Theta cycles

Daniel Disegni Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva 84105, Israel [email protected]

Abstract

We introduce ‘canonical’ classes in the Selmer groups of certain Galois representations with a conjugate-symplectic symmetry. They are images of special cycles in unitary Shimura varieties, and defined uniquely up to a scalar. The construction is a slight refinement of one of Y. Liu, based on the conjectural modularity of Kudla’s theta series of special cycles. For $2$ -dimensional representations, Theta cycles are (the Selmer images of) Heegner points. In general, they conjecturally exhibit an analogous strong relation with the Beilinson–Bloch–Kato conjecture in rank $1$ , for which we gather the available evidence.

^†^†thanks: Research supported by ISF grant 1963/20 and BSF grant 2018250. This work was partly written while the author was in residence at MSRI/SLMath (Berkeley, CA), supported by NSF grant DMS-1928930.

1 Introduction

The purpose of this largely expository note is to introduce the elements of the title and their relation to the Beilinson–Bloch–Kato (BBK) conjecture. They should play an analogous role to Heegner points on elliptic curves, in that the Bloch–Kato Selmer group $H^{1}_{f}(E,\rho)$ of a relevant Galois representation $\rho$ should be $1$ -dimensional precisely when its Theta cycle is nonzero (cf. [BST, Kim] and references therein for the case elliptic curves). Moreover, the BBK conjecture(s, reviewed in § 2) predict that the $1$ -dimensionality of the Selmer group is equivalent to the (complex or, for suitable primes, $p$ -adic) $L$ -function of $\rho$ vanishing to order $1$ at the center, and Theta cycles allow to approach this conjecture.

The following theorem summarises the state of our knowledge on the topic. Unexplained notions or loose formulations will be defined and made precise in the main body of the paper.

We denote by $\mathbf{Q}^{\textstyle\circ}\subset\overline{\mathbf{Q}}{}_{p}$ the extension of $\mathbf{Q}$ generated by all roots of unity, and we fix an embedding $\iota^{\textstyle\circ}\colon\mathbf{Q}^{\textstyle\circ}\hookrightarrow\mathbf{C}$ . We fix a rational prime $p$ and set $\Sigma:=\{\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}\ |\ \iota_{|\mathbf{Q}^{\textstyle\circ}}=\iota^{\textstyle\circ}\}$ .

Theorem A.

Let $E$ be a CM field with Galois group $G_{E}$ , and let

\rho\colon G_{E}\to\mathrm{GL}_{n}(\overline{\mathbf{Q}}{}_{p})

be an irreducible, geometric Galois representation of weight $-1$ and even dimension $n$ . Suppose that $\rho$ is conjugate-symplectic, automorphic, and has minimal regular Hodge–Tate weights.

Assume that the maximal totally real subfield $F$ of $E$ is not $\mathbf{Q}$ , and that Hypothesis 4.2 on the cohomology of unitary Shimura varieties and Hypothesis 4.3 on the modularity of generating series of special cycles hold.

1.

The construction of § 4.3 attaches to $\rho$ a pair $(\Lambda_{\rho},\Theta_{\rho})$ , well-defined up to isomorphism, consisting of a $\overline{\mathbf{Q}}{}_{p}$ -line $\Lambda_{\rho}$ together with a $\overline{\mathbf{Q}}{}_{p}$ -linear map

$\Theta_{\rho}\colon\Lambda_{\rho}\to H^{1}_{f}(E,\rho),$

whose image is spanned by classes of algebraic cycles.
2.
Suppose that $\rho$ is ‘mildly ramified’ and crystalline at $p$ -adic places.
1. (a)
  
  Assume Conjecture 5.2 on the injectivity of certain Abel–Jacobi maps, and that $p$ is unramified in $E$ . For any any $\iota\in\Sigma$ , denote by $L_{\iota}(\rho,s)$ the complex $L$ -function of $\rho$ with respect to $\iota$ . Then
  
  $\mathrm{ord}_{s=0}L_{\iota}(\rho,s)=1\ \Longrightarrow\ \Theta_{\rho}\neq 0.$
2. (b)
  
  Suppose that $E/F$ is totally split above $p$ , that $p>n$ , and that for every place $w|p$ of $E$ , the representation $\rho_{w}$ is Panchishkin–ordinary. Denote by $\mathscr{X}_{F}$ the $\overline{\mathbf{Q}}{}_{p}$ -scheme of continuous $p$ -adic characters of $G_{F}$ that are unramified outside $p$ , by $\mathfrak{m}\subset\mathscr{O}(\mathscr{X}_{F})$ the ideal of functions vanishing at $\mathbf{1}$ , and by $L_{p}(\rho)\in\mathscr{O}(\mathscr{X})$ the $p$ -adic $L$ -function of $\rho$ . Then
  
  $\mathrm{ord}_{\mathfrak{m}}L_{p}(\rho)=1\ \Longrightarrow\ \Theta_{\rho}\neq 0.$
3.

Assume that $\rho$ has ‘sufficiently large’ image. Then

$\Theta_{\rho}\neq 0\ \Longrightarrow\ \dim_{\overline{\mathbf{Q}}{}_{p}}H^{1}_{f}(E,\rho)=1.$

Examples of representations $\rho$ satisfying the general assumptions of the theorem arise from symmetric powers of elliptic curves: namely, if $A$ is a modular elliptic curve over $F$ with rational Tate module $V_{p}A$ , then by [NT] one may consider the natural representation $\rho_{A,n}$ of $G_{E}$ on ${\rm Sym}^{n-1}V_{p}A_{E}(1-n/2)$ (see [DL, § 1.3] for more details).

Part 1 of the theorem, which builds on constructions of Kudla and Y. Liu, is the main focus of this note; it is explained in § 4, after reviewing the representation-theoretic preliminaries in § 3. The construction is canonical up to a group-theoretic choice described in Remark 3.2. (However, part 3 of the theorem implies that this ambiguity is quite innocuous.)

In § 5, we state a pair of formulas for the Bloch–Beĭlinson and the Nekovář heights of Theta cycles, which are essentially reformulations of a breakthrough result of Li and Liu [LL, LL2], and of its $p$ -adic analogue by Liu and the author [DL]. They imply the assertions of Part 2, and take the shape

\langle\Theta_{\rho}(\lambda),\Theta_{\rho^{*}(1)}(\lambda^{\prime})\rangle_{\star}={c_{\star}\cdot L_{\star}^{\prime}(\rho,0)}\cdot\zeta_{\star}(\lambda,\lambda^{\prime}),

where ‘ $\star$ ’ stands for the relevant decorations, $c_{\star}$ are constants, and $\zeta_{\star}$ are canonical trivialisations of $\Lambda_{\rho}\otimes\Lambda_{\rho^{*}(1)}$ .

Part 3 is the subject of [D-euler], on which we only give some brief remarks in § 5.4; in particular, we sketch the relevance of the perspective proposed here for the results obtained there.

All the constructions and results should have analogues in the odd-dimensional case, in the symplectic case, and for more general Hodge–Tate types. We hope to return to some of these topics in future work.

Acknowledgements

It will be clear to the reader that this note is little more than an attempt to look from the Galois side, and the multiplicity-one side, at ideas of Kudla and Liu. I would like to thank Yifeng Liu for all I learned from him during and after our collaboration, and Elad Zelingher for a remark that sparked it. I am also grateful to Yannan Qiu and Eitan Sayag for helpful conversations or correspondence, and to Chao Li and Yifeng Liu for their comments on a first draft.

This text is based on a talk given at the Second JNT Biennial Conference in Cetraro, Italy, and I would like to thank the organisers for the opportunity to speak there. One of the participants reminded me of Tate’s similarly named ‘ $\theta$ -cycles’ in the theory of mod- $p$ modular forms [Jo, § 7]: besides the context, the capitalisation should also dispel any risk of confusion. Homonymous objects also occur in neuroscience, in connection with a pattern of brain activity typical of “a drowsy state transitional from wake to sleep” [McN, pp. 60-61]; I am grateful to the Cetraro audience for not indulging in this confusion either.

2 The conjecture of Beĭlinson–Bloch–Kato–Perrin-Riou

Let $E$ be a number field with Galois group $G_{E}$ , and let

\rho\colon G_{E}\to\mathrm{GL}_{n}(\overline{\mathbf{Q}}{}_{p})

be an irreducible, geometric Galois representation of weight $-1$ .

2.1 Chow and Selmer groups

A typical source of representations as above is the cohomology of algebraic varieties. In fact, define a motivation of $\rho$ to be an element of¹¹1Throughout this paper, if $R\to R^{\prime}$ is a ring map that can be understood from the context, and $X$ is an $R$ -scheme or an $R$ -module, we write $X_{R^{\prime}}:=X\otimes_{R}R^{\prime}$ .

{\mathrm{Mot}}_{\rho}:=\varinjlim_{(X,k)}\mathrm{Mot}_{\rho}(X,k),\qquad\text{where }\quad\mathrm{Mot}_{\rho}(X,k):=\mathrm{Hom}\,_{\overline{\mathbf{Q}}{}_{p}[G_{E}]}(H_{\text{\'{e}t}}^{2k-1}(X_{\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(k)),\rho),

and the limit runs over all pairs consisting of a smooth proper variety $X_{/E}$ and an integer $k\geqslant 1$ (this is a directed system by Künneth’s fromula). We refer to $(X,k)$ as a source of $f\in{\mathrm{Mot}}_{\rho}$ if $f$ is in the image of $\mathrm{Mot}_{\rho}(X,k)$ . We say that $\rho$ is motivic if ${\mathrm{Mot}}_{\rho}$ is nonzero. According to the conjecture of Fontaine–Mazur, every geometric irreducible Galois representation is motivic.

To a representation $\rho$ as above is attached its Bloch–Kato [BK] Selmer group $H^{1}_{f}(E,\rho)$ .²²2N.B.: the subscript $f$ has nothing to do with names of objects elsewhere in this text. Galois cohomology and Selmer groups are usually defined for representations with coefficients in finite extensions of $\mathbf{Q}_{p}$ . However, it is well-known that we can write $\rho=\rho_{0}\otimes_{L}\overline{\mathbf{Q}}{}_{p}$ for some finite extension $L\subset\overline{\mathbf{Q}}{}_{p}$ of $\mathbf{Q}_{p}$ and some representation $\rho_{0}\colon G_{E}\to\mathrm{GL}_{n}(L)$ (and similarly for the other representations considered in this paper). Then we define $H^{1}_{f}(E,\rho):=H^{1}_{f}(E,\rho_{0})\otimes_{L}\overline{\mathbf{Q}}{}_{p}$ . To a variety $X_{/E}$ as above is attached its Chow group $\mathrm{Ch}^{k}(X)$ of codimension- $k$ algebraic cycles on $X$ up to rational equivalence (with coefficients in $\mathbf{Q}$ ). A central object of arithmetic interest is its subgroup $\mathrm{Ch}^{k}(X)^{0}:=\mathrm{Ker}\,[\mathrm{Ch}^{k}(X)\to H^{2k}_{\text{\'{e}t}}(X_{\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(k))]$ (where the map is the cycle class). It is endowed with an Abel–Jacobi map

\mathrm{AJ}\colon\mathrm{Ch}^{k}(X)^{0}_{\overline{\mathbf{Q}}{}_{p}}\to H^{1}_{f}(E,H_{\text{\'{e}t}}^{2k-1}(X_{\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(k)))

(see [nek-height, § 5.1]). We can define an analogue of the image of $\mathrm{AJ}$ for the representation $\rho$ by

H^{1}_{f}(E,\rho)^{\mathrm{mot}}:=\sum_{f^{\prime}\in\mathrm{Mot}_{\rho}}f^{\prime}_{*}{\mathrm{AJ}}(\mathrm{Ch}^{k}(X)_{\overline{\mathbf{Q}}{}_{p}}^{0})\subset H^{1}_{f}(E,\rho),

where we have denoted by $(X,k)$ any source of the motivation $f^{\prime}$ . By an evocative abuse of nomenclature, we refer to elements of $H^{1}_{f}(E,\rho)^{\mathrm{mot}}$ as cycles.

\remaname \the\smf@thm.

If $\rho=H_{\text{\'{e}t}}^{2k_{0}-1}(X_{0,\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(k_{0}))$ for a variety $X_{0}$ and an integer $k_{0}$ , then we expect that $H^{1}_{f}(E,\rho)^{\mathrm{mot}}={\mathrm{AJ}}(\mathrm{Ch}^{k_{0}}(X_{0})_{\overline{\mathbf{Q}}{}_{p}}^{0})$ . This equality is implied by the Tate conjecture [Tat65, Conjecture 1] for $X\times X_{0}$ .

2.2 The conjecture

We say that $\rho$ is (Panchishkin-) ordinary (see [nek-height, § 6.7], [PR-htIw, § 2.3.1] for more details) if for each place $w|p$ , there is a (necessarily unique) exact sequence of De Rham $G_{E_{w}}$ -representations $0\to\rho_{w}^{+}\to\rho_{|G_{E_{w}}}\to\rho_{w}^{-}\to 0$ , such that $\mathrm{Fil}^{0}\mathbf{D}_{\mathrm{dR}}(\rho_{w}^{+})=\mathbf{D}_{\mathrm{dR}}(\rho_{w}^{-})/{\mathrm{Fil}}^{0}=0$ . For any subfield $F\subset E$ , let

\mathscr{X}_{F}:=\mathrm{Spec}\,\mathbf{Z}_{p}\llbracket\mathrm{Gal}(F_{\infty}/F)\rrbracket\otimes_{\mathbf{Z}_{p}}\overline{\mathbf{Q}}{}_{p},

where $F_{\infty}/F$ is the abelian extension with $\mathrm{Gal}(F_{\infty}/F)$ isomorphic (via class field theory) to the maximal $\mathbf{Z}_{p}$ -free quotient of $F^{\times}\backslash\mathbf{A}_{F}^{\times}/\widehat{\mathscr{O}_{F}}^{p,\times}$ .

One can conjecturally attach to $\rho$ entire $L$ -functions

L_{\iota}(\rho,s)

for ${\iota\colon L\hookrightarrow\mathbf{C}}$ and, if $\rho$ is ordinary, a $p$ -adic $L$ -function

L_{p}(\rho)\in\mathscr{O}(\mathscr{X}_{F})

interpolating suitable modifications of the $L$ -values $L_{\iota}(\rho\otimes\chi_{|G_{E}},0)$ for finite-order characters $\chi\in\mathscr{X}_{F}$ (see [PRbook], at least when taking $F=\mathbf{Q}$ ).

Denote by $\mathfrak{m}=\mathfrak{m}_{F}\subset\mathscr{O}(\mathscr{X}_{F})$ the maximal ideal of functions vanishing at the character $\mathbf{1}$ of $\mathrm{Gal}(F_{\infty}/F)$ , and by $\mathrm{ord}_{\mathfrak{m}}$ the corresponding valuation. The integer ${\mathrm{ord}}_{\mathfrak{m}}L_{p}(\rho)$ is conjecturally independent of the choice of $F$ .

\conjname \the\smf@thm (Beĭlinson, Bloch–Kato, Perrin-Riou [bei, BK, PRbook]).

Let $\rho\colon G_{E}\to\mathrm{GL}_{n}(\overline{\mathbf{Q}}{}_{p})$ be an irreducible geometric representation of weight $-1$ . Let $r\geqslant 0$ be an integer. The following conditions are equivalent:

$\text{(a)}_{\infty}$

for any $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , we have

${\mathrm{ord}}_{s=0}L_{\iota}(\rho,s)=r;$
$\text{(b)}_{\phantom{\infty}}$

$\dim_{\overline{\mathbf{Q}}{}_{p}}H^{1}_{f}(E,\rho)^{\mathrm{mot}}=\dim_{\overline{\mathbf{Q}}{}_{p}}H^{1}_{f}(E,\rho)=r$ .

If moreover $\rho$ is ordinary and $\rho_{w}^{+,*}(1)^{G_{E_{w}}}=0$ for every $w|p$ , then the above conditions are equivalent to

$\text{(a)}_{p\ }$

${\mathrm{ord}}_{\mathfrak{m}}L_{p}(\rho)=r;$

\remaname \the\smf@thm.

The first equality in (b) generalises the conjectural finiteness of the $p^{\infty}$ -torsion in the Tate–Shafarevich group of an elliptic curve. The extra condition in $\text{(a)}_{p\ }$ serves to avoid the phenomenon of exceptional zeros, cf. [Ben].

In the following pages, under some conditions on $\rho$ we will define elements in $H^{1}_{f}(E,\rho)^{\rm mot}$ whose nonvanishing is conjecturally equivalent to the conditions of Conjecture 2.2 with $r=1$ . The construction will be automorphic; in the next section, we give the representation-theoretic background.

3 Descent and theta correspondence

Suppose for the rest of this paper that $E$ is a CM field with totally real subfield $F$ . We denote by ${\rm c}\in\mathrm{Gal}(E/F)$ the complex conjugation, and by $\eta\colon F^{\times}\backslash\mathbf{A}^{\times}\to\{\pm 1\}$ be the quadratic character attached to $E/F$ .

3.1 $p$ -adic automorphic representations

We denote by $\mathbf{A}$ the adèles of $F$ ; if $S$ is a finite set of places of $F$ , we denote by $\mathbf{A}^{S}$ the adèles of $F$ away from $S$ . If $\mathrm{G}$ is a group over $F$ and $v$ is a place of $F$ , we write $G_{v}:=\mathrm{G}(F_{v})$ ; if $S$ a finite set of places of $F$ , we write $G_{S}:=\prod_{v\in S}G(F_{S})$ . (For notational purposes, we will identify a place of $\mathbf{Q}$ with the set of places of $F$ above it.) We denote by $\psi\colon F\backslash\mathbf{A}\to\mathbf{C}^{\times}$ the standard additive character with $\psi_{\infty}(x)=e^{2\pi i\mathrm{Tr}_{F_{\infty}/\mathbf{R}}x}$ , and we set $\psi_{E}:=\psi\circ\mathrm{Tr}_{E/F}$ . We view $\psi_{|\mathbf{A}^{\infty}}$ as valued in $\mathbf{Q}^{\textstyle\circ}$ via the embedding $\iota^{\textstyle\circ}$ .

Unitary groups

Fix a positive integer $n$ . For a place $v$ of $F$ , we denote by $\mathscr{V}_{v}$ be the set of isomorphism classes of (nondegenerate) $E_{v}/F_{v}$ -hermitian spaces of dimension $n$ ; this consists of one element if $v$ splits in $E$ , of two elements if $v$ is finite nonsplit, and of $n+1$ elements if $v$ is real. We denote by $\mathscr{V}^{+}$ the set of isomorphism classes of $E/F$ -hermitian spaces of dimension $n$ that are positive definite at all archimedean place, and by $\mathscr{V}^{-}$ the set of isomorphism classes of $E/F$ -hermitian spaces of dimension $n$ that are positive definite at all archimedean place but one, at which the signature is $(n-1,1)$ . We denote by $\mathscr{V}^{\circ}$ the set of isomorphism classes of $\mathbf{A}_{E}/\mathbf{A}$ -hermitian spaces of dimension $n$ such that for all but finitely many places $v$ , the Hasse–Witt invariant $\epsilon(V_{v})\coloneqq\eta_{v}((-1)^{n\choose 2}\det V_{v})=+1$ , and that $V_{v}$ is positive-define at all archimedean places. We denote $\epsilon(V):=\prod_{v}\epsilon(V_{v})$ and write $\mathscr{V}^{\circ,\epsilon}\subset\mathscr{V}^{\circ}$ for the set of spaces with $\epsilon(V)=\epsilon\in\{\pm\}$ .

We have a natural identification $\mathscr{V}^{\circ,+}=\mathscr{V}^{+}$ . We will mostly be interested in $\mathscr{V}^{\circ,-}$ , which we refer to as the set of incoherent $E/F$ -hermitian spaces, cf. [gross-incoh]. If $V\in\mathscr{V}^{\circ,-}$ , then for every archimedean place $v$ of $F$ , there exists a unique $V(v)\in\mathscr{V}^{-}$ over $F$ such that $V(v)_{w}\cong V_{w}$ if $w\neq v$ .

For $V\in\mathscr{V}$ , let $\mathrm{H}_{V}=\mathrm{U}(V)$ ; if $V\in\mathscr{V}^{\circ}$ with $\epsilon(V)=-1$ , we still use the notation $\mathrm{H}_{V}(\mathbf{A}^{S}):=\prod_{v\notin S}H_{V_{v}}$ , $H_{V_{v}}:=\mathrm{U}(V_{v})(F_{v})$ , and we refer to (the symbol)

\mathrm{H}_{V}

as an incoherent unitary group.

Suppose from now on that $n=2r$ is even. We define the quasisplit unitary group over $F$

\mathrm{G}=\mathrm{U}(W),

where $W=E^{n}$ equipped with the skew-hermitian form $\left(\begin{smallmatrix}&1_{r}\\ -1_{r}&\end{smallmatrix}\right)$ (here $1_{r}$ is the identity matrix of size $r$ ).

\definame \the\smf@thm.

1.
A relevant complex automorphic representation $\Pi$ of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ is an irreducible cuspidal automorphic representation satisfying:
1. (i)
  
  $\Pi\circ{\rm c}\cong\Pi^{\vee}$ ;
2. (ii)
  
  for every archimedean place $w$ of $E$ , the representation $\Pi_{w}$ is induced from the character ${\rm arg}^{n-1}\otimes{\rm arg}^{n-3}\otimes\ldots\otimes{\rm arg}^{1-n}$ of the torus $(\mathbf{C}^{\times})^{n}=(E_{w}^{\times})^{n}\subset\mathrm{GL}_{n}(E_{w})$ ; here ${\rm arg}(z):=z/|z|$ .
2.

A possibly relevant complex automorphic representation $\pi$ of $\mathrm{G}(\mathbf{A})$ is an irreducible cuspidal automorphic representation such that for every archimedean place $v$ of $F$ , the representation $\pi_{v}$ is the holomorphic discrete series representation of Harish-Chandra parameter $\{\tfrac{n-1}{2},\tfrac{n-3}{2},\dots,\tfrac{3-n}{2},\tfrac{1-n}{2}\}$ . We say that $\pi$ is relevant if it is possibly relevant and stable as defined at the beginning of § 3.2 below.
3.

Let $V\in\mathscr{V}^{\circ,-}$ and let $v$ be an archimedean place of $F$ . A possibly relevant complex cuspidal automorphic representation $\sigma$ of $\mathrm{H}_{V(v)}(\mathbf{A})$ is an irreducible cuspidal automorphic representation such that $\sigma_{v}$ is one of the $n$ discrete series representation of $H_{V(v)_{v}}=U(n-1,1)$ of Harish-Chandra parameter $\{\tfrac{n-1}{2},\tfrac{n-3}{2},\dots,\tfrac{3-n}{2},\tfrac{1-n}{2}\}$ , and for every other archimedean place $v^{\prime}\neq v$ of $F$ , we have $\sigma_{v^{\prime}}=\mathbf{1}$ (as a representation of $H_{V(v)_{v^{\prime}}}=U(n)$ ). We say that $\sigma$ is relevant if it is possibly relevant and stable.

\definame \the\smf@thm.

1.

A relevant $p$ -adic automorphic representation $\Pi$ of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ is a representation of $\mathrm{GL}_{n}(\mathbf{A}_{E}^{\infty})$ on a $\overline{\mathbf{Q}}{}_{p}$ -vector space, such that for every $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , the representation $\iota\Pi$ is the finite component of a (unique up to isomorphism) relevant complex automorphic representation $\Pi^{\iota}$ .
2.

A possibly relevant, respectively relevant $p$ -adic automorphic representation $\pi$ of $\mathrm{G}(\mathbf{A})$ is representation of $\mathrm{G}(\mathbf{A}^{\infty})$ on a $\overline{\mathbf{Q}}{}_{p}$ -vector space, such that for every $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , the representation $\iota\pi$ is the finite component of a (unique up to isomorphism) possibly relevant, respectively relevant, complex automorphic representation $\pi^{\iota}$ of $\mathrm{G}(\mathbf{A})$ .
3.

Let $V\in\mathscr{V}^{\circ,-}$ . A possibly relevant, respectively relevant, $p$ -adic automorphic representation $\sigma$ of $\mathrm{H}_{V}(\mathbf{A})$ is representation of $\mathrm{H}_{V}(\mathbf{A}^{\infty})$ on a $\overline{\mathbf{Q}}{}_{p}$ -vector space, such that for every $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ and every archimedean place $v$ of $F$ , the representation $\iota\sigma$ is the finite component of a (unique up to isomorphism) possibly relevant, respectively relevant, complex automorphic representation $\sigma^{\iota,(v)}$ of $\mathrm{H}_{V(v)}(\mathbf{A})$ .

3.2 Automorphic descent

For a place $v$ of $F$ , we denote by ${\rm BC}_{v}$ the base-change map from $L$ -packets of tempered representations of $G_{v}$ to tempered representations of $\mathrm{GL}_{n}(E_{v})$ , which is injective by [Mok, Lemma 2.2.1]. We denote by ${\rm BC}_{\mathrm{G}}$ and ${\rm BC}_{\mathrm{H}_{V}}$ the base-change maps from automorphic representations of the unitary groups $\mathrm{G}(\mathbf{A})$ or $\mathrm{H}_{V}(\mathbf{A})$ to automorphic representations of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ , respectively; we simply write ${\rm BC}$ when there is no risk of confusion. We say that a cuspidal automorphic representation of a unitary group is stable if its base-change is still cuspidal.

\remaname \the\smf@thm.

We have the following properties of the base-change maps.

(a)

By the explicit description given for instance in [LTXZZ, § C.3.1], if $\Pi$ is a relevant representation of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ , then: the preimage of $\Pi$ under ${\rm BC}_{\mathrm{H}_{V}}$ consists of relevant representations of $\mathrm{H}_{V}(\mathbf{A})$ ; the preimage of $\Pi$ under ${\rm BC}_{\mathrm{G}}$ contains a relevant representation of $\mathrm{G}(\mathbf{A})$ .
(b)

If $v$ is a finite place, the base-change maps may be defined for representations with coefficients over $\overline{\mathbf{Q}}{}_{p}$ , compatibly with any extensions of scalars $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ .
(c)

As a consequence of (a) and (b), ${\rm BC}$ extends to a map from relevant $p$ -adic automorphic representations of $\mathrm{G}(\mathbf{A})$ and $\mathrm{H}_{V}(\mathbf{A})$ to relevant $p$ -adic automorphic representations of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ .

Descent to a quasisplit unitary group

We fix the auxiliary choice of a Borel subgroup $\mathrm{B}\subset\mathrm{G}$ with torus $\mathrm{T}$ and unipotent radical $\mathrm{N}$ , and (the $\mathrm{T}$ -orbit of) a generic linear homomorphism $\Psi\colon\mathrm{N}(F)\backslash\mathrm{N}(\mathbf{A})\to\mathbf{C}^{\times}$ ; we call this choice $(\mathrm{N},\Psi)$ a Whittaker datum. A relevant complex or $p$ -adic automorphic representation $\pi$ of $\mathrm{G}(\mathbf{A})$ is called $\Psi$ -generic if it for every finite place, $\pi_{v}$ is $\Psi_{v}$ -generic in the sense that it has a non-vanishing $(N_{v},\Psi_{|N_{v}})$ -Whittaker functional .

\propname \the\smf@thm.

Let $\Pi$ be a relevant $p$ -adic automorphic representation of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ . Then there exists a relevant $p$ -adic automorphic representation $\pi$ of $\mathrm{G}(\mathbf{A})$ , unique up to isomorphism, which is $\Psi$ -generic and satisfies ${\rm BC}(\pi)=\Pi$ .

Proof.

By [GRS] and [Mor], for each $\iota$ there exists a relevant cuspidal automorphic representation $\pi^{\iota}$ of $\mathrm{G}(\mathbf{A})$ that is $\Psi$ -generic and satisfies ${\rm BC}(\pi^{\iota})=\Pi^{\iota}$ . By [Varma, Ato], for each finite place $v$ , each local $L$ -packet of $G_{v}$ contains a unique $\Psi$ -generic representation, which (together with the injectivity of ${\rm BC}_{v}$ ) implies that $\pi^{\iota}$ is unique up to isomorphism. Then by Remark 3.2 (b), the collection $(\pi^{\iota})$ arises from a well-defined relevant $p$ -adic automorphic representation $\pi$ of $\mathrm{G}(\mathbf{A})$ . ∎

\remaname \the\smf@thm.

Our construction of Theta cycles will be based on the descent $\pi$ ; in particular, at least a priori, it depends on the choice of the Whittaker datum $\Psi$ .

3.3 Theta correspondence

We will need to further transfer $\pi$ to a representation of unitary groups $\mathrm{H}_{V}$ for $V\in\mathscr{V}^{\circ,-}$ .

Local correspondence and duality

We first review the local theory. Let $v$ be a finite place of $F$ , and let $C$ be either $\overline{\mathbf{Q}}{}_{p}$ or $\mathbf{C}$ . For $V_{v}\in\mathscr{V}_{v}$ , let $\omega_{V_{v}}=\omega_{V_{v},\psi_{v}}$ be the Weil representation of $H_{V_{v}}\times G_{v}$ (with respect to the character $\psi_{v}$ ) over $C$ , a model of which is recalled in § 4.2 below. Whenever $\Box$ is some smooth admissible representation of a group $G^{?}$ , we denote by $\Box^{?}$ the contragredient, and by $(\,,\ )_{\Box}$ the natural pairing on $\Box\times\Box^{\vee}$ .

The first part of the following result (for nonsplit finite places) is known as theta dichotomy.

\propname \the\smf@thm.

Let $\pi_{v}$ be an tempered irreducible admissible representation of $G_{v}$ over $C=\overline{\mathbf{Q}}{}_{p}$ or $C=\mathbf{C}$ .

1.

There exists a unique $V_{v}\in\mathscr{V}_{v}$ such that

$\sigma_{v}^{\vee}:=(\pi_{v}^{\vee}\otimes\omega_{V_{v}})_{G_{v}}^{\vee}\neq 0.$
2.

The representation $\sigma_{v}^{\vee}$ is tempered and irreducible. Its contragredient $\sigma_{v}$ satisfies ${\rm BC}(\sigma_{v})={\rm BC}(\pi_{v})$ , and the space

$\mathrm{Hom}\,_{H_{V_{v}}\times G_{v}}(\sigma_{v}\otimes\pi^{\vee}_{v}\otimes\omega_{V_{v}},C)$

is $1$ -dimensional over $C$ .
3.

The representation $(\pi_{v}\otimes\omega^{\vee}_{V_{v}})_{G_{v}}$ is canonically identified with $\sigma_{v}$ .

Denote by $\vartheta$ each of the projection maps $\pi_{v}^{\vee}\otimes\omega_{V_{v}}\to\sigma_{v}^{\vee}$ , $\pi_{v}\otimes\omega^{\vee}_{V_{v}}\to\sigma_{v}$ . Then the map

\zeta_{v}(f,\varphi,\phi;f^{\prime},\varphi^{\prime},\phi^{\prime}):=(f,\vartheta(\varphi,\phi))_{\sigma_{v}}\cdot(f^{\prime},\vartheta(\varphi^{\prime},\phi^{\prime}))_{\sigma_{v}^{\vee}}

defines a canonical generator

\zeta_{v}\in\mathrm{Hom}\,_{H_{V_{v}}\times G_{v}}(\sigma_{v}\otimes\pi^{\vee}_{v}\otimes\omega_{V_{v}},\mathbf{C})\otimes_{\mathbf{C}}\mathrm{Hom}\,_{H_{V_{v}}\times G_{v}}(\sigma_{v}^{\vee}\otimes\pi_{v}\otimes\omega^{\vee}_{V_{v}},\mathbf{C}),

with the property that if $\pi_{v}$ and $\sigma_{v}$ are unramified and $f,\varphi,\phi,f^{\prime},\varphi^{\prime},\phi^{\prime}$ are spherical vectors, then

\zeta_{v}(f,\varphi,\phi;f^{\prime},\phi^{\prime},\varphi^{\prime})=(f,f^{\prime})_{\sigma_{v}}(\varphi,\varphi^{\prime})_{\pi_{v}^{\vee}}\cdot(\phi,\phi^{\prime})_{\omega_{v}^{\vee}}.

Proof.

We drop all subscripts $v$ . We start by recalling the first two statements. Consider first the case that $v$ is finite and $E$ is a field. Then $\sigma_{V}^{\vee}=(\pi^{\vee}\otimes\omega_{V})_{G}$ is the (a priori, ‘big’) theta lift of $\pi^{\vee}$ as defined in [harris, (2.1.5.1)]. By the local theta dichotomy proved in Theorem 2.1.7 (iv) ibid. and [GG11, Theorem 3.10], there is exactly one $V\in\mathscr{V}$ such that $\sigma_{V}^{\vee}$ is nonzero; we fix this $V$ and drop if from then notation. Then the other properties of $\sigma:=(\sigma^{\vee})^{\vee}$ are consequences of [GI16, Theorem 4.1] (which collects results from [GS12, GI14]). For the case $E=F\oplus F$ , see [Min08].

We now turn to the other two statements. For a character $\chi\colon F^{\times}\to C^{\times}$ , let

\displaystyle b_{n}(\chi):=\prod_{i=1}^{n}L(i,\chi\eta^{i-1}).

(3.1)

If $C=\mathbf{C}$ , then we have a canonical element

\breve{\zeta}\in\mathrm{Hom}\,_{G}(\pi^{\vee}\otimes\omega_{V},\mathbf{C})\otimes_{\mathbf{C}}\mathrm{Hom}\,_{G}(\pi\otimes\omega^{\vee}_{V},\mathbf{C})

given by

\displaystyle\breve{\zeta}(\varphi,\phi;\varphi^{\prime},\phi^{\prime}):={b_{n}(\mathbf{1})\over L(1/2,\Pi)}\int_{G}(g\varphi,\varphi^{\prime})_{\pi^{\vee}}\cdot(\omega(g)\phi,\phi^{\prime})_{\omega}\,dg,

(3.2)

where $dg$ is the measure of [DL, § 2.1 (G7)], $\Pi:={\rm BC}(\pi)$ . It is a generator by [HKS, § 6], where the regularisation of the integral is also taken care of. (For the well-known comparison between the definition in loc. cit. and the one given here, see [Sak, Lemma 3.1.2].) When $\pi$ (hence $\sigma$ ) are unramified and all the vectors are spherical, by [Yam14, Propositions 7.1, 7/2] we have

\displaystyle\breve{\zeta}(\varphi,\phi;\varphi^{\prime},\phi^{\prime})=(\varphi,\varphi^{\prime})_{\pi^{\vee}}\cdot(\phi,\phi^{\prime})_{\omega^{\vee}}

(3.3)

If $C=\overline{\mathbf{Q}}{}_{p}$ , then for any $\iota\in\Sigma$ we have a tetralinear form $\breve{\zeta}^{\iota}$ as above, and by [DL, Lemma 3.29], there is a $\breve{\zeta}\in\mathrm{Hom}\,_{G}(\pi^{\vee}\otimes\omega_{V},\overline{\mathbf{Q}}{}_{p})\otimes_{\overline{\mathbf{Q}}{}_{p}}\mathrm{Hom}\,_{G}(\pi\otimes\omega^{\vee}_{V},\overline{\mathbf{Q}}{}_{p})$ such that $\zeta\otimes_{\overline{\mathbf{Q}}{}_{p},\iota}\mathbf{C}=\zeta^{\iota}$ for every $\iota\in\Sigma$ .

Now, we may view $\breve{\zeta}$ as a map

\displaystyle\breve{\zeta}\colon(\pi^{\vee}\otimes\omega_{V})_{G}\otimes(\pi\otimes\omega^{\vee}_{V})_{G}\to C

(3.4)

that is, by inspection, invariant under the diagonal action of $H$ on both factors. It follows that $\breve{\zeta}$ gives the duality of our third statement. The fourth statement then follows from the definitions and (3.3). ∎

\remaname \the\smf@thm.

A more symmetrically defined exalinear form would be

(f,\varphi,\phi;f^{\prime},\phi^{\prime},\varphi^{\prime})\mapsto\int_{H_{V}}\int_{G}(hf,f^{\prime})_{\sigma}\cdot(g\varphi,\varphi^{\prime})_{\pi^{\vee}}\cdot(\omega(h,g)\phi,\phi^{\prime})_{\omega}\,dgdh,

where the integral in $dg$ is regularised as remarked after (3.2). If $\sigma$ is a discrete series, the integral in $dh$ converges and its value equals that of $\zeta_{v}$ , times the formal degree of $\sigma$ – for which [BP-Planch] gives a formula in terms of adjoint gamma factors. In general, regularising the integral in $dh$ amounts to regularising the inner product of two matrix coefficients of $\sigma$ . A regularisation has been proposed by Qiu [Qiu, Qiu2]; however the definition of the resulting generalised formal degree is partly conjectural, and no precise (even conjectural) formula for it appears in the literature.

Global correspondence

We have the following global variant of Proposition 3.3.

\propname \the\smf@thm.

Let $\Pi$ be a relevant $p$ -adic automorphic representation of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ , and let $\pi$ be the representation of Proposition 3.2. Then there exists a unique-up-to-isomorphism pair $(V,\sigma)$ , with $V\in\mathscr{V}^{\circ}$ and $\sigma$ a relevant $p$ -adic automorphic representation of $\mathrm{H}_{V}(\mathbf{A})$ , such that

\mathrm{Hom}\,_{\mathrm{G}(\mathbf{A}^{\infty})\times\mathrm{H}(\mathbf{A}^{\infty})}(\sigma^{\infty}\otimes\pi^{\infty,\vee}\otimes\omega_{V}^{\infty},\overline{\mathbf{Q}}{}_{p})\neq 0.

Moreover, we have $\epsilon(V)=\varepsilon(1/2,\Pi)$ .

Proof.

This follows from the explicit form of theta dichotomy in terms of the doubling epsilon factors of [harris], whose product over all places coincides with the standard central epsilon factor of $\Pi$ by [LR05]. ∎

\remaname \the\smf@thm.

The archimedean fact motivating the proposition is that if $\pi_{v}$ has Harish–Chandra parameter $\{\tfrac{n-1}{2},\tfrac{n-3}{2},\dots,\tfrac{3-n}{2},\tfrac{1-n}{2}\}$ , then it has a nonzero theta lift to some $H_{V_{v}}$ exactly for $V_{v}$ positive-definite, in which case the theta lift $\sigma_{v}$ is the trivial representation of $H_{V_{v}}$ : see [LiJ, Theorem 6.2] and [Paul].

4 Theta cycles

4.1 Assumptions on the Galois representation

Let again $\rho\colon G_{E}\to\mathrm{GL}_{n}(\overline{\mathbf{Q}}{}_{p})$ be irreducible, geometric, and of weight $-1$ . We denote by $\rho^{\mathrm{c}}\colon G_{E}\to\mathrm{GL}_{n}(\overline{\mathbf{Q}}{}_{p})$ the representation defined by $\rho^{\mathrm{c}}(g)=\rho(cgc^{-1})$ , where $c\in G_{E}$ is any fixed lift of $\mathrm{c}$ . (A different choice of lift would yield an isomorphic representation.)

We suppose from now on that the following conditions are satisfied:

1.

$\rho$ is conjugate-symplectic in the sense that there exists a perfect pairing

$\rho\otimes_{\overline{\mathbf{Q}}{}_{p}}\rho^{\rm c}\to\overline{\mathbf{Q}}{}_{p}(1)$

such that for the induced map $u\colon\rho^{c}\to\rho^{*}(1)$ and its conjugate-dual $u^{*}(1)^{\mathrm{c}}\colon\rho^{\mathrm{c}}\to\rho^{\mathrm{c},*}(1)^{\mathrm{c}}=\rho^{*}(1)$ , we have $u=-u^{*}(1)^{\mathrm{c}}$ ;
2.

$n=2r$ is even;
3.

for every $\jmath\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}_{p}$ , the $\jmath$ -Hodge–Tate weights of $\rho$ are the $n$ integers $\{-r,-r+1,\ldots,1-r\}$ (where the convention is that the cyclotomic character has weight $-1$ );
4.

$\rho$ is automorphic in the sense that for each $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , there is a cuspidal automorphic representation $\Pi^{\iota}$ of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ such that $L_{\iota}(\rho,s)=L(\Pi^{\iota},s+1/2)$ ;

Associated automorphic representations

A collection $(\Pi^{\iota})_{\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}}$ as in Condition 4 is uniquely determined up to isomorphism if it exists, by the multiplicity-one theorem for automorphic forms on $\mathrm{GL}_{n}$ ; it is conjectured to always exist. Moreover, every $\Pi^{\iota}$ is relevant in the sense of Definition 3.1.1, where Condition 1 implies property (i) in the definition, and Condition 3 implies property (ii). It is then clear that $(\Pi^{\iota})_{\iota}$ arises from a unique (up to isomorphism) relevant $p$ -adic automorphic representation

\Pi=\Pi_{\rho}

of $\mathrm{GL}_{n}(\mathbf{A}_{E})$ (Definition 3.1.1). We denote by $\pi=\pi_{\rho}$ the relevant $p$ -adic representation of $\mathrm{G}(\mathbf{A})$ associated with $\Pi$ as in Proposition 3.2, and by

(V,\sigma)=(V_{\rho},\sigma_{\rho})

the pair associated with $\pi$ as in Proposition 3.3. We also put $\mathrm{H}=\mathrm{H}_{V}$ .

4.2 Models of the representations

We now fix some concrete models of the representations $\omega$ , $\pi$ , and $\sigma$ .

Weil representations

We fix the well-known model of the representation $\omega=\otimes^{\prime}_{v\nmid\infty}\omega_{V,v}$ on $\mathscr{S}(V^{r}_{\mathbf{A}^{\infty}},\overline{\mathbf{Q}}{}_{p})$ associated with $\psi$ , on which $\mathrm{H}(\mathbf{A}^{\infty})$ acts by right translations, whereas the action of $\mathrm{G}(\mathbf{A}^{\infty})$ is recalled in [DL, §4.1 (H7)].

Denote by ${\dagger}$ the involution on $\mathrm{G}$ given by conjugation by the element $\left(\begin{smallmatrix}1_{r}&\\ &1_{r}\end{smallmatrix}\right)$ inside $\mathrm{GL}_{n}(E)$ ; it acts on any $\mathrm{G}(R)$ module for any $E$ -algebra $R$ . The representation $\omega^{{\dagger}}$ is isomorphic to the Weil representation attached to $\psi^{-1}$ , in turn isomorphic to $\omega^{\vee}$ .

Siegel-hermitian modular forms and their $q$ -expansion

The representation $\pi$ may be realised in spaces of hermitian modular forms, which we briefly review.

In [DL, § 2.2], we have defined the following objects.³³3In this discussion, most new notation will be introduced by equalities whose right-hand sides reproduce the corresponding notation in [DL].

•

A $\mathbf{C}$ -vector space $\mathscr{H}_{\mathbf{C}}=\mathscr{A}^{[r]}_{r,{\rm hol}}$ of holomorphic forms for the group $\mathrm{G}$ .
•

For any $\overline{\mathbf{Q}}{}_{p}$ -algebra $R$ , an $R$ -vector space $\mathscr{H}_{R}=\mathscr{H}^{[r]}_{r}\otimes_{\mathbf{Q}_{p}}R$ of (classical) $p$ -adic automorphic forms for $\mathrm{G}$ , such that for each $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , we have an isomorphism

$\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}}\otimes_{\iota}\mathbf{C}\to\mathscr{H}_{\mathbf{C}},\qquad\Phi\otimes 1\mapsto\Phi^{\iota}.$

In fact, only the case where $E/F$ is totally split above $p$ was considered in [DL], where $\mathscr{H}^{[r]}_{r}$ is the direct limit, over open compact subgroups $K\subset\mathrm{G}(\mathbf{A}^{\infty})$ , of subspaces of sections of a certain line bundle on a Siegel hermitian variety $\Sigma(K)_{/\mathbf{Q}_{p}}$ ; let us explain why the splitting condition is not necessary for our purposes. Define a $p$ -adic CM type of $E$ to be a set $\Phi$ of $[F:\mathbf{Q}]$ embeddings $i\colon E\hookrightarrow\overline{\mathbf{Q}}{}_{p}$ such that $i\in\Phi$ if and only if $i\circ\mathrm{c}\notin\Phi$ ; in the totally split case, the choice of a $p$ -adic CM type is equivalent to the choice of a set $\mathtt{P}_{\mathrm{C}M}$ as in [DL, §2.1 (F2)], which intervenes in the construction of $\Sigma(K)$ as a moduli scheme by fixing a signature type for test objects in the sense of [LTXZZ, Definition 3.4.3]. However, this construction, and the comparison complex Siegel hermitian varieties of [DL, Lemma 2.1], go through with any $p$ -adic CM type $\Phi$ (with the innocuous difference that, in general, $\Sigma(K)$ and $\mathscr{H}_{r}^{[r]}$ will only be defined over a finite extension of $\mathbf{Q}_{p}$ in $\overline{\mathbf{Q}}{}_{p}$ ).
•

A space ${\rm SF}(R)={\rm SF}_{r}(R)$ of formal $q$ -expansions with coefficients in the (arbitrary) ring $R$ , and a Siegel–Fourier expansion map ${\bf q}_{\infty}={\bf q}_{r}^{\rm an}\colon\mathscr{H}_{\mathbf{C}}\to{\rm SF}(\mathbf{C})$ . By the argument in the proof of [DL, Proposition 4.6] (based on Lemma 2.9 ibid.), we deduce a $\overline{\mathbf{Q}}{}_{p}$ -linear $q$ -expansion map

$\mathbf{q}_{p}\colon\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}}\to{\rm SF}(\overline{\mathbf{Q}}{}_{p})$

satisfying $\iota\mathbf{q}_{p}(\Phi)=\mathbf{q}_{\mathbf{C}}(\Phi^{\iota})$ for every $\Phi\in\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}^{\textstyle\circ}}$ and every embedding $\iota\in\Sigma$ .

By [DL, Lemma 3.13] (based on [Mok]), for a relevant $p$ -adic automorphic representation $\pi$ , the space $\mathrm{Hom}\,_{\mathrm{G}(\mathbf{A}^{\infty})}(\pi^{\prime},\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}})$ is $1$ -dimensional, and $\pi^{\vee,{\dagger}}$ is also relevant. We identify $\pi=\pi_{\rho}$ with the corresponding subspace of $\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}}$ . Then $\pi_{\rho^{*}(1)}$ is isomorphic to $\pi^{\vee,{\dagger}}$ .

Shimura varieties and their cohomology

We assume from now on that $\varepsilon(\rho)=-1$ . (The opposite case will be trivial for our purposes in Definition 4.3 below.) Then $V\in\mathscr{V}^{\circ,-}$ , and we have an inverse system

(X_{K})_{K\subset\mathrm{H}(\mathbf{A}^{\infty})}

of $n$ -dimensional smooth varieties over $E$ , with the property that for every archimedean place $w$ of $F$ , with underlying place $v$ of $F$ , the variety $X_{V,K}\times_{E,w}\mathbf{C}$ is isomorphic to the complex Shimura variety $X_{V(v),wK}$ associated with the unitary group $\mathrm{H}_{V(v)}$ and the Shimura datum attached to $w$ that is the complex conjugate to the one defined in [liu-fj, § C.1] (and thus coincides with the one specified in [LTXZZ, § 3.2] and used in [LL, DL]); see also [gross-incoh, STay]. From now on we assume that $F\neq\mathbf{Q}$ , which implies that each $X_{K}$ is projective.

Let

\displaystyle H_{\textup{\'{e}t}}^{2r-1}(X_{\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(r))

\displaystyle:=\varprojlim_{K\subset\mathrm{H}(\mathbf{A}^{\infty})}H_{\textup{\'{e}t}}^{2r-1}(X_{K,\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(r)),

where the transition maps are pushforwards. For each $K$ , we have a spherical Hecke algebra for $\mathrm{H}$ acting on $X_{K}$ ; let $\mathfrak{m}_{\rho,K}$ be the Hecke ideal denoted by $\mathfrak{m}^{\mathtt{R}}_{\pi}$ in [LL, Definition 6.8]. We denote by

M_{\rho,K}:=H_{\textup{\'{e}t}}^{2r-1}(X_{K,\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(r))_{\mathfrak{m}_{\rho,K}}

the localisation, and we set

\displaystyle M_{\rho}:=\varprojlim_{K}M_{\rho,K}\subset H_{\textup{\'{e}t}}^{2r-1}(X_{\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(r)).

We will assume the following hypothesis, which is a special case of [LL, Hypothesis 6.6] (and it is expected to be confirmed in a sequel to [KSZ]).

Hypothesis \the\smf@thm.

For each open compact $K\subset\mathrm{H}(\mathbf{A}^{\infty})$ , we have a Hecke- and Galois-equivariant decompostion

\displaystyle M_{\rho,K}\cong\bigoplus_{\sigma^{\prime}}\rho\otimes\sigma^{\prime\vee},

(4.1)

where the direct sum runs over the isomorphism classes of relevant $p$ -adic automorphic representation $\sigma^{\prime}$ of $\mathrm{H}_{V}(\mathbf{A})$ with ${\rm BC}(\sigma^{\prime})=\Pi$ .

We thus have an $\mathrm{H}(\mathbf{A}^{\infty})$ -equivariant map

\displaystyle\sigma\longrightarrow\mathrm{Hom}\,_{\overline{\mathbf{Q}}{}_{p}[G_{E}]}(H_{\textup{\'{e}t}}^{2r-1}(X_{\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(r)),\rho),

(4.2)

and we identify $\sigma$ with the image of this map. We also put $M_{\sigma,K}:=\rho\otimes\sigma^{\vee,K}\subset H_{\textup{\'{e}t}}^{2r-1}(X_{K,\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(r))$ , and

\displaystyle M_{\sigma}:=\varprojlim_{K}M_{\sigma,K}\subset M_{\rho}\subset H_{\textup{\'{e}t}}^{2r-1}(X_{\overline{E}{}},\overline{\mathbf{Q}}{}_{p}(r)).

(4.3)

Then $\sigma=\mathrm{Hom}\,_{\overline{\mathbf{Q}}{}_{p}[G_{E}]}(M_{\sigma},\rho):=\varinjlim_{K}\mathrm{Hom}\,_{\overline{\mathbf{Q}}{}_{p}[G_{E}]}(M_{\sigma,K},\rho)$ .

4.3 Construction

We proceed in four steps. The first three steps follow works of Kudla and collaborators [Kud-Duke, Kud03, KRY06] and of Liu [Liu11].

0. Special cycles in $X$

For each $x\in V^{r}_{\mathbf{A}^{\infty}}$ and each open compact $K\subset\mathrm{H}(\mathbf{A}^{\infty})$ , we have a codimension- $r$ special cycle

Z(x)_{K}\in\mathrm{Ch}^{r}(X_{K})

defined in [Liu11, § 3A]. Putting

T(x):=((x_{i},x_{j})_{V})_{ij},

where $(\,,\,)_{v}$ is the hermitian form on $V$ , we recall the definition in two basic cases. Denote by $\mathrm{Herm}_{r}(F)^{+}$ the set of $r\times r$ matrices over $E$ that satisfy $T^{\mathrm{c}}=T^{\mathrm{t}}$ and that are totally positive semidefinite. First, $Z(x)_{K}=0$ if $T(x)\notin\mathrm{Herm}_{r\times r}(F)^{+}$ . Second, assume that $T(x)\in\mathrm{Herm}_{r\times r}(F)^{+}$ is positive definite. Let $v$ be an archimedean place of $F$ , and assume further that under the isomorphism $V\otimes_{\mathbf{A}}\mathbf{A}^{\infty}\cong V(v)\otimes_{F}\mathbf{A}^{\infty}$ , the vector $x=(x_{1},\ldots,x_{r})$ corresponds to a vector $x^{(v)}\in V(v)^{r}$ . Let $V_{x}(v)\subset V(v)$ the span of $x^{(v)}_{1},\ldots,x^{(v)}_{r}$ . The embedding $\mathrm{U}(V_{x}(v)^{\perp})\hookrightarrow\mathrm{U}(V(v))$ induces a map of towers of Shimura varieties $\alpha_{x}(v)\colon X_{V_{x}(v)^{\perp}}\to X_{V(v)}$ ; then we define $Z(x(v))_{K}\in\mathrm{Ch}^{r}(X_{V(v),K})$ to be the class of the image cycle, and $Z(x)_{K}$ to be the corresponding element in $\mathrm{Ch}^{r}(X_{K})$ .

We denote by

[Z(x)_{K}]\in H_{\textup{\'{e}t}}^{2r}(X_{K},\mathbf{Q}_{p}(r))

the absolute cycle class of $Z(x)_{K}$ .

1. Theta kernel

The special cycles just defined may be assembled in a generating series. Let $\phi\in\omega$ . For any $K\subset\mathrm{H}_{V}(\mathbf{A}^{\infty})$ fixing $\phi$ , we define

{}^{\mathbf{q}}\Theta(\phi):=\mathrm{vol}(K)\sum_{{x\in K\backslash V_{\mathbf{A}^{\infty}}^{r}}}\phi(x)[Z(x)_{K}]\,q^{T(x)},

where $\mathrm{vol}(K)$ is as in [LL, Definition 3.8]. Then ${}^{\mathbf{q}}\Theta(\phi)$ is an element of $H^{2r}_{\text{\'{e}t}}(X_{K},\mathbf{Q}_{p}(r))\otimes_{\mathbf{Q}_{p}}{\rm SF}(\overline{\mathbf{Q}}{}_{p})$ , and the construction is compatible under pushforward in the tower $X_{K}$ ; in other words, we have

\Theta(\phi)\in H^{2r}_{\text{\'{e}t}}(X,\mathbf{Q}_{p}(r))\otimes_{\mathbf{Q}_{p}}{\rm SF}(\overline{\mathbf{Q}}{}_{p})

where $H^{2r}_{\text{\'{e}t}}(X,\mathbf{Q}_{p}(r)):=\varprojlim_{K}H^{2r}_{\textup{\'{e}t}}(X_{K},\mathbf{Q}_{p}(r))$ . (The reason why we prefer our $\Theta(\phi)$ to be compatible with pushforwards rather than pullbacks is that we can then pair it with elements of the automorphic representation $\sigma$ under the identification (4.2).)

The following conjecture asserts the modularity of the generating series, and from now on we will assume it holds. It is implied by the variant for Chow groups of [LL, Hypothesis 4.5]; see Remark 4.6 ibid. for comments on the supporting evidence.

Hypothesis \the\smf@thm.

For every $\phi\in\omega$ , there exists a unique

\Theta(\phi)\in H^{2r}_{\textup{\'{e}t}}(X_{V},\mathbf{Q}_{p}(r))\otimes_{\mathbf{Q}_{p}}\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}}

such that for every $g\in\mathrm{G}(\mathbf{A}^{\infty})$ , we have

\mathbf{q}_{p}(\Theta(\omega(g)\phi)={}^{\mathbf{q}}\Theta(\omega(g)\phi).

2. Arithmetic theta lifts

Denote by $\Phi\mapsto\Phi_{\pi}$ the Hecke-eigenprojection $\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}}\to\pi$ , and by $\langle\,,\,\rangle_{\pi^{\vee}}\colon\pi^{\vee}\otimes\pi\to\overline{\mathbf{Q}}{}_{p}$ the canonical duality. (We also use the same names for any base-change.)

Then for every $\varphi\in\pi^{\vee}$ , we may define

\displaystyle\Theta(\varphi,\phi):=\langle\varphi,\Theta(\phi)_{\pi}\rangle_{\pi^{\vee}}\quad\in H^{2r}_{\textup{\'{e}t}}(X,\overline{\mathbf{Q}}{}_{p}(r)).

(4.4)

By [DL, Lemma 4.11], we have in fact

\Theta(\varphi,\phi)\in H^{1}_{f}(E,M_{\sigma}),

where $M_{\sigma}=\eqref{Msg}$ . (In loc. cit. it is assumed that $\rho$ is crystalline at the $p$ -adic places, but the same proof applies using [nek-niz, Theorem B] in place of [nek-AJ].)

3. Theta cycles

For every $f\in\sigma$ , we define

\Theta_{\rho}(f,\varphi,\phi):=f_{*}\Theta(\varphi,\phi)\in H^{1}_{f}(E,{\rho}).

The following definition then satisfies the first property asserted in Theorem A.

\definame \the\smf@thm.

Let $\rho$ be a Galois representation satisfying the assumptions of § 4.1.

If $\varepsilon(\rho)=+1$ , we may put $\Lambda_{\rho}=\overline{\mathbf{Q}}{}_{p}$ and $\Theta_{\rho}:=0$ .

If $\varepsilon(\rho)=-1$ , assume that $F\neq\mathbf{Q}$ and that Hypotheses 4.2 and 4.3 hold, and let $\pi$ , $V$ , $\sigma$ be as above. Then we define

\displaystyle\Lambda_{\rho}:=(\sigma\otimes\pi^{\vee}\otimes\omega)_{\mathrm{H}(\mathbf{A}^{\infty})\times\mathrm{G}(\mathbf{A}^{\infty})},

and

	$\displaystyle\Theta_{\rho}\colon\Lambda_{\rho}$	$\displaystyle\to H^{1}_{f}(E,{\rho}),$
	$\displaystyle[(f,\varphi,\phi)]$	$\displaystyle\mapsto\Theta_{\rho}(f,\varphi,\phi).$

5 Relation to $L$ -functions and Selmer groups

We continue to denote by $\rho$ a Galois representation satisfying the assumptions of § 4.1.

5.1 Complex and $p$ -adic $L$ -functions

For every $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , and every finite-order character $\chi^{\prime}\colon G_{E}\to\overline{\mathbf{Q}}{}_{p}^{\times}$ , we have the $L$ -function

L_{\iota}(\rho\otimes\chi^{\prime},s)=L(s+1/2,\Pi^{\iota}\otimes\iota\chi^{\prime}),

which is holomorphic and has a functional equation with center at $s=0$ and sign $\varepsilon(\rho)$ .

At least under the following assumption, we also have a $p$ -adic $L$ -function.

Assumption \the\smf@thm.

The extension $E/F$ is totally split above $p$ , and for every place $w|p$ of $E$ , the representation $\rho_{w}$ is crystalline and Panchishkin-ordinary.

We need to make the auxiliary choice of an isomorphism $\alpha\colon\pi^{\vee,{\dagger}}\to\pi_{\rho^{*}(1)}$ (where $\pi=\pi_{\rho},\pi_{\rho^{*}(1)}\subset\mathscr{H}_{\overline{\mathbf{Q}}{}_{p}}$ ), which yields for each $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , an element $\mathrm{P}_{\rho,\iota}=\mathrm{P}_{\rho,\alpha,\iota}(\rho)\in\mathbf{C}^{\times}$ such that

\displaystyle\iota(\varphi_{1}^{{\dagger}},\varphi_{2})_{\pi^{\vee}}={((\alpha\varphi_{1})^{\iota,{\dagger}},\varphi^{\iota}_{2})_{\rm Pet}\over\mathrm{P}_{\rho,\iota}}

for every $\varphi_{1}\in\pi^{\vee,{\dagger}}$ , $\varphi_{2}\in\pi$ ; here

(\varphi,\varphi^{\prime})_{\rm Pet}:={\int_{\mathrm{G}(F)\backslash\mathrm{G}(\mathbf{A})}\varphi(g)\varphi^{\prime}(g)\,dg}

where $dg$ is the measure of [DL, § 2.1 (G7)].

For a character $\chi$ of $G_{F}$ , we denote $\chi_{E}:=\chi_{|G_{E}}$ , and we put $b_{n}(\chi):=\prod_{v\nmid\infty}b_{n}(\chi_{v})$ , where the factors are as in (3.1); we also define a constant

c_{\infty}=\left((-1)^{r}2^{-r^{2}-r}\pi^{r^{2}}{\Gamma(1)\cdots\Gamma(r)\over\Gamma(r+1)\cdots\Gamma(2r)}\right)^{[F:\mathbf{Q}]}.

Finally, we denote by $\mathscr{K}(\mathscr{X}_{F})$ the fraction field of $\mathscr{O}(\mathscr{X}_{F})$ .

\propname \the\smf@thm.

Suppose that $\rho$ satisfies Assumption 5.1. There is a meromorphic function

L_{p}(\rho)=L_{p,\alpha}(\rho)\quad\in\mathscr{K}(\mathscr{X}_{F})

characterised by the following property: for every finite-order character $\chi\in\mathscr{X}_{F}(\overline{\mathbf{Q}}{}_{p})$ and every embedding $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ , we have

\iota L_{p}(\rho)(\chi)=\iota e_{p}(\rho,\chi)\cdot{c_{\infty}L_{\iota}(\rho\otimes\chi_{E},0)\over b_{n}(\chi)\mathrm{P}_{\rho,\iota}}.

Here, $\iota e_{p}(\rho,\chi)=\prod_{w|v|p}\iota e_{w,\iota}(\rho,\chi)\in\iota\overline{\mathbf{Q}}{}_{p}$ , in which the product ranges over the $p$ -adic places of $E$ and of $F$ , and

\iota e_{w}(\rho,\chi):=\gamma(\iota{\rm WD}(\rho_{w}^{+}\otimes\chi_{E,w}),\psi_{E,w})^{-1}{b_{n,v}(\chi)\over L_{\iota}(\rho_{w}\otimes\chi_{E,w})}.

where the Deligne–Langlands $\gamma$ -factor and Fontaine’s functor $\iota{\rm WD}$ are as recalled in [Dplf, (1.1.4)].

Proof.

This follows by multiplying the incomplete $p$ -adic $L$ -function of [DL, Theorem 1.4] by local $L$ -factors at ramified and $p$ -adic places, as in Remark 3.38 ibid.⁴⁴4Before [DL], a $p$ -adic $L$ -function that extends $L_{p}(\rho)$ to a larger space was constructed in [EHLS]; the rationality property proved there is weaker than stated here. ∎

5.2 Pairings

Let $\rho$ be a representation satisfying the assumptions of Definition 4.3, and let $\pi_{\rho}$ , $V$ , $\sigma_{\rho}$ , $\Lambda_{\rho}$ , and $\Theta_{\rho}$ be the associated objects. We denote by $\pi_{v}$ and $\sigma_{v}$ the local components of $\pi_{\rho}$ and $\sigma_{\rho}$ at the place $v$ (which are well-defined up to isomorphism).

Dual Theta cycles

The representation $\rho^{*}(1)$ also satisfies those assumptions, and we have the corresponding map

\Theta_{\rho^{*}(1)}\colon\Lambda_{\rho^{*}(1)}\to H^{1}_{f}(E,\rho^{*}(1)).

Pairings

Let $\langle\,,\,\rangle\colon M_{\rho}\otimes M_{\rho^{*}(1)}\to\overline{\mathbf{Q}}{}_{p}(1)$ be the pairing induced by Poincaré duality. Then we define a pairing

\displaystyle(\,,\,)_{\sigma}\colon\sigma_{\rho}\otimes\sigma_{\rho^{*}(1)}\to\overline{\mathbf{Q}}{}_{p}

(5.1)

by $(f,f^{\prime})_{\sigma}\coloneqq f\circ u(f^{\prime*}(1))$ , where $f^{\prime*}(1)\colon\rho_{\sigma^{*}(1)}^{*}(1)\to M_{\rho^{*}(1)}^{*}(1)$ is the transpose, and $u\colon M_{\rho^{*}(1)}^{*}(1)\to M_{\rho}$ is the isomorphism induced by $\langle\,,\,\rangle$ . Thus $\sigma_{\rho^{*}(1)}$ is identified with $\sigma_{\rho}^{\vee}=\sigma^{\vee}$ .

We also have a canonical pairing on $\omega\otimes\omega^{{\dagger}}$ defined by

\displaystyle(\phi,\phi^{\prime})_{\mathscr{S}}=\int_{V^{r}_{\mathbf{A}^{\infty}}}\phi(x)\phi^{\prime}(x)\,dx

(5.2)

for the product of $\psi$ -selfdual measures. Thus $\omega^{{\dagger}}$ is identified with $\omega^{\vee}$ . Similarly, if we denote $\iota\Box:=\Box\otimes_{\overline{\mathbf{Q}}{}_{p},\iota}\mathbf{C}$ , and complex conjugation in $\mathbf{C}$ by a bar, we have $\overline{\iota\omega}{}=\omega^{\vee}$ .

Then:

•

for every isomorphism $\alpha\colon\pi_{\rho}^{\vee,{\dagger}}\to\pi_{\rho^{*}(1)}$ , we have a pairing

\displaystyle\zeta_{\alpha}:=\otimes_{v\nmid\infty}\zeta_{v}\circ(\,)^{{\dagger}}\circ j_{\alpha}\colon\Lambda_{\rho}\otimes\Lambda_{\rho^{*}(1)}\to\overline{\mathbf{Q}}{}_{p},

(5.3)

where $j_{\alpha}$ identifies the factor $\pi_{\rho^{*}(1)}^{\vee}$ of $\Lambda_{\rho}\otimes\Lambda_{\rho^{*}(1)}$ with $\pi_{\rho}^{{\dagger}}$ via the dual of $\alpha$ , and $(\,)^{{\dagger}}$ maps $\pi_{\rho}^{{\dagger}}\otimes\omega$ to $\pi_{\rho}\otimes\omega^{{\dagger}}=\pi_{\rho}\otimes\omega^{\vee}$ ;

•

for every $\iota\in\Sigma$ we have an identification $j_{\iota}\colon\iota{\pi_{\rho^{*}(1)}^{\vee}=\overline{\iota\pi_{\rho}}{}}$ via the restriction of $(\ ,\ )_{\rm Pet}$ to $\overline{\pi_{\rho}^{\iota}}{}\otimes\pi_{\rho^{*}(1)}$ . Then we obtain a pairing

$\zeta_{\iota}=\zeta_{v}\circ\overline{(\,)}{}\circ j_{\iota}\colon\iota\Lambda_{\rho}\otimes\iota\Lambda_{\rho^{*}(1)}\to\mathbf{C}$

where $\overline{(\,)}{}$ maps $\overline{\iota\pi_{\rho}}{}\otimes\iota\omega$ to $\iota\pi_{\rho}\otimes\overline{\iota\omega}{}=\iota\pi_{\rho}\otimes\iota\omega^{\vee}$ .

$p$ -adic height pairing

Assume that $\rho$ is Panchishkin-ordinary. Then the construction of Nekovář [nek-height] (see [DL, § 4.3] for a verification of the assumptions) yields a $p$ -adic height pairing

\langle\ ,\ \rangle\colon H^{1}_{f}(E,\rho)\otimes H^{1}_{f}(E,\rho^{*}(1))\to\Gamma_{F}\hat{\otimes}\overline{\mathbf{Q}}{}_{p}.

Complex height pairings

On the other hand, assume that $p$ is unramified in $E$ , and let $K_{p}^{\circ}=\prod_{v|p}H_{v}\subset H_{p}$ be a product ot maximal hyperspecial subgroups. Then for open compact $K^{p}\subset\mathrm{H}(\mathbf{A}^{p\infty})$ , setting $K:=K^{p}K_{p}^{\circ}\subset\mathrm{H}(\mathbf{A}^{\infty})$ , the variety $X_{K}$ has good reduction at all $p$ -adic places. Define

\mathrm{Ch}^{r}(X_{K})^{\langle p\rangle}\subset\mathrm{Ch}^{r}(X_{K})^{0}

to be the $\mathbf{Q}$ -subspace of algebraic cycles whose class in $H^{2r}(X_{K,E_{w}},\mathbf{Q}_{p}(r))$ is trivial for every finite place $w\nmid p$ of $E$ . Li and Liu [LL] observed that the construction of Beĭlinson [Bei87] unconditionally defines a height pairing

\displaystyle\langle\,,\,\rangle^{\rm BB}\colon\mathrm{Ch}^{r}(X_{K})_{\mathbf{C}}^{\langle p\rangle}\otimes_{\mathbf{C}}\mathrm{Ch}^{r}(X_{K})_{\mathbf{C}}^{\langle p\rangle}\to\mathbf{C}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}

(5.4)

that is $\mathbf{C}$ -linear in the first factor and $\mathbf{C}$ -antilinear in the second factor. (It is conjectured that the pairing takes values in $\mathbf{C}\subset\mathbf{C}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}$ ; this turns out to be the case in the application to Theta cycles.)

In order to descend this pairing to Selmer groups, we need to assume a case of a standard conjecture on the injectivity of Abel–Jacobi maps. Whenever $K\subset\mathrm{H}(\mathbf{A}^{\infty})$ is an open compact subgroup that is understood from the context, denote $\mathfrak{m}_{\rho}=\mathfrak{m}_{\rho,K}$ , $\mathfrak{m}_{\rho^{*}(1)}=\mathfrak{m}_{\rho^{*}(1),K}$

\conjname \the\smf@thm.

For $\rho^{?}\in\{\rho,\rho^{*}(1)\}$ and for each open compact $K^{p}\subset\mathrm{H}(\mathbf{A}^{p\infty})$ , the Abel–Jacobi map

\displaystyle{\rm AJ}_{p,K^{p}K_{p}^{\circ}}\colon\left(\mathrm{Ch}^{r}(X_{K^{p}K_{p}^{\circ}})_{\overline{\mathbf{Q}}{}_{p}}^{\langle p\rangle}\right)_{\mathfrak{m}_{\rho^{?}}}\to H^{1}_{f}(E,M_{\rho^{?},K^{p}K_{p}^{\circ}})

(5.5)

is injective.

Assume that $\rho$ is crystalline at all $p$ -adic places. Fix a maximal hyperspecial subgroup $K_{p}^{\circ}\subset\mathrm{H}(\mathbf{A}^{p\infty})$ , and assume that Conjecture 5.2 holds. Denote by $H^{1}_{f}(E,M_{\rho^{?},K^{p}K_{p}^{\circ}})^{X}$ the image of (5.5), and let

H^{1}_{f}(E,\rho^{?})^{X_{K_{p}^{\circ}}}:=\sum_{\sigma^{\prime},K^{p}}\sum_{f^{\prime}\in(\sigma^{\prime})^{K^{p}K_{p}}}f^{\prime}_{*}H^{1}_{f}(E,M_{\rho^{?},K^{p}K_{p}^{\circ}})^{X}

where the first sum is as in (4.1) for $\rho^{?}$ . Then for every $\iota\colon\overline{\mathbf{Q}}{}_{p}\hookrightarrow\mathbf{C}$ and every $K=K^{p}K_{p}^{\circ}$ , we have a pairing

\displaystyle\langle\ ,\ \rangle^{\iota}_{K}\colon H^{1}_{f}(E,M_{\rho,K})^{X}\otimes_{\overline{\mathbf{Q}}{}_{p}}H^{1}_{f}(E,M_{\rho^{*}(1),K})^{X}\otimes_{\overline{\mathbf{Q}}{}_{p},\iota}\mathbf{C}\to\mathbf{C}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}

(5.6)

transported from (5.4) via the maps ${\rm AJ}_{p,K^{p}K_{p}^{\circ}}\otimes_{\iota}1$ . We may deduce from it a pairing

\displaystyle\langle\ ,\ \rangle^{\iota}\colon H^{1}_{f}(E,\rho)^{X_{K_{p}^{\circ}}}\otimes_{\overline{\mathbf{Q}}{}_{p}}H^{1}_{f}(E,\rho^{*}(1))^{X_{K_{p}^{\circ}}}\otimes_{\overline{\mathbf{Q}}{}_{p},\iota}\mathbf{C}\to\mathbf{C}\otimes_{\mathbf{Q}}\mathbf{Q}_{p}

(5.7)

defined as follows.

For $i=1,2$ let

c_{i}=f_{i,*}{\rm AJ}_{p,K}c^{\prime}_{i}

for some $K=K^{p}K_{p}^{\circ}$ , some $f_{i}\in\sigma_{i}^{K}$ , and some

c_{1}^{\prime}\in\left(\mathrm{Ch}^{r}(X_{K^{p}K_{p}^{\circ}})_{\overline{\mathbf{Q}}{}_{p}}^{\langle p\rangle}\right)_{\mathfrak{m}_{\rho}},\qquad c_{2}^{\prime}\in\left(\mathrm{Ch}^{r}(X_{K^{p}K_{p}^{\circ}})_{\overline{\mathbf{Q}}{}_{p}}^{\langle p\rangle}\right)_{\mathfrak{m}_{\rho^{*}(1)}}.

If $\sigma_{1}\not\cong\sigma_{2}^{\vee}$ , we put

\displaystyle\langle c_{1},c_{2}\rangle^{\iota}:=0.

(5.8)

If $\sigma_{1}\cong\sigma_{2}^{\vee}$ , we have the pairing $(\,,\ )_{\sigma_{1}}$ of (5.1) on $\sigma_{1}\otimes\sigma_{2}$ , through which we identify $\sigma_{2}=\sigma_{1}^{\vee}$ . Let

\mathrm{t}_{K}(f_{1}\otimes f_{2})\in\mathrm{Hom}\,(\sigma_{1}^{\vee,K},\sigma_{2}^{K})=\mathrm{End}\,(\sigma_{1}^{\vee,K})=\mathrm{End}\,_{\overline{\mathbf{Q}}{}_{p}[G_{E}]}(M_{\sigma_{1},K})

be given by

\mathrm{t}_{K}(f_{1}\otimes f_{2})(v_{1})=\mathrm{vol}(K)\cdot(v_{1},f_{1})_{\sigma_{1}}\cdot f_{2},

and let

\displaystyle\mathrm{t}(f_{1}\otimes f_{2})(v_{1})=\mathrm{vol}(K)\cdot\mathrm{t}_{K}(f_{1}\otimes f_{2});

(5.9)

the normalising volume factor makes ${\mathrm{t}}$ into a well-defined map $\sigma_{1}\otimes\sigma_{1}^{\vee}\to\mathrm{End}\,_{\overline{\mathbf{Q}}{}_{p}[G_{E}]}(M_{\sigma_{1}})$ . The existence of a Hecke correspondence acting as $\mathrm{t}(f_{1}\otimes f_{2})$ implies that the action of $\mathrm{t}(f_{1}\otimes f_{2})$ on Selmer groups preserves the subspace $H^{1}_{f}(E,M_{\sigma_{1},K})^{X_{K_{p}^{\circ}}}$ . Then we define

\displaystyle\langle c_{1},c_{2}\rangle^{\iota}:=\langle\mathrm{t}(f_{1}\otimes f_{2})c_{1}^{\prime},c_{2}^{\prime}\rangle^{\iota}_{K}.

(5.10)

The definition of (5.7) in the general case follows from (5.8), (5.10) by bilinearity.

\remaname \the\smf@thm.

In the $p$ -adic case, we also have $\Gamma_{F}\hat{\otimes}\overline{\mathbf{Q}}{}_{p}$ -valued Nekovář pairings $\langle\ ,\ \rangle_{K}$ analogous to (5.6) (whose construction takes as input the pairing on $M_{\rho,K}\otimes M_{\rho^{*}(1),K}$ deduced from Poincaré duality). The analogous formula to (5.10) holds true as a consequence of the definitions and the projection formula [DZ, Lemma A.2.5].

5.3 The height formulas

We may now state the main known results on Theta cycles. They parallel those of [GZ, PR, Kol] on Heegner points.

We will say that $\rho$ is mildly ramified if $\pi_{\rho}$ (and the extension $E/F$ ) satisfy the hypotheses of [DL, Assumption 1.6], except possibly for the ones about $p$ -adic places.

\theoname \the\smf@thm.

Suppose that $\rho$ is mildly ramified and that it is crystalline at all $p$ -adic places. Assume Hypotheses 4.2, 4.3.

Assume Conjecture 5.2 and that $p$ is unramified in $E$ . Then for every $\lambda\in\Lambda_{\rho}$ , $\lambda^{\prime}\in\Lambda_{\rho^{*}(1)}$ and for every $\iota\in\Sigma$ , we have

\langle\Theta_{\rho}(\lambda),\Theta_{\rho^{*}(1)}(\lambda^{\prime})\rangle^{\iota}={c_{\infty}L_{\iota}^{\prime}(\rho,0)\over b_{n}(\mathbf{1})}\cdot\zeta_{\iota}(\lambda,\lambda^{\prime})

in $\mathbf{C}$ .

Suppose that Assumption 5.1 holds and that $p>n$ . Then for every $\lambda\in\Lambda_{\rho}$ , $\lambda^{\prime}\in\Lambda_{\rho^{*}(1)}$ and for every $\alpha\colon\pi_{\rho}^{\vee,{\dagger}}\cong\pi_{\rho^{*}(1)}$ , we have

\langle\Theta_{\rho}(\lambda),\Theta_{\rho^{*}(1)}(\lambda^{\prime})\rangle=e_{p}(\rho,\mathbf{1})^{-1}\cdot\,\mathrm{d}L_{p,\alpha}(\rho)(\mathbf{1})\cdot\zeta_{\alpha}(\lambda,\lambda^{\prime}).

in $\Gamma_{F}\hat{\otimes}\overline{\mathbf{Q}}{}_{p}=T_{\mathbf{1}}^{*}\mathscr{X}_{F}$ .

Proof.

Write $\lambda=[(f,\varphi,\phi)]$ , $\lambda^{\prime}=[(f^{\prime},\varphi^{\prime},\phi^{\prime})]$ . Consider the $p$ -adic case. By the definitions and Remark 5.2, it is equivalent to prove

\displaystyle\langle\mathrm{t}(f\otimes f^{\prime\vee})\,\Theta(\varphi,\phi),\Theta(\varphi^{\prime},\phi^{\prime})\rangle=e_{p}(\rho,\mathbf{1})^{-1}\cdot\,\mathrm{d}L_{p,\alpha}(\rho)(\mathbf{1})\cdot\breve{\zeta}_{\alpha}(\mathrm{t}(f\otimes f^{\prime\vee})\vartheta(\varphi,\phi^{\prime});\vartheta(\phi,\phi^{\prime}))

(5.11)

where the $\Theta$ ’s are the arithmetic theta liftings for $\rho$ and $\rho^{*}(1)$ as in (4.4), and

\breve{\zeta}_{\alpha}=\otimes_{v\nmid\infty}\breve{\zeta}_{v}\circ(\ )^{{\dagger}}\circ j_{\alpha}

is defined analogously to (5.3) based on the pairings (3.4). Let $T\in\mathscr{H}(\mathrm{H}(\mathbf{A}^{\infty})$ be a Hecke operator acting as $\mathrm{vol}(K)^{-1}\mathrm{t}(f\otimes f^{\prime\vee})$ on $\sigma^{\vee}$ . Then (5.11) is equivalent to [DL, Theorem 1.11] for

(\varphi,T\phi;\varphi^{\prime},\phi^{\prime}).

(Note that our definitions of the arithmetic theta lifts $\Theta(-,-)$ differ from those of [DL] by a factor $\mathrm{vol}(K)$ ; in the height formula, one factor is accounted for by (5.9), and another by the normalisation of height pairings in loc. cit..)

The complex case is similarly reduced to [LL2, Theorem 1.8]; the fact that $\langle\ ,\ \rangle^{\iota}$ is well-defined on Theta cycles follows from the definitions and [LL, Proposition 6.10 (3)]. ∎

Part 2 of Theorem A is then an immediate consequence of Theorem 5.3. For a beautiful exposition of some key aspects of the proofs of the height formulas in [LL, LL2, DL], see [Chao].

The proof of Theorem 5.3 suggests that from the point of view of height formulas, Theta cycles offer no material advantage over previous constructions. This is not so from the point of view of Euler systems, as we explain next.

5.4 An Euler system

The main technique for bounding Selmer groups is that of Euler systems, originally introduced by Kolyvagin to study Heegner points [Kol, Koly]. Roughly speaking, an Euler system for a representation $\rho$ of $G_{E}$ is a collection of integral Selmer classes defined over certain abelian extensions of $\rho$ and satisfying certain compatibility relations; the (one) class defined over $E$ itself is called the base class of the Euler system.

In a forthcoming work, Jetchev–Nekovář–Skinner theorise a variant of this notion, that we shall call a JNS Euler system. It is adapted to conjugate-symplectic representations over CM fields, where the abelian extensions are ring class fields ramified at the primes of $E$ split over the totally real subfield $F$ (see [ACR, § 8] for a summary when $F=\mathbf{Q}$ ). Their main result is that if $\rho$ has ‘sufficiently large’ image, then the existence of a JNS Euler system with nontrivial base class $z$ implies that $z$ generates the Selmer group of $\rho$ .

The following is the main result of the forthcoming [D-euler]. Granted the results of Jetchev–Nekovář–Skinner, it implies part 3 of Theorem A.

\theoname \the\smf@thm.

Let $\rho\colon G_{E}\to\mathrm{GL}_{n}(\overline{\mathbf{Q}}{}_{p})$ be a representation satisfying the assumptions of § 4.1. Then for any $\lambda\in\Lambda_{\rho}$ , there exists a JNS Euler system based on $\Theta_{\rho}(\lambda)$ .

Multiplicity-one principles are remarkably useful to prove relations between special cycles and, in particular, compatibility relations in Selmer groups – as first observed in [YZZ] and [LSZ]. The proof of Theorem 5.4 is no exception: this is the main technical advantage of having constructed a cycle depending on one parameter only.

Theta cycles

Abstract

1 Introduction

Theorem A.

Acknowledgements

2 The conjecture of Beĭlinson–Bloch–Kato–Perrin-Riou

2.1 Chow and Selmer groups

\remaname \the\smf@thm.

2.2 The conjecture

\conjname \the\smf@thm (Beĭlinson, Bloch–Kato, Perrin-Riou [bei, BK, PRbook]).

\remaname \the\smf@thm.

3 Descent and theta correspondence

3.1 pp-adic automorphic representations

Unitary groups

\definame \the\smf@thm.

\definame \the\smf@thm.

3.2 Automorphic descent

\remaname \the\smf@thm.

Descent to a quasisplit unitary group

\propname \the\smf@thm.

Proof.

\remaname \the\smf@thm.

3.3 Theta correspondence

Local correspondence and duality

\propname \the\smf@thm.

Proof.

\remaname \the\smf@thm.

Global correspondence

\propname \the\smf@thm.

Proof.

\remaname \the\smf@thm.

4 Theta cycles

4.1 Assumptions on the Galois representation

Associated automorphic representations

4.2 Models of the representations

Weil representations

Siegel-hermitian modular forms and their qq-expansion

Shimura varieties and their cohomology

Hypothesis \the\smf@thm.

4.3 Construction

0. Special cycles in XX

1. Theta kernel

Hypothesis \the\smf@thm.

2. Arithmetic theta lifts

3. Theta cycles

\definame \the\smf@thm.

5 Relation to LL-functions and Selmer groups

5.1 Complex and pp-adic LL-functions

Assumption \the\smf@thm.

\propname \the\smf@thm.

Proof.

5.2 Pairings

Dual Theta cycles

Pairings

pp-adic height pairing

Complex height pairings

\conjname \the\smf@thm.

\remaname \the\smf@thm.

5.3 The height formulas

\theoname \the\smf@thm.

Proof.

5.4 An Euler system

\theoname \the\smf@thm.

References

3.1 $p$ -adic automorphic representations

Siegel-hermitian modular forms and their $q$ -expansion

0. Special cycles in $X$

5 Relation to $L$ -functions and Selmer groups

5.1 Complex and $p$ -adic $L$ -functions

$p$ -adic height pairing