{\mathscr{G}}|_{\mathcal{S}_{G}(K^{p})}=\underline{\mathscr{A}}$ , the universal quadruple over $\mathcal{S}_{G}(K^{p})$ , and $\eta$ is the level structure in the quadruple $\underline{\mathscr{A}}$ .

Let $\underline{\omega}:=e^{\ast}\Omega_{\mathscr{G}/\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})}^{1}$ , where $e:\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})\rightarrow\mathscr{G}$ is the zero section of the semiabelian scheme $\mathscr{G}$ over $\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})$ . Then $\underline{\omega}$ is a locally free coherent $\mathscr{O}_{\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})}$ -module. The minimal compactification of $\mathcal{S}_{G}(K^{p})$ is defined to be

\mathcal{S}^{\min}_{G}(K^{p}):=\mathrm{Proj}\left(\bigoplus_{k=0}^{\infty}\mathrm{H}^{0}(\mathcal{S}_{G}^{\mathrm{tor}}(K^{p}),\det\underline{\omega}^{k})\right).

Let $\pi:\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})\rightarrow\mathcal{S}^{\min}_{G}(K^{p})$ be the natural projection.

7.4. Igusa schemes

In the following, by abuse of notations, we also denote $\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})$ and $\mathcal{S}_{G}^{\min}(K^{p})$ by their base change to $\mathbb{Z}_{p}$ via the map $\mathcal{O}_{\mathcal{K},(p)}\rightarrow\mathbb{Z}_{p}$ induced by our fixed embedding $\iota_{p}$ , and let $\mathcal{S}_{G/\mathbb{F}_{p}}^{\mathrm{tor}}(K^{p})$ and $\mathcal{S}_{G/\mathbb{F}_{p}}^{\min}(K^{p})$ be their special fibers correspondingly.

7.4.1. Hasse invariants

Let $\mathrm{Ha}\in\mathrm{H}^{0}(\mathcal{S}_{G,/\mathbb{F}_{p}}^{\mathrm{tor}}(K^{p}),(\det\underline{\omega})^{p-1})$ be the Hasse invariant defined in [Lan18, Section 6.3.1]. In particular, for each geometric point $\overline{s}$ of $\mathcal{S}_{G/\mathbb{F}_{p}}^{\mathrm{tor}}(K^{p})$ , the Hasse invariant of the corresponding semiabelian scheme $\mathscr{G}_{\overline{s}}$ is nonzero if and only if the abelian part of $\mathscr{G}_{\overline{s}}$ is ordinary. Because $\pi_{\ast}\omega$ is ample, for some $t_{E}>0$ , there exists an element in $\mathrm{H}^{0}(\mathcal{S}_{G}^{\min}(K^{p}),(\pi_{\ast}\omega)^{t_{E}(p-1)})$ lifting the $t_{E}$ -th power of the push-forward of $\mathrm{Ha}$ . We denote by $E$ the pullback under $\pi$ of any such lift, which (because $\pi^{\ast}\pi_{\ast}\omega\simeq\omega$ ) defines an element $E\in\mathrm{H}^{0}(\mathcal{S}_{G}^{\mathrm{tor}}(K^{p}),(\det\underline{\omega})^{t_{E}(p-1)})$ .

7.4.2. Ordinary locus and the Igusa tower

Consider the following moduli problem.

Definition 7.8.

Let $\mathfrak{I}_{K^{p}K_{p,n}^{1}}$ be the following category fibered in groupoids over the category of $\mathcal{O}_{E}\otimes_{\mathbb{Z}}\mathbb{Z}_{(p)}$ :

•

The objects over a scheme $S$ are quadruples $(\underline{A},\overline{\eta}^{p},j_{n})$ , where $(\underline{A},\overline{\eta}^{p})\in\mathfrak{M}^{p}(S)$ and $j_{n}$ is a level $K_{p,n}^{1}$ -structure, defined as a $K_{p,n}^{1}$ -orbit of monomorphisms as $\mathcal{O}_{\mathcal{K}}$ -schemes over $S$ :

$j_{p^{n}}:\boldsymbol{\mu}_{p^{n}}\otimes_{\mathbb{Z}}L^{+}\hookrightarrow A[p^{n}].$
•

The morphisms from $(\underline{A},\overline{\eta}^{p},j_{n})$ to $(\underline{A}^{\prime},\overline{\eta}^{p,\prime},j_{n}^{\prime})$ are given by an element

$f\in\operatorname{Hom}_{\mathfrak{M}^{p}}((\underline{A},\overline{\eta}^{p}),(\underline{A}^{\prime},\overline{\eta}^{p,\prime}))$

that is compatible with the level- $p^{n}$ structures of $A$ and $A^{\prime}$ .

Theorem 7.9.

The moduli problem $\mathfrak{I}_{K^{p}K_{p,n}^{1}}$ is relatively representable over $\mathcal{S}_{G}(K^{p})$ , and thus it is represented by a scheme $\mathcal{I}_{G}(K^{p}K_{p,n}^{1})$ over the scheme $\mathcal{S}_{G}(K^{p})$ . The scheme $\mathcal{I}_{G}(K^{p}K_{p,n}^{1})$ is called the ordinary locus of level $K_{p,n}^{1}$ .

We sometimes write $\mathcal{I}_{G}(K^{p}K_{p,n}^{1})$ simply as $\mathcal{I}_{G}(K_{p,n}^{1})$ or even $\mathcal{I}_{n}$ for short if it does not cause any confusion.

Let $\mathcal{I}_{G}^{\mathrm{tor}}(K^{p}K_{p,n}^{1})$ be the partial toroidal compactification of the ordinary locus $\mathcal{I}_{G}(K^{p}K_{p,n}^{1})$ ([Lan18, Theorem 5.2.1.1]). It is obtained by gluing to $\mathcal{I}_{G}(K^{p}K_{p,n}^{1})$ the toroidal boundary charts parameterizing degenerating families defined in [Lan18, Definition 3.4.2.0].

Let $\mathcal{I}_{G,m}^{\mathrm{tor}}(K^{p}K_{p,n}^{1})$ be the base change of $\mathcal{I}_{G}^{\mathrm{tor}}(K^{p}K_{p,n})$ to $\mathbb{Z}/p^{m}\mathbb{Z}$ . By [Lan18, Lemma 6.3.2.7], $\mathcal{S}_{G,m}^{\mathrm{tor}}(K^{p})[1/E]$ agrees with the ordinary locus in [Lan18, Theorem 5.2.1.1] for the hyperspecial level at $p$ , and by [Lan18, Corollary 5.2.2.3], the map

\pi_{m,n}:\mathcal{I}_{G,m}^{\mathrm{tor}}(K^{p}K_{p,n}^{1})\rightarrow\mathcal{S}_{G,m}^{\mathrm{tor}}(K^{p})[1/E]

forgetting the level $K_{p,n}^{1}$ -level structure is finite étale.

We gather the above objects into the following diagram, broadly speaking as the diagram of Igusa towers.

To simplify the notation, when the level group $K^{p}$ away from $p$ is clear or fixed in the context, we simply write $\mathcal{I}_{m,n}^{\mathrm{tor}}$ for $\mathcal{I}_{G,m}(K^{p}K_{p,n}^{1})$ for simplicity.

7.5. Modular forms on unitary groups

7.5.1. $p$ -adic modular forms

We define the space of mod $p^{m}$ automorphic forms on $G$ of level $n$ by

V_{G,m,n}:=\mathrm{H}^{0}(\mathcal{I}_{m,n}^{\mathrm{tor}},\mathscr{O}_{\mathcal{I}_{m,n}^{\mathrm{tor}}}).

We let $\mathscr{I}_{\mathcal{I}_{m,n}^{\mathrm{tor}}}:=\pi_{m,n}^{\ast}\mathscr{I}_{\mathcal{S}_{m}^{\mathrm{tor}}}$ , and similarly define the space of mod $p^{m}$ cuspidal automorphic forms on $G$ of level $n$ by

V^{0}_{G,m,n}:=\mathrm{H}^{0}(\mathcal{I}_{m,n}^{\mathrm{tor}},\mathscr{I}_{m,n}^{\mathrm{tor}}).

Then we define $p$ -adic automorphic forms by passing to the limit.

Definition 7.10.

We define the space of $p$ -adic automorphic forms (resp. cuspidal automorphic forms) with torsion coefficient as

\mathcal{V}_{G}:=\varinjlim_{m}\varinjlim_{n}V_{G,n,m}\quad(\text{resp. }\mathcal{V}^{0}_{G}:=\varinjlim_{m}\varinjlim_{n}V_{G,n,m}^{0}).

We define the space of $p$ -adic automorphic forms (resp. cuspidal automorphic forms) with integral coefficient as

V_{G}:=\varprojlim_{m}\varinjlim_{n}V_{G,n,m}\quad(\text{resp. }V^{0}_{G}:=\varprojlim_{m}\varinjlim_{n}V_{G,n,m}^{0}).

For any $p$ -adic ring $R$ , i.e. $R$ satisfies $R\simeq\varprojlim_{n}R/p^{n}R$ , we have $p$ -adic automorphic forms (resp. cuspidal automorphic forms) with coefficient ring $R$ defined by base change to $R$ , denoted by $V_{G}(R)$ and $V_{G}^{0}(R)$ respectively.

Recall we defined several algebraic groups $H_{\mathscr{P}}$ , $B_{\mathscr{P}}$ and $T_{\mathscr{P}}$ attached to a unitary Shimura datum previously.

Definition 7.11.

For any sufficiently large finite extension $L/\mathbb{Q}_{p}$ , we define the weight algebra with coefficient field $L$ of the unitary Shimura datum $\mathscr{P}$ as the completed group algebra $\Lambda_{\mathscr{P}}:=\mathcal{O}_{L}\llbracket{T_{\mathscr{P}}(\mathbb{Z}_{p})}\rrbracket$ .

The space of $p$ -adic automorphic forms carries many actions.

•

Recall $T_{\mathscr{P}}(\mathbb{Z}/p^{n})$ naturally identifies $K_{p,n}^{0}/K_{p,n}^{1}$ . Hence the group $T_{\mathscr{P}}(\mathbb{Z}_{p})$ actually acts on $\mathcal{V}_{G},\mathcal{V}^{0}_{G},V_{G}(R)$ , $V_{G}^{0}(R)$ , making these spaces into $\Lambda_{\mathscr{P}}$ -modules.
•

The action of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{p})$ on the Igusa tower gives an action of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{p})$ on the space of $p$ -adic automorphic forms. Let $K^{p}$ be an open compact subgroup of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{p})$ , the submodules fixed by $K^{p}$ , denoted by $V_{G}(K^{p},R),V_{G}^{0}(K^{p},R)$ and so on. These $p$ -adic automorphic forms are said to be of tame level $K^{p}$ .

Definition 7.12.

We define a $p$ -adic weight $\tau$ as a $\overline{\mathbb{Q}}_{p}$ -valued character of $\Lambda_{\mathscr{P}}$ . In other words, under the isomorphism $T_{\mathscr{P}}(\mathbb{Z}_{p})\simeq\prod_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}\mathrm{GL}_{r_{v}+s_{v}}(\mathcal{O}_{\mathcal{F},v})$ , a $p$ -adic weight $\tau$ is a collection $\{\tau_{v}\}_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}$ with each $\tau_{v}$ given by

	$\displaystyle\tau_{v}:T_{r_{v}+s_{v}}(\mathcal{O}_{\mathcal{F},v})$	$\displaystyle\rightarrow\overline{\mathbb{Q}}_{p}^{\times},$
	$\displaystyle\operatorname{diag}[a_{1},\ldots,a_{m+n}]$	$\displaystyle\mapsto\tau_{1,v}^{+}(a_{1})\cdots\tau_{r_{v},v}^{+}(a_{r_{v}})\tau_{1,v}^{-}(a_{r_{v}+1})\cdots\tau_{s_{v},v}^{-}(a_{r_{v}+s_{v}}),$

where $\tau_{i,v}^{\pm}:\mathcal{O}_{\mathcal{F},v}^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ are continuous characters. We say a $p$ -adic weight $\tau$ is arithmetic, if for each $v\in\mathscr{V}_{\mathcal{F}}^{(p)}$ , $\tau_{i,v}^{\pm}:\mathcal{O}_{\mathcal{F},v}^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ is given by a product of an algebraic character and a finite order character, i.e. $\tau_{i,v}^{\pm}(a)=\epsilon^{\pm}_{i,\tau,v}(a)a^{t_{i,\tau,v}^{\pm}}$ for finite order characters $\epsilon_{i,\tau,v}^{\pm}:\mathcal{O}_{\mathcal{F},v}^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ and $t_{i,\tau,v}^{\pm}\in\mathbb{Z}$ .

Given a $p$ -adic weight $\tau$ , we denote $\mathbb{Q}_{p}(\tau)$ the finite field extension of $\mathbb{Q}_{p}$ generated by the image of $\tau$ . We put $V_{m,n}[\tau]$ as the subspace of $V_{m,n}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{\mathbb{Q}_{p}(\tau)}$ on which $\Lambda_{\mathscr{P}}$ acts by inverse of the character $\tau$ . Similarly, we define the spaces $V_{m,n}^{0}[\tau]$ , $\mathcal{V}[\tau]$ and $\mathcal{V}^{0}[\tau]$ . These $p$ -adic automorphic forms are said to have weight $\tau$ .

7.5.2. Classical automorphic forms

Though we are free to use the space of $p$ -adic modular forms, which is larger than the space of classical automorphic forms, automorphic computations are more frequently done in the classical way. In this subsection, we consider the classical automorphic forms over unitary groups.

Let $\underline{t}:=(t_{1}^{+},\ldots,t_{r}^{+},t_{1}^{-},\ldots,t_{s}^{-})\in\mathbb{Z}^{r+s}$ be any $(r+s)$ -tuple. We define for any algebra $R$ , the space

W_{\underline{t}}(R):=\{f\in R[\mathrm{GL}_{r}\times\mathrm{GL}_{s}]:f(tn_{+}g)=k^{-1}(t)f(g),t\in T_{r}\times T_{s},n_{+}\in U_{r}\times U_{s}^{-}\},

where $R[\mathrm{GL}_{r}\times\mathrm{GL}_{s}]$ denotes the polynomial functions on $\mathrm{GL}_{r}\times\mathrm{GL}_{s}$ with coefficients in $R$ and $k$ is regarded as an algebraic character on $T_{r}\times T_{s}$ defined by

k(\operatorname{diag}[a_{1},\ldots,a_{r}],\operatorname{diag}[a_{r+1},\ldots,a_{r+s}])=a_{1}^{t_{1}^{+}}\cdots a_{r}^{t_{r}^{+}}a_{r+1}^{t_{1}^{-}}\cdots a_{r+s}^{t_{s}^{-}}.

Then $W_{\underline{t}}$ is a free $R$ -module and is the algebraic representation of $\mathrm{GL}_{r}(R)\times\mathrm{GL}_{s}(R)$ with minimal weight $-k$ with respect to $U_{r}\times U_{s}^{-}$ . We regard it as

W_{\underline{t}}=W_{\underline{t}}^{+}\boxtimes W_{\underline{t}}^{-}

as an algebraic representation of $\mathrm{GL}_{r}\times\mathrm{GL}_{s}$ .

Recall $\underline{\omega}:=e^{\ast}\Omega_{\mathscr{G}/\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})}^{1}$ with decomposition $\underline{\omega}^{+}$ (resp $\underline{\omega}^{-}$ ) be the subsheaf of $\underline{\omega}$ on which $i(b)$ acts by $b$ (resp. $\overline{b}$ ) for all $b\in\mathcal{O}_{\mathcal{K}}$ . Because $p$ is unramified in $\mathcal{K}$ , $\underline{\omega}^{+}$ (resp $\underline{\omega}^{-}$ ) is locally free of rank $r$ (resp. $s$ ) and $\underline{\omega}=\underline{\omega}^{+}\oplus\underline{\omega}^{-}$ . Set

\omega_{\underline{t}}^{+}=\underline{\mathrm{Isom}}_{\mathcal{S}^{\mathrm{tor}}_{G}}(\mathscr{O}_{\mathcal{S}^{\mathrm{tor}}_{G}}^{\oplus r},\underline{\omega}^{+})\times^{\mathrm{GL}_{r}}W_{\underline{t}}^{+},\quad\omega_{\underline{t}}^{-}=\underline{\mathrm{Isom}}_{\mathcal{S}^{\mathrm{tor}}_{G}}(\mathscr{O}_{\mathcal{S}^{\mathrm{tor}}_{G}}^{\oplus s},\underline{\omega}^{-})\times^{\mathrm{GL}_{s}}W_{\underline{t}}^{-},

and put $\omega_{\underline{t}}=\omega_{\underline{t}}^{+}\otimes\omega_{\underline{t}}^{-}$ .

Suppose for each $v\in\mathscr{V}_{\mathcal{F}}^{(p)}$ , we are given a $(r_{v}+s_{v})$ -tuple $\underline{t}_{v}$ as above and, by abuse of notation, let $\underline{t}=\{\underline{t}_{v}\}_{\mathscr{V}_{\mathcal{F}}^{(p)}}$ . Then following the above process, we have well-defined $\omega_{\underline{t}}$ provided $\underline{t}$ is a parallel weight.

Let $v\in\mathscr{V}_{\mathcal{F}}^{(p)}$ and $\underline{\epsilon}_{v}:=(\epsilon_{1,v}^{+},\ldots,\epsilon_{r,v}^{+},\epsilon_{1,v}^{-},\ldots,\epsilon_{s,v}^{-})$ be an $(r+s)$ -tuple of finite order characters $\mathcal{O}_{\mathcal{F}_{v}}^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ and let $\underline{\epsilon}=\{\underline{\epsilon}_{v}\}_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}$ . Let $F/\mathbb{Q}_{p}$ be a finite extension containing the values of all $\epsilon_{i}^{\pm}$ .

Definition 7.13.

We define the space of classical automorphic forms on $G$ of weight $\underline{t}$ , level $K^{p}K_{p,n}^{1}$ and nebentypus $\underline{\epsilon}$ as elements in the space

M_{\underline{t}}(K^{p}K_{p,n}^{1};\underline{\epsilon};F):=\mathrm{H}^{0}(\mathcal{S}_{G}^{\mathrm{tor}}(K^{p}K_{p,n}^{1}),\omega_{\underline{t}})\otimes_{\mathcal{K}}F)[\underline{\epsilon}].

Similarly we have the space of classical cuspidal automorphic forms

M_{\underline{t}}^{0}(K^{p}K_{p,n}^{1};\underline{\epsilon};F):=\mathrm{H}^{0}(\mathcal{S}_{G}^{\mathrm{tor}}(K^{p}K_{p,n}^{1}),\omega_{\underline{t}}\otimes\mathscr{I}_{\mathcal{S}_{G}^{\mathrm{tor}}(K^{p})})\otimes_{\mathcal{K}}F)[\underline{\epsilon}].

One has embeddings (see, for example, [Hsi14, Section 3.3])

	$M_{\underline{t}}(K^{p}K_{p,n}^{1};\underline{\epsilon};F)\hookrightarrow V_{G}(K,F)[\tau_{\underline{t},\underline{\epsilon}}]$	(7.23a)
and
	$M_{\underline{t}}^{0}(K^{p}K_{p,n}^{1};\underline{\epsilon};F)\hookrightarrow V_{G}^{0}(K,F)[\tau_{\underline{t},\underline{\epsilon}}],$	(7.23b)

where $\tau_{\underline{t},\underline{\epsilon}}$ is the $p$ -adic weight given by $\underline{t}$ and $\underline{\epsilon}$ following the notations in Definition 7.12.

7.6. Hecke operators on modular forms

One can turn to [Hsi14, Section 3.7] for carefully presented details. Here we only give a rush sketch. Let $K_{j}^{p}$ be open compact subgroups of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{p})$ for $j=1,2$ , such that $K_{j}^{p}\cdot G(\mathbb{Z}_{p})$ are neat.

7.6.1. Hecke operators away from $p$

Let $g\in G(\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{p})$ , we define the double coset operator

[K_{2}^{p}gK_{1}^{p}]:V_{G}(K_{1}^{p})[\tau]\rightarrow V_{G}(K_{2}^{p})[\tau]

through the action of $G(\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{p})$ on the space of modular forms

[K_{2}^{p}gK_{1}^{p}]f=\sum_{g_{j}}[g_{j}]^{\ast}f,\quad\text{where }K_{2}^{p}gK_{1}^{p}=\bigsqcup_{g_{j}}g_{j}K_{1}^{p}.

Similarly, for neat open compact subgroups $K_{j}^{p}\cdot K_{p,r}^{1}$ , we can define the corresponding double coset operators in the same way and denote them as $[K_{2,r}^{p}gK_{1,r}^{p}]$ . When $K_{1}^{p}=K_{2}^{p}$ is understood, we write $T(g)$ instead of $[K_{1}gK_{1}]$ and $T_{r}(g)$ instead of $[K_{1,r}gK_{1,r}]$ .

7.6.2. Hida’s operators at $p$

We define some particular elements $t_{j}$ in $B_{\mathscr{P}}^{+}(\mathbb{Q}_{p})$ as $t_{j}:=(1,(t_{w,j}))$ under the isomorphism (7.22), for $1\leq j\leq r+s$ , as

t_{w,j}:=\begin{cases}\operatorname{diag}[p\mathbf{1}_{j},\mathbf{1}_{r+s-j}],&\quad j\leq a_{w}\\ \operatorname{diag}[p\mathbf{1}_{a_{w}},\mathbf{1}_{r+s-j},p\mathbf{1}_{j-a_{w}}],&\quad j>a_{w}\end{cases}.

In [Hida04, Section 8.3.1], Hida has defined has defined an action of the double cosets $u_{w,j}=B_{\mathscr{P}}^{-}(\mathbb{Z}_{p})t_{w,j}B_{\mathscr{P}}(\mathbb{Z}_{p})$ on the modules of $p$ -adic modular forms and cuspforms, via correspondences on the Igusa tower. We put for any $v\in\mathscr{V}_{\mathcal{F}}^{p}$ ,

\mathbb{U}_{v}^{\prime}=\prod_{w\in\mathscr{V}_{\mathcal{K}}^{(v)}}\prod_{j=1}^{r+s}u_{w,j},\quad\text{and }\mathbb{U}_{p}^{\prime}=\prod_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}\mathbb{U}_{v}^{\prime}.

Definition 7.14.

We define a projector $\mathbb{e}:=\varinjlim_{n}(\mathbb{U}_{p}^{\prime})^{n!}$ , called Hida’s ordinary projector. For any $p$ -adic ring $R$ , we define the submodule of ordinary $p$ -adic automorphic forms (resp. ordinary $p$ -adic cuspidal automorphic forms) over $R$ as

V_{G}^{\operatorname{ord}}(K^{p},R):=\mathbb{e}V_{G}(K^{p},R),\quad(\text{resp. }V_{G}^{0,\operatorname{ord}}(K^{p},R):=\mathbb{e}V_{G}^{0}(K^{p},R)).

We can further define ordinary $p$ -adic automorphic forms of weights $\tau$ for a $p$ -adic weight $\tau$ , and adding decorations on the notations, which we shall not bother to list.

7.7. Interlude: Review on $p$ -adic measures

Next we shall introduce $p$ -adic families of ( $p$ -adic) automorphic forms. We prefer to use the setup of $p$ -adic measures, among various different but essentially equivalent perspectives. Here we briefly review the basis notions of $p$ -adic measures.

•

Let $R$ be a $p$ -adic ring. This will act as the base coefficient ring.
•

Let $M$ be a $p$ -adically complete $R$ -module.
•

Let $Y$ be a compact abelian group with totally disconnected topology, i.e. $Y$ is a profinite abelian group.

Then we denote $\mathscr{C}(Y,R)$ as the $R$ -algebra of continuous $R$ -valued functions on $Y$ . It is equipped with the topology of uniform convergence. An $M$ -valued $p$ -adic measure on $Y$ is a continuous $R$ -linear map

\mu:\mathscr{C}(Y,R)\rightarrow M,\quad f\mapsto\mu(f):=\int_{Y}f\,\mathrm{d}\mu.

The set of $M$ -valued $p$ -adic measures on $Y$ is a $p$ -adically complete $R$ -module and is denoted as $\mathrm{Meas}_{R}(Y,M)$ . We often omit the coefficient ring $R$ in the notation.

Here are some operations on $p$ -adic measures.

•

Base change: Let $R^{\prime}$ be an $R$ -algebra, which is also $p$ -adically complete, since $\mathscr{C}(Y,R^{\prime})=\mathscr{C}(Y,R)\widehat{\otimes}R^{\prime}$ , there is a natural map $\mathrm{Meas}_{R}(Y,R)\rightarrow\mathrm{Meas}_{R^{\prime}}(Y,M\widehat{\otimes}R^{\prime})$ . If the structure map $R\rightarrow R^{\prime}$ is injective, then we view $\mathrm{Meas}_{R}(Y,R)$ as a subset of $\mathrm{Meas}_{R^{\prime}}(Y,M\widehat{\otimes}R^{\prime})$ .
•

Dirac measures: Given a $y\in Y$ , we define $\delta_{y}:f\mapsto f(y)$ being a $p$ -adic measure in $\mathrm{Meas}(Y,R)$ .
•

Action of continuous functions: Let $h\in\mathscr{C}(Y,R)$ . For $\mu\in\mathrm{Meas}(Y,M)$ , we define

$h\ast\mu:f\mapsto\int_{Y}fh\,\mathrm{d}\mu.$
•

Convolution: If we further assume $Y$ is equipped with the structure of an abelian group written multiplicatively, then we can define the convolution on $\mathrm{Meas}(Y,R)$ as

$\mu_{1}\ast\mu_{2}:f\mapsto\int_{Y}\int_{Y}f(yz)\,\mathrm{d}\mu_{1}(y)\,\mathrm{d}\mu_{2}(z).$

If $f:Y\rightarrow R^{\times}$ is a continuous multiplicative character, then we have

$\int_{Y}f\,\mathrm{d}(\mu_{1}\ast\mu_{2})=\left(\int_{Y}f\,\mathrm{d}\mu_{1}\right)\left(\int_{Y}f\,\mathrm{d}\mu_{2}\right).$

We can define the convolution of measures in $\mathrm{Meas}(Y,M)$ in the same way whenever there is an appropriate “product” on $M$ .
•

Product of the test space: $Y=Y_{1}\times Y_{2}$ is a product of profinite abelian groups, then there is a natural isomorphism

$\mathrm{Meas}(Y,M)\simeq\mathrm{Meas}(Y_{1},\mathrm{Meas}(Y_{2},M)).$

Moreover, we note that $\mathrm{Meas}(Y,R)$ can be identified with the completed group algebra $R\llbracket{Y}\rrbracket$ . In practice, we often encounter the case where

•

$Y$ is a product of finite abelian group $\Delta$ with finite copies of $\mathbb{Z}_{p}$ . For example, $Y$ could be $\Gamma_{\mathcal{K}}$ , $U_{\mathcal{K},p}^{(r)}:=1+p^{r}\mathcal{O}_{K,p}$ for $r\geq 1$ , or the torus $T_{n}(\mathbb{Z}_{p})$ for some positive integer $n$ .
•

$R$ is often the the ring of integers $\mathcal{O}_{L}$ for an algebraic extension $L/\mathbb{Q}_{p}$ .
•

$M$ is often taken to be $R$ , or the space of $p$ -adic modular forms over certain reductive groups.

7.8. Hida families on unitary groups

Recall $\mathscr{P}$ is a simple unitary PEL datum given by a Hermitian space $V$ . Let $R$ be a $p$ -adic ring.

Definition 7.15.

We define a $p$ -adic family of automorphic forms over $G$ of level $K^{p}$ over $R$ to be a $p$ -adic measure $\mathbf{f}$ in

\mathrm{Meas}_{R}(T_{\mathscr{P}}(\mathbb{Z}_{p}),V_{G}(K^{p},R))

such that

\int_{T_{\mathscr{P}}(\mathbb{Z}_{p})}t\cdot\phi\,\mathrm{d}\mathbf{f}=t\cdot\int_{T_{\mathscr{P}}(\mathbb{Z}_{p})}\phi\,\mathrm{d}\mathbf{f}

(7.24)

for any $\phi\in\mathscr{C}(T_{\mathscr{P}}(\mathbb{Z}_{p}),R)$ and $t\in T_{\mathscr{P}}(\mathbb{Z}_{p})$ . A $p$ -adic family of cuspidal automorphic forms (resp.Hida family, cuspidal Hida family) over $G$ of level $K^{p}$ over $R$ is a $p$ -adic family of automorphic forms taking value in

V_{G}^{0}(K^{p},R)\quad(\text{resp. }V_{G}^{\operatorname{ord}}(K^{p},R),\quad V_{G}^{0,\operatorname{ord}}(K^{p},R)).

We denote the space of such families as $\mathcal{M}^{\bullet}_{G}(K^{p},R)$ with decorations $\bullet\in\{\emptyset,0,\operatorname{ord}\}$ .

In particular, let $\tau$ be a $p$ -adic weight and $R$ contains all the values of $\tau$ , then for $\mathbf{f}\in\mathcal{M}_{G}(K^{p},R)$ ,

\mathbf{f}_{\tau}:=\int_{T_{\mathscr{P}}(\mathbb{Z}_{p})}\tau\,\mathrm{d}\mathbf{f}\in V_{G}(K^{p},R)[\tau].

This is called the specialization of $\mathbf{f}$ at $\tau$ . We remark that $\mathcal{M}^{\bullet}_{G}(K^{p},R)$ becomes a $\Lambda_{\mathscr{P}}$ -algebra by either the action of $T_{\mathscr{P}}(\mathbb{Z}_{p})$ induced by left multiplication on $T_{\mathscr{P}}(\mathbb{Z}_{p})$ or its action on $V_{G}(K^{p},R)$ . This is well-defined by the requirement (7.24).

Recall we have a decomposition of $\mathbb{Z}_{p}$ as $\mathbb{Z}_{p}=\boldsymbol{\mu}_{p}(\mathbb{Z}_{p})\times(1+p\mathbb{Z}_{p})$ ,²³²³23Recall that we have always been assuming $p\neq 2$ . where $\Delta$ is the group of roots of unity in $\mathbb{Z}_{p}$ , a finite cyclic group of order $p-1$ . Then we have the corresponding decomposition of $T_{\mathscr{P}}(\mathbb{Z}_{p})$ into

T_{\mathscr{P}}(\mathbb{Z}_{p})=\Delta\times T^{\circ}_{\mathscr{P}}

where $\Delta=T_{\mathscr{P}}(\boldsymbol{\mu}_{p}(\mathbb{Z}_{p}))$ is the torsion subgroup of $T_{\mathscr{P}}(\mathbb{Z}_{p})$ and $T^{\circ}_{\mathscr{P}}:=T_{\mathscr{P}}(1+p\mathbb{Z}_{p})$ is the identity component of $T_{\mathscr{P}}(\mathbb{Z}_{p})$ . Regarding the decomposition above, we rewrite $\Lambda_{\mathscr{P}}=\mathcal{O}_{L}[\Delta]\llbracket{T_{\mathscr{P}}^{\circ}}\rrbracket$ . Since $\Delta$ is of order prime to $p$ , the spaces $\mathcal{M}^{\bullet}_{G}(K^{p},R)$ can be decomposed into a direct sum of isotypical pieces for the $\mathcal{O}_{L}$ -characters $\eta$ of $\Delta$ as

\mathcal{M}^{\bullet}_{G}(K^{p},R)=\oplus_{\eta\in\widehat{\Delta}}\mathcal{M}^{\bullet}_{G}(K^{p},R,\eta).

Characters $\eta$ are called branching characters. Each $\mathcal{M}^{\bullet}_{G}(K^{p},R,\eta)$ is a $\Lambda_{\mathscr{P}}^{\circ}:=\mathcal{O}_{L}\llbracket{T_{\mathscr{P}}^{\circ}}\rrbracket$ -module.

Let $\mathbb{I}$ be a normal domain over $\Lambda_{\mathscr{P}}$ (resp. $\Lambda_{\mathscr{P}}^{\circ}$ ) which is also a finite algebra over $\Lambda_{\mathscr{P}}$ (resp. $\Lambda_{\mathscr{P}}^{\circ}$ ). Then we can base change the $\Lambda_{\mathscr{P}}$ -module (resp. $\Lambda_{\mathscr{P}}^{\circ}$ -module)

\mathcal{M}^{\bullet}_{G}(K^{p},R)\quad(\text{resp. }\mathcal{M}^{\bullet}_{G}(K^{p},R,\eta))

to $\mathbb{I}$ . These elements are called $\mathbb{I}$ -adic Hida families, forming the spaces $\mathcal{M}^{\bullet}_{G}(K^{p},R;\mathbb{I})$ and $\mathcal{M}^{\bullet}_{G}(K^{p},R,\eta;\mathbb{I})$ respectively.

We can also define Hecke operators on the space of $p$ -adic families of modular forms, induced from those defined in Section 7.6. Let $\mathscr{S}$ be a finite set of places of $\mathcal{F}$ such that $K^{p}$ is maximal outside of $\mathscr{S}$ . Let $A$ be any finite torsion-free $\Lambda_{\mathscr{P}}^{\circ}$ -algebra, we define the unramified ordinary cuspidal Hecke algebra $\mathbf{h}^{0,\operatorname{ord},\mathscr{S}}(A)$ be the $A$ -subalgebra of $\operatorname{End}_{A}(\mathcal{M}^{0,\operatorname{ord}}_{G}(K^{p},A))$ generated by Hecke operators away from $\mathscr{V}_{\mathcal{F}}^{(p)}\cup\mathscr{S}$ and $\mathbf{U}^{\prime}_{v}$ -operators at places $v\in\mathscr{V}_{\mathcal{F}}^{(p)}$ .

Remark 7.16 (Other $p$ -adic families of automorphic forms).

Hida families defined in Definition 7.15 are possibly the easiest classes of $p$ -adic families of automorphic forms. There are ways of generalizing it.

•

Allowing general parabolics. Here Hida’s ordinary projector $\mathbb{e}$ is constructed via Borel double cosets. In [Hid98], Hida introduced the notion of $P$ -ordinary modular forms on a reductive group $F$ with parabolic subgroup $P$ of $G$ , generalizing the case $P=B$ in this article. A $p$ -adic automorphic form can be $P$ -ordinary without being $B$ -ordinary. There are works on this aspects, for example [EM21]. The deep work [CLW22] built up the “noncuspidal Hida theory for semiordinary modular forms” and proved the Iwasawa main conjecture of such $p$ -adic families.
•

Allowing general $p$ -slopes. Here the ordinary $p$ -adic automorphic forms are defined to be of $p$ -slope zero. There are generalizations to the $p$ -adic family of automorphic forms of finite $p$ -slope, as Coleman families.

8. The $p$ -adic family of Eisenstein series

Let $V$ be an $N$ -dimensional vector space over $\mathcal{K}$ , equipped with a non-degenerate skew-Hermitian form $\phi$ . Then $\mathsf{i}\phi$ is an Hermitian form on $V$ . We regard $V$ as an Hermitian space over $\mathcal{K}$ . In Part I, we considered spaces $V^{\sharp}$ , $V^{\heartsuit}$ and $V^{\blacklozenge}$ . We let $\mathscr{P},\mathscr{P}^{\sharp},\mathscr{P}^{\heartsuit}$ and $\mathscr{P}^{\blacklozenge}$ be the simple unitary Shimura datum attached to these Hermitian spaces, and $H,G,G^{\heartsuit},G^{\blacklozenge}$ be the unitary groups attached to them repsectively. We keep the assumptions (sgn) and (QS) in Section 2.1.1. We have algebraic groups $H_{\bullet},B_{\bullet},B^{+}_{\bullet}$ and $T_{\bullet}$ for $\bullet$ being these unitary Shimura datums. We denote them as $H_{r,s},B_{r,s},B^{+}_{r,s}$ and $T_{r,s}$ if the corresponding unitary group has signature $(r,s)$ , and similarly adopt obvious notations as such.

Remark 8.1 (On unitary groups, [Wan15, Remark 2.1]).

As recalled in Section 7, in order to have Shimura varieties for doing $p$ -adic automorphic forms and Galois representations, we need to use the general unitary groups $G$ defined over $\mathbb{Q}$ , which is smaller than the general unitary group we defined in Definition 2.1. However, this group is not convenient for local automorphic computations since we cannot treat each primes of $\mathcal{F}$ independently. So what we do implicitly is that for automorphic computations, we write down the automorphic forms on the larger general unitary groups as in Definition 2.1, and then restrict to the smaller one. For the algebraic construction, we only do the pullbacks for unitary groups instead of general unitary groups.

8.1. Setups for $p$ -adic interpolations

Following [Wan15, Definition 3.2], we define Eisenstein datums as follows.

Definition 8.2.

We define an Eisenstein datum $\mathscr{D}$ as a triple $\mathscr{D}:=(\sigma,\chi,\mathscr{S})$ , where

•

$\sigma$ is an irreducible unitary tempered cuspidal automorphic representation of $H$ ,
•

$\chi$ is a Hecke character $K^{\times}\backslash\mathbb{K}^{\times}\rightarrow\mathbb{C}^{\times}$ .
•

$\mathscr{S}$ is a finite set of primes of $\mathcal{F}$ contaning all the infinite places, primes above $p$ and places where either $\sigma$ or $\chi$ is ramified.

Given an Eisenstein datum, if they satisfy Assumptions 6.1 and 6.2, then we can define local Siegel Eisenstein sections $f_{v,s,\chi}^{\mathrm{Sieg}}$ as in Section 6, and hence define the Siegel Eisenstein series $E^{\mathrm{Sieg}}_{\mathscr{D}}(f_{v,s,\chi}^{\mathrm{Sieg}},-)$ with these Siegel Eisenstein sections. Under the pullback formula, we obtain a Klingen Eisenstein series $E^{\mathrm{Kling}}(F^{\heartsuit}(f^{\mathrm{Sieg}}_{s,\chi},\Phi;-),-)$ for cuspidal automorphic forms $\Phi\in\sigma$ .

Definition 8.3.

A $p$ -adic family of Eisenstein datums is a tuple $\mathbf{D}=(L,\mathbb{I},\mathbf{f},\chi_{0},\eta_{0})$ , where

•

$L/\mathbb{Q}_{p}$ is a finite extension.
•

$\mathbb{I}$ is a normal domain over $\Lambda_{m,n}^{\circ}$ , which is also a finite $\Lambda_{m,n}^{\circ}$ -algebra.
•

$\chi_{0}$ is a finite order Hecke character $\mathcal{K}^{\times}\backslash\mathbb{K}^{\times}\rightarrow\mathbb{C}^{\times}$ whose conductors at primes above $p$ divides $(p)$ .
•

$\eta_{0}$ is a branching character, i.e. a finite order character of $\Delta$ , the torsion part of $T_{m,n}(\mathbb{Z}_{p})$ .
•

$\mathbf{f}$ is an $\mathbb{I}$ -adic Hida family of tempered cuspidal ordinary eigenforms on $H$ , of tame level group $K_{H}^{p}\leq H(\mathbb{A}_{\mathrm{f}}^{p})$ and branching character $\eta_{0}$ . ²⁴²⁴24In [Wan15], the tempered condition is included in the assumption (TEMPERED) in [Wan15, Section 5B2]. Here we include this assumption in the definition of $p$ -adic family of Eisenstein datums.

Denote $\mathbb{I}^{{\mathrm{ur}}}$ be the normalization of an irreducible component of $\mathbb{I}\widehat{\otimes}_{\mathcal{O}_{L}}\mathcal{O}_{L}^{{\mathrm{ur}}}$ ²⁵²⁵25More precisely, there is a bijection $\{\text{ minimal primes of }\mathbb{I}\widehat{\otimes}_{\mathcal{O}_{L}}\mathcal{O}_{L}^{{\mathrm{ur}}}\}\rightarrow\{\text{ irreducible components of }\operatorname{Spec}\mathbb{I}\widehat{\otimes}_{\mathcal{O}_{L}}\mathcal{O}_{L}^{{\mathrm{ur}}}\}.$ Let $\mathfrak{P}$ be a mininal prime ideal of $\mathbb{I}\widehat{\otimes}_{\mathcal{O}_{L}}\mathcal{O}_{L}^{{\mathrm{ur}}}$ , then the corresponding irreducible component is isomorphic to $\operatorname{Spec}(\mathbb{I}\widehat{\otimes}_{\mathcal{O}_{L}}\mathcal{O}_{L}^{{\mathrm{ur}}}/\mathfrak{P})$ . The later quotient ring is an integral domain, and we take its integral closure (in its fractional field), this is the $\mathbb{I}^{{\mathrm{ur}}}$ in the text. As remarked in [Wan15, page 1957], for each such irreducible component we can make the following-up construction.. Given a $p$ -adic family of Eisenstein datum $\mathbf{D}$ , we define the corresponding Iwasawa algebra as $\Lambda_{\mathbf{D}}:=\mathbb{I}^{{\mathrm{ur}}}\llbracket{\Gamma_{\mathcal{K}}}\rrbracket$ . It is an $\Lambda_{m,n}^{\circ}$ -algebra. We also define the universal character attached to $\chi_{0}$ as the product $\boldsymbol{\chi}_{0}:=\chi_{0}\Psi_{\mathcal{K}}$ , where $\Psi_{\mathcal{K}}:\mathcal{K}^{\times}\backslash\mathbb{K}^{\times}\rightarrow\Gamma_{\mathcal{K}}\hookrightarrow\Lambda_{\mathbf{D}}^{\times}$ is the tautological character induced from the reciprocity law in class field theory under geometric normalization.

The Iwasawa algebra $\Lambda_{\mathbf{D}}$ will be used as the weight space for $p$ -adic interpolations. We define

\mathcal{X}_{\mathbb{D}}:=\operatorname{Hom}_{\mathrm{cts}}(\Lambda_{\mathbf{D}},\overline{\mathbb{Q}}_{p}^{\times})=\operatorname{Spec}\Lambda_{\mathbf{D}}(\overline{\mathbb{Q}}_{p}^{\times})

consisting of continuous $\overline{\mathbb{Q}}_{p}$ -valued charcters of $\Lambda_{\mathbf{D}}$ .

Definition 8.4.

Let $\mathsf{P}\in\mathcal{X}_{\mathbf{D}}$ .

(1)

We call $\mathsf{P}$ an arithmetic point, if it satisfies the following conditions.

•

Let $\tau_{\mathsf{P}}$ be the pullback of $\mathscr{P}$ on $\Lambda_{m,n}^{\circ}$ along canonical maps $\Lambda_{m,n}^{\circ}\hookrightarrow\mathbb{I}\rightarrow\mathbb{I}\otimes\mathcal{O}_{L}^{{\mathrm{ur}}}\rightarrow\mathbb{I}^{{\mathrm{ur}}}$ , then it is a $p$ -adic weight in the sense of Definition 7.12, with

(t_{1,\tau_{\mathsf{P}},v}^{+},\ldots,t_{m,\tau_{\mathsf{P}},v}^{+};t_{1,\tau_{\mathsf{P}},v}^{-},t_{n,\tau_{\mathsf{P}},v}^{-})=(0,\ldots,0;\kappa_{\mathsf{P}},\ldots,\kappa_{\mathsf{P}})

for any $v\in\mathscr{V}_{\mathcal{F}}^{(p)}$ , for some integer $\kappa_{\mathsf{P}}\geq m+n+2$ .

•

Let $\chi_{\mathsf{P}}:=\mathsf{P}\circ\boldsymbol{\chi}_{0}$ is the composition of $\mathsf{P}$ with the universal character $\boldsymbol{\chi}_{0}$ , i.e.

\mathcal{K}^{\times}\backslash\mathbb{K}^{\times}\mathrel{\leavevmode\hbox to13.05pt{\vbox to16.11pt{\pgfpicture\makeatletter\hbox{\hskip 0.333pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{11.97986pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{12.17986pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{6.18993pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.00002pt}{4.89409pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\boldsymbol{\chi}_{0}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\Lambda_{\mathbf{D}}\mathrel{\leavevmode\hbox to11.43pt{\vbox to16.52pt{\pgfpicture\makeatletter\hbox{\hskip 0.333pt\lower-4.86601pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{}{{}}{} {{}{}}{}{}{}{{{}{}}}{{}}{}{}{{{}{}}}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{10.36392pt}{0.0pt}\pgfsys@stroke\pgfsys@invoke{ }{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{10.56392pt}{0.0pt}\pgfsys@invoke{ }\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}{{}}}}\hbox{\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.38196pt}{-1.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}}\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{3.00002pt}{3.533pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\hbox{$\scriptstyle\mathsf{P}$}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}\overline{\mathbb{Q}}_{p}^{\times}.

Then $\chi_{\mathsf{P}}$ is a Hecke character of infinite type $(-\kappa_{\mathsf{P}}/2,\kappa_{\mathsf{P}}/2)$ .

(2)

We call $\mathsf{P}$ a classical arithmetic point, if $\mathsf{P}$ is an arithmetic point and the specialization $\mathbf{f}_{\mathsf{P}}:=\mathbf{f}_{\tau_{\mathsf{P}}}\in V_{H}^{0,\operatorname{ord}}[\tau_{\mathsf{P}}]$ is a classical automorphic form under (7.23) such that it generates an irreducible tempered unitary cuspidal automorphic representation $\sigma_{\mathbf{f}_{\mathsf{P}}}$ of $H$ . ²⁶²⁶26By the requirement $\kappa_{\mathsf{P}}\geq m+n+2$ , the archimedean part $\sigma_{\mathbf{f}_{\mathsf{P}},\infty}$ is indeed a holomorphic discrete series.
(3)

We call $\mathsf{P}$ a generic classical arithmetic point, if $\mathsf{P}$ is a classical arithmetic point such that $(\sigma_{\mathbf{f}_{\mathsf{P}}},\chi_{\mathsf{P}})$ satisfies the generic condition [Wan15, Definition 4.42].

We denote the set of such points as $\mathcal{X}_{\mathbf{D}}^{\mathrm{ari}},\mathcal{X}_{\mathbf{D}}^{\mathrm{cls}}$ and $\mathcal{X}_{\mathbf{D}}^{\mathrm{gen}}$ respectively. We know that these are all Zariski dense in $\mathcal{X}_{\mathbf{D}}$ ²⁷²⁷27This is not quite trivial, which depends on the classicality theorem in Hida theory. See [Wan15, Remark 5.2] for the justifications..

Let $\mathscr{S}_{\mathbf{f}}$ (resp. $\mathscr{S}_{\chi_{0}}$ ) be the set of finite places $v$ of $\mathcal{F}$ away from $p$ such that $K_{H,v}\neq H(\mathcal{F}_{v})$ (resp. $\chi_{0}$ is ramified). We put $\mathscr{S}_{\mathbf{D}}:=\mathscr{S}_{\mathbf{f}}\cup\mathscr{S}_{\chi_{0}}\cup\mathscr{S}_{\mathcal{F}}^{(p\infty)}$ . This is the set of “bad places” of the datum $\mathbf{D}$ . Given a point $\mathsf{P}\in\mathcal{X}_{\mathrm{cls}}$ , we have the specialization of the $p$ -adic family of Eisenstein datum $\mathbf{D}_{\mathsf{P}}=(\sigma_{\mathbf{f}_{\mathsf{P}}},\chi_{\mathsf{P}},\mathscr{S}_{\mathbf{D}})$ , We write $s_{\mathsf{P}}:=(\kappa_{\mathsf{P}}-(m+n+1))/2$ .

8.2. The $p$ -adic family of Eisenstein series

In this subsection, we recall the results on the existence of $p$ -adic families of Siegel Eisenstein series and Klingen Eisenstein series.

8.2.1. The Siegel Eisenstein family

We recall the $p$ -adic family of Siegel Eisenstein series. To do the $p$ -adic interpolation, we need a renormalization of Siegel Eisenstein series.

Let $\mathsf{P}\in\mathcal{X}_{\mathbf{D}}^{\mathrm{cls}}$ and $\mathbf{D}_{\mathsf{P}}=(\sigma_{\mathbf{f}_{\mathsf{P}}},\chi_{\mathsf{P}},\mathscr{S}_{\mathsf{P}})$ be the specialized Eisenstein datum. We define the normalization factor

B(s_{\mathsf{P}},\chi_{\mathsf{P}}):=\Omega_{\mathsf{P}}\left(\dfrac{(-2)^{-n}(2\pi\mathrm{i})^{n\kappa_{\mathsf{P}}}(2/\pi)^{n(n-1)/2}}{\prod_{j=0}^{n-1}\Gamma(\kappa_{\mathsf{P}}-j)}\right)^{-d}\cdot\prod_{v\in\mathscr{V}_{\mathcal{F}}\setminus\mathscr{S}_{\mathsf{P}}}d_{m+n+1,v}(s_{\mathsf{P}},\chi_{\mathsf{P}}),

(8.25)

where

\Omega_{\mathsf{P}}=\begin{cases}(\Omega_{p}^{\infty}/\Omega_{\infty}^{\infty})^{m\kappa_{\mathsf{P}}},&\quad n\neq 0,\\ 1,&\quad n=0,\end{cases}

with $\Omega_{p}\in(\mathbb{Z}_{p}^{{\mathrm{ur}}})^{\mathscr{V}_{\mathcal{F}}^{\infty}}$ the $p$ -adic period and $\Omega_{\infty}\in\mathbb{C}^{\mathscr{V}_{\mathcal{F}}^{\infty}}$ the CM period, and put $\Omega_{\infty}^{\infty}$ for the product of $d$ elements of $\Omega_{\infty}$ and define $\Omega_{p}^{\infty}$ similarly. ²⁸²⁸28We refer to [Hid04] for precise definitions. We define the normalized Siegel Eisenstein series as

E^{\mathrm{Sieg}}_{\chi_{\mathsf{P}}}(-)=B(s_{\mathsf{P}},\chi_{\mathsf{P}})\cdot E^{\mathrm{Sieg}}(f^{\mathrm{Sieg}}_{s_{\mathsf{P}},\chi_{\mathsf{P}}},-).

and the normalized Klingen Eisenstein series as

E^{\mathrm{Kling}}_{\mathbf{D}_{\mathsf{P}}}(-)=B(s_{\mathsf{P}},\chi_{\mathsf{P}})\cdot E^{\mathrm{Kling}}(F^{\heartsuit}(f^{\mathrm{Sieg}}_{s_{\mathsf{P}},\chi_{\mathsf{P}}},\mathbf{f}_{\mathsf{P}};-),-).

Then we have the following result.

Theorem 8.5 ([Wan15, Lemma 5.7]).

Attached to the $p$ -adic family of Eisenstein datum $\mathbf{D}$ , there exists a $p$ -adic measure

\mathbf{E}^{\mathrm{Sieg}}_{\chi_{0}}\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{G^{\blacklozenge}})

such that for any generic classical arithmetic point $\mathsf{P}\in\mathcal{X}^{\mathrm{gen}}$ , we have

\int_{\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ}}\mathsf{P}\,\mathrm{d}\mathbf{E}^{\mathrm{Sieg}}_{\chi_{0}}=E^{\mathrm{Sieg}}_{\chi_{\mathsf{P}}}.

The construction is by first interpolating the Fourier coefficients of the Siegel Eisenstein series and apply the $q$ -expansion principle. The normalization factor $B(s_{\mathsf{P}},\chi_{\mathsf{P}})$ is essential, guaranteeing that these Fourier coefficients are integral and $p$ -adically interpolatable after normalization.

8.2.2. Hecke projectors

To construct the $p$ -adic family of Klingen Eisenstein series, we recall the notion of Hecke projectors, attached to the $p$ -adic family of Eisenstein datum $\mathbf{D}$ , following [Wan15, Section 5B].

Definition 8.6 (Dual Hida family).

We first define an $\mathcal{O}_{L}$ -involution $\ddagger:\Lambda_{m,n}^{\circ}\rightarrow\Lambda_{m,n}^{\circ}$ sending any $\operatorname{diag}[a_{1},\ldots,a_{n}]$ to $\operatorname{diag}[a_{n}^{-1},\ldots,a_{1}^{-1}]$ . We define $\mathbb{I}^{\ddagger}$ to be the ring $\mathbb{I}$ but with the $\Lambda_{m,n}^{\circ}$ -algebra structure given by composing the involution $\ddagger$ with the original structure map $\Lambda_{m,n}^{\circ}\rightarrow\mathbb{I}$ . We say an $\mathbb{I}^{\ddagger}$ -adic cuspidal Hida family $\mathbf{f}^{\ddagger}$ is a dual Hida family of $\mathbf{f}$ , if for all the generic classical arithmetic point $\mathsf{P}\in\mathcal{X}_{\mathbf{D}}^{\mathrm{gen}}$ , we have $(\mathbf{f}^{\ddagger})_{\mathsf{P}}\in\sigma_{\mathbf{f}_{\mathsf{P}}}^{\vee}$ .

Definition 8.7 (Hecke projectors).

Let $\mathbf{f}$ be an $\mathbb{I}$ -adic Hida family over $H$ . We define a Hecke operator $\mathbbm{1}_{\mathbf{f}}$ in the Hecke algebra $\mathbf{h}^{0,\operatorname{ord},\mathscr{S}_{\mathbf{D}}}(\operatorname{Frac}(\mathbb{I}))$ as a Hecke projector of $\mathbf{f}$ , if for any $\mathsf{P}\in\mathcal{X}_{\mathbf{D}}^{\mathrm{gen}}$ and any $\mathbb{I}$ -adic family of automorphic forms $\mathbf{f}^{\prime}$ , the specialization of $(\mathbbm{1}_{\mathbf{f}}\circ\mathbb{e})\mathbf{f}^{\prime}$ at $\mathsf{P}$ is the projection (under the Petersson inner product) of $\mathbf{f}^{\prime}_{\mathsf{P}}$ to the one-dimensional line inside $V_{\sigma_{\mathbf{f}}}$ spanned by $\mathbf{f}_{\mathsf{P}}$ . The scalar $(\mathrm{Proj}_{\mathbf{f}})_{\mathsf{P}}(\mathbf{f}^{\prime})$ is defined to satisfy

((\mathbbm{1}_{\mathbf{f}}\circ\mathbb{e})\mathbf{f}^{\prime})_{\mathsf{P}}=(\mathrm{Proj}_{\mathbf{f}})_{\mathsf{P}}(\mathbf{f}^{\prime})\cdot\mathbf{f}_{\mathsf{P}}.

In this way, we have a well-defined map

\mathrm{Proj}_{\mathbf{f}}:\mathcal{M}_{H}(K^{p}_{H},\mathbb{I})\rightarrow\operatorname{Frac}(\mathbb{I}).

Theorem 8.8 (Wan).

Notations being as above, let $\mathbf{D}$ be a $p$ -adic family of Eisenstein datum, then the dual Hida family $\mathbf{f}^{\ddagger}$ and the Hecke projector $\mathbbm{1}_{\mathbf{f}}$ exist.

This is proved in [Wan19, Chapter 6]. More specifically, the Hecke projector is constructed as [Wan19, Equation (28)] and the dual Hida family is constructed in the proof of [Wan19, Theorem 6.8]. We remark that such existences are well-known in lower rank cases $\mathrm{U}(1,1)$ and $\mathrm{U}(2,0)$ in the classical theory before Wan’s general construction. One refers to, for example, [Wan15, Remark 5.6] for justifications.

8.2.3. The Klingen Eisenstein family

Following the recipe of the pullback formula (Proposition 2.6) and the $p$ -adic interpolation construction in [Wan15, Section 5C], we first restrict the Siegel Eisenstein family on $G^{\heartsuit}\times H$ as

\mathbf{E}^{\mathrm{Sieg}}_{\chi}|_{G^{\heartsuit}\times H}\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{G^{\heartsuit}}\otimes V_{H}).

We fix any point $\underline{A}_{\heartsuit}$ on the Igusa tower of $G^{\heartsuit}$ , we have

\mathbf{E}^{\mathrm{Sieg}}_{\chi}|_{G^{\heartsuit}\times H}(\underline{A}_{\heartsuit},-)\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{H})=\mathrm{Meas}(\Gamma_{\mathcal{K}},\mathcal{M}_{H}(K^{p,\prime}_{H})).

We shrink $K^{p}_{H}$ to be contained in $K_{H}^{p,\prime}$ if necessary, then $\mathbf{f}\in\mathcal{M}^{0,\operatorname{ord}}_{H}(K_{H}^{p,\prime})$ . As shrinking, the set $\mathscr{S}_{\mathbf{f}}$ may become larger. By abuse of notation, we still denote by $K_{H}^{p}$ the shrinked tame level group and $\mathscr{S}_{\mathbf{f}}$ the set associated to $K_{H}^{p}$ .

Then the assignment

\mathsf{P}\longmapsto\left[\underline{A}_{\heartsuit}\longmapsto\mathrm{Proj}_{\mathbf{f}^{\ddagger}}(\mathbf{E}^{\mathrm{Sieg}}_{\chi}|_{G^{\heartsuit}\times H}(\underline{A}_{\heartsuit},-))\in\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})\right]

(8.26)

gives a $p$ -adic measure

\mathbf{E}^{\mathrm{Kling}}_{\mathbf{D}}\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{G^{\heartsuit}})\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}}).

This is the Klingen Eisenstein family associated to the $p$ -adic family of Eisenstein datum $\mathbf{D}$ , satisfying that for any generic classical arithmetic points $\mathsf{P}\in\mathcal{X}^{\mathrm{gen}}$ , we have

\int_{\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ}}\mathsf{P}\,\mathrm{d}\mathbf{E}^{\mathrm{Kling}}_{\mathbf{D}}=E^{\mathrm{Kling}}_{\chi_{\mathsf{P}}}.

In the case when $H$ is a definite unitary group, that is, $H=\mathrm{U}(m,0)$ , there is another approach without using Hecke projectors, but instead using the $p$ -adic interpolation of the Petersson inner product. This approach takes the advantage that the Shimura variety of $\mathrm{U}(m,0)$ is a finite set (see, for example, [Wan15, Definition 5.1 (ii)]). This trick was invented in [Hsi14] in the case of $\mathrm{U}(2,0)$ and was generalized to the case $\mathrm{U}(m,0)$ in [Wan15, Section 5C3].

Proposition 8.9 ([Wan15, Proposition 5.9]).

Let $K^{p}$ be a neat tame level group in $H(\mathbb{A}_{\mathbb{Q},\mathrm{f}}^{p})$ . Then there exists a $\Lambda_{\mathbf{D}}$ -linear pairing

\mathbf{B}_{K^{p}}:\mathcal{M}_{H}(K^{p};\mathbb{I})\times\mathcal{M}_{H}(K^{p};\mathbb{I}^{\ddagger})\rightarrow\mathbb{I}

such that for any $\mathsf{P}\in\mathcal{X}_{\mathbf{D}}^{g}$ , we have $\mathbf{B}_{K^{p}}(\mathbf{f},\mathbf{f}^{\prime})=\left\langle{\mathbf{f}_{\mathsf{P}},\mathbf{f}^{\prime}_{\mathsf{P}}}\right\rangle_{\sigma_{\mathbf{f}_{\mathsf{P}}},\mathrm{Pet}}$ for any $\mathbf{f}\in\mathcal{M}_{H}(K^{p};\mathbb{I})$ and $\mathbf{f}^{\prime}\in\mathcal{M}_{H}(K^{p};\mathbb{I}^{\circ})$ .

Then in this case, only granting the existence of the dual Hida family $\mathbf{f}^{\ddagger}$ , the Klingen Eisenstein family can be defined by replacing (8.26) with the assignment

\mathsf{P}\longmapsto\left[\underline{A}_{\heartsuit}\longmapsto\mathbf{B}_{K^{p}_{H}}(\mathbf{E}^{\mathrm{Sieg}}_{\chi}|_{G^{\heartsuit}\times H}(\underline{A}_{\heartsuit},-),\mathbf{f}^{\ddagger})\in\mathbb{I}\right].

Here we identified

\mathbf{E}^{\mathrm{Sieg}}_{\chi}|_{G^{\heartsuit}\times H}(\underline{A}_{\heartsuit},-)\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{H})=\mathrm{Meas}(\Gamma_{\mathcal{K}},\mathcal{M}_{H}(K^{p}_{H}))

so as to apply $\mathbf{B}_{K^{p}_{H}}(-,\mathbf{f}^{\ddagger})$ . One note that in this definite case, we get an “ $\mathbb{I}$ -integral Klingen Eisenstein family”, that is,

\mathbf{E}^{\mathrm{Kling}}_{\mathbf{D}}\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{G^{\heartsuit}})\otimes\mathbb{I}.

9. The $p$ -adic interpolation of the Gan-Gross-Prasad period integral

9.1. Setups for the $p$ -adic interpolation

We define the following datum for the $p$ -adic interpolation of the Gan-Gross-Prasad period integral.

Definition 9.1.

A $p$ -adic family of Gan-Gross-Prasad datums is a tuple $\mathbf{G}=(L,\mathbb{I},\mathbb{J},\mathbf{f},\mathbf{g},\chi_{0},\eta_{0})$ , where

•

the tuple $\mathbf{D}:=(L,\mathbb{I},\mathbf{f},\chi_{0},\eta_{0})$ is a $p$ -adic family of Eisenstein datum, in the sense of Definition 8.3,
•

$\mathbb{J}$ is a normal domain over $\Lambda_{m+1,n}^{\circ}$ , which is also a finite $\Lambda_{m+1,n}^{\circ}$ -algebra, and
•

$\mathbf{g}$ is an $\mathbb{J}$ -adic Hida family of tempered cuspidal eigenform on $G$ , of tame level group $K_{G}^{p}$ and branching character $\eta_{0}$ ,

Let $\mathscr{S}_{\mathbf{g}}$ be the set of finite places $v$ of $\mathcal{F}$ away from $p$ such that $K_{G,v}\neq G(\mathcal{F}_{v})$ . Given a $p$ -adic family of Gan-Gross-Prasad datums is a tuple, we realize that $\mathbf{D}^{\prime}:=(L,\mathbb{J},\mathbf{g},\chi_{0},\eta_{0})$ is also a $p$ -adic family of Eisenstein datum, over the group $G$ . We put $\mathscr{V}_{\mathbf{G}}:=\mathscr{S}_{\mathbf{D}}\cup\mathscr{S}_{\mathbf{D}^{\prime}}$ , called the “bad places” of the datum $\mathbf{G}$ .

We define the weight space $\mathcal{Y}_{\mathbf{G}}$ as

\mathcal{Y}_{\mathbf{G}}:=\operatorname{Hom}_{\mathrm{cts}}(\mathbb{J},\overline{\mathbb{Q}}_{p}^{\times})=\operatorname{Spec}\mathbb{J}(\overline{\mathbb{Q}}_{p}^{\times})

consisting of continuous $\overline{\mathbb{Q}}_{p}$ -valued characters of $\mathbb{J}$ . We define the weight space of the $p$ -adic family of GGP datums $\mathbf{G}$ as the product space $\mathcal{X}_{\mathbf{G}}:=\mathcal{X}_{\mathbf{D}}\times\mathcal{Y}_{\mathbf{G}}$ .

Definition 9.2.

Let $\mathsf{Q}\in\mathcal{Y}_{\mathbf{G}}$ .

(1)

We call $\mathsf{Q}$ an arithmetic point if the pullback $\tau_{\mathsf{Q}}$ on $\Lambda_{m+1,n}$ along the structure map $\Lambda_{m+1,n}^{\circ}$ is an arithmetic $p$ -adic weight in the sense of Definition 7.12 with

t_{1,\tau_{\mathsf{Q}},v}^{+}\geq\ldots\geq t_{m,\tau_{\mathsf{Q}},v}^{+}\geq t_{m+1,\tau_{\mathsf{Q}},v}^{+}\geq-t_{1,\tau_{\mathsf{Q}},v}^{-}\geq-t_{n,\tau_{\mathsf{Q}},v}^{-},\quad t_{m+1,\tau_{\mathsf{Q}},v}^{+}\geq-t_{1,\tau_{\mathsf{Q}},v}^{-}+(m+n+1)

(9.27)

for any $v\in\mathscr{V}_{\mathcal{F}}^{(p)}$ .

(2)

We call $\mathsf{Q}$ a classical arithmetic point if $\mathsf{Q}$ is an arithmetic point and the specialization $\mathbf{g}_{\mathsf{Q}}:=\mathbf{g}_{\tau_{\mathsf{Q}}}\in V_{G}[\tau_{\mathsf{Q}}]$ is a classical automorphic form under (7.23) such that it generates an irreducible tempered unitary cuspidal automorphic representation $\pi_{\mathbf{g}_{\mathsf{Q}}}$ of $G$ . ²⁹²⁹29By the latter assumption in (9.27), the archimedean part $\pi_{\mathbf{g}_{\mathsf{Q}},\infty}$ is a holomorphic discrete series.

We denote the set of such points as $\mathcal{Y}_{\mathbf{G}}^{\mathrm{ari}}$ and $\mathcal{Y}_{\mathbf{G}}^{\mathrm{cls}}$ respectively. Let $\mathsf{P}\times\mathsf{Q}\in\mathcal{X}_{\mathbf{G}}$ .

(1)

We call $\mathsf{P}\times\mathsf{Q}$ an admissible point if $\mathsf{P}$ and $\mathsf{Q}$ are classical arithmetic points and $\sigma_{\mathbf{f}_{\mathsf{P}},\infty}$ and $\pi_{\mathbf{g}_{\mathsf{Q}},\infty}$ satisfies the weight interlacing assumption (Assumption 3.8).
(2)

We call $\mathsf{P}\times\mathsf{Q}$ a generic admissible point if it is a classical admissible point and $\mathsf{P}\in\mathcal{X}_{\mathbf{D}}^{\mathrm{gen}}$ .

We denote the set of such points as $\mathcal{X}_{\mathbf{G}}^{\mathrm{adm}}$ and $\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ respectively. We know that these are all Zariski dense in $\mathcal{Y}_{\mathbf{G}}$ and $\mathcal{X}_{\mathbf{G}}$ accordingly.

9.2. The $p$ -adic interpolation of GGP period integral

We use appropriate Hecke projectors to $p$ -adically interpolate the GGP period integral, and get the following theorem.

Theorem 9.3.

There exists an element

\mathbf{P}_{\mathbf{G}}\in(\Lambda_{\mathbf{D}}\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}}))\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}}),

such that for any $\mathsf{P}\times\mathsf{Q}\in\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ ,

(\mathsf{P}\times\mathsf{Q})(\mathbf{P}_{\mathbf{G}})=B(s_{\mathsf{P}},\chi_{\mathsf{P}})\mathcal{P}^{\mathrm{Kling}}(\mathbf{f}_{\mathsf{P}},\mathbf{g}_{\mathsf{Q}},\chi_{\mathsf{P}},s_{\mathsf{P}}),

with right-hand-side the GGP period integral of Klingen Eisenstein series with a cuspidal automorphic form, defined in (3.6), normalized by the factor defined in (8.25). When $n=0$ , we have $\mathbf{P}_{\mathbf{G}}\in\Lambda_{\mathbf{D}}\otimes\mathbb{J}$ .

Proof.

Recall we have the Klingen Eisenstein family

\mathbf{E}^{\mathrm{Kling}}_{\mathbf{D}}\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{G^{\heartsuit}})\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}}).

We restrict it on $G$ , obtain

\mathbf{E}^{\mathrm{Kling}}_{\mathbf{D}}|_{G}\in\mathrm{Meas}(\Gamma_{\mathcal{K}}\times T_{m,n}^{\circ},V_{G})\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})=\mathrm{Meas}(\Gamma_{\mathcal{K}},\mathcal{M}_{G}(K^{p,\prime}_{G}))\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}}).

Here similar to the construction of Klingen Eisenstein family, we shrink $K_{G}^{p}$ to be contained in $K_{G}^{p,\prime}$ if necessary, then $\mathbf{g}\in\mathcal{M}_{G}^{0}(K_{G}^{p,\prime})$ . As shrinking, $\mathscr{S}_{\mathbf{g}}$ may become larger. By abuse of notation, we still denote by $K^{p}_{H}$ the shrinked tame level group and the set $\mathscr{S}_{\mathbf{g}}$ associated to $K_{G}^{p}$ , and keep assuming that $\mathscr{S}_{\mathbf{f}}$ and $\mathscr{S}_{\mathbf{g}}$ are disjoint with every element splits in $\mathcal{K}$ . We then apply the Hecke operator $\mathbf{1}_{\mathbf{g}}$ to get

\mathrm{Proj}_{\mathbf{g}}(\mathbf{E}^{\mathrm{Kling}}_{\mathbf{D}}|_{G})\in\mathrm{Meas}(\Gamma_{\mathcal{K}},\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}}))=\operatorname{Frac}(\mathbb{J})\llbracket{\Gamma_{\mathcal{K}}}\rrbracket\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})=(\Lambda_{\mathbf{D}}\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}}))\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}}).

(9.28)

This element is the desired $\mathbf{P}_{\mathbf{D}}$ , satisfying the interpolation property in the statement. When $n=0$ , instead of using the Hecke projector $\mathrm{Proj}_{\mathbf{g}}$ , we apply $\mathbf{B}_{K_{G}^{p}}(-,\mathbf{g})$ and obtain

\mathbf{B}_{K^{p}_{G}}(\mathbf{E}^{\mathrm{Kling}}_{\mathbf{D}}|_{G},\mathbf{g})\in\mathrm{Meas}(\Gamma_{\mathcal{K}},\mathbb{J})=\mathbb{J}\llbracket{\Gamma_{\mathcal{K}}}\rrbracket\otimes\mathbb{I}=\Lambda_{\mathbf{D}}\otimes\mathbb{J},

as desired. ∎

To obtain a precise interpolation formula, we put the following “ $p$ -adically automorphic assumptions”.

Assumption 9.4 (Automorphic assumptions).

We assume that the sets $\mathscr{S}_{\mathbf{f}}$ and $\mathscr{S}_{\mathbf{g}}$ are disjoint and every place in $\mathscr{S}_{\mathbf{f}}\cup\mathscr{S}_{\mathbf{g}}$ split in $\mathcal{K}$ . We assume that there is a Zariski dense subset $\mathcal{X}_{\mathbf{G}}^{\mathrm{mul}}$ of $\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ , such that for every $\mathsf{P}\times\mathsf{Q}\in\mathcal{X}_{\mathbf{G}}^{\mathrm{mul}}$ , the cuspidal automorphic representations $\sigma_{\mathbf{f}_{\mathsf{P}}}$ and $\sigma_{\mathbf{g}_{\mathsf{Q}}}$ satisfy (BC) respectively, and hence the corresponding multiplicity one theorems.

In Part I, we have related the period integral $\mathcal{P}^{\mathrm{Kling}}(\mathbf{f}_{\mathsf{P}},\mathbf{g}_{\mathsf{Q}},\chi_{\mathsf{P}},s_{\mathsf{P}})$ with certain $L$ -values with explicit local factors at bad places. Combining Theorem 9.3 and Theorem 6.3, we have the following theorem.

Theorem 9.5.

Notations being as above. We assume Assumption 9.4. Then for any $\mathsf{P}\times\mathsf{Q}\in\mathcal{X}_{\mathbf{G}}^{\mathrm{mul}}$ , we have $(\mathsf{P}\times\mathsf{Q})(\mathbf{P}_{\mathbf{G}})^{2}$ equals

\dfrac{1}{2^{\varkappa_{\sigma_{\mathbf{f}_{\mathsf{P}}}}+\varkappa_{\pi_{\mathbf{g}_{\mathsf{Q}}}}}}\cdot\mathscr{L}^{\mathscr{V}_{\mathcal{F}}^{(p)}}(\sigma_{\mathbf{f}_{\mathsf{P}}}\times\pi_{\mathbf{g}_{\mathsf{Q}}})L^{\mathscr{S}_{\mathbf{D}}\cup\mathscr{S}_{\mathbf{g}}}\left(s_{\mathsf{P}}+\dfrac{1}{2},\pi_{\mathbf{g}_{\mathsf{Q}},v},\chi_{\mathsf{P},v}\right)L^{\mathscr{S}_{\mathbf{D}}\cup\mathscr{S}_{\mathbf{g}}}\left(s_{\mathsf{P}}+\dfrac{1}{2},\pi_{\mathbf{g}_{\mathsf{Q}},v}^{\vee},\chi_{\mathsf{P},v}\right)\\ \times\Omega_{\mathsf{P}}^{2}\left(\dfrac{(-2)^{-n}(2\pi\mathrm{i})^{n\kappa_{\mathsf{P}}}(2/\pi)^{n(n-1)/2}}{\prod_{j=0}^{n-1}\Gamma(\kappa_{\mathsf{P}}-j)}\right)^{-2d}\prod_{v\in\mathscr{S}_{\mathbf{f}}\cup\mathscr{S}_{\mathbf{g}}}\zeta_{v}C_{\sigma_{\mathbf{f}_{\mathsf{P}},v},\boldsymbol{\psi}_{v}}C_{\sigma_{\mathbf{f}_{\mathsf{P}},v}^{\vee},\boldsymbol{\psi}_{v}^{-1}}\mathcal{B}_{\pi_{\mathbf{g}_{\mathsf{Q}},v}}^{\mathrm{ess}}\mathcal{B}_{\sigma_{\mathbf{f}_{\mathsf{P}},v}}^{\mathrm{ess}}\\ \times\prod_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}\zeta_{v}\Delta_{v,\beta_{v}}^{2}\cdot\mathfrak{G}(\kappa_{v},\breve{\underline{\mu}_{v,\mathsf{P}}},\breve{\underline{\lambda}_{v,\mathsf{Q}}})\mathfrak{G}(\kappa_{v},\breve{\check{\underline{\mu}_{v,\mathsf{P}}}},\breve{\check{\underline{\lambda}_{v,\mathsf{Q}}}})\mathcal{C}_{\sigma_{\mathbf{f}_{\mathsf{P}}}}^{\operatorname{ord}}\mathcal{C}_{\pi_{\mathbf{g}_{\mathsf{Q}}}}^{\operatorname{ord}}\prod_{v\in\mathscr{S}_{\mathbf{g}}}d_{m+n+1,v}(s_{\mathsf{P}},\chi_{\mathsf{P}})^{2}\\ \times\left|\left|{\mathbf{f}_{\mathsf{P}}}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\mathbf{g}_{\mathsf{Q}}}\right|\right|_{\mathbf{g}_{\mathsf{Q}},\mathrm{Pet}}^{2}\prod_{v\in\mathscr{S}_{\mathbf{D}}\cup\mathscr{S}_{\mathbf{g}}}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{\mathbf{g}_{\mathsf{Q}},v})\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{\mathbf{g}_{\mathsf{Q}},v}^{\vee}).

Here $\underline{\mu}_{v,\mathsf{P}}$ is the tuple appearing in the local representation $\sigma_{\mathbf{f}_{P},v}$ and similarly for other notations, and $\kappa_{v}$ are auxiliary characters $\mathcal{F}_{v}^{\times}\rightarrow\overline{\mathbb{Q}}_{p}^{\times}$ that are sufficiently ramified for $v\in\mathscr{V}_{\mathcal{F}}^{(p)}$ .

Proof.

This is a simple combination of Theorem 9.3 and 6.3, with the definition of the normalization factor in (8.25). Note that the assumptions in the automorphic computations (i.e. Assumption 3.8, 3.10, 3.11, 6.1 and 6.2) are made to be satisfied in the definition of $p$ -adic families of Eisenstein datums and GGP datums (i.e. Definition 8.3 and 9.1). ∎

9.3. $p$ -adic $L$ -function of the Rankin-Selberg product of Hida families

In [Wan15], Wan constructed not only the Klingen Eisenstein families, but also $p$ -adic $L$ -functions of Hida families over unitary groups. We record his construction in our notations as follows. ³⁰³⁰30There are some typographical errors in [Wan15, Theorem 1.1 (1)]. It has been corrected in [Wan19, Theorem 6.8]. We also remark here that recently, David Marcil [Mar24] constructed the $p$ -adic $L$ -function for $P$ -ordinary Hida families over unitary groups, which is more general than Wan’s construction here.

Theorem 9.6 ([Wan15, Theorem 1.1 (1)]).

Let $\mathbf{g}$ be the Hida family of automorphic forms in the $p$ -adic family of GGP datum $\mathbf{G}$ . Then there exists an element

\mathbf{L}_{\mathbf{g},\chi_{0}}\in\Lambda_{\mathbf{D}^{\prime}}\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}}),

such that for any $\mathsf{R}\in\mathcal{X}_{\mathbf{D}^{\prime}}^{\mathrm{gen}}$ , we have

\mathsf{R}(\mathbf{L}_{\mathbf{g},\chi_{0}})=\left(\dfrac{(-2)^{-n}(2\pi\mathrm{i})^{n\kappa_{\mathsf{P}}}(2/\pi)^{n(n-1)/2}}{\prod_{j=0}^{n-1}\Gamma(\kappa_{\mathsf{P}}-j)}\right)^{-d}C_{\mathbf{g}_{\mathsf{R}}}\cdot\Omega_{\mathsf{R}}L^{\mathscr{S}_{\mathbf{D}^{\prime}}}\left(s_{\mathsf{R}}+\dfrac{1}{2},\pi_{\mathbf{g}_{\mathsf{R}},v},\chi_{\mathsf{R},v}\right),

with $C_{\mathbf{g}_{\mathsf{R}}}$ the product of remaining local factors in [Wan15, Theorem 1.1 (1)] which we shall not recall. When $n=0$ , $\mathbf{L}_{\mathbf{g},\chi_{0}}\in\Lambda_{\mathbf{D}^{\prime}}$ .

We renormalize $\mathbf{L}_{\mathbf{g},\chi_{0}}$ as

\mathbf{L}_{\mathbf{g},\chi_{0}}^{\circ}:=\left(\prod_{v\in\mathscr{S}_{\mathcal{F}}\smallsetminus(\mathscr{S}_{\mathbf{D}}\smallsetminus\mathscr{S}_{\mathbf{g}})}\mathscr{Z}^{\diamondsuit}_{v}(f^{\mathrm{Sieg}}_{v,s,\chi},\pi_{\mathbf{g}_{\mathsf{Q}},v})\right)^{-1}\prod_{v\in\mathscr{S}_{\mathbf{g}}}d_{m+n+1,v}(s_{\mathsf{P}},\chi_{\mathsf{P}})^{-1}C_{\mathbf{g}_{\mathsf{R}}}^{-1}\cdot\mathbf{L}_{\mathbf{g},\chi_{0}}.

(9.29)

Then comparing with the interpolation formula in Theorem 9.5, we have the following result, as another main result of this article.

Theorem 9.7.

Notations being as above. We assume Assumption 9.4, then there exists an element

\mathbf{L}_{\mathbf{f},\mathbf{g}}\in\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}})

such that for any $\mathsf{P}\times\mathsf{Q}\in\mathcal{X}_{\mathbf{G}}^{\mathrm{gen}}$ , we have $(\mathsf{P}\times\mathsf{Q})(\mathbf{L}_{\mathbf{f},\mathbf{g}})$ equals

\dfrac{1}{2^{\varkappa_{\sigma_{\mathbf{f}_{\mathsf{P}}}}+\varkappa_{\pi_{\mathbf{g}_{\mathsf{Q}}}}}}\times\left|\left|{\mathbf{f}_{\mathsf{P}}}\right|\right|_{\sigma,\mathrm{Pet}}^{2}\left|\left|{\mathbf{g}_{\mathsf{Q}}}\right|\right|_{\mathbf{g}_{\mathsf{Q}},\mathrm{Pet}}^{2}\cdot\prod_{v\in\mathscr{S}_{\mathbf{f}}\cup\mathscr{S}_{\mathbf{g}}}\zeta_{v}C_{\sigma_{\mathbf{f}_{\mathsf{P}},v},\boldsymbol{\psi}_{v}}C_{\sigma_{\mathbf{f}_{\mathsf{P}},v}^{\vee},\boldsymbol{\psi}_{v}^{-1}}\mathcal{B}_{\pi_{\mathbf{g}_{\mathsf{Q}},v}}^{\mathrm{ess}}\mathcal{B}_{\sigma_{\mathbf{f}_{\mathsf{P}},v}}^{\mathrm{ess}}\\ \times\prod_{v\in\mathscr{V}_{\mathcal{F}}^{(p)}}\zeta_{v}\Delta_{v,\beta_{v}}^{2}\cdot\mathfrak{G}(\kappa_{v},\breve{\underline{\mu}_{v,\mathsf{P}}},\breve{\underline{\lambda}_{v,\mathsf{Q}}})\mathfrak{G}(\kappa_{v},\breve{\check{\underline{\mu}_{v,\mathsf{P}}}},\breve{\check{\underline{\lambda}_{v,\mathsf{Q}}}})\mathcal{C}_{\sigma_{\mathbf{f}_{\mathsf{P}}}}^{\operatorname{ord}}\mathcal{C}_{\pi_{\mathbf{g}_{\mathsf{Q}}}}^{\operatorname{ord}}\cdot\mathscr{L}^{\mathscr{V}_{\mathcal{F}}^{(p)}}(\sigma_{\mathbf{f}_{\mathsf{P}}}\times\pi_{\mathbf{g}_{\mathsf{Q}}}).

Proof.

Recall (9.28) in the construction of $\mathbf{P}_{\mathbf{G}}$ , we identify

\mathbf{P}_{\mathbf{G}}\in(\Lambda_{\mathbf{D}}\otimes\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}}))\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}})=\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})\otimes(\Lambda_{\mathbf{D}^{\prime}}\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}})).

Then we define

\mathbf{L}_{\mathbf{f},\mathbf{g}}:=\dfrac{\mathbf{P}_{\mathbf{G}}^{2}}{\mathbf{L}_{\mathbf{g},\chi_{0}}^{\circ}\mathbf{L}_{\mathbf{g}^{\ddagger},\chi_{0}}^{\circ}}\in\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})\otimes(\Lambda_{\mathbf{D}^{\prime}}\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}})).

with the same interpolating formula as in the statement. Moreover, we note that the specializations along the $\mathbb{Z}_{p}$ -extension line are trivial (that is to say, no $\chi_{\mathsf{P}}$ appears in the interpolation formula), this is the purpose of renormalization (9.29). It follows that $\mathbf{L}_{\mathbf{f},\mathbf{g}}\in\operatorname{Frac}(\mathbb{I}^{{\mathrm{ur}}})\otimes\operatorname{Frac}(\mathbb{J}^{{\mathrm{ur}}})$ , as desired. ∎

The element $\mathbf{L}_{\mathbf{f},\mathbf{g}}$ can be regarded as a $p$ -adic $L$ -function of the Rankin-Selberg product of Hida families $\mathbf{f}$ over $H=\mathrm{U}(m,n)$ and $\mathbf{g}$ over $G=\mathrm{U}(m+1,n)$ . We note that there are similar recent works on this aspects in various generalizations, for instance [Liu23, HY23, Dim24].

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The Gan-Gross-Prasad period of Klingen Eisenstein families over unitary groups

Abstract.

1. Introduction

1.1. Main result: automorphic computations

Theorem A (Theorem 3.7).

Theorem B (Theorem 6.3).

1.2. Main result: pp-adic interpolations

Theorem C (Theorem 9.3).

Theorem D (Theorem 9.7).

1.3. Future works

1.4. Notations and conventions

1.4.1. Fields and places

1.4.2. Characters and automorphic representations

1.4.3. General linear groups and operation on matrices

1.4.4. Measures and pairings

Acknowledgements

Part I Automorphic Computations

2. Backgrounds

2.1. Unitary groups

Definition 2.1.

2.1.1. Unitary groups at local places

2.1.2. Hermitian basis

2.1.3. The ♯\sharp-space

2.1.4. The ♡\heartsuit-space

2.1.5. Doubling space

2.2. Eisenstein series on unitary groups

2.2.1. Klingen Eisenstein series

2.2.2. Siegel Eisenstein series

Remark 2.2.

2.3. On multiplicity one conditions

2.4. Doubling method, à la Piatetski-Shapiro and Rallis

2.4.1. Basic identity of Piatetski-Shapiro and Rallis

Theorem 2.3 (Basic identity of Piatetski-Shapiro and Rallis).

Proof.

2.4.2. Partial doubling integrals

Corollary 2.4.

Proof.

2.4.3. Local doubling integrals

Corollary 2.5.

2.5. Doubling method, à la Garrett

Proposition 2.6.

Proof.

3. The Gan-Gross-Prasad period integral

3.1. Reduce to the cuspidal GGP period integral

Proposition 3.1.

3.2. Break into local integrals

3.2.1. Contragredient, conjugation and MVW involutions

3.2.2. Ichino-Ikeda formula

Theorem 3.2 (Ichino-Ikeda formula).

3.2.3. Reduce to local Ichino-Ikeda integrals

Remark 3.3 (On complex conjugation).

3.2.4. Reduce to the doubling integrals

Theorem 3.4.

3.3. Unramified Computations

Theorem 3.5 ([Har14, Theorem 2.12]).

Theorem 3.6 ([EHLS20, Section 4.2.1]).

Theorem 3.7.

3.4. Further assumptions

Assumption 3.8 (Weight interlacing assumption).

Assumption 3.9 (Unramified assumption).

Assumption 3.10 (Splitting assumption).

Assumption 3.11 (Disjointly ramified assumption).

4. Local Ichino-Ikeda integrals at archimedean places

4.1. Weight interlacing property

Theorem 4.1.

4.2. Local Ichino-Ikeda integral

Proposition 4.2.

5. Local Ichino-Ikeda integrals at split primes

5.1. Basic representation theory of GLn​(F)\mathrm{GL}_{n}(F)

5.1.1. Whittaker models

5.1.2. The JPSS integrals and local JPSS LL-factors

Remark 5.1.

5.1.3. The naive local Rankin-Selberg LL-factor

Proposition 5.2.

Proof.

5.1.4. Essential Whittaker vectors

Remark 5.3.

5.1.5. Test vector problem

Theorem 5.4 ([JPS81, Théorème on page 208]).

Theorem 5.5.

1.2. Main result: $p$ -adic interpolations

2.1.3. The $\sharp$ -space

2.1.4. The $\heartsuit$ -space

5.1. Basic representation theory of $\mathrm{GL}_{n}(F)$

5.1.2. The JPSS integrals and local JPSS $L$ -factors

5.1.3. The naive local Rankin-Selberg $L$ -factor

5.3. Local Ichino-Ikeda integrals at $\mathscr{V}_{\mathcal{F}}^{\mathrm{ram}}$

5.3.1. The case when $\sigma$ is unramified

5.3.2. The case when $\sigma$ is ramified

5.4. Further representation theory of $\mathrm{GL}_{n}(F)$ : ordinary condition

5.5. Local Ichino-Ikeda integrals at $\mathscr{V}_{F}^{(p)}$

6.1.3. Ramified places away from places above $p$

6.1.4. Places above $p$

Part II $p$ -adic Interpolation

7.1.3. Hodge structures, lattices and level subgroups at $p$

7.5.1. $p$ -adic modular forms

7.6.1. Hecke operators away from $p$

7.6.2. Hida’s operators at $p$

7.7. Interlude: Review on $p$ -adic measures