# Algebraicity of $L$-values for $\GSp_{4}\times\GL_{2}$ and
$\GSp_{4}\times\GL_{2}\times\GL_{2}$

David Loeffler  D.L.: Mathematics Institute, University of Warwick, Coventry
CV4 7AL, United Kingdom<EMAIL_ADDRESS>and  Óscar Rivero  O.R.:
Mathematics Institute, University of Warwick, Coventry CV4 7AL, United Kingdom
<EMAIL_ADDRESS>
###### Abstract.

We prove algebraicity results for critical $L$-values attached to the group
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$, and for Gan–Gross–Prasad
periods which are conjecturally related to central $L$-values for
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}\times\operatorname{GL}_{2}$.
Our result for $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ gives a new
proof (by a very different method) of a recent result of Morimoto, and will be
used in a sequel paper to construct a new $p$-adic $L$-function for
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$. The results for
Gross–Prasad periods appear to be new. A key aspect is the computation of
certain archimedean zeta integrals, whose $p$-adic counterparts are also
studied in this note.

###### Key words and phrases:

algebraicity of $L$-values, Siegel Shimura varieties, zeta integrals

###### 2010 Mathematics Subject Classification:

11F46; 11F70.

Supported by ERC Consolidator grant “Shimura varieties and the BSD conjecture”
(D.L.) and Royal Society Newton International Fellowship NIF\R1\202208 (O.R.).
This material is based upon work supported by the National Science Foundation
under Grant No. DMS-1928930 while the authors were in residence at the
Mathematical Sciences Research Institute in Berkeley, California, during the
Spring 2023 semester.

###### Contents

  1. 1 Introduction
  2. 2 The groups $G$ and $H$ and their Shimura varieties
  3. 3 A pairing on coherent cohomology
  4. 4 Rationality results for Gan–Gross–Prasad periods
  5. 5 Novodvorsky’s zeta integral
  6. 6 Archimedean zeta integrals
  7. 7 Local zeta integrals at $p$

##  1\. Introduction

###  1.1. Critical $L$-values for
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$

Let $\pi\times\sigma$ be an automorphic representation of
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$. We can attach to $\pi$
and $\sigma$ a degree 8 $L$-function $L(\pi\times\sigma,s)$, associated to the
tensor product of the natural degree 4 (spin) and degree 2 (standard)
representations of the $L$-groups of $\operatorname{GSp}_{4}$ and
$\operatorname{GL}_{2}$.

If $\pi$ and $\sigma$ are algebraic, then this $L$-function is expected to
correspond to a motive, and in particular we can ask whether it has critical
values. More precisely, suppose that (the $L$-packet of) $\pi$ corresponds to
a holomorphic Siegel modular form of weight $(k_{1},k_{2})$, with
$k_{1}\geqslant k_{2}\geqslant 2$; and that $\sigma$ corresponds to a modular
form of weight $\ell\geqslant 1$. Then we expect there to exist motives
$M(\pi)$ (of motivic weight $k_{1}+k_{2}-3$) and $M(\sigma)$ (of motivic
weight $\ell-1$) such that

$L(\pi\times\sigma,s)=L\left(M(\pi)\otimes
M(\sigma),s+\tfrac{w}{2}\right),\qquad w=k_{1}+k_{2}+\ell-4.$

For $L(\pi\times\sigma,s)$ to be a critical value, we must have
$s=-\tfrac{w}{2}\bmod\mathbf{Z}$, and $s$ must satisfy various inequalities
depending on how the weights of $\pi$ and $\sigma$ interlace. For fixed
$(k_{1},k_{2})$, the allowed pairs $(s,\ell)$ form three disjoint polygonal
regions in the plane (all symmetric about the line $s=\tfrac{1}{2}$); for
compatibility with the conjectures of [LZ20b] on Gross–Prasad periods (see
below), we denote these by $(A)$, $(D)$ and $(F)$. The inequalities defining
these regions are given in Table 1.

The first main goal of this paper is to prove an algebraicity result for
weights in region $(D)$. As we shall shortly recall, many cases of this
algebraicity are already known by work of Morimoto, Böcherer–Heim and Saha.
However, our method is independent of these works and uses rather different
methods; it gives some cases of algebraicity which are not already known, and
(perhaps more importantly) will serve as the starting-point for results on
interpolation of region $(D)$ $L$-values in $p$-adic families, which will be
treated in a sequel paper.

Table 1. Critical regions for
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$

Region | Range of $\ell$ | Range of $s$
---|---|---
$(A)$ | $k_{1}+k_{2}-1$ | $\leqslant\ell\phantom{\leqslant}$ |  | $|2s-1|\leqslant\ell-(k_{1}+k_{2}-1)$
$(D)$ | $k_{1}-k_{2}+3$ | $\leqslant\ell\leqslant$ |  $k_{1}+k_{2}-3$ | $|2s-1|\leqslant\min\left(k_{1}+k_{2}-3-\ell,\ell-(k_{1}-k_{2}+3)\right)$
$(F)$ | $1$ | $\leqslant\ell\leqslant$ |  $k_{1}-k_{2}+1$ | $|2s-1|\leqslant k_{1}-k_{2}+1-\ell$

(In the “missing” cases $\ell=k_{1}\pm(k_{2}-2)$, the $L$-function has no
critical values.)

###  1.2. Gross–Prasad periods for
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}\times\operatorname{GL}_{2}$

The second problem we consider here is to study the _Gross–Prasad period_
attached to a cuspidal automorphic representation $\pi$ of
$\operatorname{GSp}_{4}$ and two cuspidal automorphic representations
$\sigma_{1},\sigma_{2}$ of $\operatorname{GL}_{2}$, which is defined by

$\mathcal{P}(\varphi,\psi_{1},\psi_{2})=\int_{[H]}\varphi(h)\psi_{1}(h_{1})\psi_{2}(h_{2})\mathrm{d}h,$

for forms $\varphi\in\pi$ and $\psi_{i}\in\sigma_{i}$. Here $H$ denotes the
group
$\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$,
and $[H]$ the adelic symmetric space
$\mathbf{R}^{\times}H(\mathbf{Q})\backslash H(\mathbf{A})$. (We may assume
$\chi_{\pi}\chi_{\sigma_{1}}\chi_{\sigma_{2}}=1$, since the integral is
trivially zero otherwise.) The Gross–Prasad period vanishes unless the central
characters satisfy the condition
$\varepsilon(\pi_{v}\times\sigma_{1,v}\times\sigma_{2,v})=+1$ for all places
$v$. If this condition holds, the Gross–Prasad conjecture for
$\operatorname{SO}_{4}\times\operatorname{SO}_{5}$ predicts that
$\mathcal{P}(-)$ is non-zero if and only if
$L(\pi\times\sigma_{1}\times\sigma_{2},\tfrac{1}{2})\neq 0$, and Ichino and
Ikeda [II10] have formulated a precise conjecture relating
$|\mathcal{P}(\varphi,\psi_{1},\psi_{2})|^{2}$ to the central $L$-value. (More
precisely, the original works of Gross–Prasad and Ichino–Ikeda apply when
$\chi_{\pi}=1$, so $\pi$ factors through $\operatorname{SO}_{5}$ and
$\sigma=\sigma_{1}\boxtimes\sigma_{2}$ through its subgroup
$\operatorname{SO}_{4}$; the more general formulation we use here is due to
Emory [Emo19].) We shall prove here results for the periods $\mathcal{P}(-)$,
which are not logically dependent on the conjectured relation to $L$-values,
but nonetheless the Gross–Prasad and Ichino–Ikeda conjectures are the key
motivation for studying these periods.

If we take $\pi$ to have weight $(k_{1},k_{2})$, as above, and
$\sigma_{1},\sigma_{2}$ to correspond to holomorphic cuspforms of weights
$c_{1},c_{2}$, then we have 9 different cases
$\\{a,a^{\prime},b,b^{\prime},c,d,d^{\prime},e,f\\}$ depending on the
inequalities satisfied by $(c_{1},c_{2})$ and $(k_{1},k_{2})$. See Figure 2 of
[LZ20b] for a diagram illustrating these. In cases $\\{b,b^{\prime},e\\}$ the
Gross–Prasad period is automatically zero. In the remaining six cases one
expects a rationality result for the Gross–Prasad period, whose formulation
will depend on which case we consider; this is implicit in Conjecture 4.1.2 of
[LZ20b].

The relation between these periods and the
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ problem above is via
Novodvorsky’s integral formula for the
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ $L$-function [Nov79].
Novodvorsky’s integral can be seen as a “degenerate case” of the Gross–Prasad
period, in which the cuspidal $\operatorname{GL}_{2}$-representation on the
first factor of $H$ is replaced by a space of Eisenstein series. This relation
allows the two rationality problems to be treated in parallel. If $s$ is a
critical value for the $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$
$L$-function, then the Eisenstein automorphic representation playing the role
of $\sigma_{1}$ has weight $1+|2s-1|$; and the cases $(A)$, $(D)$, $(F)$ of
the previous section correspond to requiring that the pair
$(c_{1},c_{2})=(1+|2s-1|,\ell)$ should satisfy the inequalities $(a),(d),(f)$
respectively.

###  1.3. Known results for
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$

Let $\pi$, $\sigma$ be as in Section 1.1. We assume that $\ell\neq
k_{1}\pm(k_{2}-2)$, so that critical values exist and we are in one of the
cases $(A)$, $(D)$, $(F)$. We may suppose without loss of generality that
$\pi$ is _tempered_ , since Arthur’s classification of the discrete spectrum
[Art04, GT18] shows that for non-tempered $\pi$, the $L$-function can be
expressed as a product of automorphic $L$-functions for
$\operatorname{GL}_{2}$ and
$\operatorname{GL}_{2}\times\operatorname{GL}_{2}$, and the algebraicity
properties of these $L$-values are well-understood.

Let $m$ denote the sum of the four smallest Hodge numbers of $M(\pi)\otimes
M(\sigma)$, so that

$m=\begin{cases}2k_{1}+2k_{2}-6&\text{case $(A)$},\\\
k_{1}+k_{2}+\ell-4&\text{case $(D)$},\\\ 2k_{2}+2\ell-6&\text{case
$(F)$}.\end{cases}$

######  Conjecture 1.1.

  1. (1)

There exists a period $\Omega(\pi\times\sigma)\in\mathbf{C}^{\times}$ with the
following property: for every $j\in\mathbf{Z}$ such that $s=-\tfrac{w}{2}+j$
is a critical value, we have

$\frac{L(\pi\times\sigma,-\tfrac{w}{2}+j)}{(-2\pi
i)^{4j-m}\Omega(\pi\times\sigma)}\in\overline{\mathbf{Q}}.$

  2. (2)

For $\chi$ a Dirichlet character we have
$\Omega(\pi\times\sigma\times\chi)=\Omega(\pi\times\sigma)$ mod
$\overline{\mathbf{Q}}^{\times}$.

Of course, one would also like to have a tolerably explicit form for the
period $\Omega(\pi\times\sigma)$. For cases (A), (D), Morimoto [Mor18,
Proposition 2.1] has shown that Deligne’s general algebraicity conjectures
[Del79] imply an explicit form for the periods $\Omega(\pi\times\sigma)$ in
terms of Petersson norms of holomorphic eigenforms.

####  1.3.1. Known results: case $(F)$

For case $F$, the conjecture is known in full: it is proved in [LPSZ19],
building on Harris’ study of “occult periods” for the degree 4 $L$-function of
$\operatorname{GSp}_{4}$ in [Har04]. In this case, the period
$\Omega(\pi\times\sigma)$ depends only on $\pi$ (not on $\sigma$), and is
defined using the $\overline{\mathbf{Q}}$-structure on $H^{2}$ of coherent
automorphic sheaves on a toroidal compactification of the Siegel modular
threefold.

####  1.3.2. Known results: case $(D)$

There are a number of papers in the literature establishing Conjecture 1.1 for
weights in region (D) under various additional hypotheses on the weights
and/or levels.

  * •

Böcherer–Heim [BH06] consider the case $k_{1}=k_{2}=2k_{G}$ and $\ell=2k_{h}$
for some integers $k_{G},k_{h}$, and $\pi$ and $\sigma$ are unramified at all
finite primes, generated by holomorphic eigenforms $G$ for
$\operatorname{Sp}_{4}(\mathbf{Z})$ and $h$ for
$\operatorname{SL}_{2}(\mathbf{Z})$ respectively. For such $\pi$, $\sigma$,
they prove part (1) of 1.1 for the explicit period

$\Omega(\pi\times\sigma)^{\mathrm{BH}}\coloneqq\langle G,G\rangle\cdot\langle
h,h\rangle,$

where $\langle-,-\rangle$ denotes the Petersson scalar product (and $G$ and
$h$ are normalised to have their Fourier coefficients in
$\overline{\mathbf{Q}}$). Their result assumes that the first Fourier–Jacobi
coefficient of $G$ is non-vanishing.

  * •

Saha [Sah10] proves an analogous result with slightly stronger assumptions on
the weight (he assumes that $k_{1}=k_{2}=\ell\geqslant 6$), but less
restrictive conditions on the level (allowing $\pi$ and $\sigma$ to be either
the Steinberg representation, or its unramified quadratic twist, at some
finite places). This refines earlier works of Furusawa, Pitale–Schmidt, and
Saha himself. As in Böcherer–Heim, Saha assumes a non-vanishing hypothesis for
a certain Fourier coefficient (depending on the levels of $\pi$ and $\sigma$);
and the period he uses is again $\langle G,G\rangle\cdot\langle h,h\rangle$.

  * •

Morimoto [Mor18] considers the more general case where $k_{1}$ need not be
equal to $k_{2}$ (so $\pi$ is generated by vector-valued, rather than scalar-
valued, holomorphic Siegel modular forms), and makes no assumption on the
levels at all. However, his result excludes certain critical values close to
the centre of the functional equation.

####  1.3.3. Case $(A)$

For case $(A)$, the relevant period $\Omega(\pi\times\sigma)$ should be
independent of $\pi$ and given by $\langle g,g\rangle^{2}$, where $g$ is the
normalised newform generating $\sigma$ as before. This has been proved, under
some auxiliary hypotheses on the weights and levels, by Furusawa–Morimoto
[FM16].

###  1.4. Our results

In this paper, we focus on case $(D)$, or more precisely on the following sub-
case:

######  Definition 1.2.

Say we are in case $(D^{-})$ if $(k_{1},k_{2},\ell,s)$ satisfy the
inequalities of case $(D)$ together with the additional condition
$\ell\leqslant k_{1}$.

For weights in this region we prove the following:

######  Theorem 1.3.

Assume the weights of $(\pi,\sigma)$ are in case $(D^{-})$, and the following
local hypothesis holds:

  * •

For each prime $\ell$ (if any) such that $\sigma_{\ell}$ is supercuspidal, the
central character of $\pi_{\ell}$ is a square in the group of characters of
$\mathbf{Q}_{\ell}^{\times}$.

Then 1.1 holds for $\pi\times\sigma$, with a period of the form
$\Omega(\pi\times\sigma)=\Omega^{(1)}(\pi)\cdot\langle h,h\rangle$, where
$\langle h,h\rangle$ denotes the Petersson norm of the normalised newform
generating $\sigma$, and $\Omega^{(1)}(\pi)$ is a period depending only on
$\pi$ (defined using $H^{1}$ of coherent sheaves on the compactified Siegel
threefold).

The local hypothesis is required in order to show that Novodvorsky’s zeta
integral computes the correct $L$-factors at the bad primes. It is, of course,
vacuous if $\pi$ has trivial central character. We hope that future work will
allow this assumption to be removed.

Our methods are very different from the works cited above. Where those works
use integral formulae involving holomorphic (or nearly-holomorphic) Siegel
cusp forms, and Eisenstein series restricted from a much larger group (namely
$U(3,3$), we instead work with a non-holomorphic cusp form for
$\operatorname{GSp}_{4}$ belonging to the unique _globally generic_
representation in the same $L$-packet as $\pi$, and an Eisenstein series
pushed forward from $\operatorname{GL}_{2}\times\operatorname{GL}_{2}$. In
particular, our results are unconditional, not depending on any non-vanishing
assumptions on Fourier coefficients; and they cover all critical values of the
relevant $L$-function (even the near-central ones excluded in [Sah10] and
[Mor18]).

######  Remark 1.3.

Comparing our results with those of Morimoto cited above, we can deduce a
consequence for the arithmetic of $\pi$ alone: if $\pi$ has weight
$(k_{1},k_{2})$ and $k_{2}$ is sufficiently large that Theorem 1 of _op.cit._
applies for some $\ell$ (it suffices to take $k_{2}>6$), then the $H^{1}$
Whittaker period $\Omega^{(1)}(\pi)$ is a
$\overline{\mathbf{Q}}^{\times}$-multiple of $\langle G,G\rangle$, for an
appropriately normalised Siegel eigenform $G$ generating $\pi$. This period
relation is not at all obvious a priori.

If $k_{2}>3$, then the set of $(s,\ell)$ satisfying $(D^{-})$ is a strict
subset of the $(s,\ell)$ satisfying $(D)$; and the methods used in this
article do not apply for the remaining cases. It seems likely that these
remaining cases will require a theory of Maass–Shimura-type differential
operators acting on $H^{1}$ of Siegel threefolds; this is beyond the scope of
the present work.

###  1.5. Connection with other works

In a sequel to this paper [LR23], we build on the algebraicity results
developed here in order to define a $p$-adic $L$-function interpolating
$L$-values along the “edge” of region $(D)$ (that is, with
$s=\frac{k_{1}+k_{2}-2-\ell}{2}$).

This computation requires the evaluation of a certain local zeta integral at
the $p$-adic place (which gives the Euler factor
$\mathcal{E}^{(D)}(\pi\times\sigma)$ appearing in the interpolation property
of our $p$-adic $L$-function). We carry out this local computation here,
rather than in the sequel paper, since its proof has much in common with the
local Archimedean computation needed to prove Theorem 1.3 (and little in
common with the $p$-adic interpolation computations which form the bulk of the
sequel paper).

### Acknowledgements

The authors would like to thank Sarah Zerbes for informative conversations
related to this work, and Kazuki Morimoto for patiently explaining his results
in [Mor18] to us.

##  2\. The groups $G$ and $H$ and their Shimura varieties

###  2.1. Groups and parabolics

We denote by $G$ the group scheme $\operatorname{GSp}_{4}$ (over
$\mathbf{Z}$), defined with respect to the anti-diagonal matrix
$J=\left(\begin{smallmatrix}&&&1\\\ &&1\\\ &-1\\\ -1\end{smallmatrix}\right)$;
and we let $\nu$ be the multiplier map $G\to\mathbf{G}_{m}$. We define
$H=\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$,
which we embed into $G$ via the embedding

$\iota\,:\,\left[\begin{pmatrix}a&b\\\
c&d\end{pmatrix},\begin{pmatrix}a^{\prime}&b^{\prime}\\\
c^{\prime}&d^{\prime}\end{pmatrix}\right]\mapsto\begin{pmatrix}a&&&b\\\
&a^{\prime}&b^{\prime}\\\ &c^{\prime}&d^{\prime}\\\ c&&&d\end{pmatrix}.$

We sometimes write $H_{i}$ for the $i$-th $\operatorname{GL}_{2}$ factor of
$H$. We write $T$ for the diagonal torus of $G$, which is contained in $H$ and
is a maximal torus in either $H$ or $G$.

We write $B_{G}$ for the upper-triangular Borel subgroup of $G$, and
$P_{\operatorname{Si}}$ and $P_{\operatorname{Kl}}$ for the standard Siegel
and Klingen parabolics containing $B$, so

$P_{\operatorname{Si}}=\left(\begin{smallmatrix}\star&\star&\star&\star\\\
\star&\star&\star&\star\\\ &&\star&\star\\\
&&\star&\star\end{smallmatrix}\right),\qquad
P_{\operatorname{Kl}}=\left(\begin{smallmatrix}\star&\star&\star&\star\\\
&\star&\star&\star\\\ &\star&\star&\star\\\ &&&\star\end{smallmatrix}\right).$

We write $B_{H}=\iota^{-1}(B_{G})=\iota^{-1}(P_{\operatorname{Si}})$ for the
upper-triangular Borel of $H$.

In this paper $P_{\operatorname{Si}}$ will be much more important than
$P_{\operatorname{Kl}}$ (in contrast to [LPSZ19]). We have a Levi
decomposition
$P_{\operatorname{Si}}=M_{\operatorname{Si}}N_{\operatorname{Si}}$, with
$M_{\operatorname{Si}}\cong\operatorname{GL}_{2}\times\operatorname{GL}_{1}$,
identified as a subgroup of $G$ via

$(A,u)\mapsto\begin{pmatrix}A\\\ &uA^{\prime}\end{pmatrix},\qquad
A^{\prime}\coloneqq\begin{pmatrix}&1\\\
1\end{pmatrix}\cdot{}^{t}\\!A^{-1}\cdot\begin{pmatrix}&1\\\ 1\end{pmatrix}.$

The intersection $B_{M}\coloneqq M\cap B_{G}$ is the standard Borel of $M$;
its Levi factor is $T$.

###  2.2. Flag varieties and Bruhat cells

We write $\operatorname{FL}_{G}$ for the Siegel flag variety $P\backslash G$,
with its natural right $G$-action. There are four orbits for the Borel $B_{G}$
acting on $\operatorname{FL}_{G}$, the _Bruhat cells_ , represented by a
subset of the Weyl group of $G$, the _Kostant representatives_ , which are the
smallest-length representatives of the quotient $W_{M}\backslash W_{G}$. We
denote these by $w_{0},\dots,w_{3}$; see [LZ21] for explicit matrices. Note
that the cell $C_{w_{i}}=P\backslash Pw_{i}B_{G}\subset\operatorname{FL}_{G}$
has dimension $\ell(w_{i})=i$.

Analogously, for the $H$-flag variety $\operatorname{FL}_{H}=B_{H}\backslash
H$, we have 4 Kostant representatives $w_{00}=\mathrm{id}$,
$w_{10}=\left(\begin{pmatrix}0&1\\\ -1&0\end{pmatrix},\mathrm{id}\right)$,
similarly $w_{01}$, $w_{11}$ (with the cell $C_{w_{ij}}$ having dimension
$i+j$). (This is the whole of the Weyl group of $H$, since the Levi subgroup
of $M_{H}=T$ is trivial.)

###  2.3. Representations

We retain the conventions about algebraic weights and roots of [LZ21]. In
particular, we identify characters of $T$ with triples of integers
$(r_{1},r_{2};c)$, with $r_{1}+r_{2}=c$ modulo 2 corresponding to
$\operatorname{diag}(st_{1},st_{2},st_{2}^{-1},st_{1}^{-1})\mapsto
t_{1}^{r_{1}}t_{2}^{r_{2}}s^{c}$. With our present choices of Borel subgroups,
a weight $(r_{1},r_{2};c)$ is dominant for $H$ if $r_{1},r_{2}\geqslant 0$,
dominant for $M_{G}$ if $r_{1}\geqslant r_{2}$, and dominant for $G$ if both
of these conditions hold. (We frequently omit the central character $c$ if it
is not important in the context.)

###  2.4. Models of Shimura varieties

Let $K$ be a neat open compact subgroup of
$\operatorname{GSp}_{4}(\mathbf{A}_{\operatorname{f}})$. Denote by
$Y_{G,\mathbf{Q}}$ the canonical model over $\mathbf{Q}$ of the level $K$
Shimura variety. It is a smooth quasiprojective threefold, whose complex
points are canonicallly identified with a double-coset space of
$G(\mathbf{A})$, as discussed in [LPSZ19, §2.3]. We write $Y_{H,\mathbf{Q}}$
for the canonical $\mathbf{Q}$-model of the Shimura variety for $H$ of level
$K_{H}=K\cap H(\mathbf{A}_{\operatorname{f}})$, which is a moduli space for
ordered pairs of elliptic curves with level structure. Further, there is a
morphism of algebraic varieties $\iota\,:\,Y_{H,\mathbf{Q}}\rightarrow
Y_{G,\mathbf{Q}}$.

After choosing a suitable combinatorial datum (a rational polyhedral cone
decomposition), we can define a smooth compactification $X_{G,\mathbf{Q}}$ of
$Y_{G,\mathbf{Q}}$. This depends on the choice of cone decomposition, but we
shall not indicate this in the notation, since the choice of cone
decomposition will remain fixed throughout. As usual, we denote by $D$ the
boundary divisor.

The cone decomposition for $G$ naturally determines a cone decomposition for
$H$ and hence a compactification $X_{H,\mathbf{Q}}$ of $Y_{H,\mathbf{Q}}$, and
the embedding $\iota$ extends to a finite morphism $X_{H,\mathbf{Q}}\to
X_{G,\mathbf{Q}}$ (which we also denote by $\iota$). One can in fact always
choose the cone decomposition in such a way that this map of toroidal
compactifications is a closed immersion [Lan19], although we do not need this
here.

######  Remark 2.0.

If $K_{H}$ is the product of subgroups $K_{H,1}\times K_{H,2}$, then
$Y_{H,\mathbf{Q}}$ is a product of two modular curves. Each of these has a
canonical compactification given by adjoining finitely many cusps; but
$X_{H,\mathbf{Q}}$ may not be the product of these compactified modular curves
(depending on the choice of the toroidal boundary data). In general
$X_{H,\mathbf{Q}}$ will be obtained from the product of compactified modular
curves by a finite sequence of blow-ups concentrated above points of the form
$(\text{cusp})\times(\text{cusp})$.

###  2.5. Coefficient sheaves

We adopt the conventions recalled in [LZ21, §2.5.1]. For our further use,
recall that the Weyl group acts on the group of characters $X^{*}(T)$ via
$(w\cdot\lambda)(t)=\lambda(w^{-1}tw)$. As discussed in loc. cit., we can
define explicitly $w_{G}^{\max}$, the longest element of the Weyl group, as
well as $\rho=(2,1;0)$, which is half the sum of the positive roots for $G$.

There is a functor from representations of $P_{G}$ to vector bundles on
$X_{G,\mathbf{Q}}$; and we let $\mathcal{V}_{\kappa}$, for $\kappa\in
X^{\bullet}(T)$ that is $M_{G}$-dominant, be the image of the irreducible
$M_{G}$-representation of highest weight $\kappa$. Given an integral weight
$\nu\in X^{\bullet}(T)$ such that $\nu+\rho$ is dominant, we define

$\kappa_{i}(\nu)=w_{i}(\nu+\rho)-\rho,\quad 0\leqslant i\leqslant 3,$

where as usual $\rho$ is half the sum of the positive roots. These are the
weights $\kappa$ such that representations of infinitesimal character
$\nu^{\vee}+\rho$ contribute to
$R\Gamma(S_{K}^{G,\operatorname{tor}},\mathcal{V}_{\kappa})$; if $\nu$ is
dominant (i.e. $r_{1}\geqslant r_{2}\geqslant 0$), they are the weights which
appear in the _dual BGG complex_ computing de Rham cohomology with
coefficients in the algebraic $G$-representation of highest weight $\nu$. See
[LZ21, §2.5.2] for explicit formulae.

###  2.6. Cohomology

According to the previous discussion, if $V$ is an algebraic representation of
$P_{S}$ over $\mathbf{Z}_{(p)}$, we have a vector bundle $\mathcal{V}$ on
$X_{G}$ defined by $\mathcal{V}:=V\times^{P_{S}}\mathcal{T}_{G}$, where
$\mathcal{T}_{G}$ is the canonical $P_{S}$-torsor over $X_{G}$.

The Zariski cohomology groups $H^{i}(X_{G},\mathcal{V})$ and
$H^{i}(X_{G},\mathcal{V}(-D))$ are independent, up to canonical isomorphism,
of the choice of cone decomposition $\Sigma$ for the compactification, and
have actions of prime-to-$p$ Hecke operators $[KgK]$, for $g\in
G(\mathbf{A}_{\operatorname{f}}^{p})$. The same is true for $H$ in place of
$G$, and hence there are morphisms of sheaves

(1)
$H^{i}(X_{G},\mathcal{V})\xrightarrow{\iota^{*}}H^{i}(X_{H},\mathcal{V}_{B_{H}})$

and also

(2)
$H^{i}(X_{H},\mathcal{V}_{B_{H}}\otimes\alpha_{G/H}^{-1})\xrightarrow{\iota_{*}}H^{i+1}(X_{G},\mathcal{V})$

for $0\leqslant i\leqslant 2$ and any $P_{S}$-representation $V$. Here,
$\alpha_{G/H}$ denotes the character $(1,1;0)$ of $B_{H}$, the Borel subgroup
of $H$. These maps will play a key role later for defining the appropriate
pairings involved in the main constructions of the note.

Let $\pi$ (resp. $\sigma$) be a cuspidal automorphic representation of $G$
(resp. $\operatorname{GL}_{2}$), and consider the arithmetic normalisation of
the finite part,
$\pi_{\operatorname{f}}^{\prime}:=\pi_{\operatorname{f}}\otimes\|\cdot\|^{-(r_{1}+r_{2})/2}$
(resp.
$\sigma_{\operatorname{f}}^{\prime}:=\sigma_{\operatorname{f}}\otimes\|\cdot\|^{-c_{1}/2}$.
Let $L_{1}$ stand for the irreducible $M_{S}$-representation with highest
weight $L_{1}\,:\,\lambda(r_{1}+3,1-r_{2})$. Similarly, $L_{2}$ is the
irreducible $M_{S}$-representation with highest weight
$L_{2}\,:\,\lambda(r_{2}+2,-r_{1})$. The following result is a consequence of
Arthur’s classification:

######  Theorem 2.1.

Let $i=1,2$. If $\pi$ is of general type or of Yoshida type, then
$\pi_{\operatorname{f}}^{\prime}$ appears with multiplicity one as a
Jordan–Hölder factor of the $G(\mathbf{A}_{\operatorname{f}})$-representations

$H^{3-i}(X_{G,\mathbf{Q}},\mathcal{V}_{\kappa_{i}(\nu)}(-D))\otimes\mathbf{C}\quad\text{
and }\quad
H^{3-i}(X_{G,\mathbf{Q}},\mathcal{V}_{\kappa_{i}(\nu)})\otimes\mathbf{C}.$

Moreover, it appears as a direct summand of both representations, and the map
between the two is an isomorphism on this summand. If $L$ is any irreducible
representation of $M_{S}$ which is not isomorphic to $V_{\kappa_{i}(\nu)}$,
then the localisations of $H^{3-i}(X_{G,\mathbf{Q}},[L](-D))$ and
$H^{3-i}(X_{G,\mathbf{Q}},[L])$ at the maximal ideal of the spherical Hecke
algebra associated to the $L$-packet of $\pi$ are zero for all $i$.∎

We write $H^{i}(\pi_{\operatorname{f}})$ for the
$\pi_{\operatorname{f}}^{\prime}$-isotypical component of
$H^{i}\left(X_{G,E},[L_{1}](-D)\right)$, for some number field $E$ over which
$\pi_{\operatorname{f}}^{\prime}$ is definable.

##  3\. A pairing on coherent cohomology

In this section we define a pairing between coherent cohomology groups which
we will later use to study $L$-values for region $(D^{-})$.

###  3.1. Automorphic forms as coherent cohomology classes

We fix a weight $\nu=(r_{1},r_{2};c)=(k_{1}-3,k_{2}-3;c)$, for integers
$k_{1}\geqslant k_{2}\geqslant 2$. Then there are two discrete-series
representations of $\operatorname{GSp}_{4}(\mathbf{R})$ of infinitesimal
character $\nu^{\vee}+\rho$: a holomorphic discrete series $\pi_{\infty}^{H}$,
corresponding classically to holomorphic Siegel modular forms of weight
$(k_{1},k_{2})$, which contributes to cohomology in degrees 0 and 3; and a
generic discrete series $\pi_{\infty}^{W}$, which contributes in degrees 1 and
2.

More canonically, we can write this as follows. Let
$K_{\infty}=\mathbf{R}^{\times}\cdot U_{2}(\mathbf{R})$ denote the maximal
compact-mod-centre subgroup of $G(\mathbf{R})_{+}$. The representation
$\pi_{\infty}^{W}$ has two direct summands as a
$G(\mathbf{R})_{+}$-representation,
$\pi_{\infty}^{W}=\pi_{\infty,1}\oplus\pi_{\infty,2}$, which have minimal
$K_{\infty}$-types $\tau_{1}=(r_{1}+3,-r_{2}-1)$ and
$\tau_{2}=(r_{2}+1,-r_{1}-3)$, respectively. Since the minimal
$K_{\infty}$-type in an irreducible discrete series has multiplicity 1, we
have $\dim\operatorname{Hom}_{K_{\infty}}(\tau_{i},\pi_{\infty})=1$ for
$i=1,2$. Then, for each automorphic representation $\pi$ whose Archimedean
component is $\pi_{\infty}^{W}$, we have a canonical isomorphism of
irreducible smooth $G(\mathbf{A}_{\operatorname{f}})$-representations

$\operatorname{Hom}_{K_{\infty}}(\tau_{i},\pi)\\{\frac{r_{1}+r_{2}}{2}\\}\cong
H^{3-i}(X_{G,\mathbf{Q}},\mathcal{V}_{\kappa_{i}}(-D))_{\mathbf{C}}[\pi_{\mathrm{f}}].$

######  Remark 3.0.

For weights $(k_{1},k_{2})$ sufficiently far from the walls of the Weyl
chamber (so there are no non-tempered representations contributing to the
cohomology), this is proved in [HK92]. It follows from the results of [Su19],
together with Arthur’s classification of discrete-series representations of
$\operatorname{GSp}_{4}$, that the result in fact applies for all weights.

Given $\xi\in
H^{3-i}(X_{G,\mathbf{Q}},\mathcal{V}_{\kappa_{i}}(-D))_{\mathbf{C}}[\pi_{\mathrm{f}}]$,
we denote by $F_{\xi}$ the corresponding homomorphism $\tau_{i}\to\pi$; we may
consider $F_{\xi}$ as a harmonic vector-valued cusp form on $G$, taking values
in the representation $\tau_{i}^{\vee}$.

We use similar notations for $\operatorname{GL}_{2}$: for any $k\geqslant 1$,
the space $H^{0}(X_{\operatorname{GL}_{2}},\mathcal{V}_{(-k)})$ is isomorphic
to weight $k$ modular forms; and
$H^{1}(X_{\operatorname{GL}_{2}},\mathcal{V}_{(k-2)})$ is isomorphic to the
space of anti-holomorphic modular forms of weight $-k$ (i.e. complex
conjugates of holomorphic forms of weight $k$). We write $\omega\to
F_{\omega}$ and $\eta\to F_{\eta}$ for these isomorphisms.

###  3.2. Whittaker periods

Suppose $\pi$ is a cuspidal automorphic representation with Archimedean
component $\pi_{\infty}^{W}$, and which is globally generic. Let $E$ be the
coefficient field of $\pi$.

If we choose $j\in\\{1,2\\}$ then there is a canonical basis of
$\operatorname{Hom}_{K_{\infty}}(\tau_{3-j},\mathcal{W}(\pi_{\infty}^{W}))$
computed by Moriyama (which we shall recall in more detail below). From this
we obtain two $E$-rational structures on
$\operatorname{Hom}_{K_{\infty}}(\tau_{j},\pi)$: one via the isomorphism to
coherent $H^{j}$, and one via tensoring Moriyama’s basis vector at $\infty$
with the canonical $E$-structure on the Whittaker model of
$\pi_{\operatorname{f}}$. Since $\pi_{\operatorname{f}}$ is irreducible, these
must differ by a constant in $\mathbf{C}^{\times}/E^{\times}$.

######  Notation.

We denote this scalar factor by
$\Omega^{(j)}_{\pi}\in\mathbf{C}^{\times}/E^{\times}$, the _$H^{j}$ Whittaker
period_ of $\pi$.

The period denoted by $\Omega^{W}_{\pi}$ in [LPSZ19] (appearing in rationality
results for case $(F)$) is the $H^{2}$ Whittaker period. In the present work,
it is the $H^{1}$ Whittaker period which appears instead. (It is far from
clear _a priori_ how these periods are related to each other.)

Similar considerations apply to holomorphic cuspidal $\operatorname{GL}_{2}$
representations $\sigma$. In this case we obtain a period for $H^{0}$ and a
period for $H^{1}$. We may choose our standard Whittaker functions at $\infty$
so that the Whittaker-rational classes are those whose $q$-expansions have
coefficients in $E$. By the $q$-expansion principle the $H^{0}$ Whittaker
period is just 1. On the other hand, since the Serre duality pairing on
coherent cohomology preserves the $E$-rational structures, and this duality
pairing corresponds to the Petersson product on automorphic forms, the $H^{1}$
Whittaker period must be given by the Petersson norm of the normalised newform
generating $\sigma$.

###  3.3. Pullback to $H$

Now let $(c_{1},c_{2})$ be a pair of integers, with
$c_{1}+c_{2}=k_{1}+k_{2}\bmod 2$, and satisfying the inequalities

$1\leqslant c_{1},\quad k_{1}-k_{2}+2\leqslant c_{2}-c_{1},\quad
c_{2}\leqslant k_{1}$

defining the region $(D^{-})$. We denote by $\lambda$ the weight
$(-c_{1},c_{2}-2)$ for $H$. We want to define a pairing

(3) $H^{1}\left(X_{G,\mathbf{Q}},\mathcal{V}^{G}_{\kappa_{2}}(-D)\right)\times
H^{1}\left(X_{H,\mathbf{Q}},\mathcal{V}^{H}_{\lambda}\right)\to\mathbf{Q}.$

Let us define $t=\frac{(c_{2}-c_{1})-(k_{1}-k_{2}+2)}{2}\geqslant 0$.

###  3.4. The $t=0$ case

Let us first suppose that we have $c_{2}-c_{1}=k_{1}-k_{2}+2$; this is the
situation studied in §2.5 of [HK92]. We have
$\kappa_{2}(\nu)=(k_{2}-4,-k_{1})$ and one computes that the pullback of
$\mathcal{V}_{\kappa_{2}(\nu)}$ to $H$ is given by the direct sum

$\bigoplus_{0\leqslant j\leqslant
k_{1}+k_{2}-4}\mathcal{V}^{H}_{(k_{2}-4-j,j-k_{1})}.$

Since $c_{2}\leqslant k_{1}$, this implies that one of the direct summands in
$\iota^{*}(\mathcal{V}_{\kappa_{2}(\nu)})$ is
$\mathcal{V}^{H}_{(c_{1}-2,-c_{2})}$; so we obtain a pullback map

$\displaystyle\iota^{*}:H^{1}(X_{G,\mathbf{Q}},\mathcal{V}^{G}_{\kappa_{2}(\nu)}(-D))$
$\displaystyle\to
H^{1}(X_{H,\mathbf{Q}},\mathcal{V}^{H}_{(c_{1}-2,-c_{2})}(-D))$
$\displaystyle=\left[H^{1}(X_{H,\mathbf{Q}},\mathcal{V}^{H}_{(-c_{1},c_{2}-2)})\right]^{\vee}\quad\text{(by
Serre duality)},$

defining a pairing between the groups (3), which can be explicitly expressed
as the integral

$\langle\xi,\omega\boxtimes\eta\rangle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbf{R}^{\times}H(\mathbf{Q})\backslash
H(\mathbf{A})}F_{\xi}(v_{c_{1},-c_{2}})(\iota(h))F_{\omega}(h_{1})F_{\eta}(h_{2})\,\mathrm{d}h,$

where $v_{c_{1},-c_{2}}$ is the weight $(c_{1},-c_{2})$ standard basis vector
of $\tau_{2}$. Note that the integrand is invariant under $K_{\infty}$. (It is
also invariant under a finite-index subgroup of $\mathbf{A}^{\times}$; in
practice we will be interested in the case when $\xi$, $\omega$, and $\eta$
have product 1, so we may instead take the integral over
$\mathbf{A}^{\times}H(\mathbf{Q})\backslash H(\mathbf{A})$.)

###  3.5. General $t\geqslant 0$

In general, let us write $c_{1}^{\prime}=c_{1}+2t=c_{2}-(k_{1}-k_{2}+2)$, so
that $(c_{1}^{\prime},c_{2})$ satisfies the assumptions of the previous
section. Then the _Maass–Shimura derivative_ $\delta^{t}$ sends the
holomorphic form $F_{\omega}$ to a non-holomorphic automorphic form
$F^{(t)}_{\omega}$ of $K_{\infty}$-type $-c_{1}^{\prime}$ (i.e. a nearly-
holomorphic form of weight $c_{1}^{\prime}$ in the sense of Shimura). So we
may form the more general integral

$\langle\xi,\delta^{t}(\omega)\boxtimes\eta\rangle=\frac{1}{(2\pi
i)^{2}}\int_{\mathbf{R}^{\times}H(\mathbf{Q})\backslash
H(\mathbf{A})}F_{\xi}(v_{c_{1}^{\prime},-c_{2}})(\iota(h))\delta^{t}F_{\omega}(h_{1})F_{\eta}(h_{2})\,\mathrm{d}h,$

which _a priori_ defines a pairing between the groups in (3) after extension
to $\mathbf{C}$; our goal is to show that it respects the rational structures.

As in many previous works (e.g. [Urb14, DR14]), we can interpret
$F^{(t)}_{\omega}$ as a section of a larger vector bundle
$\widetilde{\mathcal{V}}_{(-c1^{\prime})}\supseteq\mathcal{V}_{(-c_{1}^{\prime})}$,
corresponding to a reducible representation of $B_{\operatorname{GL}_{2}}$.
Arguing as in [LPSZ19], we can find a sheaf
$\widetilde{\mathcal{V}}^{G}_{\kappa_{2}}\twoheadrightarrow\mathcal{V}^{G}_{\kappa_{2}}$
(defined as a subquotient of the Hodge filtration on the sheaf attached to the
algebraic representation of weight $\nu$), and a pairing
$\iota^{*}\left(\widetilde{\mathcal{V}}^{G}_{\kappa_{2}}\right)\times\widetilde{\mathcal{V}}^{H}_{(-c1^{\prime},c_{2}-2)}\to\Omega^{1}_{X_{H}}$,
which is compatible with the obvious pairing
$\iota^{*}\left(\mathcal{V}^{G}_{\kappa_{2}}\right)\times\mathcal{V}^{H}_{(-c1^{\prime},c_{2}-2)}\to\Omega^{1}_{X_{H}}$.
Moreover, the map on $H^{1}(X_{G,\mathbf{Q}},-)$ induced by the quotient map
$\widetilde{\mathcal{V}}^{G}_{\kappa_{2}}\twoheadrightarrow\mathcal{V}^{G}_{\kappa_{2}}$
is an isomorphism on the $\pi_{\operatorname{f}}$-eigenspace. So we can
interpret $\langle\xi,\delta^{t}(\omega)\boxtimes\eta\rangle$ as a cup-product
in the cohomology of these larger sheaves; in particular, it respects the
$E$-structures on the cohomology groups, where $E$ is the rationality field of
$\pi_{\operatorname{f}}$.

######  Remark 3.0.

If $(c_{1},c_{2})$ lies in region $(D)$, but not in the subregion $(D^{-})$,
then the above construction does not work, because the $K^{H}_{\infty}$-type
$(c_{1}^{\prime},-c_{2})$ no longer appears in $\tau_{2}$. It seems possible
that this can be “repaired” by applying differential operators to $F_{\xi}$,
rather than to the two $\operatorname{GL}_{2}$ factors; we hope to investigate
this further in a future work.

##  4\. Rationality results for Gan–Gross–Prasad periods

Let $\pi$ be as in the previous section; and let $\sigma_{1},\sigma_{2}$ be
cuspidal automorphic representations of $\operatorname{GL}_{2}$, generated by
holomorphic cuspidal modular forms of weights $c_{1}$ and $c_{2}$
respectively, such that $(c_{1},c_{2})$ lies in region $(D^{-})$ of our
diagram. We suppose that $\omega$ and $\eta$ are in the $\sigma_{1}$, resp.
$\sigma_{2}$, isotypic part of the cohomology groups.

######  Theorem 4.1.

If all three classes $\xi,\omega,\eta$ are defined over some number field $E$
as coherent cohomology classes, then the period
$\langle\xi,\delta^{t}(\omega)\boxtimes\eta\rangle$ is in $E$.

Since the relation between rationality as coherent cohomology classes and
rationality in the Whittaker model is given by the periods defined above, we
can reformulate this as follows:

######  Corollary 4.2.

If $\xi$, $\omega$ and $\eta$ are defined over $E$ in the respective Whittaker
models, then we have

$\frac{\langle\xi,\delta^{t}(\omega)\boxtimes\eta\rangle}{\Omega^{(1)}_{\pi}\cdot\langle
h,h\rangle}\in E,$

where $h$ is the normalised holomorphic newform generating $\sigma_{2}$.

We briefly recall the relation between this period and central $L$-values. If
$\pi$ and $\sigma_{1}\otimes\sigma_{2}$ have trivial central characters (and
thus factor through $\operatorname{SO}_{5}$ and $\operatorname{SO}_{4}$
respectively), and the local root numbers
$\varepsilon_{v}(\pi_{v}\times\sigma_{1,v}\times\sigma_{2,v})$ are $+1$ for
all finite places, then the Ichino–Ikeda conjecture [II10] predicts a formula
for the absolute value of the global period. This has the form

$\frac{|\langle\xi,\delta^{t}(\omega)\boxtimes\eta\rangle|^{2}}{\|F_{\xi}\|^{2}\cdot\|F_{\delta^{t}(\omega)\boxtimes\eta}\|^{2}}=(\star)\cdot\frac{L(\pi\times\sigma_{1}\times\sigma_{2},\tfrac{1}{2})}{L(\operatorname{ad}\pi,1)L(\operatorname{ad}\sigma_{1},1)L(\operatorname{ad}\sigma_{2},1)}\prod_{v}c_{v},$

where $(*)$ is an explicit factor, and $c_{v}$ are local matrix coefficients
(which are nonzero, and equal to 1 for all but finitely many places). So if
the Ichino–Ikeda conjecture holds, then Theorem 4.1 determines
$L(\pi\times\sigma_{1}\times\sigma_{2},\tfrac{1}{2})$ up to an algebraic
factor (although making this explicit would involve computing the local matrix
coefficient $c_{v}$, which is a nontrivial task, particularly for $v=\infty$).

##  5\. Novodvorsky’s zeta integral

We now return to the case considered in the introduction, so $\pi\times\sigma$
is an automorphic representation of
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ and the weights
$(k_{1},k_{2},\ell)$ satisfy the inequalities $(D)$ of Table 1. We consider
Novodvorsky’s zeta integral

$Z(F_{0},\Phi_{1},F_{2};s)=\int_{Z_{H}(\mathbf{A})H(\mathbf{Q})\backslash
H(\mathbf{A})}F(h)E^{\Phi_{1}}(h_{1};\chi,s)F^{\prime}(h_{2})\mathrm{d}h,$

where $F$ and $F^{\prime}$ are automorphic forms in $\pi$ and $\sigma$
respectively, and $E^{\Phi_{1}}(h_{1};\chi,s)$ is an Eisenstein series,
depending on a choice of Schwartz function
$\Phi_{1}\in\mathcal{S}(\mathbf{A}^{2})$.

###  5.1. Expression via coherent cohomology

We first show how this zeta integral can be interpreted as a coherent cup
product of the type considered in the previous sections, but with $\omega$ an
Eisenstein, rather than cuspidal, form. We take $F_{0}=F_{\xi}$ and
$F_{1}=F_{\eta}$, for coherent cohomology classes $\xi$, $\eta$ contributing
to $H^{1}(X_{G})$ and $H^{1}(X_{\operatorname{GL}_{2}})$, as before (so we are
taking $c_{2}=\ell$). We suppose $\ell\leqslant k_{1}$ (so we are in case
$(D^{-})$).

As explained in [LPSZ19], for suitable choices of parameters
$E^{\Phi_{1}}(h_{1};\chi,s)$ is a nearly-holomorphic Eisenstein series.

We take
$\Phi_{1,\infty}(x,y)=2^{1-c_{1}^{\prime}}(x+iy)^{c_{1}^{\prime}}\exp(-\pi(x^{2}+y^{2}))$,
where $c_{1}^{\prime}=\ell-(k_{1}-k_{2}+2)\geqslant 1$; and let $\Phi$ be any
Schwartz function of the form
$\Phi_{1,\operatorname{f}}\times\Phi_{1,\infty}$, for this particular
$\Phi_{1,\infty}$ and any Schwartz function on
$\mathbf{A}_{\operatorname{f}}^{2}$. The condition for $s$ to be a critical
value is precisely that $s=\tfrac{c_{1}^{\prime}}{2}\bmod\mathbf{Z}$ and
$|2s-1|\leqslant c_{1}^{\prime}-1$; and for $s$ satisfying this,
$E^{\Phi_{1}}(-;\chi,s)$ is nearly-holomorphic (of weight $c_{1}^{\prime}$.
Moreover, if we let $c_{1}=1+|2s-1|$ and
$t=\tfrac{c_{1}^{\prime}-c_{1}}{2}\in\mathbf{Z}_{\geqslant 0}$, then there is
a holomorphic Eisenstein series of weight $c_{1}$ whose image under
$\delta^{t}$ is $E^{\Phi_{1}}(-;\chi,s)$. (For $\Phi_{1,\operatorname{f}}$ of
a certain specific form this is proved in [LLZ14, Corollary 5.2.1]; the
general proof is no different.)

######  Remark 5.0.

One can check that $\Phi_{1,\operatorname{f}}$ takes values in our fixed
number field $E$, then $E^{\Phi_{1}}(-;\chi_{1},s)$ is defined over that
number field (as a coherent cohomology class).

######  Corollary 5.1.

If $\Phi_{\operatorname{f}}$ is $E$-valued, and $\xi$, $\eta$ are defined over
$E$ as coherent cohomology classes, then $Z(F_{0},\Phi_{1},F_{2};s)$ lies in
$(2\pi i)^{2}E$.

###  5.2. Eulerian factorisation

As explained in [LPSZ19], if $\pi$ and $\sigma$ are generic, then we can write
this as an integral in terms of the Whittaker functions $W_{0}$ and $W_{2}$
associated to $F_{0}$ and $F_{2}$. If the data $W_{0},\Phi_{1},W_{2}$ are
factorisable as products of local data, then the global integral has a
corresponding factorisation as $\prod_{v}Z_{v}(W_{0,v},\Phi_{1,v},F_{2,v};s)$,
where

$Z_{v}(W_{0,v},\Phi_{1,v},F_{2,v};s)=\int_{(Z_{H}N_{H}\backslash
H)(\mathbf{Q}_{v})}W_{0,v}(h)f^{\Phi_{1,v}}(h_{1};\chi_{v},s)W_{2,v}(h_{2})\,\mathrm{d}h.$

For all but finitely many places, the local integral $Z_{v}$ is equal to the
$L$-factor $L(\pi_{v}\times\sigma_{v},s)$, so we can write

$Z(F_{0},\Phi_{1},F_{2};s)=L(\pi\times\sigma,s)\cdot
Z_{\infty}(W_{0,\infty},\Phi_{1,\infty},W_{2,\infty};s)\cdot\prod_{\ell\in
S}\frac{Z_{v}(W_{0,\ell},\Phi_{1,\ell},W_{2,\ell};s)}{L(\pi_{\ell}\times\sigma_{\ell},s)},$

where $S$ is a finite set of (finite) primes.

For $v=\ell$ a finite prime in $S$, the ratio
$\dfrac{Z_{\ell}(W_{0,\ell},\Phi_{1,\ell},W_{2,\ell};s)}{L(\pi_{\ell}\times\sigma_{\ell},s)}$
is a rational function in $\ell^{-s}$ and $\ell^{s}$. The local assumption on
$\pi\times\sigma$ in Theorem 1.3 implies that this function is actually a
polynomial, and the ideal generated by these polynomials (as
$W_{0,\ell},\Phi_{1,\ell},W_{2,\ell}$ vary) is the unit ideal. One can check
that if $s=\tfrac{w}{2}\bmod\mathbf{Z}$, and the data
$(W_{0,\ell},\Phi_{1,\ell},W_{2,\ell})$ are defined over $E$, then the zeta
integral is itself $E$-valued.

If we choose test data of this form at the local places, and equal to the
standard Moriyama test data at $\infty$, then we can conclude that

$Z(F_{0},\Phi_{1},F_{2};s)\in E^{\times}\cdot Z_{\infty}\cdot
L(\pi\times\sigma,s)$

where $Z_{\infty}$ is the local zeta integral for Moriyama’s standard test
data at $\infty$; but we also have

$Z(F_{0},\Phi_{1},F_{2};s)\in E^{\times}\cdot(2\pi i)^{2}\langle
h,h\rangle\Omega^{(1)}_{\pi}.$

Thus, in order to prove Theorem 1.3, it remains to compute the Archimedean
local integral $Z_{\infty}$.

##  6\. Archimedean zeta integrals

###  6.1. Zeta-integral generalities

For $\Pi$ a smooth representation of $\operatorname{GSp}_{4}(F)$, where $F$ is
a local field (archimedean or not), we have the two-parameter
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ zeta-integral

$Z(W,\Phi_{1},\Phi_{2};\chi_{1},\chi_{2},s_{1},s_{2}).$

If we write this in terms of Bessel models, it is

(4) $\int_{DN_{H}\backslash
H}B_{W}(h;s_{1}-s_{2}+\tfrac{1}{2})f^{\Phi_{1}}(h_{1};s_{1},\chi_{1})f^{\Phi_{2}}(h_{2};s_{2},\chi_{2})\mathrm{d}h.$

We have the Iwasawa decomposition $H=B_{H}K_{H}$, where $K_{H}$ is the maximal
compact. If the data are chosen so that the integrand is $K$-invariant, then
(using the fact that the $f^{\Phi}$’s live in principal-series
representations, so have a known transformation property under $B_{H}$) we
obtain

$Z(W,\Phi_{1},\Phi_{2},s_{1},s_{2})=f^{\Phi_{1}}(1;\chi_{1},s_{1})f^{\Phi_{2}}(1;\chi_{2},s_{2})\int_{F^{\times}}B_{W}(\left(\begin{smallmatrix}t\\\
&t\\\ &&1\\\
&&&1\end{smallmatrix}\right);s_{1}-s_{2}+\tfrac{1}{2})|t|^{(s_{1}+s_{2}-2)}\,\mathrm{d}^{\times}t.$

If $F=\mathbf{R}$, then we have an explicit formula for
$\int_{F^{\times}}B_{W}(\dots)(\dots)$ due to Moriyama, which we recall below.

We want to use this to study
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ zeta-integrals. For
$c\in\mathbf{Z}_{\geqslant 1}$ there is a (limit-of-)discrete-series
representation $D_{c}^{+}$ of $\operatorname{SL}_{2}(\mathbf{R})$,
corresponding to weight $c$ holomorphic modular forms, whose lowest $K$-type
is $\left(\begin{smallmatrix}\cos\theta&\sin\theta\\\
-\sin\theta&\cos\theta\end{smallmatrix}\right)\mapsto e^{ic\theta}$. The
Whittaker function of the lowest $K$-type vector is given along the torus by

$W^{(k)}(\left(\begin{smallmatrix}t&0\\\
0&1\end{smallmatrix}\right))=t^{c/2}e^{-2\pi t}$

(up to an arbitrary scalar, but if we want our Whittaker functions to match up
with the conventional notion of $q$-expansions then this is clearly the good
normalisation).

We can embed $D_{c}^{+}$ into a principal-series representation
$I(|\cdot|^{s-1/2},|\cdot|^{1/2-s}\chi^{-1})$, taking $\chi=(\text{sign})^{c}$
and $s=\tfrac{c}{2}$. Then the principal series is reducible, with
$D_{c}^{+}\oplus D_{c}^{-}$ as a subrepresentation and a finite-dimensional
representation as quotient.

If we consider the function
$\Phi^{(c)}(x,y)=2^{1-c}(x+iy)^{c}e^{-\pi(x^{2}+y^{2})}$, then
$f^{\Phi^{(c)}}(-;\operatorname{sign}^{c},s)$ has the same $K$-type, and it
lives in $I(|\cdot|^{s-1/2},|\cdot|^{1/2-s}\operatorname{sign}^{c})$.
Specializing at $s=\tfrac{c}{2}$,
$f^{\Phi^{(c)}}(-;\operatorname{sgn}^{c},\tfrac{c}{2})$ must land in the
$D_{c}^{+}$ subrepresentation, since its $K$-type does not appear in any of
the other factors. Hence, its image under the Whittaker transform,
$W^{\Phi^{(c)}}(-,\operatorname{sgn}^{c},c/2)$, must be a scalar multiple of
the above Whittaker function.

If we evaluate the Whittaker function
$W^{\Phi^{(c)}}(-,\operatorname{sgn}^{c},c/2)$ at the identity, we are led to
a rather nasty definite integral, which eventually turns out to be
$e^{-2\pi}$. This is the value at 1 of the normalised Whittaker function
above, so our normalisations are compatible (the archimedean analogue of the
compatibility noted in [LPSZ19, p4097]). That is, if we substitute
$\Phi_{2}=\Phi^{(c_{2})}$, $s_{2}=\tfrac{c_{2}}{2}$, and
$\chi_{2}=\operatorname{sgn}^{c_{2}}$ in Moriyama’s formulae, and let
$s=s_{1}$, we obtain a formula for the
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ zeta integral
$Z(W,\Phi_{1},W^{(c_{2})};s)$.

###  6.2. Choosing the parameters

Let $(r_{1},r_{2})$ be integers with $r_{1}\geqslant r_{2}\geqslant-1$. This
determines an $L$-packet of representations of
$\operatorname{GSp}_{4}(\mathbf{R})$, which are discrete-series if
$r_{2}\geqslant 0$ and limit-of-discrete-series if $r_{2}=-1$, as usual.

######  Notation.

In the notations of Moriyama’s paper [Mor04], let
$(\lambda_{1},\lambda_{2})=(r_{1}+3,-1-r_{2})$; and set
$d=\lambda_{1}-\lambda_{2}=r_{1}+r_{2}+4$.

Then we have the inequalities $1-\lambda_{1}\leqslant\lambda_{2}\leqslant 0$
that Moriyama requires (and in fact strict inequality holds). Attached to
$(\lambda_{1},\lambda_{2})$, Moriyama defines a pair of discrete / limit-of-
discrete series $\operatorname{Sp}_{4}(\mathbf{R})$-representations
$D_{(\lambda_{1},\lambda_{2})}$ and $D_{(-\lambda_{2},-\lambda_{1})}$, with
$D_{(\lambda_{1},\lambda_{2})}$ contributing to coherent $H^{1}$, and
$D_{(-\lambda_{2},-\lambda_{1})}$ to coherent $H^{2}$. Note that the Whittaker
functions of $D_{(-\lambda_{2},-\lambda_{1})}$ are supported on
$\operatorname{GSp}_{4}^{+}(\mathbf{R})$, while the Whittaker functions of the
dual representation are supported on the non-identity component.

We let $\Pi_{\infty}$ be the unique representation of
$\operatorname{GSp}_{4}(\mathbf{R})$ whose restriction to
$\operatorname{Sp}_{4}(\mathbf{R})$ is $D_{\lambda_{1},\lambda_{2}}\oplus
D_{-\lambda_{2},-\lambda_{1}}$, and whose central character is trivial on
$\mathbf{R}_{>0}$.

######  Remark 6.0.

Our parametrisation of the $K$-types follows [HK92], and unfortunately the
conventions of Harris–Kudla and Moriyama are not the same; so in our
notations, $D_{(\lambda_{1},\lambda_{2})}$ has minimal $K$-type
$(-\lambda_{2},-\lambda_{1})$ and $D_{(-\lambda_{2},-\lambda_{1})}$ has
highest $K$-type $(\lambda_{1},\lambda_{2})$ (sic!).

###  6.3. Moriyama’s result

Moriyama defines explicit Whittaker functions $W_{k}$, for $0\leqslant
k\leqslant\lambda_{1}-\lambda_{2}$, giving a basis of the minimal $K$-type of
$D_{(-\lambda_{2},-\lambda_{1})}$. In Proposition 8 of _op.cit._ he states a
formula for the integral

$Z(s,y_{1};W)=\int_{y\in\mathbf{R}^{\times}}\int_{x\in\mathbf{R}}W\left(\begin{pmatrix}uy\\\
&y\\\ &x&1&\\\
&&&u^{-1}\end{pmatrix}\right)|y|^{s-3/2}\,\mathrm{d}x\,\mathrm{d}^{\times}y,$

for $u>0$ an auxiliary parameter. This involves a Mellin inversion integral,
and we get rid of this by taking the forward Mellin integral to get a formula
for the Mellin transform of the Bessel function along the torus. (NB: Moriyama
uses the other model of $\operatorname{GSp}_{4}$, with the last two rows &
last two columns of the matrix switched.) Unravelling this, we get the
following formula: for $0\leqslant k\leqslant d$, if $W_{k}$ is the vector of
$K_{H}^{\circ}$-type $(-r_{1}-3+k,r_{2}+1-k)$, then111Note there is a typo on
p. 4108 of [LPSZ19], we erroneously flipped the two components of the
$K_{H}^{\circ}$-type. Accordingly, the value of $k$ given there is wrong and
should be replaced with $d-k$.

$\int_{\mathbf{R}^{\times}}B_{W_{k}}(\left(\begin{smallmatrix}u\\\ &u\\\
&&1\\\
&&&1\end{smallmatrix}\right);s_{1}-s_{2}+\tfrac{1}{2})|u|^{s_{1}+s_{2}-2}\,\mathrm{d}u=C\cdot\frac{(-1)^{k}L(\Pi_{\infty},s_{1}-s_{2}+\tfrac{1}{2})L(\Pi_{\infty},s_{1}+s_{2}-\tfrac{1}{2})}{\pi^{s_{1}+s_{2}-\tfrac{1}{2}}\Gamma(s_{1}+\tfrac{r_{1}+3-k}{2})\Gamma(s_{2}+\tfrac{-1-r_{2}+k}{2})},$

where $C$ is some constant (depending on $(r_{1},r_{2})$ but not on any of the
other data). This is [LPSZ19, Theorem 8.21].

###  6.4. Region $(F)$ case revisited

For region $(F)$, we apply this to compute
$Z(W_{k},\Phi^{(c_{1})},W^{(c_{2})};s)$ for integers $(c_{1},c_{2})$ with
$c_{i}\geqslant 1$ and $c_{1}+c_{2}=r_{1}-r_{2}+2$. We take
$k=r_{1}+3-c_{1}=r_{2}+1-c_{2}$. Then the $K$-type of $W_{k}$ is
$(-c_{1},-c_{2})$, meaning it can pair nontrivially with a pair of holomorphic
forms of weights $c_{1}$ and $c_{2}$.

Then we get

$C\cdot\frac{(-1)^{k}L(\Pi_{\infty},s_{1}-s_{2}+\tfrac{1}{2})L(\Pi_{\infty},s_{1}+s_{2}-\tfrac{1}{2})}{\pi^{s_{1}+s_{2}-\tfrac{1}{2}}\Gamma(s_{1}+\tfrac{c_{1}}{2})\Gamma(s_{2}+\tfrac{c_{2}}{2})}\cdot
f^{\Phi^{(c_{1})}}(1,s_{1})f^{\Phi^{(c_{2})}}(1,s_{2})\Bigg{|}_{(s_{1},s_{2})=(s,\tfrac{c_{2}}{2})}.$

An explicit computation gives

$f^{\Phi^{(c)}}(1,s)=2^{1-c}i^{c}\pi^{-(s+c/2)}\Gamma(s+\tfrac{c}{2}),$

so up to factors which don’t depend on the $c_{i}$ and hence can be absorbed
into $C$.

Moreover, for region $(F)$ we have

$L(\Pi_{\infty},s_{1}-s_{2}+\tfrac{1}{2})L(\Pi_{\infty},s_{1}+s_{2}-\tfrac{1}{2})=L(\Pi_{\infty}\times\Sigma_{\infty},s).$

So we get $(\text{const})\cdot(-1)^{c_{2}}\cdot
L(\Pi_{\infty}\times\Sigma_{\infty},s)$, which is (by definition of critical
values) non-zero in the critical range.

###  6.5. Region $(D)$

Now we are going to take $c_{1},c_{2}$ with $c_{1}\geqslant 1$ and
$c_{2}-c_{1}=r_{1}-r_{2}+2$; and we choose

$k=r_{1}+3+c_{1}=c_{2}+r_{2}+1.$

The constraint $k\leqslant d=r_{1}+r_{2}+4$ corresponds to $c_{2}\leqslant
r_{1}+3$, which is the inequality required for the “bottom half” of region
$(D)$. Our test data will be

$Z(W_{k},\left(\begin{smallmatrix}-1&\\\
&1\end{smallmatrix}\right)\Phi^{(c_{1})},W^{(c_{2})}),$

and since acting by $\left(\begin{smallmatrix}-1&\\\
&1\end{smallmatrix}\right)$ does not change the values of $f^{\Phi}$ along the
torus, we can write this as

$C\cdot\frac{(-1)^{k}L(\Pi_{\infty},s_{1}-s_{2}+\tfrac{1}{2})L(\Pi_{\infty},s_{1}+s_{2}-\tfrac{1}{2})}{\pi^{s_{1}+s_{2}-\tfrac{1}{2}}\Gamma(s_{1}-\tfrac{c_{1}}{2})\Gamma(s_{2}+\tfrac{c_{2}}{2})}\cdot
f^{\Phi^{(c_{1})}}(1,s_{1})f^{\Phi^{(c_{2})}}(1,s_{2})\Bigg{|}_{(s_{1},s_{2})=(s,\tfrac{c_{2}}{2})}.$

Note the change from $s_{1}+\tfrac{c_{1}}{2}$ to $s_{1}-\tfrac{c_{1}}{2}$ in
the denominator.

On the other hand, in this case the numerator is not
$L(\Pi_{\infty}\times\Sigma_{\infty},s)$ any more; one computes that

$L(\Pi_{\infty},s+c_{2}-\tfrac{1}{2})L(\Pi_{\infty},s-c_{2}+\tfrac{1}{2})=L(\Pi_{\infty}\times\Sigma_{\infty},s)\cdot(2\pi)^{c_{1}}\frac{\Gamma(s-\tfrac{c_{1}}{2})}{\Gamma(s+\tfrac{c_{1}}{2})}.$

So the zeta-integral computes to

$\text{(const)}\cdot(-2\pi)^{c_{2}}\cdot
L(\Pi_{\infty}\times\Sigma_{\infty},s),$

for some constant depending only on $(r_{1},r_{2})$; and we may choose our
normalisation of the Archimedean Whittaker function so that this constant is
1.

##  7\. Local zeta integrals at $p$

###  7.1. Nonarchimedean $L$-factors

We can extend the previous computations to non-archimedean primes. For that
purpose, let $F$ be a non-archimedean field, and write $L(\pi\times\sigma,s)$
for the local $L$-factor associated to $\pi\otimes\sigma$ via Shahidi’s
method, as in [GT05, §4]. We recall the following result from [LPSZ19, Thm.
8.9].

######  Proposition 7.1.

The vector space of functions on $\mathbf{C}$ spanned by the
$Z(W,\Phi_{1},\Phi_{2};\chi_{1},\chi_{2},s_{1},s_{2})$, as the data
$(W,\Phi_{1},\Phi_{2})$ vary, is a fractional ideal of $\mathbf{C}[q^{\pm s}]$
containing the constant functions. If at least one of $\pi$ and $\sigma$ is
unramified, this fractional ideal is generated by the $L$-factor
$L(\pi\otimes\sigma,s)$. Further, there exists a canonical normalization for
which the zeta integral is precisely $L(\pi\otimes\sigma,s)$.

The same considerations presented in [LPSZ19, §8.3.1] regarding the
rationality of the Whittaker models also apply.

###  7.2. A special $H$-orbit on $\operatorname{FL}_{G}$

From now on, let $p$ be a fixed prime. In this section, we discuss how to
compute the local integrals at the prime $p$ for our choice of local
conditions, and how this recovers the expected Euler factor.

######  Lemma 7.2.

Let $\tau=\left(\begin{smallmatrix}1\\\ 1&1\\\ &&1\\\
&&-1&1\end{smallmatrix}\right)\in M_{\operatorname{Si}}$, and let
$\hat{\tau}=\tau w_{1}$. Then $H\hat{\tau}P$ is open in $G$, and
$H\cap\hat{\tau}P\hat{\tau}^{-1}$ is a copy of $\operatorname{GL}_{2}$,
embedded in $H$ via

$\left(\begin{pmatrix}a&b\\\ c&d\end{pmatrix},\begin{pmatrix}a&-b\\\
-c&d\end{pmatrix}\right),$

and in $G$ via

$\left(\begin{pmatrix}a&b\\\ c&d\end{pmatrix},\begin{pmatrix}a&-b\\\
-c&d\end{pmatrix}\right)\mapsto\begin{pmatrix}a&b&&b\\\ c&d&c\\\ &&a&-b\\\
&&-c&d\end{pmatrix}.$

###### Proof.

Elementary computation. ∎

######  Lemma 7.3.

Suppose $\bar{n}\in\overline{N}(\mathbf{Z}_{p})$ is congruent to 1 modulo
$p^{k}$, for $k\geqslant 1$. Then we may write
$\hat{\tau}\bar{n}=h\hat{\tau}p$ for some $h\in H(\mathbf{Z}_{p})$ and $p\in
P(\mathbf{Z}_{p})$, with both $h$ and $p$ congruent to 1 modulo $p^{k}$.

###### Proof.

This follows from the matrix identity

$h(x,y,z)\hat{\tau}\left(\begin{smallmatrix}1\\\ &1\\\ x&y&1\\\
z&x&&1\end{smallmatrix}\right)=\hat{\tau}p(x,y,z),$

where

$h(x,y,z)=\left(\begin{pmatrix}(x+1)-\tfrac{yz}{x+1}&\tfrac{y}{x+1}\\\
-z&1\end{pmatrix}\begin{pmatrix}1&\\\ &x+1\end{pmatrix}\right),\qquad
p(x,y,z)=\left(\begin{smallmatrix}x+1&y&0&y/(x+1)\\\ 0&x+1&0&0\\\
&&1&-y/(x+1)\\\ &&&1\end{smallmatrix}\right).\qed$

###  7.3. Siegel Jacquet modules

Now let $\pi$ be an irreducible smooth generic representation of
$\operatorname{GSp}_{4}(\mathbf{Q}_{p})$. We suppose $\pi$ is is the
normalised induction of a representation $\rho\times\lambda$ of
$M(\mathbf{Q}_{p})$. Note that we have

$L(\pi,s)=L(\lambda,s)L(\rho\otimes\lambda,s)L(\omega_{\rho}\lambda,s),$

and the modulus character $\delta_{P}$ is $(A,c)\mapsto|\det(A)/c|^{3}$.

######  Lemma 7.4.

The normalised Jacquet module with respect to $\overline{P}$,
$J_{\overline{P}}(\pi)=\pi_{\overline{N}}\otimes\delta_{P}^{1/2}$, contains a
unique subrepresentation isomorphic to $\rho\times\lambda$.

###### Proof.

Bernstein’s second adjointness theorem shows that
$\operatorname{Hom}_{M}(\rho\times\lambda,J_{\overline{P}}(\pi))=\operatorname{Hom}_{G}(\pi,\pi)\cong\mathbf{C}$.
∎

Thus the unnormalised Jacquet module
$\pi_{\overline{N}}=\pi/\pi(\overline{N})$ contains a canonical
$M$-subrepresentation isomorphic to
$(\rho\times\lambda)\otimes\delta_{P}^{-1/2}$. We write $\pi[\lambda]$ for the
preimage in $\pi$ of this subrepresentation of $\pi_{\overline{N}}$. For any
$v\in\pi[\lambda]$, we have
$\operatorname{diag}(1,1,x,x)v=|x|^{3/2}\lambda(x)v\bmod\pi(\overline{N})$.

######  Note 7.4.

If $\rho$ and $\lambda$ are unramified, then a vector invariant under the
depth $t$ Siegel parahoric $K_{G,\operatorname{Si}}(p^{t})$ has this property
if and only if it lies in the $U_{1}^{\prime}=\alpha$ eigenspace (modulo the
zero generalised eigenspace), where $\alpha=p^{3/2}\lambda(p)$ and
$U_{1}^{\prime}$ is the Hecke operator given by the double coset of
$\operatorname{diag}(1,1,p,p)$.

###  7.4. Trilinear forms and Siegel Jacquet modules

Let $\pi$ be as above, and let $\sigma_{1},\sigma_{2}$ be irreducible
$G$-representations with
$\omega_{\pi}\omega_{\sigma_{1}}\omega_{\sigma_{2}}=1$. One knows that
$\operatorname{Hom}_{H(\mathbf{Q}_{p})}(\pi\times\sigma_{1}\times\sigma_{2},\mathbf{C})$
has dimension $\leqslant 1$. We suppose it is nonzero, and choose a basis
vector $\mathfrak{z}$.

######  Remark 7.4.

If one or more of the $\sigma_{i}$ is principal-series, we can construct
$\mathfrak{z}$ using Novodvorsky’s
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ zeta integral.

######  Proposition 7.5.

For $x\in\pi$, $y_{i}\in\sigma_{i}$, we consider the sequence of elements
$(z_{k}(x,y_{1},y_{2}))_{k\geqslant 0}$ defined by

$z_{k}(x,y_{1},y_{2})=\left(\frac{\lambda(p)}{p^{3/2}}\right)^{-k}\mathfrak{z}\Big{(}\hat{\tau}s_{k}x,y_{1},\left(\begin{smallmatrix}-1&\\\
&1\end{smallmatrix}\right)y_{2}\Big{)},\qquad
s_{k}=\operatorname{diag}(1,1,p^{k},p^{k}).$

If $x\in\pi[\lambda]$, then $(z_{k})$ is eventually constant, and its limiting
value depends only on the image of $x$ in $\pi[\lambda]/\pi_{\overline{N}}$.

###### Proof.

We first show that if $n\in\overline{N}(\mathbf{Q}_{p})$, then we have

$z_{k}(\overline{n}x,y_{1},y_{2})=z_{k}(x,y_{1},y_{2})\qquad\forall k\gg 0$

(for any fixed $x,y_{1},y_{2}$). Since the elements
$s_{k}\overline{n}s_{k}^{-1}$ approach the identity as $k\to\infty$, for all
$k\gg 0$ we can write

$\hat{\tau}s_{k}\overline{n}s_{k}^{-1}=h_{k}\hat{\tau}p_{k},\qquad h_{k}\in
H(\mathbf{Z}_{p}),p_{k}\in P(\mathbf{Z}_{p}),$

with $h_{k}$ and $p_{k}$ tending to 1 as $k\to\infty$. For all sufficiently
large $k$, $h_{k}$ will fix $y_{1}\otimes\left(\begin{smallmatrix}-1&\\\
&1\end{smallmatrix}\right)y_{2}$, so we have

$z_{k}(\overline{n}x,y_{1},y_{2})=\left(\frac{\lambda(p)}{p^{3/2}}\right)^{-k}\mathfrak{z}\Big{(}\hat{\tau}s_{k}\gamma_{k}x,y_{1},\left(\begin{smallmatrix}-1&\\\
&1\end{smallmatrix}\right)y_{2}\Big{)},\qquad\gamma_{k}=s_{k}^{-1}p_{k}s_{k}.$

Since conjugation by $s_{k}$ acts trivially on $M(\mathbf{Q}_{p})$, and
shrinks $N(\mathbf{Q}_{p})$, the fact that $p_{k}\to 1$ certainly implies
$\gamma_{k}\to 1$; so $\gamma_{k}x=x$ for sufficiently large $k$. This proves
the claim.

Since
$z_{k+1}(x,y_{1},y_{2})=\frac{p^{3/2}}{\lambda(p)}z_{k}(s_{1}x,y_{1},y_{2})$
by definition, and for $x\in\pi[\lambda]$ we have
$s_{1}x=\frac{\lambda(p)}{p^{3/2}}x\bmod\pi_{\overline{N}}$, it follows that
$z_{k}$ is eventually constant for $x\in\pi[\lambda]$. ∎

######  Definition 7.6.

For
$\mathfrak{z}\in\operatorname{Hom}(\pi\times\sigma_{1}\times\sigma_{2},\mathbf{C})$,
we write $\partial_{\operatorname{Si}}(\mathfrak{z})$ for the trilinear form
on $\pi[\lambda]\times\sigma_{1}\times\sigma_{2}$ mapping $(x,y_{1},y_{2})$ to
$\lim_{k\to\infty}z_{k}(x,y_{1},y_{2})$.

One checks that for $A=\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{GL}_{2}$, one has

$\partial_{\operatorname{Si}}(\mathfrak{z})\left(\begin{pmatrix}A&\\\
&\det(A)A^{\prime}\end{pmatrix}x,Ay_{1},Ay_{2}\right)=\partial_{\operatorname{Si}}(x,y_{1},y_{2}).$

So we have defined a map

(5)
$\partial_{\operatorname{Si}}:\operatorname{Hom}_{H}(\pi\times\sigma_{1}\times\sigma_{2},\mathbf{C})\to\operatorname{Hom}_{\operatorname{GL}_{2}}(\sigma_{0}\times\sigma_{1}\times\sigma_{2},\mathbf{C}),$

where $\sigma_{0}=\rho\otimes\lambda$ is the restriction of
$(\rho\times\lambda)\otimes\delta_{P}^{-1/2}$ to $\operatorname{GL}_{2}$,
embedded in $M$ via $A\mapsto(A,\det(A))$. Note that both source and target of
this map have dimension $\leqslant 1$.

######  Proposition 7.7.

Suppose $x,y_{1},y_{2}$ are invariant under the principal congruence subgroup
modulo $p^{t}$ for some $t\geqslant 1$, and we have $U^{\prime}_{1}x=\alpha
x$, where $\alpha=p^{3/2}\lambda(p)$. Then $x\in\pi[\lambda]$, and we have
$\mathfrak{z}_{k}(x,y_{1},y_{2})=\mathfrak{z}_{0}(x,y_{1},y_{2})$ for all
$k\geqslant 0$.

###### Proof.

The relation $U^{\prime}_{1}x=\alpha x$ translates into $p^{3}s_{1}x=\alpha
x\bmod\pi(\overline{N})$, since $U_{1}^{\prime}$ is the composite of $s_{1}$
and a sum over $p^{3}$ coset representatives all lying in $\overline{N}$. Thus
$x\in\pi[\lambda]$.

A similar argument shows that
$\alpha^{k}\mathfrak{z}_{0}(x,y_{1},y_{2})=\mathfrak{z}_{0}((U^{\prime}_{1})^{k}x,y_{1},y_{2})=p^{3k}\mathfrak{z}_{0}(s_{k}x,y_{1},y_{2})$,
so $z_{k}(x,y_{1},y_{2})=z_{0}(x,y_{1},y_{2})$. ∎

###  7.5. Relating the zeta-integrals

We now identify $\pi$, $\sigma_{1}$, and $\sigma_{2}$ with their Whittaker
models. More precisely, as in [LPSZ19] we take Whittaker models for $\pi$ with
respect to some additive character $\Psi$, and for $\sigma_{1}$ and
$\sigma_{2}$ with respect to $\Psi^{-1}$.

If $\pi$, the $\sigma_{i}$, and $\Psi$ are all unramified, then there is a
canonical spherical vector in the Whittaker model of each, and a unique
$H$-invariant trilinear form $\mathfrak{z}^{\operatorname{sph}}$ satisfying
$\mathfrak{z}^{\operatorname{sph}}(W_{0}^{\operatorname{sph}},W_{1}^{\operatorname{sph}},W_{2}^{\operatorname{sph}})=1$.
Similarly, there is a spherical $\operatorname{GL}_{2}$-invariant trilinear
form $\mathfrak{y}^{\operatorname{sph}}$ on
$\sigma_{0}\times\sigma_{1}\times\sigma_{2}$. We identify
$J_{\overline{P}}(W(\pi))$ with $W(\sigma_{0})$ via mapping the normalised
$U^{\prime}_{1}$-eigenvector $W^{\prime,\operatorname{Si}}_{\alpha}$ to the
normalised spherical vector.

######  Proposition 7.8.

In this unramified setting we have

$\partial_{\operatorname{Si}}(\mathfrak{z}^{\operatorname{sph}})=\Delta\cdot\mathfrak{y}^{\operatorname{sph}},\qquad\Delta\coloneqq\frac{p^{2}}{(p^{2}-1)}\cdot
L(\omega_{\rho}\lambda\times\sigma_{1}\times\sigma_{2},\tfrac{1}{2})^{-1}.$

Note that $\omega_{\pi}=\lambda^{2}\omega_{\rho}$, so

$L(\omega_{\rho}\lambda\times\sigma_{1}\times\sigma_{2},s)=L(\lambda^{\vee}\times\sigma_{1}^{\vee}\times\sigma_{2}^{\vee},\tfrac{1}{2}).$

###### Proof.

Since $\mathfrak{y}^{\operatorname{sph}}$ is 1 at the spherical data, and the
spherical vector of $\sigma_{0}$ is the image of $W^{\prime}_{\alpha}$, it
suffices to calculate

$\partial_{\operatorname{Si}}(\mathfrak{z}^{\operatorname{sph}})\left(W^{\prime,\operatorname{Si}}_{\alpha},W_{1}^{\operatorname{sph}},W_{2}^{\operatorname{sph}}\right)=\mathfrak{z}^{\operatorname{sph}}\left(\hat{\tau}W^{\prime,\operatorname{Si}}_{\alpha},W_{1}^{\operatorname{sph}},\left(\begin{smallmatrix}-1&\\\
&1\end{smallmatrix}\right)W_{2}^{\operatorname{sph}}\right)=\mathfrak{z}^{\operatorname{sph}}\left(\hat{\tau}W^{\prime,\operatorname{Si}}_{\alpha},W_{1}^{\operatorname{sph}},W_{2}^{\operatorname{sph}}\right).$

Since $\hat{\tau}$ lies in the same $H(\mathbf{Z}_{p})$-orbit on
$G/P_{\operatorname{Si}}$ as the element $\eta J$ appearing in [LZ20a], we can
apply the formulae of _op.cit._ for Iwahori-level Shintani functions to obtain
the stated result. In the notation of _op.cit._ for the Hecke parameters, we
have

$L(\lambda^{\vee}\times\sigma_{1}^{\vee}\times\sigma_{2}^{\vee},\tfrac{1}{2})=\left(1-\frac{p^{2}}{\alpha\mathfrak{a}_{1}\mathfrak{a}_{2}}\right)\left(1-\frac{p^{2}}{\alpha\mathfrak{a}_{1}\mathfrak{b}_{2}}\right)\left(1-\frac{p^{2}}{\alpha\mathfrak{b}_{1}\mathfrak{a}_{2}}\right)\left(1-\frac{p^{2}}{\alpha\mathfrak{b}_{1}\mathfrak{b}_{2}}\right).\qed$

###  7.6. Expansion along $\sigma_{2}$

We now perform a second “reduction along a Jacquet module” argument.

######  Proposition 7.9.

Let $W_{0},W_{1}\in\mathcal{W}(\sigma_{0}),\mathcal{W}(\sigma_{1})$, and
choose $t\geqslant 1$ such that $W_{0},W_{1}$ are fixed by
$\left(\begin{smallmatrix}1&\\\
p^{t}\mathbf{Z}_{p}&1\end{smallmatrix}\right)$. Then the value

$\mathfrak{y}^{\operatorname{sph}}\left(W_{0},W_{1},W^{\prime}_{\mathfrak{a}_{2}}[\ell]\right)$

is independent of $\ell\geqslant t$. Its limiting value is equal to the value
at $s=\tfrac{1}{2}$ of the function defined for $\Re(s)\gg 0$ by

$\Delta^{\prime}_{s}\cdot\int_{\mathbf{Q}_{p}^{\times}}W_{0}\left(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right)\right)W_{1}\left(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right)\right)\tau(x)|x|^{s-1}\,\mathrm{d}x,$

which has analytic continuation as a polynomial in $p^{\pm s}$; here $\tau$ is
the unramified character sending $x$ to $p^{-1/2}\mathfrak{b}_{2}$, and

$\Delta_{s}^{\prime}=\frac{p}{(p+1)}L(\sigma_{0}\times\sigma_{1}\times\tau,s)^{-1}=\frac{p}{(p+1)}\left(1-\frac{p^{2-s}}{\beta\mathfrak{a}_{1}\mathfrak{a}_{2}}\right)\left(1-\frac{p^{2-s}}{\beta\mathfrak{b}_{1}\mathfrak{a}_{2}}\right)\left(1-\frac{p^{2-s}}{\gamma\mathfrak{a}_{1}\mathfrak{a}_{2}}\right)\left(1-\frac{p^{2-s}}{\gamma\mathfrak{b}_{1}\mathfrak{a}_{2}}\right).$

###### Proof.

One can write down an explicit formula for $\mathfrak{y}^{\operatorname{sph}}$
as the leading term of the $\operatorname{GL}_{2}\times\operatorname{GL}_{2}$
Rankin–Selberg zeta-integral (corresponding to deforming $\mathfrak{a}_{2}$
and $\mathfrak{b}_{2}$ to $\mathfrak{a}_{2}|\cdot|^{1/2-s}$ and
$\mathfrak{b}_{2}|\cdot|^{s-1/2}$):

$\mathfrak{y}^{\operatorname{sph}}\left(W_{0},W_{1},W_{2}\right)=\lim_{s\to\tfrac{1}{2}}\frac{\langle
f_{2}(g;s),R(g;s)\rangle}{L(\sigma_{0}\times\sigma_{1}\times\tau,s)},$

where $f_{2}$ denotes the Siegel section corresponding to $W_{2}$,
$\langle-,-\rangle$ denotes integration over $B_{G}\backslash
G\cong\mathbf{P}^{1}$, and

$R(g;s)\coloneqq\int_{\mathbf{Q}_{p}^{\times}}W_{0}\left(\left(\begin{smallmatrix}x&\\\
&1\end{smallmatrix}\right)g\right)W_{1}\left(\left(\begin{smallmatrix}x&\\\
&1\end{smallmatrix}\right)g\right)\tau(x)|x|^{s-1}\,\mathrm{d}^{\times}x.$

For the particular choice of $W_{2}$ above, $f_{2}$ takes the value $p^{\ell}$
on the preimage of the identity in $\mathbf{P}^{1}(\mathbf{Z}/p^{\ell})$, and
its value is 0 elsewhere. Since the measure of this neighbourhood is
$\tfrac{1}{p^{\ell-1}(p+1)}$, the integral over $\mathbf{P}^{1}$ is simply
$\tfrac{p}{p+1}R(1)$. ∎

###  7.7. Conclusions

Note that the product of the two $L$-factors
$\Delta\cdot\Delta^{\prime}_{s}|_{s=1/2}$ appearing in Proposition 7.8 and
Proposition 7.9 is a degree 8 factor of the degree 16 $L$-factor
$L(\pi\times\sigma_{1}\times\sigma_{2},\tfrac{1}{2})$, and thus corresponds to
an 8-dimensional direct summand of the 16-dimensional Weil–Deligne
representation associated to $\pi\times\sigma_{1}\times\sigma_{2}$; and the
8-dimensional subrepresentation which we obtain is precisely the one giving
the “Panchishkin subrepresentation” of the Galois representation when the
weights lie in region $(D)$. So the $L$-factor is the one denoted
$\mathcal{E}^{(d)}$ in [LZ20b], that here we call $\mathcal{E}^{(D)}$ to be
consistent with our prevoius notation.

######  Proposition 7.10.

Let $W_{1}^{\operatorname{dep}}\in\mathcal{W}(\sigma_{1})$ be the normalised
$p$-depleted vector (so that $W_{1}(\left(\begin{smallmatrix}x&\\\
&1\end{smallmatrix}\right))=\operatorname{ch}_{\mathbf{Z}_{p}^{\times}}(x)$).
Then for any $\ell\geqslant 2$ we have

$\mathfrak{z}^{\mathrm{sph}}\left(\hat{\tau}W^{\prime}_{\alpha,\operatorname{Si}}[\ell],W_{1}^{\operatorname{dep}},W^{\prime}_{\mathfrak{a}_{2}}[\ell]\right)=\frac{p^{3}}{(p+1)^{2}(p-1)}\mathcal{E}^{(D)}.$

In an ideal world (i.e. if we had a comprehensive version of “Siegel-parabolic
higher Hida theory” available to us), the above formula would presumably be
the right one to use for the interpolation property of our $p$-adic
$L$-functions. However, for technical reasons we are constrained to work at
Iwahori level, so we need a variant of this formula.

######  Proposition 7.11.

For any $\ell\geqslant 2$ we have

$\mathfrak{z}^{\mathrm{sph}}\left(w_{01}^{-1}\tau w_{2}\cdot
W^{\prime,\operatorname{Iw}}_{\alpha,\beta}[\ell],\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)W_{1}^{\operatorname{dep}},W_{\mathfrak{a}_{2}}\right)=\left(\tfrac{p^{2}}{\beta\mathfrak{b}_{2}}\right)^{t}\cdot\frac{p^{3}}{(p+1)^{2}(p-1)}\cdot\mathcal{E}^{(D)}.$

###### Proof.

Since $w_{2}=w_{1}\cdot w_{\operatorname{Si}}$, where $w_{\operatorname{Si}}$
is the long Weyl element of $M_{\operatorname{Si}}$, we can calculate this
quantity as

$\mathfrak{z}^{\mathrm{sph}}\left(\hat{\tau}\cdot
w_{\operatorname{Si}}W^{\prime,\operatorname{Iw}}_{\alpha,\beta}[\ell],\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)W_{1}^{\operatorname{dep}},wW_{\mathfrak{a}_{2}}\right).$

Everything in sight is invariant under the principal congruence subgroup mod
$p^{\ell}$, so we may apply Proposition 7.7 to express this as

$\Delta\cdot\mathfrak{y}^{\operatorname{sph}}\left(w\cdot
W^{\prime}_{\beta/p},\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)W_{1}^{\operatorname{dep}},wW_{\mathfrak{a}_{2}}\right),$

where both $w$’s denote $\left(\begin{smallmatrix}&1\\\
-1&\end{smallmatrix}\right)\in\operatorname{GL}_{2}$, and
$W^{\prime}_{\gamma/p}$ is a $U^{\prime}$-eigenvector of eigenvalue $\beta/p$
in $\mathcal{W}(\sigma_{0})$. We have We have
$wW_{\mathfrak{a}_{2}}=\mathfrak{b}_{2}^{-\ell}\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)W^{\prime}_{\mathfrak{a}_{2}}[\ell]$; and similarly
$w\cdot
W^{\prime}_{\beta/p}[\ell]=\left(\tfrac{p^{2}}{\beta}\right)^{\ell}W_{\beta/p}$.
So we obtain

$\Delta\cdot\left(\tfrac{p^{2}}{\beta\mathfrak{b}_{2}}\right)^{\ell}\mathfrak{y}^{\operatorname{sph}}\left(\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)\cdot
W_{\beta/p},\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)W_{1}^{\operatorname{dep}},\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)W^{\prime}_{\mathfrak{a}_{2}}[\ell]\right).$

After cancelling the $\left(\begin{smallmatrix}p^{\ell}&\\\
&1\end{smallmatrix}\right)$ terms using the
$\operatorname{GL}_{2}$-equivariance, we can now conclude using Proposition
7.9. ∎

######  Remark 7.11.

If, in place of $W_{1}^{\operatorname{dep}}$, we use the Iwahori eigenvector
$W_{\mathfrak{a}_{2}}$, then this corresponds to deleting the degree-one
factor $\left(1-\frac{p^{2}}{\gamma\mathfrak{b}_{1}\mathfrak{a}_{2}}\right)$
from $\mathcal{E}^{(D)}$. This gives exactly the “greatest common divisor” of
$\mathcal{E}^{(D)}$ and $\mathcal{E}^{(E)}$.

## References

  * [Art04] J. Arthur, _Automorphic representations of $\operatorname{GSp}(4)$_, Contributions to automorphic forms, geometry, and number theory, Johns Hopkins Univ. Press, 2004, pp. 65–81. MR 2058604.
  * [BH06] S. Böcherer and B. E. Heim, _Critical values of $L$-functions on ${\rm GSp}_{2}\times{\rm GL}_{2}$_, Math. Z. 254 (2006), no. 3, 485–503. MR 2244361.
  * [DR14] H. Darmon and V. Rotger, _Diagonal cycles and Euler systems I: a $p$-adic Gross–Zagier formula_, Ann. Sci. École Norm. Sup. (4) 47 (2014), no. 4, 779–832. MR 3250064.
  * [Del79] P. Deligne, _Valeurs de fonctions $L$ et périodes d’intégrales_, Automorphic forms, representations and $L$-functions (Corvallis, 1977), Proc. Sympos. Pure Math., vol. 33.2, Amer. Math. Soc., Providence, RI, 1979, with an appendix by N. Koblitz and A. Ogus, pp. 313–346. MR 546622.
  * [Emo19] M. Emory, _On the global Gan–Gross–Prasad conjecture for general spin groups_, preprint, 2019, arXiv:1901.01746.
  * [FM16] M. Furusawa and K. Morimoto, _On special values of certain $L$-functions, II_, Amer. J. Math. 138 (2016), no. 4, 1117–1166. MR 3538153.
  * [GT18] T. Gee and O. Taïbi, _Arthur’s multiplicity formula for $\operatorname{GSp}_{4}$ and restriction to $\operatorname{Sp}_{4}$_, preprint, 2018, arXiv:1807.03988.
  * [GT05] A. Genestier and J. Tilouine, _Systèmes de Taylor–Wiles pour $\operatorname{GSp}_{4}$_, Formes automorphes. II. Le cas du groupe $\operatorname{GSp}(4)$, Astérisque, vol. 302, Soc. Math. France, 2005, pp. 177–290. MR 2234862.
  * [Har04] M. Harris, _Occult period invariants and critical values of the degree four $L$-function of $\operatorname{GSp}(4)$_, Contributions to automorphic forms, geometry, and number theory (in honour of J. Shalika), Johns Hopkins Univ. Press, 2004, pp. 331–354. MR 2058613.
  * [HK92] M. Harris and S. Kudla, _Arithmetic automorphic forms for the nonholomorphic discrete series of $\operatorname{GSp}(2)$_, Duke Math. J. 66 (1992), no. 1, 59–121. MR 1159432.
  * [II10] A. Ichino and T. Ikeda, _On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture_, Geom. Funct. Anal. 19 (2010), no. 5, 1378–1425. MR 2585578.
  * [Lan19] K.-W. Lan, _Closed immersions of toroidal compactifications of Shimura varieties_, preprint, 2019\.
  * [LLZ14] A. Lei, D. Loeffler, and S. L. Zerbes, _Euler systems for Rankin–Selberg convolutions of modular forms_, Ann. of Math. (2) 180 (2014), no. 2, 653–771. MR 3224721.
  * [LPSZ19] D. Loeffler, V. Pilloni, C. Skinner, and S. L. Zerbes, _Higher Hida theory and $p$-adic $L$-functions for $\operatorname{GSp}(4)$_, preprint, 2019, arXiv:1905.08779.
  * [LR23] D. Loeffler and O. Rivero, _On $p$-adic $L$-functions for $\operatorname{GSp}(4)\times\operatorname{GL}(2)$_, preprint, 2023.
  * [LZ20a] D. Loeffler and S. L. Zerbes, _On some zeta-integrals for unramified representations of $\operatorname{GSp}(4)$_, in preparation, 2020\.
  * [LZ20b] by same author, _$p$ -adic L-functions and diagonal cycles for $\operatorname{GSp}(4)\times\operatorname{GL}(2)\times\operatorname{GL}(2)$_, in preparation, 2020.
  * [LZ21] by same author, _On the Bloch–Kato conjecture for $\operatorname{GSp}(4)\times\operatorname{GL}(2)$_, preprint, 2021, arXiv:2106.14511.
  * [Mor18] K. Morimoto, _On tensor product $L$-functions for quaternion unitary groups and $\rm GL(2)$ over totally real number fields: mixed weight cases_, Adv. Math. 337 (2018), 317–362. MR 3853053.
  * [Mor04] T. Moriyama, _Entireness of the spinor $L$-functions for certain generic cusp forms on $\operatorname{GSp}(2)$_, Amer. J. Math. 126 (2004), no. 4, 899–920. MR 2075487.
  * [Nov79] M. E. Novodvorsky, _Automorphic $L$-functions for the symplectic group $\operatorname{GSp}(4)$_, Automorphic forms, representations and $L$-functions (Corvallis, 1977), Proc. Sympos. Pure Math., vol. 33.2, Amer. Math. Soc., 1979, pp. 87–95. MR 546610.
  * [Sah10] A. Saha, _Pullbacks of Eisenstein series from $\rm GU(3,3)$ and critical $L$-values for $\rm GSp(4)\times GL(2)$_, Pacific J. Math. 246 (2010), no. 2, 435–486. MR 2652263.
  * [Su19] J. Su, _Coherent cohomology of Shimura varieties and automorphic forms_, preprint, 2019.
  * [Urb14] E. Urban, _Nearly overconvergent modular forms_, Iwasawa Theory 2012: State of the Art and Recent Advances (Berlin), Contrib. Math. Comput. Sci., vol. 7, Springer, 2014, pp. 401–441.