On $p$-adic $L$-functions for symplectic representations of $\GL(N)$ over number fields

Let us state our main result more precisely. Let $F$ be an arbitrary number field, $G\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Res}_{F/\mathbf{Q}}(\operatorname{GL}_{N})$, and $\pi$ a RASCAR of $G(\mathbf{A})$. This forces $N=2n$ to be even, and $\pi\cong\pi^{\vee}\otimes\eta$ to be essentially self-dual, for a Hecke character $\eta$ of $F^{\times}\backslash\mathbf{A}_{F}^{\times}$. Further, $\pi$ is symplectic if and only if it admits a Shalika model, if and only if it is a functorial transfer from $\mathrm{Res}_{F/\mathbf{Q}}\mathrm{GSpin}_{2n+1}$ [FJ93, AS06, AS14]. In this setting, the automorphic realisation of Deligne’s conjecture was recently proved – including the period relations at infinity – in work of Jiang–Sun–Tian [JST].

Let $p$ be a prime, and assume that $\pi$ is spherical at all primes $\mathfrak{p}$ of $F$ above $p$. Then $\pi_{\mathfrak{p}}=\operatorname{Ind}_{B}^{G}\theta_{\mathfrak{p}}$ is an unramified principal series, the normalised induction of some unramified character $\theta_{\mathfrak{p}}=(\theta_{\mathfrak{p},1},...,\theta_{\mathfrak{p},2n})$ (where $B$ is the upper-triangular Borel). Let $Q\subset G$ be the standard parabolic subgroup with Levi $H\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Res}_{F/\mathbf{Q}}(\operatorname{GL}_{n}\times\operatorname{GL}_{n})$, and let $J_{\mathfrak{p}}\subset\operatorname{GL}_{2n}(F_{\mathfrak{p}})$ be the parahoric subgroup of type $Q$. If $\mathfrak{p}|p$, a regular $Q$-refinement to level $J_{\mathfrak{p}}$ is a choice of simple Hecke eigenvalue $\alpha_{\mathfrak{p}}$ on $\pi_{\mathfrak{p}}^{J_{\mathfrak{p}}}$, defined precisely in Definition 3.3. We assume this choice is spin (Definition 3.3), in that it interacts well with the Shalika model; as explained in the introduction of [BGW], we expect this is essential for constructions of $p$-adic $L$-functions via Shalika models. Write $\tilde{\pi}$ for $\pi$ with such a choice of regular spin $Q$-refinement at each $\mathfrak{p}|p$. We assume $\tilde{\pi}$ to be non-$Q$-critical (Definition 7.1). This is satisfied if the integrally normalised eigenvalues $\alpha_{\mathfrak{p}}^{\circ}$, defined in §7.1, have non-$Q$-critical slope. In particular if $\tilde{\pi}$ is ordinary (i.e. each $v_{p}(\alpha_{\mathfrak{p}}^{\circ})=0$) then it is non-$Q$-critical.

We say a Hecke character $\chi:F^{\times}\backslash\mathbf{A}_{F}^{\times}\to\mathbf{C}^{\times}$ is critical for $\pi$ if $s=1/2$ is a Deligne-critical value of $L(\pi\times\chi,s)$, and then we define $L(\pi,\chi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L(\pi\times\chi,1/2)$. The critical characters are determined by their infinity type and the weight $\lambda$ of $\pi$. For general $F$, this $L$-function can have many different critical regions; see e.g. the pictures in [LLZ15, p.1605] or [BDP13, §4.1]. We focus here on the ‘standard’ critical range¹¹1Constructions of $p$-adic $L$-functions interpolating other ranges would be extremely interesting; see e.g. [BDP13], which gives an example for $\operatorname{GL}_{2}$ over an imaginary quadratic field, assuming $\pi$ is base-change., corresponding to the ‘balanced weight’ condition in [JST] (and, when $F$ is imaginary quadratic, region $\Sigma^{(1)}$ in [LLZ15]). We describe this range in Lemma 2.4. For example, when $F$ is imaginary quadratic, this gives a square of standard critical infinity types (see §7.4). Crucially, these standard critical values admit a representation-theoretic interpretation via branching laws for $H\subset G$, described in Lemma 2.6.

We write $\mathrm{Crit}_{p}(\pi)$ for the set of standard critical characters that have $p$-power conductor. Let $\operatorname{Gal}_{p}$ be the Galois group of the maximal abelian extension of $F$ unramified outside $p\infty$. Attached to any $\chi\in\mathrm{Crit}_{p}(\pi)$ is a canonical character $\chi_{[p]}$ on $\operatorname{Gal}_{p}$ (see §2.5).

We construct a $p$-adic $L$-function attached to $\tilde{\pi}$, in the following sense:

The growth condition is described in Definition 7.4. In particular, if $\tilde{\pi}$ is ordinary, then $L_{p}^{i_{p}}(\tilde{\pi})$ is a bounded distribution, that is, a $p$-adic measure. Here $L^{(p)}$ is the $L$-function with the Euler factors at $p$ removed, $\tau(\chi_{f})$ is a Gauss sum, $q_{\mathfrak{p}}=N_{F/\mathbf{Q}}(\mathfrak{p})$, and $\varpi_{\mathfrak{p}}$ is a uniformiser. This agrees exactly with the conjectures of Coates–Perrin-Riou/Panchishkin [Pan94, Conj. 6.2], except possibly the term $\Omega_{\pi,\chi_{\infty}}$, which we comment on in Remark 3.8.

If $F$ is imaginary quadratic, and $\tilde{\pi}$ has non-$Q$-critical slope, then (a) and (b) determine $L_{p}^{i_{p}}(\tilde{\pi})$ uniquely by [Loe14]. Analogous unicity results for $F$ totally real are [BDW, Prop. 6.25].

To our knowledge, Theorem A gives the first such construction beyond the special cases of $\operatorname{GL}_{2}$ (over any $F$) or $F$ totally real; see §1.3 for a summary of previous results. The totally real case was handled in [DJR20, BDW]. The construction we give is inspired by those works, particularly the overconvergent approach of [BDW]. There are some additional features in the general number field setting, which we briefly summarise.

1. (1)

   Notably, whilst Leopoldt’s conjecture predicts that $L_{p}(\tilde{\pi})$ is 1-variabled when $F$ is totally real, in general our $p$-adic $L$-functions can be many-variabled. For example, when $F=\mathbf{Q}$ we have $\operatorname{Gal}_{p}\cong\mathbf{Z}_{p}^{\times}$, and one only has cyclotomic variation; when $F$ is imaginary quadratic, one has a two-variable $p$-adic $L$-function, with both cyclotomic and anticyclotomic variation.

2. (2)

   The method uses automorphic cycles/modular symbols. The totally real (resp. $\operatorname{GL}_{2}$) constructions relied on a certain numerical coincidence; that these cycles have dimension equal to the top degree (resp. bottom degree) in which cuspidal cohomology for $G$ contributes. This rendered some relevant cohomology groups 1-dimensional, making certain choices unique up to scalar. In the general number field case, we work in the middle of the cuspidal range, where the analogous groups are never 1-dimensional.

We handle (1) by blending the methods of [BDW] with earlier work [Wil17, BW19] of the author and Barrera for $\operatorname{GL}_{2}$. Problem (2) is more serious, and for this we exploit works of Lin–Tian [LT20] and Jiang–Sun–Tian [JST]. They nailed down good elements in these higher-dimensional cohomology groups and proved not only the non-vanishing hypothesis of attached zeta integrals at infinity, but also the expected period relations at infinity, at least in the cyclotomic direction (see Remark 3.8). We hope that the period relations in all directions can be extracted via similar methods, but do not address that here, instead focusing on the $p$-adic interpolation.

We imagine the following special case might be of particular interest. Let $\mathcal{F}$ be a cohomological Siegel modular form on $\mathrm{GSp}_{4}/\mathbf{Q}$, and base-change it to an imaginary quadratic field $F$. Under appropriate assumptions, we may apply our construction to the Langlands transfer of $\mathcal{F}/F$ to $\operatorname{GL}_{4}/F$ (via [AS06]), giving a two-variable $p$-adic spin $L$-function for $\mathcal{F}/F$, including anticyclotomic variation. For $\operatorname{GL}_{2}$, such constructions for base-change modular forms have had important arithmetic consequences, such as proofs of one divisibility of the Iwasawa Main Conjecture [SU14].

When $\tilde{\pi}$ is ordinary at each $\mathfrak{p}|p$, then as mentioned above, our construction yields a $p$-adic measure. As an immediate application, we get the following generalisation of a result of Dimitrov–Januszewski–Raghuram (who in [DJR20] treated the case $F$ totally real). Let $\lambda=(\lambda_{\sigma})_{\sigma\in\Sigma}$ denote the weight of $\pi$, where $\lambda_{\sigma}=(\lambda_{\sigma,1},...,\lambda_{\sigma,2n})\in\mathbf{Z}^{2n}$ is dominant and $\Sigma$ is the set of embeddings $\sigma:F\hookrightarrow\mathbf{C}$ (see §2.3).

Here $Q$-ordinarity is in the sense of [Hid98, §1.1]. The unitary assumption ensures $s=1/2$ is a Deligne-critical value of $L(\pi,s)$.

Given Theorem A, the proof of Theorem B is simple. The argument is literally identical to [DJR20, §4.4], so we give only a sketch, and refer the reader there for full details.

If one drops assumption (1.1), the theorem instead becomes: if there exists a $\chi$ as in the theorem such that $L(\pi\times(\chi\circ N_{F/\mathbb{Q}}),1/2)\neq 0$, then there are infinitely many such $\chi$.

In the $\operatorname{GL}_{4}$ case, after transferring via [AS06], this gives non-vanishing of many twisted central values of spin $L$-functions of (very regular weight, cohomological, Klingen-ordinary) genus 2 Siegel modular forms over number fields.

This work generalises (and is visibly inspired by) many other constructions. We summarise a few. In the case of $\operatorname{GL}(2)$, i.e. $n=1$, every RACAR is a RASCAR, and our main results/methods specialise exactly to those of [BW19]; and the results/methods of that paper in turn followed the earlier works [PS11, Bar18, Wil17] (over $\mathbf{Q}$, totally real, and imaginary quadratic base fields respectively). These methods were later used in [Bel12, BDJ22, BH, BW21a, BW21b] to vary $p$-adic $L$-functions in families.

In the case of general $\operatorname{GL}(2n)$, when $F$ is totally real our main result specialises exactly to [BDW, Thm. 6.23]. Earlier constructions in the ordinary situation were given in [AG94, Geh18, DJR20]. This construction was then used in [BDW, BDG⁺] to construct and study $p$-adic families, and – under some technical hypotheses – to vary $p$-adic $L$-functions over these families.

The variation of the present construction in families would be very interesting. The methods of these earlier papers – which worked in either top or bottom cohomological degree – do not apply. Fundamental difficulties include: controlling the families; constructing classes in the correct cohomological degree; and showing that these classes interpolate the ‘good’ elements from [JST].

Rohrlich [Roh89] proved a $\operatorname{GL}_{2}$ analogue of Theorem B. The case $F$ totally real is [DJR20]. For $\operatorname{GL}_{4}/\mathbf{Q}$, an analogous result for all weights and unitary CARs was recently proved in [RY].

Finally, in Theorems A and B it should be possible to weaken the assumption that $\pi_{\mathfrak{p}}$ is spherical at all $\mathfrak{p}|p$ to $Q$-parahoric-spherical using forthcoming work of Dimitrov–Jorza [DJ].

This paper would not exist without my previous collaboration with Daniel Barrera and Mladen Dimitrov, and I thank them wholeheartedly for several years’ worth of discussions. I also thank Andy Graham and Andrei Jorza for many highly relevant discussions during our follow-up collaboration, and David Loeffler for his comments and corrections on an earlier draft. This research was supported by EPSRC Postdoctoral Fellowship EP/T001615/2.

Let $F$ be a number field of degree $d=r+2s$, where $F$ has $r$ real embeddings and $s$ pairs of complex embeddings. Let $\Sigma\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\{\sigma:F\hookrightarrow\mathbf{C}\}=\Sigma_{\mathbf{R}}\sqcup\Sigma_{\mathbf{C}}$ be the union of the real and complex embeddings. Each $\sigma\in\Sigma$ has a conjguate $c\sigma$, and $\sigma=c\sigma$ if and only if $\sigma(F)\subset\mathbf{R}$. Fix a rational prime $p$ and an isomorphism $i_{p}:\mathbf{C}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\overline{\mathbf{Q}}_{p}$. Let $F^{p\infty}$ be the maximal abelian extension of $F$ unramified outside $p\infty$, and let $\operatorname{Gal}_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Gal}(F^{p\infty}/F)$ be its Galois group. Let $\psi$ be the standard non-trivial additive character of $F\backslash\mathbf{A}_{F}$ (as e.g. fixed in [DJR20, §4.1]).

If $v$ is a place of $F$, we write $F_{v}$ for the completion of $F$ at $v$, $\mathcal{O}_{v}$ for its ring of integers, and $\mathbf{F}_{v}$ for its residue field. We fix a choice of uniformiser $\varpi_{v}\in\mathcal{O}_{v}$.

Let $G=\mathrm{Res}_{F/\mathbf{Q}}\operatorname{GL}_{2n}$, $B=TN$ be the upper-triangular Borel sugroup in $G$, $T$ the diagonal torus and $N$ the unipotent. Let $H=\mathrm{Res}_{F/\mathbf{Q}}(\operatorname{GL}_{n}\times\operatorname{GL}_{n})$, with $\iota:H\hookrightarrow G$, $(h_{1},h_{2})\mapsto\left(\begin{smallmatrix}h_{1}&\\ &h_{2}\end{smallmatrix}\right)$. We may abuse notation and write e.g. $B(F_{v})$ for the upper-triangular matrices in $\operatorname{GL}_{2n}(F_{v})$.

Let $K_{\infty}=C_{\infty}Z_{\infty}\subset G(\mathbf{R})$, where $Z_{\infty}$ is the center and $C_{\infty}=\mathrm{O}_{2n}(\mathbf{R})^{r}\times\mathrm{U}_{2n}(\mathbf{C})^{s}$ is the maximal compact subgroup of $G(\mathbf{R})$. If $A$ is a reductive real Lie group, then $A^{\circ}$ denotes the connected component of the identity.

Let $Q=HN_{Q}$ be the standard parabolic with Levi $H$. For a prime $\mathfrak{p}|p$ of $G$, let $J_{\mathfrak{p}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\{g\in\operatorname{GL}_{2n}(\mathcal{O}_{\mathfrak{p}}):g\hskip 2.0pt(\mathrm{mod}\hskip 2.0pt\mathfrak{p})\in Q(\mathbf{F}_{\mathfrak{p}})\}\subset\operatorname{GL}_{2n}(F_{\mathfrak{p}})$ be the parahoric subgroup of type $Q$, and $J_{p}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\prod_{\mathfrak{p}|p}J_{\mathfrak{p}}\subset G(\mathbf{Q}_{p})$.

Let $\pi$ be a regular algebraic cuspidal automorphic representation (RACAR) of $G(\mathbf{A})$. If $\pi$ is essentially-self-dual, i.e. there exists some Hecke character $\eta$ such that $\pi^{\vee}\cong\pi\otimes\eta^{-1}$, then $L(\pi\times\pi^{\vee},s)=L(\pi,\mathrm{Sym}^{2}\otimes\eta^{-1},s)L(\pi,\wedge^{2}\otimes\eta^{-1},s)$. This has a simple pole at $s=1$, which occurs either in the symmetric or exterior square $L$-function. Then:

Recall $\pi$ has a $(\eta,\psi)$-Shalika model if there is an intertwining $\mathcal{S}_{\psi}^{\eta}:\pi\hookrightarrow\operatorname{Ind}_{\mathcal{S}(\mathbf{A})}^{G(\mathbf{A})}(\eta\otimes\psi)$, where $\mathcal{S}=\big{\{}s(h,X)=\left(\begin{smallmatrix}h&\\ &h\end{smallmatrix}\right)\left(\begin{smallmatrix}1_{n}&X\\ &1_{n}\end{smallmatrix}\right):h\in\operatorname{GL}_{n},X\in\mathrm{M}_{n}\big{\}}$ is the Shalika group and $(\eta\otimes\psi)(s(h,X))=\eta(\det(h))\cdot\psi(\mathrm{Tr}(X))$. Our conventions are summarised in [BDW, §2.6].

In the definition, (1)$\iff$(3) was substantially proved in [JS90], and (1)$\iff$(2) in [AS06, AS14]. For the equivalence in the exact form above, see [JST, Prop. 2.5].

Let $X^{*}(T)$ be the set of algebraic weights for $T$. A general $\lambda\in X^{*}(T)$ has form $\lambda=(\lambda_{\sigma})_{\sigma\in\Sigma}$, where $\lambda_{\sigma}=(\lambda_{\sigma,1},...,\lambda_{\sigma,2n})\in\mathbf{Z}^{2n}$. Let $X_{+}^{*}(T)$ be the set of dominant weights, where $\lambda_{\sigma,i}\geqslant\lambda_{\sigma,i+1}$ for all $\sigma$ and $i$. Attached to any $\lambda\in X_{+}^{*}(T)$ is an algebraic representation $V_{\lambda}$ of $G$ of highest weight $\lambda$, with dual $V_{\lambda}^{\vee}$. We can decompose $V_{\lambda}=\otimes_{\sigma\in\Sigma}V_{\lambda,\sigma}$, $V_{\lambda}^{\vee}=\otimes_{\sigma\in\Sigma}V_{\lambda,\sigma}^{\vee}$, where $V_{\lambda,\sigma}$ is the algebraic $\operatorname{GL}_{2n}$-representation of highest weight $\lambda_{\sigma}$, with $G$ acting via $\sigma$.

Let $\pi$ be RASCAR of $G(\mathbf{A})$. Let $\mathfrak{g}_{\infty}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\mathrm{Lie}(G(\mathbf{R}))$. As in [Clo90], there is a unique $\lambda\in X^{*}_{0}(T)$, the weight of $\pi$, such that

Here $\lambda$ is pure by [Clo90, Lem. 4.9]. Since $\pi$ is a RASCAR, $\lambda$ is further restricted: by [JST, Prop. 2.17], for all $\sigma\in\Sigma$ there exists ${\sf w}_{\sigma}\in\mathbf{Z}$ such that $\lambda_{\sigma,i}+\lambda_{\sigma,2n+1-i}={\sf w}_{\sigma}$ (i.e. each $\lambda_{\sigma}$ is individually pure). We have ${\sf w}_{\sigma}={\sf w}$ for $\sigma\in\Sigma_{\mathbf{R}}$, and ${\sf w}_{\sigma}+{\sf w}_{c\sigma}=2{\sf w}$ for $\sigma\in\Sigma_{\mathbf{C}}$. We let $\underline{{\sf w}}=({\sf w}_{\sigma})_{\sigma\in\Sigma}\in\mathbf{Z}^{\Sigma}$.

Let $\pi$ be a RASCAR with standard $L$-function $L(\pi,s)$. For a Hecke character $\chi:F^{\times}\backslash\mathbf{A}_{F}^{\times}\to\mathbf{C}^{\times}$, let $\pi\times\chi$ denote the RASCAR with $G(\mathbf{A})$-action twisted by $\chi$. We write $L(\pi,\chi)\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=L(\pi\times\chi,1/2)$.

Crucially $\mathrm{Crit}(\pi)$ also admits a representation-theoretic description. For $\mathbf{j}_{1},\mathbf{j}_{2}\in\mathbf{Z}^{\Sigma}$, let $V_{\mathbf{j}_{1},\mathbf{j}_{2}}^{H}$ be the $H$-representation of highest weight $(\mathbf{j}_{1},\mathbf{j}_{2})$, that is, $\det_{1}^{\mathbf{j}_{1}}\cdot\det_{2}^{\mathbf{j}_{2}}$. The following is proved in the same way as [GR14, Prop. 6.3.1] and [JST, Prop. 2.20], via [Kna01, Thm. 2.1].

We will interpolate the standard critical $L$-values ‘at $p$’. As such, let

where $p^{\beta_{0}}\mathrel{\vbox{\hbox{\scriptsize.}\hbox{\scriptsize.}}}=\prod_{\mathfrak{p}|p}\mathfrak{p}^{\beta_{0,\mathfrak{p}}}$, for $\beta_{0}=(\beta_{0,\mathfrak{p}})\in\mathbf{Z}_{\geqslant 0}^{\{\mathfrak{p}|p\}}$ a multiexponent.

Let $\chi$ be a Hecke character with infinity type $\mathbf{j}\in\mathbf{Z}[\Sigma]$. We have an algebraic homomorphism $w^{\mathbf{j}}:F^{\times}\longrightarrow\overline{\mathbf{Q}}^{\times}$ given by $w^{\mathbf{j}}(x)=\prod_{\sigma\in\Sigma}\sigma(x)^{j_{\sigma}}.$ This then induces maps

Recall $\operatorname{Gal}_{p}$ is the Galois group for the maximal abelian extension of $F$ unramified outside $p\infty$. The structure of this group is described in detail in [BDW, §6.1]; in particular, the Artin reciprocity map induces an isomorphism $\mathrm{Cl}_{F}^{+}(p^{\infty})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\operatorname{Gal}_{p}$, and we have an exact sequence

Via reciprocity, we may consider $\chi_{[p]}$ as a character of $\operatorname{Gal}_{p}$.

If $K\subset G(\mathbf{A}_{f})$ is open compact, the locally symmetric space of level $K$ is

If $M$ is a left $G(\mathbf{Q})$-module, then we have an attached ‘archimedean’ local system $\mathcal{M}$ on $S_{K}$ given by the locally constant sections of

where $\gamma(g,m)kz=(\gamma gkz,\gamma\cdot m)$.

Similarly, if $M$ is a left $K$-module, then we have an attached ‘$p$-adic’ local system on $S_{K}$, given by the local constant sections of (2.5), but with action $\gamma(g,m)kz=(\gamma gkz,k^{-1}\cdot m)$.

If $M$ is a left $G(\mathbf{Q}_{p})$-module, then $G(\mathbf{Q})$ acts via $G(\mathbf{Q})\subset G(\mathbf{Q}_{p})$, and $K$ acts via projection to $G(\mathbf{Q}_{p})$. We get two attached local systems $\mathcal{M}$ and $\mathscr{M}$, and these are isomorphic via the map $(g,m)\mapsto(g_{p}^{-1}\cdot m)$ of local systems. For more detail see [BDW, §2.3].

By [Clo90, p.120], if $\pi$ has weight $\lambda$ then the $(\mathfrak{g}_{\infty},K_{\infty}^{\circ})$-cohomology $\mathrm{H}^{i}(\mathfrak{g}_{\infty},K_{\infty}^{\circ};\pi_{\infty}\otimes V_{\lambda}^{\vee}(\mathbf{C}))$ is non-zero exactly when

Via cuspidal cohomology (see [Clo90]), for any open compact $K\subset G(\mathbf{A}_{f})$ there is an injective and Hecke-equivariant map

Composing with $i_{p}:\mathbf{C}\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}\overline{\mathbf{Q}}_{p}$, and the isomorphism from §2.6, we get an injective map

First we recap [FJ93, §3] (see also [JST, Prop. 2.10]). Let $v$ be a finite place, and fix a Haar measure $dg_{v}$ on $\operatorname{GL}_{n}(F_{v})$ such that $\operatorname{GL}_{n}(\mathcal{O}_{v})$ has volume 1. The local Friedberg–Jacquet integral is