a $\mathcal{U}_{p,\operatorname{Kl}}$-equivariant direct-sum decomposition as
the sum of $(\Pi_{p}^{\prime})^{\operatorname{Kl}(p)}$, on which
$\mathcal{U}_{p,\operatorname{Kl}}$ is invertible, and a complementary
subspace on which $\mathcal{U}_{p,\operatorname{Kl}}$ is nilpotent.

Similar statements hold for the dual $\Pi^{\vee}$ in place of $\Pi$, and the
transpose of $\mathcal{U}_{p,\operatorname{Kl}}^{\prime}$ with respect to the
natural pairing
$(\Pi_{p}^{\prime})^{\operatorname{Kl}(p^{m})}\times(\Pi_{p}^{\prime\vee})^{\operatorname{Kl}(p^{m})}$
is given by $\chi_{\Pi}(p)\cdot\mathcal{U}_{p,\operatorname{Kl}}$.

######  Definition 5.8.

If $\eta\in H^{2}(\Pi_{\mathrm{f}})^{\operatorname{Kl}(p)}$, and $m\geqslant
1$, then we define $\eta_{m}\in
H^{2}(\Pi_{\mathrm{f}})^{\operatorname{Kl}(p^{m})}$ to be the unique vector
such that the linear functional $\langle-,\eta_{m}\rangle$ on
$H^{1}(\Pi_{\mathrm{f}}^{\vee})^{\operatorname{Kl}(p^{m})}$ has the following
properties:

  * •

it vanishes on the $\mathcal{U}_{p,\operatorname{Kl}}=0$ generalised
eigenspace,

  * •

it agrees with $\eta$ on the image of the $\operatorname{Kl}(p)$-invariants.

It is clear that the $\eta_{m}$ are compatible under the “normalised trace”
maps,

$\eta_{m}=\tfrac{1}{p^{3}}\sum_{\gamma\in\operatorname{Kl}(p^{m})/\operatorname{Kl}(p^{m+1})}\gamma\eta_{m+1}.$

We have the following reasonably concrete formula:

######  Proposition 5.9.

If $\eta_{1}=\eta$ lies in the
$\mathcal{U}_{p,\operatorname{Kl}}^{\prime}=\lambda$ eigenspace, then we have

$\eta_{m}=\left(\frac{p^{3-r_{2}}}{\lambda}\right)^{m-1}\sum_{a\in\mathbf{Z}/p^{m-1}}\left(\begin{smallmatrix}1\\\
&1\\\ &&1\\\ p^{m}a&&&1\end{smallmatrix}\right){\left(\begin{smallmatrix}1\\\
&p\\\ &&p\\\ &&&p^{2}\end{smallmatrix}\right)}^{m-1}\eta.$

###### Proof.

Let $\eta^{\prime}_{m}$ denote the right-hand side of the above formula. For
$m=1$ we evidently have $\eta_{1}^{\prime}=\eta_{1}$.

Since the operator $\mathcal{U}_{p,\operatorname{Kl}}^{\prime}$ at level
$\operatorname{Kl}(p^{m})$ (for any $m\geqslant 1$) has a set of single-coset
representatives which are lower-triangular and commute with
$\left(\begin{smallmatrix}1\\\ &1\\\ &&1\\\
p^{m}a&&&1\end{smallmatrix}\right)$, it follows easily that
$\eta_{m}^{\prime}$ is a
$\mathcal{U}_{p,\operatorname{Kl}}^{\prime}$-eigenvector at level
$\operatorname{Kl}(p^{m})$ with the same eigenvalue as $\eta_{1}$. Moreover,
we can group the coset representatives in the normalised trace map for
$\operatorname{Kl}(p^{m-1})/\operatorname{Kl}(p^{m})$ in such a way that the
inner sum becomes $\mathcal{U}^{\prime}_{p,\operatorname{Kl}}$ acting on
$\eta_{1}$, and using the eigenvector property of $\eta_{1}$ gives the trace-
compatibility of the $\eta_{m}^{\prime}$. Hence $\eta_{m}^{\prime}=\eta_{m}$
for all $m$. ∎

######  Definition 5.10.

Choose a place $v$ of $E$ above $p$. We say that $\Pi$ is _Klingen-ordinary_
at $p$ (with respect to $v$) if the operator
$\mathcal{U}_{p,\operatorname{Kl}}$ on
$(\Pi_{p}^{\prime})^{\operatorname{Kl}(p)}$ has an eigenvalue $\lambda$ which
is a $p$-adic unit.

It follows from the proof of Lemma 3.22 above that the unit eigenvalue
$\lambda$ is unique if it exists (and the corresponding eigenspace is
1-dimensional). In particular, $\lambda$ lies in $E_{v}$, rather than in some
finite extension.

###  5.6. Interpolation of cup-products

We now apply the above constructions to interpolate cup-products in families.
Let $L$ be the completion of $E$ at our place $v\mid p$, and $\mathcal{O}$ its
ring of integers. We shall assume that $r_{2}\geqslant 1$ and $\Pi$ is
Klingen-ordinary at $p$ (with respect to the place $v$); and we choose some
class $\eta\in H^{2}\left(\Pi_{\mathrm{f}}\right)$, lying in the ordinary
eigenspace for $\mathcal{U}_{p,\operatorname{Kl}}^{\prime}$. Thus $\eta$
defines a linear functional

$e_{\operatorname{Kl}}\cdot
H^{1}\left(X_{G,\operatorname{Kl}}(p)_{L},[L_{1}]\right)\to L.$

We also choose a flat $p$-adic $\mathcal{O}$-algebra $R$, and
$\tau_{1},\tau_{2}:\mathbf{Z}_{p}^{\times}\to R^{\times}$ continuous
characters such that $\tau_{1}+\tau_{2}=r_{1}-r_{2}+2$. With these notations,
(4.1) gives a map

$\displaystyle\iota_{\star}:\mathcal{S}_{(\tau_{1},\tau_{2})}(R)$
$\displaystyle\to H^{1}\left(\mathfrak{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p),\mathfrak{F}_{R}(3+r_{1},1-r_{2})(-D)\right)$
$\displaystyle=R\mathop{\hat{\otimes}}_{\mathbf{Z}_{p}}H^{1}\left(\mathfrak{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p),\mathfrak{F}_{\mathbf{Z}_{p}}(3+r_{1},1-r_{2})(-D)\right).$

######  Definition 5.11.

Given $\mathcal{E}\in\mathcal{S}_{(\tau_{1},\tau_{2})}(R)$, we define an
element
$\left\langle\iota_{\star}\left(\mathcal{E}\right),\eta\right\rangle\in
R[1/p]$ as follows: it is the image of $\iota_{\star}(\mathcal{E})$ under the
composition of maps

$\displaystyle
R[1/p]\mathop{\hat{\otimes}}_{\mathbf{Z}_{p}}H^{1}\left(\mathfrak{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p),\mathfrak{F}(3+r_{1},1-r_{2})(-D)\right)$
$\displaystyle\xrightarrow{e_{\operatorname{Kl}}}R[1/p]\mathop{\hat{\otimes}}_{\mathbf{Z}_{p}}e_{\operatorname{Kl}}\cdot
H^{1}\left(\mathfrak{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p),\mathfrak{F}(3+r_{1},1-r_{2})(-D)\right)$
$\displaystyle\xrightarrow{\cong}R[1/p]\otimes_{\mathbf{Q}_{p}}e_{\operatorname{Kl}}\cdot
H^{1}\left(X_{G,\operatorname{Kl}}(p)_{\mathbf{Q}_{p}},[L_{1}](-D)\right)$
$\displaystyle\xrightarrow{\varpi}R[1/p]\otimes_{\mathbf{Q}_{p}}e_{\operatorname{Kl}}\cdot
H^{1}\left(X_{G,\operatorname{Kl}}(p)_{\mathbf{Q}_{p}},[L_{1}]\right)$
$\displaystyle\xrightarrow{\langle-,\eta\rangle}R[1/p].$

Here the second map is the isomorphism of Theorem 3.6, and the map $\varpi$ is
the“forget cuspidality” map, given by the natural inclusion of sheaves
$[L_{1}](-D)\hookrightarrow[L_{1}]$.

######  Proposition 5.12.

Let $\phi:R\to L$ be a continuous ring homomorphism such that

  * •

the composites $\phi\circ\tau_{i}:\mathbf{Z}_{p}^{\times}\to L^{\times}$ are
the algebraic characters $x\mapsto x^{t_{i}}$, for some integers
$t_{i}\geqslant 1$ with $t_{1}+t_{2}=r_{1}-r_{2}+2$;

  * •

the $p$-adic modular form
$\mathcal{E}_{\phi}\in\mathcal{S}_{(t_{1},t_{2})}(K^{p}_{H};L)$ given by
specialising $\mathcal{E}$ is the restriction to
$\mathcal{X}_{H}^{\mathrm{ord}}$ of a classical holomorphic form $E_{\phi}\in
M_{(t_{1},t_{2})}(p,L)$.

Then we have

$\phi\left(\left\langle\iota_{\star}\left(\mathcal{E}\right),\eta\right\rangle\right)=\left\langle\iota_{1,\star}(E_{\phi}),\eta\right\rangle.$

###### Proof.

The construction of the pairing $\langle-,-\rangle$ is compatible with base-
change in $R$, so it suffices to assume that $R=\mathcal{O}$, $\phi$ is the
identity map, and $\mathcal{E}$ is the $p$-adic modular form associated to a
classical modular form $E$.

If $E$ is cuspidal as a classical modular form, then the result is essentially
a restatement of the commutativity of the diagrams in the previous section (in
the simple case where $m=1$ and the $\chi_{i}$ are trivial). However, we are
working in a slightly greater degree of generality, where we allow $E$ to be a
“fake cusp form” in the sense of §4.4 above, so it can be non-vanishing at
some components of $D_{H}$ lying outside the ordinary locus; and this is
rather more delicate.

We first summarise why the obvious argument does not work. Let
$(k_{1},k_{2})=(r_{1}+3,1-r_{2})$. From $E$ we can form the following two
cohomology classes:

  * •

An “algebraic” class $z^{\mathrm{alg}}=\iota_{1,\star}(E)\in
H^{1}\left(X_{G,\operatorname{Kl}}(p)_{L},\omega_{G}(k_{1},k_{2})\right)$.

  * •

A “analytic” class $z^{\mathrm{an}}=\iota_{\star}(\mathcal{E})\in
H^{1}\left(\mathcal{X}_{G,\operatorname{Kl}}^{\geqslant
1}(p),\mathcal{F}_{G}(k_{1},k_{2})(-D_{G})\right)$.

These are compatible, in the sense that the restriction of $z^{\mathrm{alg}}$
to $\mathcal{X}_{G,\operatorname{Kl}}^{\geqslant 1}(p)$ coincides with the
image of $z^{\mathrm{an}}$ in
$H^{1}\left(\mathcal{X}_{G,\operatorname{Kl}}^{\geqslant
1}(p),\mathcal{F}_{G}(k_{1},k_{2})\right)$ under the “forget cuspidality” map
$\varpi$.

The classicity theorem for ordinary cohomology, used in the construction of
our pairing, shows that there is some $z^{\mathrm{alg}}_{\mathrm{cusp}}\in
H^{1}\left(X_{G,\operatorname{Kl}}(p)_{L},\omega_{G}(k_{1},k_{2})(-D)\right)$
whose restriction to $\mathcal{X}_{\operatorname{Kl}}^{\geqslant 1}(p)$ is
$e_{\operatorname{Kl}}\cdot z^{\mathrm{an}}$. If we can show that the image of
$z^{\mathrm{alg}}_{\mathrm{cusp}}$ under the “forget cuspidality” map $\varpi$
coincides with $e_{\operatorname{Kl}}\cdot z^{\mathrm{alg}}$, then we are
done. However, all that we know is that
$\varpi(z^{\mathrm{alg}}_{\mathrm{cusp}})$ and $e_{\operatorname{Kl}}\cdot
z^{\mathrm{alg}}$ have the same image in
$H^{1}\left(\mathcal{X}_{G,\operatorname{Kl}}^{\geqslant
1}(p),\mathcal{F}_{G}(k_{1},k_{2})\right)$, and since we have no analogue of
Theorem 3.6 for the non-cuspidal $p$-adic cohomology, we do not know that the
restriction map
$H^{1}\left(X_{G,\operatorname{Kl}}(p)_{L},\omega_{G}(k_{1},k_{2})\right)\to
H^{1}\left(\mathcal{X}_{G,\operatorname{Kl}}^{\geqslant
1}(p),\mathcal{F}_{G}{(k_{1},k_{2})}\right)$ is injective on the ordinary
part.

We work around this by using _overconvergent_ cohomology. Theorem 3.14 gives a
classicity result for both of the spaces

$H^{1}\left(\mathcal{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p)^{\dagger},\omega_{G}(k_{1},k_{2})\right)\quad\text{and}\quad
H^{1}\left(\mathcal{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p)^{\dagger},\omega_{G}(k_{1},k_{2})(-D)\right).$

Performing the same construction as before for the overconvergent spaces, we
obtain a class $z^{{\dagger}}\in H^{1}\left(\mathcal{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p)^{\dagger},\omega_{G}(k_{1},k_{2})(-D)\right)$ with
the following properties: the restriction of $z^{{\dagger}}$ to the $p$-rank
$\geqslant 1$ locus (forgetting the overconvergence) is $z^{\mathrm{an}}$; and
the image of $z^{{\dagger}}$ in $H^{1}\left(\mathcal{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p)^{\dagger},\omega_{G}(k_{1},k_{2})\right)$
(forgetting the cuspidality) coincides with the restriction of
$z^{\mathrm{alg}}$.

The classes $\varpi(z^{\mathrm{alg}}_{\mathrm{cusp}})$ and
$e_{\operatorname{Kl}}\cdot z^{\mathrm{alg}}$ have the same image in
$H^{1}\left(\mathcal{X}^{\geqslant
1}_{G,\operatorname{Kl}}(p)^{\dagger},\omega_{G}(k_{1},k_{2})\right)$ (namely
$e_{\operatorname{Kl}}\cdot\varpi(z^{{\dagger}})$). Using the classicity
theorem for this overconvergent cohomology, we conclude that
$\varpi(z^{\mathrm{alg}}_{\mathrm{cusp}})$ and $e_{\operatorname{Kl}}\cdot
z^{\mathrm{alg}}$ coincide. ∎

###  5.7. Higher level specialisations

We also have a version of this theorem for specialisations that are classical
of higher $p$-power levels, using the vectors $\eta_{m}$ at level
$\operatorname{Kl}(p^{m})$ constructed from $\eta$ as in §5.5.

######  Theorem 5.13.

Let $\phi:R\to L$ be a continuous ring homomorphism such that
$\phi\circ\tau_{i}$ has the form $x\mapsto x^{t_{i}}\chi_{i}(x)$, for some
$t_{i}$ as before and finite-order characters $\chi_{1},\chi_{2}$ with
$\chi_{1}\chi_{2}=1$. Suppose that $\phi(\mathcal{E})$ is the image of a
classical modular form $E_{\phi}\in M_{(t_{1}+1,t_{2}+1)}(K_{H}(p^{m}),L)$,
for some $m\gg 0$.

Then we have

$\phi\left(\langle\iota_{\star}(\mathcal{E}),\eta\rangle\right)=\left\langle\iota_{m,\star}(E_{\phi}),\eta_{m}\right\rangle.$

###### Proof.

By the same arguments as above,
$\phi\left(\langle\iota_{\star}(\mathcal{E}),\eta\rangle\right)$ is the image
of $\iota_{m,\star}(E_{\phi})$ under the unique linear functional on
$H^{1}(X_{G,\operatorname{Kl}}(p^{m})_{L},[L_{1}])$ which factors through the
ordinary idempotent $e_{\operatorname{Kl}}$ and agrees with
$\langle-,\eta\rangle$ on the image of
$H^{1}(X_{G,\operatorname{Kl}}(p)_{L},[L_{1}])$. This is exactly the
definition of $\eta_{m}$. ∎

##  6\. “Nearly” coherent cohomology

It is well-known that nearly-holomorphic modular forms can be considered as
$p$-adic modular forms. In this section, we shall formulate an analogous
statement for our $p$-adic cohomology spaces for $\operatorname{GSp}_{4}$, and
show that the $\Lambda$-adic pushforward map constructed in the previous
section is compatible with this additional structure.

###  6.1. “Nearly” sheaves

In this section, we’ll consider coherent cohomology with coefficients in
certain indecomposable representations of $P_{\operatorname{S}}$. This is
needed in order to study pushforwards of non-holomorphic Eisenstein series; we
regard it as an analogue for $H^{1}$ of Siegel varieties of Shimura’s theory
of nearly-holomorphic modular forms.

######  Definition 6.1.

If $V$ is a finite-dimensional algebraic representation of
$P_{\operatorname{S}}$, we let $\operatorname{Fil}^{n}V$ denote the direct sum
of the eigenspaces of $W$ on which the torus $\left(\begin{smallmatrix}x\\\
&x\\\ &&1\\\ &&&1\end{smallmatrix}\right)$ acts with weights $\geqslant n$
(i.e. the weight spaces of weight $\lambda(r_{1},r_{2};c)$ with
$r_{1}+r_{2}+c\geqslant 2n$).

This filtration is stable under $P_{\operatorname{S}}$, and the
$P_{\operatorname{S}}$-action on the graded pieces
$\operatorname{Gr}^{n}V=\operatorname{Fil}^{n}V/\operatorname{Fil}^{n+1}V$
factors through $M_{\operatorname{S}}$. We shall apply this to
$V=V(r_{1},r_{2};r_{1}+r_{2})$, for some integers $r_{1}\geqslant
r_{2}\geqslant 0$; in this case the non-zero graded pieces are in degrees
$0\leqslant n\leqslant r_{1}+r_{2}$.

######  Definition 6.2.

  * •

Let $\tilde{L}_{1}$ denote the representation
$V/\operatorname{Fil}^{r_{1}+1}V\otimes W(3,1;0)$, and $L_{1}^{\prime}$ the
smallest filtration subspace $\operatorname{Gr}^{r_{1}}V\otimes W(3,1;0)$ of
$\tilde{L}_{1}$.

  * •

Dually, let $\tilde{L}_{2}$ denote the $P_{\operatorname{S}}$-representation
$\operatorname{Fil}^{r_{2}}V\otimes W(2,0;0)$, and $L_{2}^{\prime}$ the top
filtration quotient $\operatorname{Gr}^{r_{2}}V\otimes W(2,0;0)$ of
$\tilde{L}_{2}$.

There is a natural projection $L_{1}^{\prime}\twoheadrightarrow L_{1}$, and a
natural inclusion $L_{2}\hookrightarrow L_{2}^{\prime}$ (both of which are
split, but we do not need this).

######  Proposition 6.3.

The natural maps

$H^{2}\left(X_{G,E},[L_{2}](-D)\right)\to
H^{2}\left(X_{G,E},[L^{\prime}_{2}](-D)\right)$

and

$H^{2}\left(X_{G,E},[\tilde{L}_{2}](-D)\right)\to
H^{2}\left(X_{G,E},[L^{\prime}_{2}](-D)\right),$

induced by the maps of $P_{\operatorname{S}}$-representations
$\tilde{L}_{2}\twoheadrightarrow L^{\prime}_{2}\hookleftarrow L_{2}$, are both
isomorphisms on the $\Pi_{\mathrm{f}}^{\prime}$ generalised eigenspace (and
similarly without $(-D)$).

###### Proof.

This follows readily from the final statement of Theorem 5.2, since the kernel
of $\tilde{L}_{2}\twoheadrightarrow L^{\prime}_{2}$ and the cokernel of
$L_{2}\hookrightarrow L^{\prime}_{2}$ do not contribute to the
$\Pi_{\mathrm{f}}^{\prime}$-eigenspace of $H^{2}$. ∎

We write $\tilde{\eta}$ for the unique class in
$H^{2}\left(X_{G,E},[\tilde{L}_{2}](-D)\right)[\Pi_{\mathrm{f}}]$
corresponding to $\eta$ under the above isomorphisms. The linear functional
$\langle-,\tilde{\eta}\rangle$ is thus a homomorphism

$H^{1}\left(X_{G,E},[\tilde{L}_{1}]\right)\to E,$

characterised as follows:

  * •

it factors through the $\Pi_{\mathrm{f}}^{\vee}$-isotypical part;

  * •

its restriction to $H^{1}\left(X_{G,E},[L_{1}^{\prime}]\right)$ is the
composite of the projection $L_{1}^{\prime}\twoheadrightarrow L_{1}$ and
pairing with $\eta$.

This extended homomorphism will play a role analogous to Shimura’s
“holomorphic projection” operator in the $\operatorname{GL}_{2}$ theory.

###  6.2. Pullback and pushforward of “nearly” sheaves

######  Proposition 6.4 (cf. [LSZ17, Proposition 4.3.1]).

Suppose $(r_{1},r_{2};c)$ is a $G$-dominant weight. The restriction to $H$ of
the irreducible representation $V_{G}(r_{1},r_{2};c)$ of $G$ is a direct sum
of distinct irreducible $H$-representations. The representation
$V_{H}(t_{1},t_{2};c)$ appears as a direct summand if and only if the integers
$t_{1},t_{2}$ satisfy $t_{1}+t_{2}=r_{1}+r_{2}\bmod 2$ and

$r_{1}-r_{2}\leqslant t_{1}+t_{2}\leqslant
r_{1}+r_{2},\quad|t_{1}-t_{2}|\leqslant r_{1}-r_{2}.\qed$

Since the cocharacter used to define the filtration is strictly dominant with
respect to $B_{H}$, any representation of $B_{H}$ has a canonical
$B_{H}$-invariant filtration, with the action on the graded pieces factoring
through $T$; and this is compatible with the above branching from $G$ to $H$.

We shall be particularly interested in $H$-subrepresentations of the form
$V_{H}(t_{1},t_{2};c)$ with $t_{1}+t_{2}=r_{1}-r_{2}$; note that there are
precisely $r_{1}-r_{2}+1$ such subrepresentations. We shall call these
subrepresentations _small_.

Recall now the larger sheaves $\tilde{L}_{2}\twoheadrightarrow
L_{2}^{\prime}\hookleftarrow L_{2}$ defined above. Since
$V_{H}(t_{1},t_{2};r_{1}+r_{2})$ is a direct summand of $V$, we have a
projection map

$\tilde{L}_{2}|_{B_{H}}=\left(\operatorname{Fil}^{r_{2}}V\otimes
W(2,0;0)\right)|_{B_{H}}\to V_{H}\otimes\lambda(1,1;0)$

given by the tensor product of the projections $V|_{H}\to V_{H}$ and
$W(2,0;0)|_{H}\to\lambda(1,1;0)$. This determines a homomorphism

$\iota_{\text{nearly}}^{\star}:H^{2}\left(X_{G,\mathbf{Q}},[\tilde{L}_{2}](-D)\middle)\to
H^{2}\middle(X_{H,\mathbf{Q}},[V_{H}]\otimes\omega_{H}(1,1;0)(-D)\right).$

Note that the composition of $\iota_{\text{nearly}}^{\star}$ with the
projection $V_{H}\to V_{H}/\operatorname{Fil}^{r_{2}+1}V_{H}\cong
W_{H}(1-t_{1},1-t_{2};r_{1}+r_{2})$ factors through
$H^{2}\left(X_{G,\mathbf{Q}},[L_{2}^{\prime}](-D)\right)$, and we can (and do)
normalise such that this composite agrees with $\iota^{\star}$ on the image of
$L_{2}\hookrightarrow L_{2}^{\prime}$.

Dually, we obtain a map

$\iota^{\text{nearly}}_{\star}:H^{0}\left(X_{H,\mathbf{Q}},[V_{H}]\otimes\omega_{H}(1,1;0)\middle)\to
H^{1}\middle(X_{G,\mathbf{Q}},[\tilde{L}_{1}]\right).$

The source of this map is the space of _nearly-holomorphic_ modular forms for
$H$ of weight $(1+t_{1},1+t_{2})$. We thus have an extended period pairing

$H^{0}\big{(}X_{H,E},[V_{H}]\otimes\omega_{H}(1,1;0)\big{)}\otimes
H^{2}(\Pi_{\mathrm{f}})\to
E,\quad(f,\eta)\mapsto\langle\iota^{\text{nearly}}_{\star}(f),\tilde{\eta}\rangle.$

From the construction of $\tilde{\eta}$, we see that the restriction of this
pairing to the space of holomorphic forms agrees with the period pairing of
§5.3 above.

###  6.3. Archimedean theory: the Hodge splitting

We can compute $H^{2}(X_{G,\mathbf{C}},[\tilde{L}_{2}](-D))$ in terms of
automorphic forms, using results of Su [Su19]. As above, let $K_{\infty}$ be
the standard maximal compact subgroup of $G(\mathbf{R})_{+}$, fixing the point
$h=iI_{2}\in\mathcal{H}_{2}$. The Shimura cocharacter determines a
decomposition of $\operatorname{Lie}(G)_{\mathbf{C}}$ as
$\mathfrak{k}\oplus\mathfrak{p}^{+}\oplus\mathfrak{p}^{-}$, where
$\mathfrak{k}=\operatorname{Lie}(K_{\infty})_{\mathbf{C}}$, and
$\mathfrak{p}^{+}$ (resp. $\mathfrak{p}^{-}$) is identified with the
holomorphic (resp. antiholomorphic) tangent space of $\mathcal{H}_{2}$ at $h$.
The parabolic $P_{h}\subset G(\mathbf{C})$ with Lie algebra
$\mathfrak{p}=\mathfrak{k}\oplus\mathfrak{p}^{-}$ is a conjugate of the Siegel
parabolic $P_{\operatorname{S}}$, so we may identify any
$P_{\operatorname{S}}$-representation $V$ with a representation of
$P_{\infty}$.

The main theorem of _op.cit._ gives a canonical and Hecke-equivariant
isomorphism

$H^{*}(X_{G,\mathbf{C}},[V])\cong
H^{*}\\!\left(\mathfrak{p},K_{\infty};\mathcal{A}(G)^{K}\otimes V\right),$

for any level $K$ and any algebraic representation $V$ of
$P_{\operatorname{S}}$, where $\mathcal{A}(G)$ is the space of automorphic
forms on $G$ (twisted by an appropriate power of the norm character so that
the central characters match). The relative Lie algebra cohomology on the
right-hand side can be computed as the cohomology of the complex with $j$-th
term

$\operatorname{Hom}_{K_{\infty}}\left(\textstyle{\bigwedge^{j}}(\mathfrak{p}^{-})\otimes
V^{\vee},\mathcal{A}(G)^{K}\right).$

######  Lemma 6.5.

Let $\Pi_{\infty,2}$ be the $G(\mathbf{R})_{+}$-submodule of $\Pi_{\infty}$
described in §5.4 above. Then we have

$\operatorname{Hom}_{K_{\infty}}\left(\textstyle{\bigwedge^{j}}(\mathfrak{p}^{-})\otimes(\tilde{L}_{2})^{\vee},\
\Pi_{\infty,2}\right)=0$

for all $j\neq 2$, and for $j=2$ this space is 1-dimensional and maps
isomorphically to its image with $L_{2}$ in place of $\tilde{L}_{2}$.

###### Proof.

An explicit description of the $K_{\infty}$-types appearing in
$\Pi_{\infty,2}$ is given in [Sch17]. The minimal $K_{\infty}$-type is
$\tau_{2}=(r_{2}+1,-r_{1}-3)$, and the other $K_{\infty}$-types lie in the
convex cone $\\{\tau_{2}+m\cdot(1,0)+n\cdot(-1,-1):m,n\geqslant 0\\}$.

On the other hand, the $K_{\infty}$-types appearing in
$(\tilde{L}_{2})^{\vee}$ are all contained in a different convex cone,
$\\{(r_{2}+2,-r_{1})+m\cdot(1,1)+n\cdot(-1,1):m,n\geqslant 0\\}$. So the
weights of $(\tilde{L}_{2})^{\vee}\otimes\bigwedge^{j}(\mathfrak{p}^{-})$ are
contained in the translate of this cone by the highest weight of
$\bigwedge^{j}(\mathfrak{p}^{-})$; this translation is by $(0,-2)$ if $j=1$,
by $(-1,-3)$ if $j=2$, and by $(-3,-3)$ if $j=3$. So if $j\neq 2$, these
regions have empty intersection, whereas if $j=2$ the intersection consists
only of $\tau_{2}$, which appears in both representations with multiplicity 1.
∎

Thus the $\Pi_{\mathrm{f}}$-isotypical part of the
$(\mathfrak{p},K_{\infty})$-cohomology is represented by a complex
concentrated in degree 2, giving a a canonical space of Dolbeault differential
forms representing
$H^{2}\big{(}X_{G,\mathbf{C}},[\tilde{L}_{2}]\big{)}[\Pi_{\mathrm{f}}]$: the
vector-valued automorphic forms whose coordinate projections lie in the
minimal $K_{\infty}$-type of $\Pi_{\mathrm{f}}\otimes\Pi_{\infty,2}$.
Moreover, these differentials in fact take values in $[L_{2}]$, regarded as a
subsheaf of $[\tilde{L}_{2}]$ via the Hodge splitting: the projections of
these differentials to the other graded pieces of $[\tilde{L}_{2}]$ are
trivial, since the corresponding $K_{\infty}$-types do not appear in
$\Pi_{\infty,2}$. Since $\Pi$ is cuspidal, these differential forms are
rapidly-decreasing.

Now let $t_{1},t_{2}$ be integers with $t_{1}+t_{2}=r_{1}-r_{2}$, as before.
The analogue of this $K_{\infty}$-equivariant splitting for $H$ in place of
$G$ is the _Hodge splitting_ in the category of $C^{\infty}$ vector bundles,

$[V_{H}]\otimes\omega_{H}(1,1;0)\to\omega_{H}(1+t_{1},1+t_{2}),$

which allows nearly-holomorphic modular forms to be interpreted as scalar-
valued real-analytic functions on $H(\mathbf{Q})\backslash
H(\mathbf{A}_{\mathrm{f}})$, transforming under $K_{H,\infty}$ via the
character $(1+t_{1},1+t_{2})$. Evidently these two splittings are compatible
(since they are both given by projection to eigenspaces for the action of the
centre of $K_{\infty}$, which is contained in $K_{H,\infty}$), so we deduce
the following theorem:

######  Theorem 6.6.

Let $E\in H^{0}\left(X_{H,\mathbf{C}},[V_{H}]\otimes\omega_{H}(1,1;0)\right)$
and let $E_{\infty}\in
H^{0}\left(X_{H,\mathbf{C}},\omega(t_{1}+1,t_{2}+1)_{C^{\infty}}\right)$ be
its image under the Hodge splitting. If $\eta$ and $F_{\eta}$ are as in
Theorem 5.6, then we have

$\langle\iota^{\text{nearly}}_{\star}(E),\tilde{\eta}\rangle=\frac{1}{(2\pi
i)^{3}}\int_{\mathbf{R}^{\times}H(\mathbf{Q})\backslash
H(\mathbf{A})}F_{\eta}(v_{t_{1},t_{2}})E_{\infty}(h)\,\mathrm{d}h.$

###### Proof.

We know that the $[\tilde{L}_{1}]$-valued current representing
$\iota^{\text{nearly}}_{\star}(E)$ is the direct sum of
$\iota_{\star}(E_{\infty})$ and some other terms lying in other graded pieces
of the sheaf $[\tilde{L}_{1}]$. By the previous lemma, $\tilde{\eta}$ pairs to
zero with the latter; on the other hand, we clearly have
$\langle\iota^{\text{nearly}}_{\star}(E),\tilde{\eta}\rangle=\langle\iota_{\star}(E_{\infty}),\eta\rangle$
since $\tilde{\eta}$ maps to $\eta$ in $[L_{2}]$. ∎

######  Remark 6.7.

This construction is strongly analogous to one appearing in Harris’ work
[Har04], which also considers period integrals of automorphic forms for $G$
multiplied by nearly-holomorphic Eisenstein series on $H$. However, our
treatment differs from Harris’ in the following point: in defining
$\tilde{\eta}$, we used $G(\mathbf{A}_{\mathrm{f}})$-equivariance to split a
cohomology exact sequence on $G$ (relating the cohomology of $[L_{2}]$ and
$\tilde{L}_{2}]$). Harris uses instead a splitting on $H$, characterised by
equivariance for the action of $H(\mathbf{A}_{\mathrm{f}})$ (see the proof of
Proposition 1.10.3 of _op.cit_). It is not clear to us whether these
constructions agree in general.

###  6.4. A $p$-adic splitting

We now define a partial $p$-adic analogue of the Hodge splitting. Let us
briefly recall the definition of the torsor $\mathcal{T}_{G}$ defined in
§2.3.3 above. Over the integral model of the open Shimura variety $Y_{G}$ of
prime-to-$p$ level $K^{p}\cdot G(\mathbf{Z}_{p})$, we have a locally free
sheaf $\mathcal{H}^{1}_{\mathrm{dR}}(A)$ of rank 4, with a rank 2 locally free
subsheaf $\operatorname{Fil}^{1}\mathcal{H}^{1}_{\mathrm{dR}}(A)$. There is a
canonical extension of $\mathcal{H}^{1}_{\mathrm{dR}}(A)$ to a locally free
sheaf $\mathcal{H}^{1}_{\mathrm{dR}}(A)^{\mathrm{can}}$ on $X_{G}$, fitting
into a short exact sequence of locally free sheaves

$0\to\omega_{A^{\Sigma}}\to\mathcal{H}^{1}_{\mathrm{dR}}(A)^{\mathrm{can}}\to\omega_{A^{\Sigma}}^{\vee}\to
0,$

where $A^{\Sigma}$ is the semiabelian variety over $X_{G}$ extending $A$, and
$\omega_{A^{\Sigma}}$ its conormal sheaf at the identity section. Moreover, if
$\mathfrak{IG}_{G}(p^{\infty})^{\mathrm{ord}}$ denotes the preimage in
$\mathfrak{IG}_{G}(p^{\infty})$ of the ordinary locus
$\mathfrak{X}_{G}^{\mathrm{ord}}\subset\mathfrak{X}_{G}$, then over
$\mathfrak{X}_{G}^{\mathrm{ord}}$ there is a splitting

$\mathcal{H}^{1}_{\mathrm{dR}}(A)^{\mathrm{can}}\cong\omega_{A^{\Sigma}}\oplus\mathcal{U},$

where $\mathcal{U}$ is the _unit root subsheaf_ [Liu19, §3.12].

Over the Igusa tower $\mathfrak{IG}_{G}(p^{\infty})$, we have a morphism of
$p$-divisible groups $\alpha:\mu_{p^{\infty}}\hookrightarrow
A^{\Sigma}[p^{\infty}]$, and hence a canonical map
$\omega_{A_{G}^{\Sigma}}\to\mathcal{O}_{\mathfrak{IG}_{G}(p^{\infty})}$, i.e.
a class $[\alpha]\in
H^{0}(\mathfrak{IG}_{G}(p^{\infty}),\omega_{A^{\Sigma}}^{\vee})$.

######  Proposition 6.8.

There exists a unique lifting of $[\alpha]$ to a class
$[\alpha]^{\mathrm{ur}}\in
H^{0}(\mathfrak{IG}_{G}(p^{\infty}),\mathcal{H}^{1}_{\mathrm{dR}}(A)^{\mathrm{can}})$
with the following property: its restriction to
$\mathfrak{IG}_{G}(p^{\infty})^{\mathrm{ord}}$ takes values in the subsheaf
$\mathcal{U}$.

###### Proof.

By construction, there is a unique class $[\alpha]^{\mathrm{ord}}\in
H^{0}(\mathfrak{IG}_{G}(p^{\infty})^{\mathrm{ord}},\mathcal{H}^{1}_{\mathrm{dR}}(A)^{\mathrm{can}})$
lifting the restriction of $[\alpha]$ and taking values in the unit-root
subsheaf $\mathcal{U}$. Since $\mathfrak{IG}_{G}(p^{\infty})^{\mathrm{ord}}$
is dense in $\mathfrak{IG}_{G}(p^{\infty})$, it follows that $[\alpha]^{ur}$
is unique if it exists.

Let $\mathfrak{IG}_{G}(p^{\infty})^{\circ}$ denote the preimage in
$\mathfrak{IG}_{G}(p^{\infty})$ of the open Shimura variety $Y_{G}\pmod{p}$.
Since the complement of
$\mathfrak{IG}_{G}(p^{\infty})^{\circ}\cup\mathfrak{IG}_{G}(p^{\infty})^{\mathrm{ord}}$
has codimension $\geqslant 2$ in $\mathfrak{IG}_{G}(p^{\infty})$ (and the
sheaf is locally free), it suffices to find a second section
$[\alpha]^{\circ}$ over $\mathfrak{IG}_{G}(p^{\infty})^{\circ}$ which
coincides with $[\alpha]^{\mathrm{ord}}$ where both are defined.

We shall carry this out using a generalisation of the construction of the
unit-root splitting given in [Iov00]. Recall that if $A/S$ is an abelian
scheme over an arbitrary base $S$, then there exists a _universal vectorial
extension_ of $A$, which is universal among short exact sequences of $S$-group
schemes

$0\to V\to I\to A\to 0$

where $V$ is a vector group. Moreover, the space of invariant differentials
$\operatorname{Inv}(I/S)$ is isomorphic to $H^{1}_{\mathrm{dR}}(A/S)$, with
the subspace $\operatorname{Inv}(A/S)$ corresponding to
$\operatorname{Fil}^{1}$. Taking associated formal groups, we have a short
exact sequence $0\to\hat{V}\to\hat{I}\to\hat{A}\to 0$.

If we are given a morphism
$\alpha:\hat{\mathbf{G}}_{m}\hookrightarrow\hat{A}$, and $\hat{I}_{\alpha}$
denotes the pullback of this extension along $\alpha$, we have a diagram of
exact sequences

${0}$${\hat{V}}$${\hat{I}}$${\hat{A}}$${0}$${0}$${\hat{V}}$${\hat{I}_{\alpha}}$${\hat{\mathbf{G}}_{m}}$${0}$

However, any extension of a formal multiplicative group by an additive one
must be split, so there is a (necessarily unique) map
$\hat{\mathbf{G}}_{m}\to\hat{I}_{\alpha}$ splitting the lower sequence. Thus
the composite

$v_{\alpha}:\operatorname{Inv}(I/S)\to\operatorname{Inv}(I_{\alpha}/S)\to\operatorname{Inv}\left(\hat{\mathbf{G}}_{m}/S\right)$

is an extension of the pullback map
$\operatorname{Inv}(A/S)\to\operatorname{Inv}\left(\hat{\mathbf{G}}_{m}/S\right)\cong\mathcal{O}_{S}$
to $H^{1}_{\mathrm{dR}}(A/S)$. Moreover, if $p$ is topologically nilpotent on
$S$ and $A$ is ordinary, then $\hat{A}$ is itself of multiplicative type.
Hence we have a splitting of the top row, which is clearly compatible with the
splitting of the bottom row; and the main result of [Iov00] shows that this
construction recovers the unit-root splitting. This gives the required section
$[\alpha]^{\circ}$. ∎

######  Notation 6.9.

Let $Q$ be the subgroup
$P_{\operatorname{S}}\cap\overline{P}_{\operatorname{Kl}}$ of $G$, where
$P_{\operatorname{S}}$ is the standard Siegel parabolic subgroup and
$\overline{P}_{\operatorname{Kl}}=J^{-1}P_{\operatorname{Kl}}J$ is the lower-
triangular Klingen.

Note that $T$ is a Levi subgroup of $Q$, so any weight
$\lambda(r_{1},r_{2};c)$ can be regarded as a representation of $Q$.

######  Definition 6.10.

Let $\mathcal{T}^{\operatorname{Kl}}$ denote the sheaf over
$\mathfrak{X}_{G,\operatorname{Kl}}^{\geqslant 1}(p^{\infty})$ parametrising
bases $f_{1},\dots,f_{4}$ of $\mathcal{H}^{1}_{\mathrm{dR}}(A)^{\mathrm{can}}$
compatible with the filtration and polarisation, and with the additional
property that the pullback of $f_{4}$ to
$\mathfrak{IG}_{G,\operatorname{Kl}}^{\geqslant 1}(p^{\infty})$ is a scalar
multiple of $[\alpha]^{\mathrm{ur}}$. This is is a reduction of structure of
the $P_{\operatorname{S}}$-torsor $\mathcal{T}$ to a $Q$-torsor over
$\mathfrak{X}_{G,\operatorname{Kl}}^{\geqslant 1}(p^{\infty})$.

One can interpret the comparison morphism of §3.2 using this torsor. Let
$k_{1}\geqslant k_{2}$ be integers. We shall see in Lemma 6.11(i) below that
the inclusion of the highest weight space $\lambda(k_{1},k_{2})$ into
$W_{G}(k_{1},k_{2})$ has a $Q$-equivariant splitting. If we pull back further
to $\mathfrak{IG}_{G}(p^{\infty})$, the sheaf $[\lambda(k_{1}-k_{2},0)]$ has a
canonical trivialisation, so we obtain a morphism of sheaves

$[W_{G}(k_{1},k_{2})]\to[W_{G}(k_{2},k_{2})]=(\mathop{det}\nolimits\omega_{A})^{k_{2}}.$

Pushing back down to $\mathfrak{X}_{G,\operatorname{Kl}}^{\geqslant 1}(p)$,
this gives us a morphism

$[W(k_{1},k_{2})]\to\left(\pi_{\star}\mathcal{O}_{\mathfrak{IG}_{G}(p^{\infty})}\right)^{\Gamma=k_{1}-k_{2}}\otimes(\mathop{det}\nolimits\omega_{A})^{k_{2}},$

which is the sought-after comparison morphism. The advantage of this new
interpretation is that we can easily see how to extend the comparison morphism
to larger coefficient sheaves, using some simple Lie-theoretic computations.

######  Lemma 6.11.

  1. (i)

Let $k_{1}\geqslant k_{2}$. Then there is a unique (up to scalars)
$Q$-equivariant map

$W(k_{1},k_{2};c)\to\lambda(k_{1},k_{2};c).$

  2. (ii)

Let $r_{1}\geqslant r_{2}\geqslant 0$. Then there is a unique (up to scalars)
$Q$-equivariant map

$V(r_{1},r_{2};r_{1}+r_{2})\to\lambda(r_{1},-r_{2};r_{1}+r_{2}).$

This map factors through $V/\operatorname{Fil}^{r_{1}+1}V$.

  3. (iii)

The restriction of the homomorphism (ii) to $\operatorname{Gr}^{r_{1}}V$ is
non-trivial, and factors as the composite of the projection from
$\operatorname{Gr}^{r_{1}}V$ to its unique $M_{\operatorname{S}}$-summand
isomorphic to $W(r_{1},-r_{2};r_{1}+r_{2})$, composed with the map from (i)
for this subrepresentation.

###### Proof.

Part (i) is clear, since the image of $Q$ in the Levi quotient of
$P_{\operatorname{S}}$ is the lower-triangular Borel. For part (ii), we argue
analogously, using the fact that $Q$ is contained in a Weyl-group conjugate of
the standard Borel subgroup of $G$, and $\lambda(r_{1},-r_{2};c)$ is the
lowest-weight vector for this conjugate Borel. The compatibility (iii) is
clear by considering the weights appearing in each factor. ∎

######  Proposition 6.12.

The morphism of $Q$-representations

$\operatorname{Gr}^{r_{1}}V\otimes W(3,1;0)\longrightarrow
W(r_{1}+3,1-r_{2};r_{1}+r_{2})\longrightarrow\lambda(r_{1}+3,1-r_{2};r_{1}+r_{2})$

has a unique $Q$-equivariant extension to the whole of $\tilde{L}_{1}$.

###### Proof.

This is an easy consequence of Lemma 6.11(iii). ∎

In terms of sheaves, this gives the following.

######  Definition 6.13.

For $r_{1}\geqslant r_{2}\geqslant 0$, we define a map

$[\widetilde{L}_{1}]\to\left(\pi_{\star}\mathcal{O}_{\mathfrak{IG}_{G}(p^{\infty})}\right)^{\Gamma=r_{1}+r_{2}+2}\otimes(\mathop{det}\nolimits\omega_{A})^{1-r_{2}}$

using the $Q$-equivariant morphism of Proposition 6.12.

######  Remark 6.14.

If $r_{2}=0$, so that
$[V]=\operatorname{Sym}^{r_{1}}\mathcal{H}^{1}_{\mathrm{dR}}(A)$, this
morphism has a simple explicit description: it is given by restriction to
$\operatorname{Sym}^{r_{1}}\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}_{1})$ and
the trivialisation $\mathcal{G}_{1}\cong\mu_{p^{\infty}}$ over
$\mathfrak{IG}_{G}(p^{\infty})$.

######  Corollary 6.15 (“Unit root splitting”).

For any $r_{1}\geqslant r_{2}\geqslant 0$, we have a diagram of cohomology
groups

${H^{1}\left(\mathfrak{X}_{G,\operatorname{Kl}}(p)^{\geqslant
1},[\tilde{L}_{1}](-D)\right)}$${H^{1}\left(\mathfrak{X}_{G,\operatorname{Kl}}(p)^{\geqslant
1},[L_{1}](-D)\right)}$${H^{1}\left(\mathfrak{X}_{G,\operatorname{Kl}}(p)^{\geqslant
1},\mathfrak{F}(r_{1}+3,1-r_{2})(-D)\right).}$

###  6.5. Branching to $H$

Recall that we have defined a homomorphism of formal schemes

$\iota_{\infty}:\mathfrak{X}_{H}^{\mathrm{ord}}(p^{\infty})\to\mathfrak{X}_{G,\operatorname{Kl}}^{\geqslant
1}(p^{\infty}).$

The space $\mathfrak{X}_{H}^{\mathrm{ord}}(p^{\infty})$ classifies pairs of
_ordinary_ elliptic curves $(E_{1},E_{2})$ with prime-to-$p$ level structure
and isomorphisms

$\gamma:E_{1}[p^{\infty}]^{\circ}\xrightarrow{\ \cong\
}E_{2}[p^{\infty}]^{\circ}.$

Hence there is a reduction of
$\iota_{\infty}^{*}(\mathcal{T}^{\operatorname{Kl}})$ to a torsor for the
group $\\{\operatorname{diag}(x,x,y,y):x,y\in\mathcal{O}^{\times}\\}$.

######  Theorem 6.16.

Let $V_{H}$ be a small subrepresentation of $V$ given by a pair
$(t_{1},t_{2})$ as above. Then the composite

$H^{0}(\mathfrak{X}_{H}^{\mathrm{ord}},\omega^{1,1}\otimes[V_{H}](-D))\to
H^{1}(\mathfrak{X}_{G}^{\geqslant 1},\tilde{L}_{1}(-D))\to
H^{1}\left(\mathfrak{X}_{G}^{\geqslant
1},\mathfrak{F}(r_{1}+3,1-r_{2})(-D)\right)$

coincides with the composite of the unit-root splitting

$H^{0}(\mathfrak{X}_{H}^{\mathrm{ord}},\omega^{1,1}\otimes[V_{H}](-D))\to
H^{0}(\mathfrak{X}_{H}^{\mathrm{ord}},\omega^{(t_{1}+1,t_{2}+1)}(-D))$

and the pushforward

$H^{0}(\mathfrak{X}_{H}^{\mathrm{ord}},\omega^{(t_{1}+1,t_{2}+1)}(-D))\to
H^{1}(\mathfrak{X}_{G}^{\geqslant 1},L_{1}(-D))\to
H^{1}\left(\mathfrak{X}_{G}^{\geqslant
1},\mathfrak{F}(r_{1}+3,1-r_{2})(-D)\right).$

###### Proof.

We need to compare two maps of sheaves on
$\mathfrak{X}_{H}^{\mathrm{ord}}(p^{\infty})$. The first is the composite

$V_{H}(t_{1},t_{2})\to\iota_{\infty}^{*}(V/\operatorname{Fil}^{r_{1}+1})\to\iota_{\infty}^{*}\left([\lambda(r_{1},-r_{2})]\right),$

where the second map is given by the splitting of Corollary 6.15. The second
is the map

$V_{H}(t_{1},t_{2})\to\omega^{t_{1},t_{2}}\to\iota^{*}\left([\lambda(r_{1},-r_{2})]\right),$

where the first map comes from the unit-root splitting on $H$. Via our torsor
formalism we are reduced to checking the equality of two morphisms of
representations of the group $\\{\left(\begin{smallmatrix}x\\\ &x\\\ &&y\\\
&&&y\end{smallmatrix}\right):x,y\in\mathcal{O}^{\times}\\}$, and this is
obvious. ∎

######  Corollary 6.17.

Theorem 5.13 holds as stated if we assume only that $\phi(\mathcal{E})$ is the
image under the $p$-adic unit-root splitting of a classical nearly-holomorphic
modular form.∎

######  Remark 6.18.

If $V_{H}$ is any irreducible $H$-subrepresentation of $V$ (not necessarily
small), then the same argument as above extends to show that the composite
$H^{0}(\mathfrak{X}_{H}^{\mathrm{ord}},\omega^{1,1}\otimes[V_{H}](-D))\to
H^{1}(\mathfrak{X}_{G}^{\geqslant 1},\tilde{L}_{1}(-D))\to
H^{1}\left(\mathfrak{X}_{G}^{\geqslant
1},\mathfrak{F}(r_{1}+3,1-r_{2})(-D)\right)$ factors through projection to
$\omega^{1,1}\otimes[\operatorname{Gr}^{r_{1}}V_{H}]$, regarded as a subsheaf
of $[V_{H}]$ over $\mathfrak{X}_{H}^{\mathrm{ord}}$ via the unit root
splitting. The small subrepresentations are precisely those where
$\operatorname{Gr}^{r_{1}}V_{H}$ is the smallest nonzero filtration step.

##  7\. Families of Eisenstein series

The theory developed in the previous sections allows us to define $p$-adic
measures interpolating the cohomological periods
$\langle\iota_{m,\star}(E),\eta\rangle$, as $E$ varies over the
specialisations of some family of cuspidal $p$-adic modular forms
$\mathcal{E}$ on $H$. In this section, we shall specify the particular
families $\mathcal{E}$ which we shall consider, which will be built up from
Eisenstein series and cusp forms for $\operatorname{GL}_{2}$. We shall prove
in the remaining sections that the resulting periods
$\langle\iota_{m,\star}(E),\eta\rangle$ are special values of $L$-functions.

###  7.1. Non-holomorphic Eisenstein series

Let $\Phi_{\mathrm{f}}\in\mathcal{S}(\mathbf{A}_{\mathrm{f}}^{2},\mathbf{C})$
be a Schwartz function.

######  Definition 7.1.

For $k\geqslant 1$, $\tau=x+iy$ in the upper half-plane, and $s\in\mathbf{C}$
with $\operatorname{Re}(s)\geqslant 1$, we define

$E^{(k,\Phi_{\mathrm{f}})}(\tau;s)\coloneqq\frac{\Gamma(s+\tfrac{k}{2})}{(-2\pi
i)^{k}\pi^{s-\tfrac{k}{2}}}\sum_{(m,n)\in\mathbf{Q}^{2}-(0,0)}\frac{\Phi_{\mathrm{f}}(m,n)y^{(s-\tfrac{k}{2})}}{(m\tau+n)^{k}|m\tau+n|^{2s-k}}.$

This can be extended to all $s\in\mathbf{C}$ by analytic continuation in $s$.

######  Remark 7.2.

If $\Phi_{\mathrm{f}}$ is the indicator function of
$(0,\alpha)+\hat{\mathbf{Z}}^{2}$, this series is
$E^{(k)}_{\alpha}(\tau,s-\tfrac{k}{2})$ in the notation of [LLZ14].

If $\chi$ is a Dirichlet character, and $\widehat{\chi}$ the corresponding
adelic character as in §2.2, we define

$R_{\chi}(\Phi_{\mathrm{f}})\coloneqq\int_{a\in\widehat{\mathbf{Z}}^{\times}}\widehat{\chi}(a)\left(\left(\begin{smallmatrix}a&0\\\
0&a\end{smallmatrix}\right)\cdot\Phi_{\mathrm{f}}\right)\,\mathrm{d}^{\times}a,$

the projection of $\Phi_{\mathrm{f}}$ to the $\widehat{\chi}^{-1}$-isotypical
subspace for $\hat{\mathbf{Z}}^{\times}$, and we set
$E^{(k,\Phi_{\mathrm{f}})}(\tau;\chi,s)\coloneqq
E^{(k,R_{\chi}(\Phi_{\mathrm{f}}))}(\tau;s).$ Note that
$E^{(k,\Phi_{\mathrm{f}})}(-;\chi,s)$, vanishes if $(-1)^{k}\chi(-1)\neq 1$.

We can interpret $E^{(k,\Phi_{\mathrm{f}})}(-;s)$ and
$E^{(k,\Phi_{\mathrm{f}})}(-;\chi,s)$ as $C^{\infty}$ sections of a line
bundle on the $\operatorname{GL}_{2}$ Shimura variety. More precisely, the
$C^{\infty}$ sections of the automorphic line bundle $\omega(k)$ are the
smooth functions
$f:\operatorname{GL}_{2}(\mathbf{A}_{\mathrm{f}})\times\mathcal{H}\to\mathbf{C}$
satisfying

$f(g,\tau)=(ad-bc)(c\tau+d)^{-k}f\left(\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)g,\tfrac{a\tau+b}{c\tau+d}\right)\quad\text{for
all}\left(\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\right)\in\operatorname{GL}_{2}^{+}(\mathbf{Q}).$

With these notations,
$E^{(k,\Phi_{\mathrm{f}})}(s)\coloneqq(g,\tau)\mapsto\|\mathop{det}\nolimits
g\|^{s+1-k/2}E^{(k,g\cdot\Phi)}(\tau;s)$ is a $C^{\infty}$ section of
$\omega(k)$, and similarly with $\chi$; and
$E^{(k,\Phi_{\mathrm{f}})}(\chi,s)$ transforms under the centre of
$\operatorname{GL}_{2}(\mathbf{A}_{\mathrm{f}})$ by
$\widehat{\chi}^{-1}\|\cdot\|^{2-k}$.

###  7.2. Nearly holomorphic specialisations

######  Proposition 7.3.

For integers $j\in[0,k-1]$, the $C^{\infty}$ section
$E^{(k,\Phi_{\mathrm{f}})}(\tfrac{k}{2}-j)$ of $\omega(k)$ is nearly-
holomorphic; and if $\Phi_{\mathrm{f}}$ takes values in a number field $E$,
this section is defined over $E$.

######  Notation 7.4.

We let $\psi$ denote the unique additive character
$\mathbf{A}/\mathbf{Q}\to\mathbf{C}^{\times}$ satisfying
$\psi(x_{\infty})=\exp(-2\pi ix_{\infty})$ for $x_{\infty}\in\mathbf{R}$. For
$\Phi_{\mathrm{f}}\in\mathcal{S}(\mathbf{A}_{\mathrm{f}})$, we write
$\Phi_{\mathrm{f}}^{\prime}(u,v)$ for the Fourier transform in the second
variable only:

$\Phi^{\prime}_{\mathrm{f}}(u,v)\coloneqq\int_{\mathbf{A}_{\mathrm{f}}}\Phi_{\mathrm{f}}(u,w)\psi(vw)\,\mathrm{d}w.$

If we define the “$q$-expansion” of a nearly-holomorphic form to be the
Fourier expansion of its holomorphic part, then for $n>0$ we have

$a_{n}\left(E^{(k,\Phi_{\mathrm{f}})}(\tfrac{k}{2}-j)\right)=\sum_{\begin{subarray}{c}(u,v)\in(\mathbf{Q}^{\times})^{2}\\\
uv=n\end{subarray}}u^{j}v^{(k-1-j)}\operatorname{sgn}(u)\Phi_{\mathrm{f}}^{\prime}(u,v).$

The constant term
$a_{0}\left(E^{(k)}_{\Phi_{\mathrm{f}}}(\tfrac{k}{2}-j)\right)$ is zero unless
$j=0$ or $j=k-1$, in which case it is given by a special value of a linear
combination of Hurwitz zeta functions. For our purposes it suffices to note
that if $\Phi_{\mathrm{f}}(-,0)=\Phi_{\mathrm{f}}(0,-)=0$, then the constant
term is zero for all $j$.

###  7.3. Families of p-adic Eisenstein series

We shall now define a family of local Schwartz functions $\Phi_{p,\mu,\nu}$,
depending on a choice of two Dirichlet characters $\mu$ and $\nu$ of $p$-power
conductor.

######  Definition 7.5.

We define $\Phi_{p,\mu,\nu}\in\mathcal{S}(\mathbf{Q}_{p}^{2},\mathbf{C})$ as
the unique function such that

$\Phi^{\prime}_{p,\mu,\nu}(x,y)=\begin{cases}\mu(x)\nu(y)&\text{if
$x,y\in\mathbf{Z}_{p}^{\times}$,}\\\ 0&\text{otherwise.}\end{cases}$

(The values of $\Phi_{p,\mu,\nu}$ can be made explicit in terms of Gauss sums,
but we do not need this.) Note that $\Phi_{p,\mu,\nu}$ transforms under
$\left(\begin{smallmatrix}a&0\\\ 0&d\end{smallmatrix}\right)$,
$a,d\in\mathbf{Z}_{p}^{\times}$, by
$\mu(a)\nu^{-1}(d)=\widehat{\mu}^{-1}(a)\widehat{\nu}(d)$.

Now let $\Phi^{(p)}$ be a Schwartz function on
$(\mathbf{A}_{\mathrm{f}}^{p})^{2}$ with values in a number field $E$, and
$\chi^{(p)}$ a Dirichlet character of prime-to-$p$ conductor such that
$\Phi^{(p)}$ satisfies $\left(\begin{smallmatrix}a&0\\\
0&a\end{smallmatrix}\right)\cdot\Phi^{(p)}=\widehat{\chi}^{(p)}(a)^{-1}\Phi^{(p)}$
for $a\in(\widehat{\mathbf{Z}}^{(p)})^{\times}$.

Let $L$ be the completion of $E$ at some prime above $p$. If $\mu,\nu$ are
Dirichlet characters of $p$-power conductor as above, and we set
$\chi=\chi^{(p)}\mu\nu^{-1}$, then the function
$\Phi_{\mu,\nu}=\Phi^{(p)}\Phi_{p,\mu,\nu}$ is in the image of the idempotent
$R_{\chi}$.

######  Theorem 7.6.

Let $R=\Lambda_{L}(\mathbf{Z}_{p}^{\times}\times\mathbf{Z}_{p}^{\times})$,
with its two canonical characters $\kappa_{1},\kappa_{2}$. For each
$\Phi^{(p)}$ taking values in $L$, there exists an element
$\mathcal{E}^{\Phi^{(p)}}\left(\kappa_{1},\kappa_{2};\chi^{(p)}\right)\in\mathcal{S}_{\kappa_{1}+\kappa_{2}+1}(R)$,
whose specialisation at $(a+\mu,b+\nu)$, for integers $a,b\geqslant 0$, is the
$p$-adic modular form associated to the algebraic nearly-holomorphic form

$(g,\tau)\mapsto\widehat{\nu}(\mathop{det}\nolimits g)^{-1}\cdot
E^{(a+b+1,\Phi_{\mu,\nu})}\left(g,\tau;\chi^{(p)}\mu\nu^{-1},\tfrac{b-a+1}{2}\right)\in
M_{a+b+1}^{\mathrm{nh}}.$

For $g\in\operatorname{GL}_{2}(\mathbf{A}_{\mathrm{f}}^{p})$, we have
$g\cdot\mathcal{E}^{\Phi^{(p)}}=\|\mathop{det}\nolimits
g\|^{1-\kappa_{1}}\cdot\mathcal{E}_{g\cdot\Phi^{(p)}}$. The $q$-expansion of
$\mathcal{E}^{\Phi^{(p)}}$ at $\infty$ is

$\sum_{\begin{subarray}{c}u,v\in(\mathbf{Z}_{(p)}^{\times})^{2}\\\
uv>0\end{subarray}}\operatorname{sgn}(u)u^{\kappa_{1}}v^{\kappa_{2}}(\Phi^{(p)})^{\prime}(u,v)q^{uv}.$

The Eisenstein series $\mathcal{E}_{\Phi^{(p)}}(-;\chi^{(p)})$ is identically
0 on the components of $\operatorname{Spec}R$ where
$\kappa_{1}(-1)\kappa_{2}(-1)\neq-\chi^{(p)}(-1)$.

This result is essentially a restatement of Katz’s theory of the Eisenstein
measure; we have stated it in a slightly unusual form in order to spell out
precisely the
$\operatorname{GL}_{2}\left(\mathbf{A}_{\mathrm{f}}^{p}\right)$-equivariance
properties of the construction. Note that the factor
$\widehat{\nu}(\mathop{det}\nolimits g)^{-1}$ is required in order that the
right-hand side be invariant under $\left(\begin{smallmatrix}1&*\\\
&*\end{smallmatrix}\right)\subset\operatorname{GL}_{2}(\mathbf{Z}_{p})$, which
is a prerequisite for it to be the specialisation of a $p$-adic modular form.

###  7.4. Input to the machine: the case of $\operatorname{GSp}_{4}$

Suppose we are given an automorphic representation $\Pi$ as before, which is
globally generic and cohomological with coefficients in $V(r_{1},r_{2})$, and
unramified and Klingen-ordinary at $p$. We set $d=r_{1}-r_{2}\geqslant 0$, and
$\chi_{\Pi}$ the Dirichlet character such that $\Pi$ has central character
$\widehat{\chi}_{\Pi}$.

Let $R=\Lambda_{L}(\mathbf{Z}_{p}^{\times}\times\mathbf{Z}_{p}^{\times})$ with
its two canonical characters $\mathbf{j}_{1},\mathbf{j}_{2}$. Given
$\Phi_{1}^{(p)},\Phi_{2}^{(p)}$, we consider the family of $p$-adic modular
forms for $H$ given by

$\mathcal{E}^{\Phi_{1}^{(p)}}(d-\mathbf{j}_{1},\mathbf{j}_{2};\chi_{\Pi})\boxtimes\mathcal{E}^{\Phi_{2}^{(p)}}(0,\mathbf{j}_{1}-\mathbf{j}_{2};\mathrm{id}).$

By construction, the specialisation of this family at
$(a_{1}+\rho_{1},a_{2}+\rho_{2})$, for integers $a_{1},a_{2}$ such that
$d\geqslant a_{1}\geqslant a_{2}\geqslant 0$ and Dirichlet characters
$\rho_{i}$ of $p$-power conductor, is the product of two Eisenstein series of
the form

$\widehat{\nu}_{i}(\mathop{det}\nolimits
g)^{-1}E^{(k_{i},\Phi_{i})}(\chi_{i},s_{i}),$

where the parameters are given by

$\displaystyle k_{1}$ $\displaystyle=1+d-a_{1}+a_{2},$ $\displaystyle k_{2}$
$\displaystyle=1+a_{1}-a_{2},$ $\displaystyle\chi_{1}$
$\displaystyle=\rho_{1}^{-1}\rho_{2}^{-1}\chi_{\Pi},$ $\displaystyle\chi_{2}$
$\displaystyle=\rho_{1}^{-1}\rho_{2},$ $\displaystyle s_{1}$
$\displaystyle=\tfrac{1-d+a_{1}+a_{2}}{2},$ $\displaystyle s_{2}$
$\displaystyle=\tfrac{1+a_{1}-a_{2}}{2}.$

The Schwartz functions $\Phi_{\mathrm{f},i}$ are given by
$\Phi_{\mathrm{f},i}=\Phi_{i}^{(p)}\times\Phi_{p,\mu_{i},\nu_{i}}$, where
$\Phi_{i}^{(p)}$ are chosen arbitrarily, and the characters $\mu_{i},\nu_{i}$
are given by

$\displaystyle\mu_{1}$ $\displaystyle=\rho_{1}^{-1},$ $\displaystyle\mu_{2}$
$\displaystyle=\mathrm{id},$ $\displaystyle\nu_{1}$ $\displaystyle=\rho_{2},$
$\displaystyle\nu_{2}$ $\displaystyle=\rho_{1}\rho_{2}^{-1}.$

######  Remark 7.7.

Note that our choices are made such that the Eisenstein series on the second
factor of $H$ is holomorphic, although the one on the first factor is only
nearly-holomorphic in general.

We also choose a vector $\eta\in H^{2}(\Pi_{\mathrm{f}})$, lying in the
ordinary $\mathcal{U}_{p,\operatorname{Kl}}^{\prime}$-eigenspace at level
$\operatorname{Kl}(p)$. From the theory of §4, we obtain a measure
$\mathcal{L}\in R$, whose specialisation at $(a_{1}+\rho_{1},a_{2}+\rho_{2})$
interpolates the period integral of $\eta$ against the specialisation of the
above family.

We shall see in the following chapters that if
$(-1)^{a_{1}}\rho_{1}(-1)\neq(-1)^{a_{2}}\rho_{2}(-1)$, this period integral
will be (up to various elementary factors) equal to the product of $L$-values

$L\left(\Pi\otimes\rho_{1}^{-1},\frac{1-d}{2}+a_{1}\right)\cdot
L\left(\Pi\otimes\rho_{2}^{-1},\frac{1-d}{2}+a_{2}\right).$

On the other hand, if $(-1)^{a_{1}}\rho_{1}(-1)=(-1)^{a_{2}}\rho_{2}(-1)$,
then both of the Eisenstein series will be identically 0. This shows that we
can only interpolate products of twisted $L$-values of opposite parity, a
condition which is familiar from the setting of Kato’s $\operatorname{GL}_{2}$
Euler system.

###  7.5. Input to the machine: the case of
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$

Now let us suppose we are given an auxiliary automorphic representation
$\sigma$ of $\operatorname{GL}_{2}$, corresponding to a holomorphic modular
form of weight $\ell\geqslant 1$, and suppose that

$d^{\prime}=r_{1}-r_{2}+1-\ell\geqslant 0.$

For $\lambda$ a holomorphic modular form in the space of $\sigma$, we consider
the family of $p$-adic modular forms over
$R=\Lambda_{L}(\mathbf{Z}_{p}^{\times})$ given by

$\mathcal{E}^{\Phi^{(p)}}(d^{\prime}-\mathbf{j},\mathbf{j};\chi_{\Pi}\chi_{\sigma})\boxtimes\lambda.$

Note that at $\mathbf{j}=a+\rho$ with $0\leqslant a\leqslant d^{\prime}$, this
specialises to

$\widehat{\rho}(\mathop{det}\nolimits
h)^{-1}E^{(\Phi_{\mathrm{f}},d^{\prime}+1)}\left(h_{1};\rho^{-2}\chi_{\Pi}\chi_{\sigma},\tfrac{1-d^{\prime}}{2}+a\right)\lambda(h_{2})$

where $\Phi_{\mathrm{f}}=\Phi^{(p)}\Phi_{(p,\rho^{-1},\rho)}$. We shall see in
the next chapter that the cup-product of this series with $\eta$ gives a zeta-
integral computing

$L\left(\Pi\otimes\sigma\otimes\rho^{-1},\tfrac{1-d^{\prime}}{2}+a\right).$

######  Remark 7.8.

One checks that for $\rho$ any Dirichlet character, the critical values of
$L(\Pi\otimes\rho,s)$ are exactly the values $s=\tfrac{1-d}{2}+a$, for
integers $0\leqslant a\leqslant d$. Similarly, the critical values of
$L(\Pi\otimes\sigma\otimes\rho,s)$ are the $s=\tfrac{1-d^{\prime}}{2}+a$ for
$0\leqslant a\leqslant d^{\prime}$. So in each case we hit the full interval
of critical values of the relevant $L$-function (and not any others!).

If we have $\ell\geqslant r_{1}-r_{2}+2$, then the $L$-function for
$\Pi\otimes\sigma$ may still have some critical values; but we do not see them
by this method.

##  8\. Integral formulae for L-functions I: local theory

In this section and the next, we recall a general formula which will be used
to relate the cohomological periods studied above to critical values of
$L$-functions, based on work of Novodvorsky and others. In this section, we
let $F$ be an arbitrary local field, and $\psi_{F}$ a non-trivial additive
character $F\to\mathbf{C}^{\times}$. We use $\psi_{F}$ to define a character
$\psi:N(F)\to\mathbf{C}$ by by

$\psi(n)=\psi_{F}(x+y),\ \ n=\left(\left(\begin{smallmatrix}1&x&*&*\\\
&1&y&*\\\ &&1&-x\\\ &&&1\end{smallmatrix}\right)\right).$

###  8.1. Siegel sections

If $\Phi$ is a Schwartz function on $F^{2}$, $\chi$ a smooth unitary character
of $F^{\times}$, and $g\in\operatorname{GL}_{2}(F)$, we define

$f^{\Phi}(g;\chi,s)\coloneqq|\mathop{det}\nolimits
g|^{s}\int_{F^{\times}}\Phi((0,a)g)\chi(a)|a|^{2s}\mathrm{d}^{\times}a.$

This integral converges absolutely for $\operatorname{Re}(s)>0$ and defines an
element of the principal series representation
$I(|\cdot|^{s-\frac{1}{2}},\chi^{-1}|\cdot|^{\frac{1}{2}-s})$ of
$\operatorname{GL}_{2}(F)$, the normalized induction of the representation
$\left(\begin{smallmatrix}x&*\\\
&y\end{smallmatrix}\right)\mapsto|x/y|^{s-\frac{1}{2}}\chi(y)^{-1}$ of the
Borel. It has meromorphic continuation in $s$; if $v$ is a finite place, it is
even a rational function of $q^{s}$ (where $q$ is the cardinality of the
residue field, as usual).

######  Remark 8.1.

Note that this is an _un-normalised_ Siegel section, and hence is not
necessarily entire.

We define the _Whittaker transform_ $W^{\Phi}(g;\chi,s)$ of
$f^{\Phi}(g;\chi,s)$ to be the function

$W^{\Phi}(g;\chi,s)=\int_{F}f^{\Phi}\left(\eta\left(\begin{smallmatrix}1&x\\\
0&1\end{smallmatrix}\right)g;\chi,s\right)\psi_{F}(x)\mathrm{d}x,\ \
\eta=\left(\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\right).$

If $s$ is such that
$I(|\cdot|^{s-\frac{1}{2}},\chi^{-1}|\cdot|^{\frac{1}{2}-s})$ is irreducible,
then every vector in this representation is $f^{\Phi}$ for some $\Phi$, and
the map $f^{\Phi}\mapsto W^{\Phi}$ gives the isomorphism from
$I(|\cdot|^{s-\frac{1}{2}},\chi^{-1}|\cdot|^{\frac{1}{2}-s})$ to its Whittaker
model (with respect to the inverse character $\bar{\psi}_{F}$).

For fixed $g$ and $\Phi$, $W^{\Phi}(g;\chi,s)$ is entire as a function of $s$;
if $v$ is nonarchimedean, $\chi$ and $\psi_{F}$ are unramified, and $\Phi$ is
the characteristic function of $\mathcal{O}_{F}^{2}$, then
$W^{\Phi}(1;\chi,s)$ is identically 1. These functions satisfy a functional
equation: if $\hat{\Phi}$ is the 2-variable Fourier transform defined by

$\hat{\Phi}(x,y)=\int_{F^{2}}\Phi(u,v)\psi_{F}(xv-
yu)\,\mathrm{d}u\,\mathrm{d}v,$

then we compute that

(8.1) $W^{\hat{\Phi}}(g;\chi^{-1},1-s)=\chi(\mathop{det}\nolimits
g)W^{\Phi}(g;\chi,s).$

######  Remark 8.2.

The values of $W^{\Phi}$ on the diagonal torus can be given in terms of the
“partial” Fourier transform

(8.2) $\Phi^{\prime}(x,y)\coloneqq\int_{F}\Phi(x,w)\psi_{F}(yw)\,\mathrm{d}w.$

With this notation, we have

$W^{\Phi}(\left(\begin{smallmatrix}x&\\\
&y\end{smallmatrix}\right);\chi,s)=\chi(-1)|x|^{s}|y|^{s-1}\int_{a}\Phi^{\prime}\left(xa,\tfrac{1}{ya}\right)\chi(a)|a|^{2s-1}\,\mathrm{d}^{\times}a.$

###  8.2. Definition of the local integrals

Let $\pi$ be an irreducible smooth representation of $G(F)$, with unitary
central character $\chi_{\pi}$. We assume $\pi$ is _generic_ , i.e. that $\pi$
is isomorphic to a space of functions $G(F)\to\mathbf{C}$ satisfying
$W(ng)=\psi(n)W(g)$ for $n\in N(F)$. Such a model is unique (see [Rod73, Thm.
3] if $F$ is nonarchimedean, and [Wal83, Thm. 8.8(1)] if $F$ is archimedean).
We fix a choice of isomorphism between $\pi$ and its Whittaker model, and for
$\varphi\in\pi$, we write $W_{\varphi}$ for the corresponding Whittaker
function.

#### The two-parameter $\operatorname{GSp}_{4}$ integral

Let $\chi_{1},\chi_{2}$ be smooth characters of $F^{\times}$ such that
$\chi_{1}\chi_{2}=\chi_{\pi}$.

######  Definition 8.3.

We define

$Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})\coloneqq\int_{Z_{G}(F)N_{H}(F)\backslash
H(F)}W_{\varphi}(h)f^{\Phi_{1}}(h_{1};\chi_{1},s_{1})W^{\Phi_{2}}(h_{2};\chi_{2},s_{2})\,\mathrm{d}h,$

where $\varphi\in\pi$ and each $\Phi_{i}$ is a Schwartz function on $F^{2}$.

Substituting in the definition of $W^{\Phi_{2}}(-)$ as an integral, we obtain
the following alternative formula:

######  Proposition 8.4.

For $\varphi\in\pi$, the integral

$B_{\varphi}(g,s)\coloneqq\int_{F^{\times}}\int_{F}W_{\varphi}\left(\left(\begin{smallmatrix}a\\\
&a\\\ &x&1\\\
&&&1\end{smallmatrix}\right)w_{2}g\right)|a|^{s-\tfrac{3}{2}}\chi_{2}^{-1}(a)\,\mathrm{d}x\,\mathrm{d}^{\times}a,\qquad
w_{2}=\left(\begin{smallmatrix}1\\\ &&1\\\ &-1\\\
&&&1\end{smallmatrix}\right)$

converges for $\operatorname{Re}(s)\gg 0$ and has meromorphic continuation to
all values of $s$; and we have

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

where $D$ denotes the torus $\\{\operatorname{diag}(p,q,p,q):p,q\in
F^{\times}\\}$.

The function $B_{\varphi}(g;s)$ is an element of the _Bessel model_ of the
representation $\pi$: it transforms by a character under left-translation by
the Bessel subgroup $DN_{\operatorname{S}}$. On the other hand, the integral
defining $B_{\varphi}$ in terms of $W_{\varphi}$ is an instance of
Novodvorsky’s local zeta integral for $L(\pi\otimes\chi_{2}^{-1},s)$ [Nov79].
These two facts will be crucial in our analysis of the two-parameter integral
$Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})$.

#### The $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ integral

Similarly, if $\pi$ is as before and $\sigma$ is a generic representation of
$\operatorname{GL}_{2}(F)$, we let $\chi=\chi_{\pi}\chi_{\sigma}$ and define

$Z(\varphi,\lambda,\Phi,s)\coloneqq\int_{Z_{G}(F)N_{H}(F)\backslash
H(F)}W_{\varphi}(h)f^{\Phi}(h_{1};\chi,s)W_{\lambda}(h_{2})\,\mathrm{d}h,$

where $\varphi\in\pi$ and $\lambda\in\sigma$, and $W_{\lambda}$ denotes the
image of $\lambda$ in the Whittaker model of $\sigma$, again with respect to
the opposite character $\bar{\psi}_{F}$.

######  Remark 8.5.

Note that if $\chi_{2}|\cdot|^{2s_{2}-1}\notin\\{|\cdot|^{\pm 1}\\}$ (so that
$\sigma=I(|\cdot|^{s_{2}-1/2},|\cdot|^{1/2-s_{2}}\chi_{2}^{-1})$ is
irreducible), we can regard the first integral as a special case of the
second, taking $\lambda=f^{\Phi_{2}}(-;\chi_{2},s_{2})$. However, it is
convenient to treat the first integral separately in order to understand the
reducible cases, and the variation in $s_{2}$.

Essentially the same analysis of the local zeta integrals as in [Sou84] and
[GPSR87] yields the following:

######  Proposition 8.6.

  * (i)

There exists a positive real number $R>0$, depending on $\pi$ and $\sigma$,
such that the local zeta integral $Z(\varphi,\lambda,\Phi,s)$ converges
absolutely for $\operatorname{Re}(s)>R$, for all choices of
$(\varphi,\lambda,\Phi)$. Furthermore, each $Z(\varphi,\lambda,\Phi,s)$ has a
meromorphic continuation in $s$, and is a rational function of $q^{s}$ if $F$
is nonarchimedean.

  * (ii)

There exists a positive real number $R>0$, depending on $\pi$ and $s_{2}$,
such that the local zeta integral $Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})$
converges absolutely for $\operatorname{Re}(s_{1})>R$, for all choices of
$(\varphi,\Phi_{1},\Phi_{2})$. Furthermore, each
$Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})$ extends to a meromorphic function
of $(s_{1},s_{2})\in\mathbf{C}^{2}$, and is a rational function of
$(q^{s_{1}},q^{s_{2}})$ if $F$ is nonarchimedean.∎

###  8.3. Nonarchimedean $L$-factors

In this section we assume $F$ is nonarchimedean.

######  Definition 8.7.

Let $L(\pi,s)$ and $L(\pi\otimes\sigma,s)$ denote the local $L$-factors
associated to $\pi$ and to $\pi\otimes\sigma$ via Shahidi’s method, as in
[GT11a, §4].

By construction, these $L$-factors coincide with the Artin $L$-factors of the
Weil–Deligne representations $\mathrm{rec}(\pi)$ and
$\mathrm{rec}(\pi)\otimes\mathrm{rec}(\sigma)$ respectively, where
“$\mathrm{rec}$” denotes the local Langlands correspondence of _op.cit._.

######  Theorem 8.8.

(i) The vector space of functions on $\mathbf{C}^{2}$ spanned by the
$Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})$, as the data
$(\varphi,\Phi_{1},\Phi_{2})$ vary, is a fractional ideal of
$\mathbf{C}[q^{\pm s_{1}},q^{\pm s_{2}}]$ containing the constant functions.
This fractional ideal of $\mathbf{C}[q^{\pm s_{1}},q^{\pm s_{2}}]$ is
generated by the product of $L$-factors

$L\left(\pi,s_{1}+s_{2}-\tfrac{1}{2}\middle)L\middle(\pi\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2}\right).$

(ii) If $\pi$, $\chi_{i}$ and $\psi_{F}$ are unramified, $\varphi^{0}\in\pi$
is the unique spherical vector such that $W_{\varphi^{0}}(1)=1$, and
$\Phi^{0}_{1}=\Phi^{0}_{2}=\operatorname{ch}(\mathcal{O}_{F}^{2})$, then

$Z(\varphi^{0},\Phi^{0}_{1},\Phi^{0}_{2},s_{1},s_{2})=L\left(\pi,s_{1}+s_{2}-\tfrac{1}{2}\middle)L\middle(\pi\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2}\right).$

###### Proof.

It suffices to prove the analogous result result for fractional ideals in
$\mathbf{C}[q^{\pm s_{1}}]$, for a fixed value of $s_{2}$. It follows from the
computations of Takloo-Bighash [TB00] that the “lowest common denominator” of
Novodvorsky’s zeta integrals agrees with Shahidi’s definition of the
$L$-factor. That is, if we define

$\widetilde{B}_{\varphi}(g)=L(\pi\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2})^{-1}\cdot
B_{\varphi}(g),$

then $\widetilde{B}_{\varphi}(g)\in\mathbf{C}[q^{\pm s_{1}}]$ for all $g$, and
there is no $s_{1}$ such that $\widetilde{B}_{\varphi}(g)$ vanishes for all
$g$ and $\varphi$.

It follows from the theory summarized in [RS16] that the space of functions
$\\{\widetilde{B}_{\varphi}(-):\varphi\in\pi\\}$, is the (unique) _split
Bessel model_ of $\pi$, with respect to the character of the Bessel torus
given by $(|\cdot|^{s_{2}-s_{1}}\chi_{2},|\cdot|^{s_{1}-s_{2}}\chi_{1})$. The
integral $\int_{DN\backslash
H}\widetilde{B}_{\varphi}(h)f^{\Phi_{1}}(h_{1})f^{\Phi_{2}}(h_{2})\,\mathrm{d}h$
is precisely Piatetski-Shapiro’s zeta integral [PS97] for the local $L$-factor
$L(\pi,s)$, with $s=s_{1}+s_{2}-\tfrac{1}{2}$. So to prove (i) we are reduced
to the question of whether the Piaetski-Shapiro’s zeta integral always agrees
with the Shahidi and Novodvorsky definitions of $L(\pi,s)$, independently of
the choice of character used in the definition of the Bessel model. This is
exactly the main result of [RW17].

The unramified computation (ii) now follows from the corresponding computation
of Novodvorsky and Piatetski-Shapiro’s integrals in the unramified case. ∎

For the $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ integral we have a
less complete result:

######  Theorem 8.9.

(i) The vector space of functions on $\mathbf{C}$ spanned by the
$Z(\varphi,\lambda,\Phi,s)$, as the data $(\varphi,\lambda,\Phi)$ vary, is a
fractional ideal of $\mathbf{C}[q^{\pm s}]$ containing the constant functions.
If _either_

  * •

$\sigma$ is principal series, or

  * •

$\pi$ arises from a pair $(\tau_{1},\tau_{2})$ of generic irreducible
representations of $\operatorname{GL}_{2}$ with the same central character
(via the theta-lifting from $\operatorname{GSO}_{2,2}$), and if $\sigma$ is
supercuspidal, then neither of the $\tau_{i}$ is isomorphic to an unramified
twist of $\sigma^{\vee}$,

then this fractional ideal is generated by the $L$-factor
$L(\pi\otimes\sigma,s)$. In particular, this holds if at least one of $\pi$
and $\sigma$ is unramified.

(ii) If $\pi$, $\sigma$ and $\psi_{F}$ are all unramified, $\varphi^{0}\in\pi$
and $\lambda^{0}\in\sigma$ are the spherical vectors normalised such that
$W_{\varphi^{0}}(1)=W_{\lambda^{0}}(1)=1$, and
$\Phi^{0}=\operatorname{ch}(\mathcal{O}_{F}^{2})$, then

$Z(\varphi^{0},\lambda^{0},\Phi^{0},s)=L(\pi\otimes\sigma,s).$

######  Remark 8.10.

The representations of $\operatorname{GSp}_{4}(F)$ which are theta-lifts from
the split orthogonal group
$\operatorname{GSO}_{2,2}(F)\cong(\operatorname{GL}_{2}(F)\times\operatorname{GL}_{2}(F))/\\{(z,z^{-1}):z\in
F^{\times}\\}$ are tabulated in [GT11b]. Note that this class of
representations includes all irreducible principal series, and all
representations irreducibly induced from supercuspidal representations of the
Siegel Levi subgroup.

###### Proof.

Firstly, we note that if $\sigma$ is an irreducible principal series
representation, then after replacing $\pi$ and $\sigma$ with $\pi\otimes\beta$
and $\sigma\otimes\beta^{-1}$ for a suitable character $\beta$, we can arrange
that $\sigma=I(|\cdot|^{s_{2}-1/2},|\cdot|^{1/2-s_{2}}\chi_{2}^{-1})$ for some
$s_{2}$. Then we have
$L(\pi\otimes\sigma,s)=L\left(\pi,s_{1}+s_{2}-\tfrac{1}{2}\middle)L\middle(\pi\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2}\right)$
and part (i) in this case follows from the previous theorem. Part (ii) follows
similarly, noting that the Whittaker transform of
$\Phi^{0}=\operatorname{ch}(\mathcal{O}_{F}^{2})$ satisfies
$W^{\Phi^{0}}(1)=1$, so the two natural normalisations of the spherical test
data coincide.

The assertion concering $\pi$ lifted from $\operatorname{GSO}_{2,2}$ follows
from the results of [Sou84]. Soudry shows that in this case the fractional
ideal of values of $Z(\varphi,\lambda,\Phi,s)$ is generated by the product of
$\operatorname{GL}_{2}\times\operatorname{GL}_{2}$ Rankin–Selberg $L$-factors,
$L(\tau_{1}\otimes\sigma,s)L(\tau_{2}\otimes\sigma,s)$; since the Gan–Takeda
local Langlands correspondence is (by construction) compatible with theta-
lifting from $\operatorname{GSO}_{2,2}$, this Rankin–Selberg $L$-factor
coincides with the Shahidi $L$-factor.

It remains to check that the fractional ideal contains 1 in all cases. This
follows from the “$P_{3}$ theory” of [RS07, §2.5], which can be used to
construct $\varphi$ such that the restriction of $W_{\varphi}$ to the Klingen
parabolic is non-zero, but has arbitrarily small support modulo the centre and
the unipotent radical of the Borel. (Compare the proof of Proposition 2.6.4 of
_op.cit._.) We can then choose the Schwartz function $\Phi$ such that
$f^{\Phi}(h_{1})$ is supported (modulo the Borel of $H$) in a small
neighbourhood of the identity in $\mathbf{P}^{1}$, cf. the proof of Lemma
14.7.5 of [Jac72]. ∎

####  8.3.1. Rationality

Let us suppose $\pi$ is definable over a number field $E$, i.e. we have
isomorphisms $\pi\otimes_{\mathbf{C},\sigma}\mathbf{C}\cong\pi$ for all
$\sigma\in\operatorname{Aut}(\mathbf{C}/E)$. From the uniqueness of the
Whittaker model it follows easily that $\mathcal{W}(\pi)$ is the base-
extension to $\mathbf{C}$ of the $E$-vector space

$\mathcal{W}(\pi)_{E}=\\{W\in\mathcal{W}(\pi):W(g)^{\sigma}=W\left(w(\kappa_{\ell}(\sigma))g\right)\
\forall\sigma\in\operatorname{Aut}(\mathbf{C}/E)\\},$

where $w(x)=\operatorname{diag}(x^{3},x^{2},x,1)\in G(F)$, and $\kappa_{\ell}$
is the $\ell$-adic cyclotomic character. Note that if $\pi$ is unramified, the
normalised spherical vector $W_{\varphi^{0}}$ of Theorem 8.8(ii) is in
$\mathcal{W}(\pi)_{E}$.

######  Proposition 8.11.

In the above setting, suppose $\chi_{1},\chi_{2}$ take values in $E^{\times}$,
and $\chi_{2}$ is unramified. If $W_{\varphi}\in\mathcal{W}(\pi)_{E}$, and
$\Phi_{1},\Phi_{2}$ are $E$-valued, then
$Z(\varphi,\Phi_{1},\Phi_{2};s_{1},s_{2})\in
E[q^{\pm(s_{1}+s_{2})},q^{\pm(s_{1}-s_{2})}]$.

###### Proof.

This is a routine check from the definition of the zeta-integral. ∎

It follows that $L(\pi,s-\tfrac{1}{2})=P(q^{-s})$ for a polynomial $P$ with
coefficients in $E$; and moreover, we can find a vector in
$\mathcal{W}(\pi)_{E}\otimes\mathcal{S}(F^{2},E)\otimes\mathcal{S}(F^{2},E)$
whose image under $Z(-;s_{1},s_{2})$ is exactly
$L\left(\pi,s_{1}+s_{2}-\tfrac{1}{2}\middle)L\middle(\pi\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2}\right)$.
This will be used in §10 below to show that our $L$-values lie in $E$ after
renormalising by a suitable period.

(A similar statement can be formulated for the
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ zeta integral, using
$E$-rational Whittaker models of both $\pi$ and $\sigma$; we leave the details
to the reader.)

###  8.4. Local calculations at $p$

We carry out an evaluation of the local zeta integral for particular choices
of (ramified) data. This calculation will be used to identify the Euler
factors at $p$ in our final formula for the values of the $p$-adic
$L$-function. To simplify notation we assume that $F=\mathbf{Q}_{p}$, and that
$\psi_{F}$ is unramified (i.e. trivial on $\mathbf{Z}_{p}$ but not on
$p^{-1}\mathbf{Z}_{p}$). We also assume that the local representation $\pi$ of
$G(\mathbf{Q}_{p})$ is an unramified principal series.

In this section (_only_) we shall deal exclusively with the Klingen parabolic
$P_{\operatorname{Kl}}$ and its Levi $M_{\operatorname{Kl}}$, so we shall drop
the subscripts and denote them simply by $P$ and $M$. We identify $M$ with
$\operatorname{GL}_{2}\times\operatorname{GL}_{1}$ by

$\left(\begin{smallmatrix}\lambda\\\ &A\\\
&&\mathop{det}\nolimits(A)/\lambda\end{smallmatrix}\right)\mapsto(A,\lambda).$

Under this identification, the modulus character $\delta_{P}$ is the character
of $\operatorname{GL}_{2}\times\operatorname{GL}_{1}$ given by
$(A,\lambda)\mapsto|\lambda^{4}/\mathop{det}\nolimits(A)^{2}|$. (See e.g.
[RS07, §2.2]). In particular, we have

$\delta_{P}(\left(\begin{smallmatrix}\mathop{det}\nolimits(A)\\\ &A\\\
&&1\end{smallmatrix}\right))=|\mathop{det}\nolimits A|^{2}.$

####  8.4.1. The vectors $\phi_{r}\in\pi_{p}$

Since $\pi$ is a principal series representation, it is in particular induced
from the Klingen parabolic. Let us choose an unramified principal series
$\tau$ of $\operatorname{GL}_{2}(\mathbf{Q}_{p})$, and a character
$\theta:\mathbf{Q}_{p}^{\times}\to\mathbf{C}^{\times}$, such that $\pi$ is
isomorphic to the induced representation
$\operatorname{Ind}_{P}^{G}(\tau\boxtimes\theta)$ (normalised induction from
the Klingen parabolic). With these notations, the central character of $\pi$
is given by $\chi_{\pi}=\theta\chi_{\tau}$ (where $\chi_{\tau}$ is the central
character of $\tau$), and the spin $L$-factor of $\pi$ is given by

$L(\pi,s)=L(\tau,s)L(\tau\otimes\theta,s).$

For $r\geqslant 0$, let $K_{r}\subset G(\mathbf{Z}_{p})$ be the depth $r$
Klingen parahoric subgroup:

$K_{r}\coloneqq\\{g\in G(\mathbf{Z}_{p})\ :\ g\bmod{p^{r}}\in
P(\mathbf{Z}_{p}/p^{r}\mathbf{Z}_{p})\\}.$

If we identify the quotient $P(\mathbf{Q}_{p})\backslash G(\mathbf{Q}_{p})$
with $\mathbf{P}^{3}$, then $K_{r}$ is precisely the stabiliser of the point
$(0:0:0:1)\in\mathbf{P}^{3}(\mathbf{Z}_{p}/p^{r}\mathbf{Z}_{p})$, so we have
the following characterisation of the coset $P(\mathbf{Q}_{p})K_{r}$:

######  Lemma 8.12.

Suppose $r\geqslant 1$ and let $g=\left(\begin{smallmatrix}*&*&*&*\\\
b^{\prime}&*&*&*\\\ c^{\prime}&*&*&*\\\ a&b&c&d\end{smallmatrix}\right)\in
G(\mathbf{Q}_{p})$. Then $g\in P(\mathbf{Q}_{p})K_{r}$ if and only if
$d\in\mathbf{Q}_{p}^{\times}$ and $d^{-1}a,d^{-1}b,d^{-1}c\in
p^{r}\mathbf{Z}_{p}$. ∎

######  Definition 8.13.

For $r\geqslant 1$, let $\phi_{r}\in\pi_{p}^{K_{r}}$ be the function with
support $P(\mathbf{Q}_{p})K_{r}$, and such that $\phi_{r}(1)=p^{3r}$.

Note that the functions $\phi_{r}$ are trace-compatible, i.e. we have

$\frac{1}{[K_{r}:K_{r+1}]}\sum_{k\in K_{r}/K_{r+1}}k\phi_{r+1}=\phi_{r}.$

Moreover, the restriction of $\phi_{r}$ to $M(\mathbf{Q}_{p})$ is independent
of $r$ up to scaling: it is given by $(A,\lambda)\mapsto
p^{3r}|\lambda^{2}/\mathop{det}\nolimits(A)|\xi(A)\theta(\lambda)$, where
$\xi$ is the normalised spherical vector of the unramified
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$-representation $\tau$.

A straightforward explicit computation gives the following:

######  Lemma 8.14.

Let $r\geqslant 1$, and let $t_{P}=\operatorname{diag}(1,p,p,p^{2})$. Then

  1. (i)

The vector $\phi_{r}$ is an eigenvector for the operator $[K_{r}tK_{r}]$ on
$\pi_{p}^{K_{r}}$, with eigenvalue $p^{2}\chi_{\tau}(p)$.

  2. (ii)

We have

$\phi_{r}=\left(\tfrac{p}{\chi_{\tau}(p)}\right)^{r-1}\sum_{a\in
p^{r}\mathbf{Z}_{p}/p^{2r-1}\mathbf{Z}_{p}}\left(\begin{smallmatrix}1&&&\\\
&1&&\\\ &&1&\\\ a&&&1\end{smallmatrix}\right)t_{P}^{r-1}\cdot\phi_{1}.\qed$

Conversely, given any eigenvalue of $[K_{1}tK_{1}]$ on $\pi_{p}^{K_{1}}$, we
can always write $\pi_{p}$ in the form
$\operatorname{Ind}_{P}^{G}(\tau\boxtimes\theta)$ such that
$p^{2}\chi_{\tau}(p)$ is the given eigenvalue.

####  8.4.2. Statement of the formula

Let $\sigma$ be any generic representation333We allow the case where $\sigma$
is a reducible principal series, in which case we simply _define_
$\mathcal{W}(\sigma)$ to be the image of the Whittaker transform. of
$\operatorname{GL}_{2}(\mathbf{Q}_{p})$, and $\mathcal{W}(\sigma)$ its
Whittaker model with respect to $\psi^{-1}$. Let
$\chi=\chi_{\pi}\chi_{\sigma}$. The goal of this section is the following
computation:

######  Proposition 8.15.

Let $\Phi_{1}\in\mathcal{S}(\mathbf{Q}_{p}^{2})$ and $\lambda\in\sigma$. Then
there is some $R$ (depending on $\Phi_{1}$ and $\lambda$) such that the zeta
integral

(8.3)
$Z_{p}(\gamma\cdot\phi_{r},\lambda,\Phi_{1},s)=\int_{(Z_{G}N_{H}\backslash
H)(\mathbf{Q}_{p})}W_{\phi_{r}}(h\gamma)f^{\Phi_{1}}(h_{1};\chi,s)W_{\lambda}(h_{2})\,\mathrm{d}h,\quad\gamma=\left(\begin{smallmatrix}1\\\
1&1\\\ &&1\\\ &&-1&1\end{smallmatrix}\right)$

is independent of $r\geqslant R$. If $\Phi_{1}^{\prime}(0,0)=0$, where
$\Phi_{1}^{\prime}$ denotes the partial Fourier transform as in (8.2), then
this limiting value is given by

$\frac{L(\tau\times\sigma,s)}{L(\tau^{\vee}\times\sigma^{\vee},1-s)\epsilon(\tau\times\sigma,s)}\int_{\mathbf{Q}_{p}^{\times}}W_{\xi}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))W^{\Phi_{1}}\left(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right);\chi,s\right)W_{\lambda}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))\frac{\theta(x)}{|x|}\,\mathrm{d}^{\times}x.$

######  Remark 8.16.

  1. (i)

The assumption on $\Phi_{1}$ is probably not needed, but it is true in the
cases of interest, and it simplifies the proof.

  2. (ii)

In the present paper, we shall apply this formula with the data $\Phi_{1}$ and
$\lambda$ chosen such that the integrand is a multiple of the characteristic
function of $\mathbf{Z}_{p}^{\times}$, so the integral is just the product of
the three Whittaker functions at the identity. However, we shall need to
consider more general choices of the local data in a sequel to the present
paper (in preparation).

  3. (iii)

Since $\tau$ is assumed to be unramified, we have

$\epsilon(\tau\otimes\sigma,s)=\epsilon(\sigma)^{2}\cdot(p^{-2s}\chi_{\tau}(p))^{\mathfrak{f}(\sigma)}$

where $\mathfrak{f}(\sigma)$ is the conductor of $\sigma$, and
$\epsilon(\sigma)=\epsilon(\sigma,0)$. In our applications, $\sigma$ will be a
principal-series representation, so $\epsilon(\sigma)$ will be a product of
Gauss sums, as in §2.2. However, $\chi_{\tau}(p)$ is a more sophisticated
invariant: we shall choose $\tau$ such that
$\chi_{\tau}(p)=p^{-(r_{1}+2)}\lambda$, where $\lambda$ is the $p$-adic unit
eigenvalue of the operator $\mathcal{U}_{p,\operatorname{Kl}}$ acting on
$\Pi_{p}$.

We shall first give the argument assuming the slightly stronger condition that
$\Phi_{1}(0,x)=0$ for all $x$.

####  8.4.3. Reduction to a $\operatorname{GL}_{2}\times\operatorname{GL}_{2}$
integral

In this section, we shall use explicit formulae for the Whittaker transform to
express the integral (8.3) as a Rankin–Selberg integral for
$\operatorname{GL}_{2}\times\operatorname{GL}_{2}$. For brevity we shall write
$f_{1}(-)=f^{\Phi_{1}}(-;\chi,s)$. Recall that $\Phi_{1}$ is such that
$f_{1}(1)=0$, so $f_{1}$ is supported in the “big cell”
$B_{\operatorname{GL}_{2}}\eta N_{\operatorname{GL}_{2}}$, where
$\eta=\left(\begin{smallmatrix}0&1\\\ -1&0\end{smallmatrix}\right)$ as before.
Moreover, the function $x\mapsto
f_{1}\left(\eta\left(\begin{smallmatrix}1&x\\\
0&1\end{smallmatrix}\right)\right)$ is compactly supported.

######  Lemma 8.17.

The integral (8.3) equals

$\int_{a\in\mathbf{Q}_{p}}f_{1}(\eta\left(\begin{smallmatrix}1&a\\\
0&1\end{smallmatrix}\right))\int_{h\in
N_{\operatorname{GL}_{2}}\backslash\operatorname{GL}_{2}}W_{\phi_{r}}\left(\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\
&h&\\\
&&&1\end{smallmatrix}\right)u_{a}\gamma\right)|\mathop{det}\nolimits(h)|^{s-1}W_{\lambda}(h)\,\mathrm{d}h\,\mathrm{d}a,$

where $u_{a}=\iota(\eta\left(\begin{smallmatrix}1&a\\\
0&1\end{smallmatrix}\right),1)$.

###### Proof.

Note that if $(h_{1},h_{2})\in H$ with $h_{1}$ in the big Bruhat cell, then
there exists $h\in\operatorname{GL}_{2}$, uniquely determined modulo
$N_{\operatorname{GL}_{2}}$, and $a\in\mathbf{Q}_{p}$ such that

$N_{H}Z_{G}\cdot(h_{1},h_{2})=N_{H}Z_{G}\cdot\left(\left(\begin{smallmatrix}\mathop{det}\nolimits
h&\\\
&1\end{smallmatrix}\right),h\right)\cdot\left(\eta\left(\begin{smallmatrix}1&a\\\
&1\end{smallmatrix}\right),1\right).$

Making this change of variables in the integral, and using the fact that
$f_{1}$ transforms on the left under
$\left(\begin{smallmatrix}\mathop{det}\nolimits h&\\\
&1\end{smallmatrix}\right)$ by $(\mathop{det}\nolimits h)^{s}$, the lemma
follows. (The $s-1$ in the exponent comes from the modulus character, because
we are writing $N_{H}Z_{G}\backslash H$ as the product of two factors that do
not commute.) ∎

The map from the model of $\pi_{p}=\operatorname{Ind}_{B}^{G}(\alpha)$ as an
induced representation to the Whittaker model $\mathcal{W}(\pi_{p})$ is given
explicitly as follows: for $\phi\in\pi_{p}$ we have

$W_{\phi}(g)=\int_{N_{B}(\mathbf{Q}_{p})}\phi_{r}(Jng)\psi(-m-x)\,\mathrm{d}n,\
\ n=\left(\begin{smallmatrix}1&&&\\\ &1&m&\\\ &&1&\\\
&&&1\end{smallmatrix}\right)\left(\begin{smallmatrix}1&x&y&z\\\ &1&&y\\\
&&1&-x\\\ &&&1\end{smallmatrix}\right)=n(m)n(x,y,z).$

Here $J$ is as in §2.1. We have
$n(x,y,z)\left(\begin{smallmatrix}\mathop{det}\nolimits h\\\ &h\\\
&&1\end{smallmatrix}\right)=\left(\begin{smallmatrix}\mathop{det}\nolimits
h\\\ &h\\\ &&1\end{smallmatrix}\right)n(u,v,w)$, where $(u,v)=(x,y)\cdot
h(\mathop{det}\nolimits h)^{-1}$, $w=z/\mathop{det}\nolimits h$; thus
$x=\langle(u,v),(0,1)h\rangle$, where
$\langle,\rangle:\mathbf{Q}_{p}^{2}\times\mathbf{Q}_{p}^{2}\rightarrow\mathbf{Q}_{p}$
is the pairing $\langle(a,b),(c,d)\rangle=ad-bc$. Hence we have

$\begin{split}W_{\phi_{r}}&(\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\
&h&\\\ &&&1\end{smallmatrix}\right)u_{a}\gamma)\\\
&=|\mathop{det}\nolimits(h)|^{2}\int_{N_{B}}\phi_{r}\left(Jn(m)\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\
&h&\\\
&&&1\end{smallmatrix}\right)n(u,v,w)u_{a}\gamma\right)\psi(-m-\langle(u,v),(0,1)h\rangle)\,\mathrm{d}m\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w.\end{split}$

We note the following matrix identity. Let
$u(m)=\left(\begin{smallmatrix}1&m\\\ 0&1\end{smallmatrix}\right)$. Then

$Jn(m)\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\ &h&\\\
&&&1\end{smallmatrix}\right)n(u,v,w)u_{a}\gamma=\\\
=\iota(\eta,\eta)\cdot\iota(1,u(m))\cdot\iota(\left(\begin{smallmatrix}\mathop{det}\nolimits
h&\\\ &1\end{smallmatrix}\right),h)\cdot n(u,v,w)\cdot\iota(\eta
u(a),1)\cdot\gamma\\\ =\left(\begin{smallmatrix}-1&&\\\ &\eta u(m)h&\\\
&&&-\mathop{det}\nolimits(h)\end{smallmatrix}\right)\cdot\iota(\eta^{-1},1)\cdot
n(u,v,w)\cdot\iota(\eta u(a),1)\gamma.$

One verifies that

$\iota(\eta^{-1},1)\cdot n(u,v,w)\cdot\iota(\eta
u(a),1)\cdot\gamma=\left(\begin{smallmatrix}1&&-a&a\\\ &1&a&-a\\\ &&1\\\
&&&1\end{smallmatrix}\right)\cdot\left(\begin{smallmatrix}wa+1&-ua&-(u+w)a^{2}+(1-v)a&(u+w)a^{2}\\\
-wa+(-v+1)&ua+1&(u+w)a^{2}+(2v-2)a&(-u-w)a^{2}+(-v+1)a\\\ u&0&-ua+1&ua\\\
u-w&u&wa+(v-1)&-wa+1\\\ \end{smallmatrix}\right)$

If $k_{a}(u,v,w)$ denotes the last matrix above, then we deduce that

$Jn(m)\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\ &h&\\\
&&&1\end{smallmatrix}\right)n(u,v,w)u_{a}\gamma\in
P(\mathbf{Q}_{p})k_{a}(u,v,w).$

It follows from the definition of $\phi_{r}$ that
$\phi_{r}(Jn(m)\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\ &h&\\\
&&&1\end{smallmatrix}\right)n(u,v,w)u_{a}\gamma)=0$ unless $k_{a}(u,v,w)\in
PK_{r}$.

######  Lemma 8.18.

Let $C=\max(-v_{p}(a),0)$. If $r\geqslant\max(2C,C+1)$, then $k_{a}(u,v,w)\in
P(\mathbf{Q}_{p})K_{r}$ implies $k_{a}(u,v,w)\in K_{r}$.

###### Proof.

Suppose that $k_{a}(u,v,w)\in PK_{r}$. By Lemma 8.12, we must have

$1-aw\in\mathbf{Q}_{p}^{\times}\quad\text{and}\quad\tfrac{u}{1-aw},\tfrac{w}{1-aw},\tfrac{v-1+aw}{1-aw}\in
p^{r}\mathbf{Z}_{p}.$

If $\tfrac{w}{1-aw}=p^{r}b$ for $b\in\mathbf{Z}_{p}$, then
$w(1+p^{r}ab)=p^{r}b$. Since $p^{r}a\in p\mathbf{Z}_{p}$ by assumption, we can
conclude that $(1+p^{r}ab)$ is a unit and hence $w\in p^{r}\mathbf{Z}_{p}$.
Thus $1-aw$ is also a unit, and we deduce that $u\in p^{r}\mathbf{Z}_{p}$.
Since $v=1-aw\bmod p^{r}$ we have $v=1\bmod p^{r-C}$. In particular,
$ua^{2}$,$wa^{2}$ and $(v-1)a$ all have valuation at least $r-2C\geqslant 0$;
hence $k_{a}(u,v,w)$ is integral (and even congruent to the identity modulo
$p^{r-2C}$). ∎

We suppose that $r\geqslant\max(2C,C+1)$, and let $R(a,r)$ denote the set of
$(u,v,w)$ such that $k_{a}(u,v,w)\in K_{r}$. Then we obtain the formula

$W_{\phi_{r}}(\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\ &h&\\\
&&&1\end{smallmatrix}\right)u_{a}\gamma)\\\
=|\mathop{det}\nolimits(h)|^{2}\int_{R(a,r)}\int_{m}\phi_{r}\left(\left(\begin{smallmatrix}-1&&\\\
&\eta u(m)h&\\\
&&&-\mathop{det}\nolimits(h)\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&-a&a\\\
&1&a&-a\\\ &&1\\\
&&&1\end{smallmatrix}\right)\right)\psi(-m-\langle(u,v),(0,1)h\rangle)\,\mathrm{d}m\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w.$

Since the argument of $\phi_{r}$ lies in $P(\mathbf{Q}_{p})$, we have

$\phi_{r}\left(\left(\begin{smallmatrix}-1&&\\\ &\eta u(m)h&\\\
&&&-\mathop{det}\nolimits(h)\end{smallmatrix}\right)\left(\begin{smallmatrix}1&&-a&a\\\
&1&a&-a\\\ &&1\\\
&&&1\end{smallmatrix}\right)\right)=p^{3r}|\mathop{det}\nolimits
h|^{-1}\xi\Big{(}\eta u(m)hu(a)\Big{)}$

where $\xi$ is the spherical vector of $\tau$, as before. (Note that
$\theta(-1)=1$, since $\theta$ is assumed to be unramified.) Thus the inner
integral over $m$ reduces to the Whittaker transform of $\xi$, evaluated at
$hu(a)$; so we can write this as

$W_{\phi_{r}}(\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\ &h&\\\
&&&1\end{smallmatrix}\right)u_{a}\gamma)=p^{3r}|\mathop{det}\nolimits(h)|\int_{(u,v,w)\in
R(a,r)}W_{\xi}(hu(a))\psi(-\langle(u,v),(0,1)h\rangle)\,\mathrm{d}u\,\mathrm{d}v\,\mathrm{d}w.$

If $\Phi_{a,r}$ denotes the Schwartz function
$\Phi_{a,r}(u,v)=\operatorname{vol}\\{w:(u,v,w)\in R(a,r)\\}$, then we can
collapse this down to

$W_{\phi_{r}}(\left(\begin{smallmatrix}\mathop{det}\nolimits(h)&&\\\ &h&\\\
&&&1\end{smallmatrix}\right)u_{a}\gamma)=p^{3r}|\mathop{det}\nolimits(h)|\cdot
W_{\xi}(hu(a))\cdot\hat{\Phi}_{a,r}((0,1)h),$

where the Fourier transform for Schwartz functions is defined as above.

######  Remark 8.19.

The function $\Phi_{a,r}(u,v)$ is rather explicit, and depends only on the
valuation of $a$. If $C$ is as above, then the set $R(a,r)$ is given by
$\\{u=1\bmod p^{r},v=1\bmod p^{r-C},w=a^{-1}(1-v)\bmod p^{r+C}\\}$, so
$\Phi_{a,r}=p^{-(r+C)}\operatorname{ch}((p^{r}\mathbf{Z}_{p})\times(1+p^{r-C}\mathbf{Z}_{p}))$.

Putting things together we have:

######  Proposition 8.20.

Suppose $\Phi_{1}(1,-)$ is supported in $p^{-N}\mathbf{Z}_{p}$, and let
$r\geqslant\max(2N,N+1)$. Then we have

$\eqref{eq:localintegral}=p^{3r}\int_{a\in\mathbf{Q}_{p}}f_{1}(\eta\left(\begin{smallmatrix}1&a\\\
0&1\end{smallmatrix}\right))\int_{h\in
N_{\operatorname{GL}_{2}}\backslash\operatorname{GL}_{2}}W_{u(a)\xi}(h)\cdot
w_{2}(h)\cdot\hat{\Phi}_{a,r}((0,1)h)|\mathop{det}\nolimits(h)|^{s}\,\mathrm{d}h\,\mathrm{d}a,$

a finite linear combination of Rankin–Selberg integrals.∎

####  8.4.4. Application of the functional equation

By the functional equation for Rankin–Selberg integrals, due to Jacquet
[Jac72, Theorem 14.7 (3)], for any $w_{1}\in\mathcal{W}(\xi)$ and
$w_{2}\in\mathcal{W}(\sigma)$, and any
$\Phi\in\mathcal{S}(\mathbf{Q}_{p}^{2})$, we have

$\int_{N_{2}\backslash\operatorname{GL}_{2}}w_{1}(h)w_{2}(h)\hat{\Phi}((0,1)h)|\mathop{det}\nolimits(h)|^{s}\,\mathrm{d}h=\\\
\frac{1}{\gamma(\tau\times\sigma,s)}\int_{N_{2}\backslash\operatorname{GL}_{2}}\frac{w_{1}(h)w_{2}(h)}{(\chi_{\tau}\chi_{\sigma})(\mathop{det}\nolimits
h)}\Phi((0,1)h)|\mathop{det}\nolimits(h)|^{1-s}\,\mathrm{d}h,$

where

$\gamma(\tau\times\sigma,s)=\frac{L(\tau^{\vee}\times\sigma^{\vee},1-s)}{L(\tau\times\sigma,s)}\epsilon(\tau\times\sigma,s)$

is Jacquet’s local $\gamma$-factor. (More precisely, the integrals on the left
and right sides are convergent for $\operatorname{Re}(s)\gg 0$, resp
$\operatorname{Re}(s)\ll 0$, and both have meromorphic continuation to all $s$
and are equal as meromorphic functions.) Noting that
$\chi_{\tau}\chi_{\sigma}=\chi/\theta$, and taking $\Phi=\Phi_{a,r}$, we have

$\eqref{eq:localintegral}=\frac{p^{3r}}{\gamma(\tau\times\sigma,s)}\int_{a\in\mathbf{Q}_{p}}f_{1}(\eta\left(\begin{smallmatrix}1&a\\\
0&1\end{smallmatrix}\right))\int_{h\in
N_{2}\backslash\operatorname{GL}_{2}}W_{u(a)\xi}(h)w_{2}(h)\Phi_{a,r}((0,1)h)(\theta/\chi)(\mathop{det}\nolimits
h)|\mathop{det}\nolimits(h)|^{1-s}\,\mathrm{d}h\,\mathrm{d}a.$

The Schwartz function $\Phi_{a,r}$ has total integral $p^{-3r}$, and as
$r\to\infty$, its support becomes concentrated in smaller and smaller
neighbourhoods of $(0,1)$. Hence the limiting value is given by

$\eqref{eq:localintegral}=\tfrac{1}{\gamma(\tau\times\sigma,s)}\int_{a\in\mathbf{Q}_{p}}f_{1}(\eta\left(\begin{smallmatrix}1&a\\\
0&1\end{smallmatrix}\right))\int_{x\in\mathbf{Q}_{p}^{\times}}W_{u(a)\xi}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))w_{2}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))\frac{\theta(x)}{|x|^{s}\chi(x)}\,\mathrm{d}^{\times}x\,\mathrm{d}a.$

We have $W_{u(a)\xi}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))=\psi(xa)W_{\xi}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))$, and we compute

$\int_{a}f_{1}(\eta\left(\begin{smallmatrix}1&a\\\
0&1\end{smallmatrix}\right))\psi(xa)\,\mathrm{d}a=\chi(x)|x|^{s-1}W^{\Phi_{1}}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right)).$

This concludes the proof of Proposition 8.15 assuming that $\Phi_{1}(0,-)$ is
identically 0.

####  8.4.5. Conclusion of the proof

We have proved the formula of Proposition 8.15 for functions $\Phi_{1}$
satisfying $\Phi_{1}(0,x)=0$ for all $x$, which is a more restrictive
condition than stated in the proposition. We now reduce the general case to
this.

Consider the auxiliary zeta integral

$\widetilde{Z}_{p}(\varphi,\lambda,\Phi,s)\coloneqq\int_{(Z_{G}N_{H}\backslash
H)(\mathbf{Q}_{p})}W_{\varphi}(h)f^{\Phi_{1}}(h_{1};\chi^{-1},s)W_{\lambda_{v}}(h_{2})\chi(\mathop{det}\nolimits
h)^{-1}\,\mathrm{d}h.$

As before, this has meromorphic continuation to all $s$; and the functions
$Z(\dots)$ and $\widetilde{Z}(\dots)$ are related by the following local
functional equation [Sou84]:

$Z_{p}(\varphi,\lambda,\Phi,s)=\gamma(\pi\times\sigma,s)^{-1}\widetilde{Z}_{p}(\varphi,\lambda,\hat{\Phi},1-s),$

where

$\gamma(\pi\times\sigma,s)=\gamma(\tau\times\sigma,s)\gamma(\tau\times\sigma\times\theta,s).$

Since $\widetilde{Z}_{p}$ can be obtained from $Z_{p}$ by replacing the
representation $\sigma$ with $\sigma\otimes\chi^{-1}$, we deduce that if
$\hat{\Phi}(0,x)=0$ for all $x$, then

$\displaystyle\widetilde{Z}_{p}(\varphi,\lambda,\hat{\Phi},1-s)$
$\displaystyle=\frac{1}{\gamma(\tau\times\sigma\times\chi^{-1},1-s)}\int_{\mathbf{Q}_{p}^{\times}}W_{\xi}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))W^{\hat{\Phi}_{1}}\left(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right);\chi^{-1},1-s\right)W_{\lambda}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))\tfrac{\theta(x)}{\chi(x)|x|}\,\mathrm{d}^{\times}x$
$\displaystyle=\frac{1}{\gamma(\tau\times\sigma\times\chi^{-1},1-s)}\int_{\mathbf{Q}_{p}^{\times}}W_{\xi}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))W^{\Phi_{1}}\left(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right);\chi,s\right)W_{\lambda}(\left(\begin{smallmatrix}x&0\\\
0&1\end{smallmatrix}\right))\tfrac{\theta(x)}{|x|}\,\mathrm{d}^{\times}x$

where the second equality comes from the functional equation (8.1) relating
$W^{\Phi_{1}}$ and $W^{\hat{\Phi}_{1}}$. Since
$\chi=\chi_{\tau}\chi_{\sigma}\theta$, we have
$\gamma(\tau\times\sigma\times\theta,s)\gamma(\tau\times\sigma\times\chi^{-1},1-s)=1$.
So we have shown that the formula of Proposition 8.15 holds for all $\Phi_{1}$
such that $\hat{\Phi}_{1}(0,x)=0$ for all $x$. Since

$(\hat{\Phi}_{1})^{\prime}(x,y)=\Phi_{1}^{\prime}(y,x),$

this is equivalent to requiring that $\Phi_{1}^{\prime}(x,0)=0$ for all $x$.
Since any Schwartz function $h$ on $\mathbf{Q}_{p}^{2}$ such that $h(0,0)=0$
is a linear combination of functions vanishing identically along
$\\{0\\}\times\mathbf{Q}_{p}$ and $\mathbf{Q}_{p}\times\\{0\\}$, this shows
that our formula holds for all $\Phi_{1}$ such that
$\Phi_{1}^{\prime}(0,0)=0$, completing the proof of Proposition 8.15.∎

###  8.5. Local integrals at $\infty$

We now consider the local integral at a real infinite place $v=\infty$. For
simplicity, we shall now assume that $\psi_{F}(x)=\exp(-2\pi ix)$ for
$x\in\mathbf{R}$.

####  8.5.1. Siegel sections

We shall need to compute the functions $f^{\Phi}$ where
$\Phi=\Phi_{\infty}^{(k)}\in\mathcal{S}(\mathbf{R}^{2},\mathbf{C})$ is the
function given by

$\Phi_{\infty}^{(k)}(x,y)\coloneqq 2^{1-k}(x+iy)^{k}e^{-\pi(x^{2}+y^{2})}$

for some $k\in\mathbf{Z}_{\geqslant 1}$. (The factor $2^{1-k}$ is convenient
for comparisions with algebraic Eisenstein series, as we shall shortly see.)
We find readily that

$f^{\Phi}\left(\left(\begin{smallmatrix}a&\\\
&a^{-1}\end{smallmatrix}\right);\chi,s\right)=\begin{cases}0&\text{if
$\chi(-1)\neq(-1)^{k}$},\\\
2^{1-k}i^{k}\Gamma(s+\tfrac{k}{2})\pi^{-(s+k/2)}a^{2s}&\text{otherwise}.\end{cases}$

####  8.5.2. Moriyama’s formula

A formula for the archimedean Whittaker function of a vector in the lowest
$K$-type subspace of a generic discrete-series representation $\Pi_{\infty}$
is given in [Mor04, §3.2]. Let $(\lambda_{1},\lambda_{2})$ be integers such
that $1-\lambda_{1}<\lambda_{2}<0$. Then there is a unique generic discrete-
series representation of $\operatorname{GSp}_{4}(\mathbf{R})$ whose central
character has finite order and which is isomorphic as an
$\operatorname{Sp}_{4}(\mathbf{R})$-representation to the direct sum
$D_{(\lambda_{1},\lambda_{2})}\oplus D_{(-\lambda_{2},-\lambda_{1})}$, in
Moriyama’s notation. For $0\leqslant k\leqslant d$, we let $v_{k}$ be the
$k$-th standard basis vector of the lowest $K$-type of
$D_{(-\lambda_{2},-\lambda_{1})}$.

What we shall need here is Proposition 8 of _op.cit._ , which gives a formula
for the image of $v_{k}$ under the canonical transformation from the Whittaker
model to the Bessel model of $\Pi_{\infty}^{W}$. Recall that we have defined

$B_{v_{k}}(g,s)\coloneqq\int_{\mathbf{R}^{\times}}\int_{\mathbf{R}}W_{v_{k}}\left(\left(\begin{smallmatrix}a\\\
&a\\\ &x&1\\\
&&&1\end{smallmatrix}\right)w_{2}g\right)|a|^{s-\tfrac{3}{2}}\chi_{2}(a)^{-1}\,\mathrm{d}x\,\mathrm{d}^{\times}a,\qquad
w_{2}=\left(\begin{smallmatrix}1\\\ &&1\\\ &-1\\\
&&&1\end{smallmatrix}\right).$

######  Theorem 8.21 (Moriyama).

For $\operatorname{Re}(t)$ sufficiently large, we have

$\int_{0}^{\infty}B_{v_{k}}\left(\left(\begin{smallmatrix}y\\\ &y\\\ &&1\\\
&&&1\end{smallmatrix}\right),s\right)y^{t-\tfrac{3}{2}}\,\mathrm{d}^{\times}y\\\
=C\cdot\frac{(-1)^{k}L(\Pi_{\infty},s)L(\Pi_{\infty},t)}{\pi^{t}\Gamma(\tfrac{t-s+\lambda_{2}+k+1}{2})\Gamma(\tfrac{t+s+\lambda_{1}-k}{2})},$

where $C$ is a constant depending on $\lambda_{1}$ and $\lambda_{2}$ (but not
on $k,s,t$), and
$L(\Pi_{\infty},s)=\Gamma_{\mathbf{C}}(s+\tfrac{\lambda_{1}+\lambda_{2}-1}{2})\Gamma_{\mathbf{C}}(s+\tfrac{\lambda_{1}-\lambda_{2}-1}{2})$,
$\Gamma_{\mathbf{C}}(s)=2(2\pi)^{-s}\Gamma(s)$.

###### Proof.

Moriyama states his formula slightly differently, in the form of an inverse
Mellin transform. The result above follows by applying the “forward” Mellin
transform to both sides of Moriyama’s formula, in the same way as the proof of
Proposition 5.12 of [Lem17] (which is essentially the same argument as ours
but with a different normalisation of the Bessel function). Moriyama also has
a factor $\binom{d}{k}$, which does not appear in our formulae, owing to a
different convention for the standard basis of the lowest $K$-type. ∎

We shall apply this with $(\lambda_{1},\lambda_{2})=(r_{1}+3,-r_{2}-1)$, so
that $d=r_{1}+r_{2}+4$. The vector $v_{k}$ is then an eigenvector for the
action of the diagonal torus in $K_{G,\infty}^{\circ}$, which is just
$K_{H,\infty}^{\circ}$, with weight $(r_{2}+1-k,-r_{1}-3+k)$. We shall let
$(k_{1},k_{2})$ be integers $\geqslant 1$ summing to $r_{1}-r_{2}+2$, and take
$k=r_{1}+3-k_{2}=r_{2}+1+k_{1}$; thus $v_{k}$ has $K_{H}^{\circ}$-type
$(-k_{1},-k_{2})$, meaning it can pair non-trivially against a pair of
holomorphic modular forms of weight $(k_{1},k_{2})$. Finally, we shall take
$s=s_{1}-s_{2}+\tfrac{1}{2}$ and $t=s_{1}+s_{2}-\tfrac{1}{2}$. Note that the
right-hand side of Moriyama’s formula is now

$(-1)^{k_{1}+r_{2}+1}C\cdot\frac{L(\Pi_{\infty},s_{1}-s_{2}+\tfrac{1}{2})L(\Pi_{\infty},s_{1}+s_{2}-\tfrac{1}{2})}{\pi^{s_{1}+s_{2}+\tfrac{1}{2}}\Gamma(s_{1}+\tfrac{k_{1}}{2})\Gamma(s_{2}+\tfrac{k_{2}}{2})}.$

On the other hand, if one chooses $\Phi_{i,\infty}=\Phi_{\infty}^{(k_{i})}$,
then the integrand in the local Bessel-model integral is right
$K_{H,\infty}$-invariant; so by the Iwasawa decomposition, $H=N_{H}T_{H}K_{H}$
where $T_{H}$ is the diagonal torus, we conclude that the left-hand side of
Moriyama’s formula is exactly the integral
$Z_{\infty}\left(v_{k},\Phi_{\infty}^{(k_{1})},\Phi_{\infty}^{(k_{2})},s_{1},s_{2}\right)$
up to a sign and the factor $f^{\Phi_{1}}(1)f^{\Phi_{2}}(1)$, which
corresponds precisely to the Gamma-factors in the denominator above.

If we instead consider the $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$
integral, then the Whittaker function associated to the holomorphic vector
$\lambda\in\sigma_{\infty}$ coincides with the Whittaker function of
$\Phi_{\infty}^{(\ell)}$ at $s_{2}=\tfrac{\ell}{2}$. So we see that in this
case the local integral at $\infty$ is again equal to the expected value,
namely $L(\Pi_{\infty}\times\sigma_{\infty},s)$.

##  9\. Integral formulae for L-functions II: global theory

We now let $F$ be an arbitrary number field, with adèle ring $\mathbf{A}$. Let
$\psi_{F}=\prod_{v}\psi_{F_{v}}$ be a non-trivial additive character of
$\mathbf{A}/F$, which we regard as a character of $N(\mathbf{A})$ as before.

###  9.1. Globally generic $\pi$

Let $\pi$ be a cuspidal automorphic represenation of $G(\mathbf{A})$. For each
$\varphi\in\pi$ we define its _Whittaker transform_ with respect to $\psi$:

$W_{\varphi}(g)=\int_{N_{G}(F)\backslash
N_{G}(\mathbf{A})}\varphi(ng)\psi(n^{-1})\mathrm{d}n.$

The representation $\pi$ is globally generic if some $W_{\varphi}(g)$ is non-
zero. If $\pi$ is globally generic, it can thus be modeled as a space of
functions $W:G(\mathbf{A})\to\mathbf{C}$ satisfying $W(ng)=\psi(n)W(g)$ for
all $n\in N_{G}(\mathbf{A})$.

We fix an identification $\pi=\sideset{}{{}^{\prime}}{\bigotimes_{v}}\pi_{v}$
of $\pi$ with a restricted tensor product of irreducible local
representations. If $\pi$ is globally generic then each $\pi_{v}$ is clearly
also generic, that is, $\pi_{v}$ can also be modelled in a space of functions
$W:G(F_{v})\to\mathbf{C}$ satisfying $W(ng)=\psi(n)W(g)$ for all $n\in
N_{G}(F_{v})$.

######  Remark 9.1.

Note that our definition of “globally generic” is strictly stronger than
requiring that each $\pi_{v}$ be generic, and depends on the realisation of
$\pi$ as a space of functions on $G(F)\backslash G(\mathbf{A})$ rather than on
the abstract isomorphism class of $\pi$.

Assuming $\pi$ is globally generic, we fix Whittaker models
$\pi_{v}\stackrel{{\scriptstyle\sim}}{{\to}}\mathcal{W}_{v}$, $\phi_{v}\mapsto
W_{\phi_{v}}$ for all places $v$. These choices can and shall be made in such
a way that:

  * •

If $\varphi=\otimes_{v}\varphi_{v}$, then we have
$W_{\varphi}(g)=\prod_{v}W_{\phi_{v}}(g_{v})$.

  * •

If $v$ is a finite place such that $\pi_{v}$ and $\psi_{F_{v}}$ are
unramified, and $\phi^{0}_{v}\in\pi_{v}$ is the basis of the unramified
vectors that was implicitly fixed in the definition of the restricted tensor
product $\pi=\sideset{}{{}^{\prime}}{\bigotimes_{v}}\pi_{v}$, then
$W_{\phi^{0}_{v}}(1)=1$.

###  9.2. Global Eisenstein series

For a Schwartz function $\Phi:\mathbf{A}^{2}\to\mathbf{C}$, and $\chi$ a
unitary Grössencharacter, we define a global Siegel section by

$f^{\Phi}(g;\chi,s)\coloneqq\|\mathop{det}\nolimits
g\|^{s}\int_{\mathbf{A}^{\times}}\Phi_{v}((0,a)g)\chi(a)\|a\|^{2s}\,\mathrm{d}^{\times}a,$

We may then form the series

$E^{\Phi}(g;\chi,s):=\sum_{\gamma\in
B(F)\backslash\operatorname{GL}_{2}(F)}f^{\Phi}(\gamma g;\chi,s).$

(cf. [Jac72, §19]). The sum converges absolutely and uniformly on any compact
subset of $\\{s:\operatorname{Re}(s)>1\\}$, and defines a function on the
quotient $\operatorname{GL}_{2}(F)\backslash\operatorname{GL}_{2}(\mathbf{A})$
that transforms under the center by $\chi^{-1}$. It has a meromorphic
continuation in $s$.

######  Remark 9.2.

Note that $g\cdot E^{\Phi}(-;\chi,s)=\|\mathop{det}\nolimits
g\|^{s}E^{g\cdot\Phi}(-;\chi,s)$.

We define the Whittaker transform of $E^{\Phi}(g;\chi,s)$ with respect to
$\bar{\psi}_{F}$ to be

$W^{\Phi}(g;\chi,s):=\int_{N_{\operatorname{GL}_{2}}(F)\backslash
N_{\operatorname{GL}_{2}}(\mathbf{A})}E^{\Phi}(ng;\chi,s)\psi_{F}(x)\mathrm{d}n,\
\ n=\left(\begin{smallmatrix}1&x\\\ 0&1\end{smallmatrix}\right).$

If $\operatorname{Re}(s)>1$ this unfolds to

$W^{\Phi}(g;\chi,s)=\int_{N_{\operatorname{GL}_{2}}(\mathbf{A})}f^{\Phi}(\eta
ng;\chi,s)\psi_{F}(x)\mathrm{d}n,\ \ \eta=\left(\begin{smallmatrix}0&1\\\
-1&0\end{smallmatrix}\right).$

If $\Phi(g)=\prod_{v}\Phi_{v}(g_{v})$ with $\Phi_{v}$ a Schwartz function on
$F_{v}^{2}$ and $\Phi_{v}$ the characteristic function $\Phi_{v}^{0}$ of
$\mathcal{O}_{F_{v}}^{2}$ for almost all finite $v$, then we have
factorisations

$f^{\Phi}(g;\chi,s)=\prod_{v}f^{\Phi_{v}}(g_{v};\chi_{v},s),\qquad
W^{\Phi}(g;\chi,s)=\prod_{v}W^{\Phi_{v}}(g_{v};\chi_{v},s)$

where the local integrals $f^{\Phi_{v}}$ and $W^{\Phi_{v}}$ are as defined
above. Furthermore, the local Whittaker transforms
$W^{\Phi_{v}}(g;\chi_{v},s)$ converge for all values of $s$ and are
holomorphic as functions of $s$, and (for a given $g$ and $\Phi$) all but
finitely many of these are 1.

###  9.3. The global integrals

Let $\chi_{1},\chi_{2}$ be two unitary Grössencharacters of
$\mathbf{A}^{\times}$ such that $\chi_{1}\chi_{2}=\chi_{\pi}$. For
$\varphi\in\pi$ and $\Phi_{1},\Phi_{2}\in\mathcal{S}(\mathbf{A}^{2})$, define

$Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})\coloneqq\int_{Z_{G}(\mathbf{A})H(F)\backslash
H(\mathbf{A})}\varphi(\iota(h))E^{\Phi_{1}}(h_{1};\chi_{1},s_{1})E^{\Phi_{2}}(h_{2};\chi_{2},s_{2})\,\mathrm{d}h,$

where
$H=\operatorname{GL}_{2}\times_{\operatorname{GL}_{1}}\operatorname{GL}_{2}$
and the $E^{\Phi_{i}}(h_{i};\chi_{i},s_{i})$ are Eisenstein series as in
Section 9.2. This integral converges absolutely and is holomorphic at all
values of $s_{1}$ and $s_{2}$ such that neither Eisenstein series has a pole.
This is because the restriction of the cuspform $\varphi$ to $H(\mathbf{A})$
is rapidly decreasing.

We also consider a second integral associated to $\pi$ and an auxiliary
cuspidal automorphic representation $\sigma$ of
$\operatorname{GL}_{2}(\mathbf{A})$. We now let
$\chi=\chi_{\pi}\chi_{\sigma}$, and for $\varphi\in\pi$ and
$\lambda\in\sigma$, we set

$Z(\varphi,\lambda,\Phi,s)\coloneqq\int_{Z_{G}(\mathbf{A})H(F)\backslash
H(\mathbf{A})}\varphi(\iota(h))E^{\Phi}(h_{1};\chi,s)\lambda(h_{2})\,\mathrm{d}h.$

As before, this is holomorphic away from the poles of the Eisenstein series
$E^{\Phi}$.

We now suppose that the test data $(\varphi,\Phi_{1},\Phi_{2})$ for the first
integral, and $(\varphi,\lambda,\Phi)$ for the second, are products of local
data $\varphi=\bigotimes_{v}\varphi_{v}$ etc; and for each place $v$ of $F$,
we write $Z_{v}(\dots)$ for the local integrals of the previous section at the
place $v$. As a straightforward consequence of Proposition 8.6 and Langlands’
general results on the convergence of automorphic $L$-functions, we have the
following Euler product factorisation:

######  Proposition 9.3.

(i) Let $s_{2}$ be fixed. There exists a positive real number $R^{\prime}$,
possibly depending on $\pi$ and $s_{2}$, such that for
$\operatorname{Re}(s_{1})>R^{\prime}$, the product
$\prod_{v}Z_{v}(\varphi_{v},\Phi_{1,v},\Phi_{2,v},s_{1},s_{2})$ converges
absolutely. For such $s_{1}$ we have

$\begin{split}Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})&=\prod_{v}Z_{v}(\varphi_{v},\Phi_{1,v},\Phi_{2,v},s_{1},s_{2})\\\
&=L^{S}(\pi_{v},s_{1}+s_{2}-\tfrac{1}{2})L^{S}(\pi_{v}\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2})\prod_{v\in
S}Z_{v}(\varphi_{v},\Phi_{1,v},\Phi_{2,v},s_{1},s_{2}),\end{split}$

where $S$ is any finite set of places of $F$ including all the archimedean
places and all finite places where $\pi$ or the $\chi_{i}$ are ramified, and
is such that $\Phi_{i,v}=\Phi_{i,v}^{0}$, $\varphi_{v}=\varphi^{0}_{v}$ for
all $v\notin S$.

(ii) There exists a positive real number $R^{\prime}$, possibly depending on
$\pi$ and $\sigma$, such that for $\operatorname{Re}(s)>R^{\prime}$ the
product $\prod_{v}Z_{v}(\varphi_{v},\lambda_{v},\Phi_{v},s)$ converges
absolutely, and for such $s$ we have

$\begin{split}Z(\varphi,\lambda,\Phi,s)&=\prod_{v}Z_{v}(\varphi_{v},\lambda_{v},\Phi_{v},s)\\\
&=L^{S}(\pi_{v}\otimes\sigma_{v},s)\prod_{v\in
S}Z_{v}(\varphi_{v},\lambda_{v},\Phi_{v},s),\end{split}$

for $S$ any sufficiently large finite set of primes as before.

###### Proof.

For part (ii), see [Nov79, Theorem 7], or the introduction of [Sou84]. Part
(i) follows similarly, taking an Eisenstein series in place of the cusp form
$\lambda$. ∎

As $Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})$,
$L^{S}(\pi_{v},s_{1}+s_{2}-\tfrac{1}{2})$ and
$L^{S}(\pi_{v}\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2})$ have meromorphic
continuations in $s_{1}$ (and are even meromorphic in both $s_{1}$ and
$s_{2}$), for any fixed $s_{2}$ the equality

$Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})=L^{S}(\pi_{v},s_{1}+s_{2}-\tfrac{1}{2})L^{S}(\pi_{v}\otimes\chi_{2}^{-1},s_{1}-s_{2}+\tfrac{1}{2})\prod_{v\in
S}Z_{v}(\varphi_{v},\Phi_{1,v},\Phi_{2,v},s_{1},s_{2})$

is an equality of meromorphic functions in $s_{1}$ (and similarly for the
$\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$ integral).

######  Remark 9.4.

The proof of the preceding proposition also shows that if $\pi$ is not
globally generic, then the integrals
$Z(\varphi,\Phi_{1},\Phi_{2},s_{1},s_{2})$ and $Z(\varphi,\lambda,\Phi,s)$ are
identically zero, for all choices of the test data.

##  10\. Construction of the $p$-adic $L$-function

###  10.1. Comparing algebraic and real-analytic Eisenstein series

We now assume the number field $F$ is $\mathbf{Q}$. A routine unravelling of
notations gives the following:

######  Proposition 10.1.

Suppose $\Phi\in\mathcal{S}(\mathbf{A}^{2},\mathbf{C})$ has the form
$\Phi=\Phi_{\mathrm{f}}\cdot\Phi_{\infty}^{(k)}$, for some $k\geqslant 1$ and
$\Phi_{\mathrm{f}}\in\mathcal{S}(\mathbf{A}_{\mathrm{f}}^{2},\mathbf{C})$, and
$\widehat{\chi}$ is the adelic character associated to the Dirichlet character
$\chi$ as in §2.2.

Then the Eisenstein series $E^{\Phi}(-;\widehat{\chi},s)$ on
$\operatorname{GL}_{2}(\mathbf{Q})\backslash\operatorname{GL}_{2}(\mathbf{A})$
of §9.2 is related to the Eisenstein series
$E^{k,\Phi_{\mathrm{f}}}(-;\chi,s)$ on
$\operatorname{GL}_{2}(\mathbf{A}_{\mathrm{f}})\times\mathcal{H}$ of §7.1 by

$E^{\Phi}\left(g_{\mathrm{f}}\left(\begin{smallmatrix}y&x\\\
0&1\end{smallmatrix}\right);\widehat{\chi},s\right)=y^{k/2}\|\mathop{det}\nolimits
g_{\mathrm{f}}\|^{s}\cdot E^{k,\Phi_{\mathrm{f}}}(g_{\mathrm{f}},x+iy;\chi,s)$

for $x+iy\in\mathcal{H}$ and
$g_{\mathrm{f}}\in\operatorname{GL}_{2}(\mathbf{A}_{\mathrm{f}})$.∎

###  10.2. Periods and algebraicity

######  Definition 10.2.

Let $E$ be a subfield of $\overline{\mathbf{Q}}$. An element
$W\in\operatorname{Ind}_{N(\mathbf{A}_{\mathrm{f}})}^{G(\mathbf{A}_{\mathrm{f}})}(\psi)$
(i.e. a $\psi$-Whittaker function on $G$) is said to be _defined over $E$_ if
it takes values in $E(\mu_{\infty})$ and satisfies

$W(g)^{\sigma}=W\left(w(\kappa(\sigma))g\right)$

for all $g\in G(\mathbf{A}_{\mathrm{f}})$ and
$\sigma\in\operatorname{Gal}(\overline{\mathbf{Q}}/E)$, where
$w(x)=\operatorname{diag}(x^{3},x^{2},x,1)$ for
$x\in\hat{\mathbf{Z}}^{\times}$, and
$\kappa:\operatorname{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})\to\widehat{\mathbf{Z}}^{\times}$
is the adelic cyclotomic character.

This is the global counterpart of the local definitions in §8.3.1. We now take
$E$ to be the field of definition of the arithmetic twist
$\Pi_{\mathrm{f}}^{\prime}$, as in §5 above. Given $\eta\in
H^{2}(\Pi_{\mathrm{f}})\otimes\mathbf{C}$, we say $\eta$ is _Whittaker
$E$-rational_ if the Whittaker functions of the coordinate projections of the
corresponding element of Harris’ space $\mathscr{H}_{L_{2}}$ are the product
of an $E$-defined Whittaker function on $G(\mathbf{A}_{\mathrm{f}})$ and the
_standard Whittaker function_ at $\infty$, which is the unique function that
makes the constant $C$ in Moriyama’s formula equal 1.

It follows easily from the definitions that the space of
Whittaker-$E$-rational classes is exactly $\Omega^{W}(\Pi)\cdot
H^{2}(\Pi_{\mathrm{f}})$, for some nonzero constant
$\Omega^{W}(\Pi)\in\mathbf{C}^{\times}$, the _Whittaker period_ of $\Pi$,
well-defined modulo $E^{\times}$.

######  Notation 10.3.

For the remainder of this section, we resume the notations of §5.6, so $p>2$
is an unramified prime, and $L$ the completion of $E$ at a prime above $p$
with respect to which $\Pi_{p}$ is Klingen ordinary.

###  10.3. The $\operatorname{GSp}_{4}$ $L$-function

######  Proposition 10.4.

Let $R=\Lambda_{L}(\mathbf{Z}_{p}^{\times}\times\mathbf{Z}_{p}^{\times})$.
There is an element $\mathcal{L}_{p}(\Pi,\mathbf{j}_{1},\mathbf{j}_{2})\in R$
whose specialisation at a locally-algebraic point
$x=(a_{1}+\rho_{1},a_{2}+\rho_{2})$ of $\operatorname{Spec}R$, such that
$d=r_{1}-r_{2}\geqslant a_{1}\geqslant a_{2}\geqslant 0$ and
$(-1)^{a_{1}+a_{2}}\rho_{1}(-1)\rho_{2}(-1)=-1$, is given by

$\frac{R_{p}(\Pi,\rho_{1},a_{1})R_{p}(\Pi,\rho_{2},a_{2})\Lambda(\Pi\otimes\rho_{1}^{-1},\tfrac{1-d}{2}+a_{1})\Lambda(\Pi\otimes\rho_{2}^{-1},\tfrac{1-d}{2}+a_{2})}{\Omega^{W}(\Pi)},$

where

$R_{p}(\Pi,a,\rho)=(-1)^{a}\cdot\begin{cases}\left(1-\tfrac{p^{a+r_{2}+1}}{\alpha}\right)\left(1-\tfrac{p^{a+r_{2}+1}}{\beta}\right)\left(1-\frac{\gamma}{p^{a+r_{2}+2}}\right)\left(\frac{\delta}{p^{a+r_{2}+2}}\right)&\text{if
$\rho$ is trivial},\\\\[8.61108pt]
G(\rho)^{2}\left(\frac{p^{2a+2r_{2}+2}}{\alpha\beta}\right)^{m}&\text{if
$\rho$ has conductor $p^{m}>1$},\end{cases}$

where $\alpha,\beta,\gamma,\delta$ are the roots of the Hecke polynomial such
that $v_{p}(\alpha\beta)=r_{2}+1$, and $G(\rho)$ is the Gauss sum as in §2.2.

###### Proof.

If we choose Schwartz functions $\Phi_{1}^{(p)}$ and $\Phi_{2}^{(p)}$ on
$(\mathbf{A}_{\mathrm{f}}^{p})^{2}$, then we can use these to build $p$-adic
families $\mathcal{E}(\Phi_{1}^{(p)},\Phi_{2}^{(p)})$ of Eisenstein series on
$H$, as in §7.4 above. If we also choose $\eta\in H^{2}(\Pi_{\mathrm{f}})$
which is invariant under $\operatorname{Kl}(p)$ and lands in the ordinary
eigenspace, then we can form a $p$-adic analytic function interpolating the
cup-products of $\eta$ with specialisations of the family
$\mathcal{E}(\Phi_{1}^{(p)},\Phi_{2}^{(p)})$.

Specialising at $(a_{1}+\rho_{1},a_{2}+\rho_{2})$, the cup-product is given by
a zeta-integral $Z(\dots)$, but with $\Pi$ replaced by the twisted
representation $\Pi\otimes\rho_{1}^{-1}$. This factorises as a product of
local integrals as above. For finite primes $\ell\neq p$ we may choose a
finite linear combination of triples ($\phi_{v},\Phi_{1,v},\Phi_{2,v})$ such
that the corresponding local zeta integral is the $L$-factor (and the
rationality statements of §8.3.1 allow us to choose the data such that the
resulting coherent cohomology classes are defined over $E$). For the place
$\infty$, Moriyama’s results show that the specific Archimedean data we have
chosen give the Archimedean $\Gamma$-factor and the sign $(-1)^{a_{1}-a_{2}}$.
For the place $p$, we have chosen specific local data and evaluated the local
integrals explicitly, which gives the local correction factors
$R_{p}(\Pi,\rho_{i},a_{i})$. ∎

######  Lemma 10.5.

If $r_{1}-r_{2}\geqslant 1$, then for all finite-order characters $\rho$ of
$\mathbf{Z}_{p}^{\times}$, we have
$\Lambda(\Pi\otimes\rho^{-1},\tfrac{1+d}{2})\neq 0$.

###### Proof.

If $r_{1}-r_{2}\geqslant 2$ this is obvious from the convergence of the Euler
product. The case $r_{1}-r_{2}=1$ follows from results of Jacquet–Shalika
[JS76] on non-vanishing of $L$-functions for $\operatorname{GL}_{n}$ on the
abcissa of convergence (the “prime number theorem” for $\operatorname{GL}_{n}$
$L$-functions), applied to the automorphic representation of
$\operatorname{GL}_{4}$ lifted from $\Pi$. ∎

If $r_{1}=r_{2}$ then we cannot directly establish the non-vanishing, so we
must impose it as a hypothesis:

######  Hypothesis 10.6.

There exist finite-order characters $\rho_{+},\rho_{-}$ of
$\mathbf{Z}_{p}^{\times}$, with $\rho_{+}(-1)=1$ and $\rho_{-}(-1)=-1$, such
that $\Lambda(\Pi\otimes\rho_{+}^{-1},\tfrac{1}{2})$ and
$\Lambda(\Pi\otimes\rho_{-}^{-1},\tfrac{1}{2})$ are non-zero.

######  Theorem 10.7.

(Theorem A) Suppose that $r_{1}-r_{2}\geqslant 1$ or that Hypothesis 10.6
holds. Then there exist constants $\Omega^{+}_{\Pi}$ and $\Omega^{-}_{\Pi}$,
uniquely determined modulo $E^{\times}$ and satisfying
$\Omega^{+}_{\Pi}\Omega^{-}_{\Pi}=\Omega_{\Pi}^{W}\pmod{E^{\times}}$; and an
element $\mathcal{L}(\Pi,\mathbf{j})\in\Lambda_{L}(\mathbf{Z}_{p}^{\times})$,
such that

$\mathcal{L}(\Pi,a+\rho)=(-1)^{a}R_{p}(\Pi,\rho,a)\frac{\Lambda(\Pi\otimes\rho^{-1},\tfrac{1-d}{2}+a)}{\Omega_{\Pi}^{\pm}}$

for all $(a,\rho)$ with $0\leqslant a\leqslant d$, where the sign $\pm$ is
given by $(-1)^{a}\rho(-1)$.

###### Proof.

Let $\rho_{+}$ and $\rho_{-}$ be any two Dirichlet characters of $p$-power
conductor such that $\rho_{+}(-1)=(-1)^{d}$, $\rho^{-}(-1)=-(-1)^{d}$. From
the preceding lemma, we see that
$\Lambda(\Pi\otimes\rho_{+}^{-1},\tfrac{1+d}{2})$ and
$\Lambda(\Pi\otimes\rho_{-}^{-1},\tfrac{1+d}{2})$ are both nonzero. If $d=1$
then we assume also that both characters $\rho_{\pm}$ are ramified at $p$,
which ensures that the local Euler factors $R(\rho_{\pm},d)$ are non-zero.

Then we may define

$\displaystyle\Omega^{+}(\Pi)$
$\displaystyle=\Omega^{W}(\Pi)/[R_{p}(\Pi,\rho_{-},d)\Lambda(\Pi\otimes\rho_{-}^{-1},\tfrac{1+d}{2})]$
$\displaystyle\Omega^{-}(\Pi)$
$\displaystyle=\Omega^{W}(\Pi)/[R_{p}(\Pi,\rho_{+},d)\Lambda(\Pi\otimes\rho_{+}^{-1},\tfrac{1+d}{2})].$

We can then define
$\mathcal{L}(\Pi,\mathbf{j})=\mathcal{L}(\Pi,\mathbf{j},d+\rho^{-})$ on the
component of weight space where $(-1)^{\mathbf{j}}=+1$, and
$\mathcal{L}(\Pi,\mathbf{j},d+\rho^{+})$ on the other component; and the
interpolation formula follows easily from the corresponding property of the
two-variable $L$-function $\mathcal{L}(\Pi,\mathbf{j}_{1},\mathbf{j}_{2})$. ∎

###  10.4. The $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$
$L$-function

A similar analysis for the $\operatorname{GSp}_{4}\times\operatorname{GL}_{2}$
integral formula gives the following:

######  Theorem 10.8.

(Theorem B) Let $R=\Lambda_{L}(\mathbf{Z}_{p}^{\times})$. There is an element
$\mathcal{L}_{p}(\Pi\times\sigma,\mathbf{j})\in R$ whose specialisation at a
locally-algebraic point $x=a+\rho$ of $\operatorname{Spec}R$, with $0\leqslant
a\leqslant d^{\prime}=r_{1}-r_{2}-\ell+1$, is given by

$R_{p}(\Pi\times\sigma,a,\rho)\frac{\Lambda^{S}(\Pi\otimes\sigma\otimes\rho^{-1},\tfrac{1-d^{\prime}}{2}+a)}{\Omega_{\Pi}^{W}},$

where $S$ is the set of finite places $\ell$ such that neither $\Pi_{\ell}$
nor $\sigma_{\ell}$ is principal series, and

$R_{p}(\Pi\times\sigma,a,\rho)=\frac{1}{\epsilon(\tau_{p}\otimes\sigma_{p}\otimes\widehat{\rho}_{p}^{-1},\tfrac{1-d^{\prime}}{2}+a)}L(\tau_{p}^{\vee}\otimes\sigma_{p}^{\vee}\otimes\widehat{\rho}_{p},\tfrac{1+d^{\prime}}{2}-a)^{-1}L(\tau_{p}\otimes\sigma_{p}\otimes\theta_{p}\widehat{\rho}_{p}^{-1},\tfrac{1-d^{\prime}}{2}+a)^{-1}$

where we have written
$\Pi_{p}=\operatorname{Ind}_{P_{\operatorname{Kl}}}^{G}(\tau_{p}\boxtimes\theta_{p})$
for some $\tau_{p}$ and $\theta_{p}$ as in §8.4.∎

(The factor $R_{p}(\Pi\times\sigma,a,\rho)$ can also be written explicitly in
terms of Satake parameters and Gauss sums, as in Proposition 10.4; we leave
the details to the reader.)

######  Remark 10.9.

It would be interesting to attempt to define a canonical normalisation for the
periods $\Omega^{\mathrm{W}}(\Pi)$ and $\Omega^{\pm}(\Pi)$ up to $p$-adic
units, analogous to the results of [Vat99] in the $\operatorname{GL}_{2}$
case.

For $\Omega^{\mathrm{W}}(\Pi)$, one possibility would be to use the
$\operatorname{GSp}(4)$ new-vector theory of Okazaki [Oka19]. This singles out
a uniquely-determined level group $K\subset G(\mathbf{A}_{\mathrm{f}})$ such
that $\Pi_{\mathrm{f}}^{K}$ is one-dimensional, and such that
$\mathcal{W}(\Pi_{\mathrm{f}})^{K}$ has a canonical normalised generator
$w^{\mathrm{new}}$ satisfying $w^{\mathrm{new}}(1)=1$. We then have two
natural $\mathcal{O}_{E,(v)}$-lattices in
$\mathcal{W}(\Pi_{\mathrm{f}})^{K}\otimes\mathbf{C}$: one given by Whittaker
functions whose values at 1 are integral, and one given by the image under the
comparison map of the lattice in $H^{2}(\Pi_{\mathrm{f}})^{K}$ generated by
the cohomology of the $\mathbf{Z}_{p}$-model $X_{G}(K)$ (or the $K$-invariants
of cohomology at a suitable higher level, since this $K$ may not be neat).
Using the ratio of these two lattices, we obtain a canonical normalisation for
$\Omega^{\mathrm{W}}(\Pi)$ up to multiplication by
$\mathcal{O}_{E,(v)}^{\times}$.

However, since we do not know if the test data for the zeta-integrals
$Z_{\ell}(\dots,s)$ at the bad primes $\ell$ can be chosen “integrally”, even
with this normalised period it is not clear if the $p$-adic $L$-functions of
Proposition 10.4 and Theorem B will lie in the Iwasawa algebra with
$\mathcal{O}$-coefficients.

The situation for the signed periods $\Omega^{\pm}(\Pi)$ is still more
mysterious, and there seems to be no natural way of defining a canonical
normalisation for these periods directly on $\operatorname{GSp}_{4}$ (rather
than via functorial lifting to $\operatorname{GL}_{4}$ as in [DJR18]).

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