for any $x\in L^{\times,+}$, with $(x,p)=1$. Thus they induce endomorphisms
$T_{\ell}$ and $S_{\ell}$ of $\mathfrak{w}^{G}_{k,\alpha,I}$ which preserve
the cuspforms.

###### Proof.

Left to the reader. ∎

We now explain how to descend the $U_{\mathfrak{P}_{i}}$ and
$V_{\mathfrak{P}_{i}}$ operators to $\mathfrak{w}^{G}_{k,\alpha,I}$. The
operator $U_{\mathfrak{P}_{i}}$ commutes with the $\Gamma$-action and descends
to an operator
$U_{\mathfrak{P}_{i}}\colon{g_{x_{i}^{-1}\mathfrak{P}_{i}\mathfrak{c}}}_{*}\mathfrak{w}^{G,x_{i}^{-1}\mathfrak{P}_{i}\mathfrak{c}}_{k,\alpha,I}\to{g_{\mathfrak{c}}}_{*}\mathfrak{w}^{G,\mathfrak{c}}_{k,\alpha,I}$.
However, this operator is dependent on the choice of $x_{i}$, and hence does
not commute with the action of $\text{Princ}(L)^{+,(p)}$. Similarly,
$V_{\mathfrak{P}_{i}}$ descends to an operator
$V_{\mathfrak{P}_{i}}\colon{g_{x_{i}\mathfrak{P}_{i}^{-1}\mathfrak{c}}}_{*}\mathfrak{w}^{G,x_{i}\mathfrak{P}_{i}^{-1}\mathfrak{c}}_{k,\alpha,I}\to{g_{\mathfrak{c}}}_{*}\mathfrak{w}^{G,\mathfrak{c}}_{k,\alpha,I}$
which also depends on the choice of $x_{i}$. We make these canonical in the
following way. Let $(v_{\text{un}},w_{\text{un}})$ be the universal weight on
$\mathcal{W}^{G}_{p}$.

######  Definition 6.16.

  1. (1)

Define the operator
$\mathbf{U}_{\mathfrak{P}_{i}}\colon{g_{x_{i}^{-1}\mathfrak{P}_{i}\mathfrak{c}}}_{*}\mathfrak{w}^{G,x_{i}^{-1}\mathfrak{P}_{i}\mathfrak{c}}_{k,\alpha,I}\to{g_{\mathfrak{c}}}_{*}\mathfrak{w}^{G,\mathfrak{c}}_{k,\alpha,I}$
as $\mathbf{U}_{\mathfrak{P}_{i}}\mathrel{\mathop{\mathchar
58\relax}}=(\varpi_{i}/x_{i})^{v_{\text{un}}}U_{\mathfrak{P}_{i}}$.

  2. (2)

Define the operator
$\mathbf{V}_{\mathfrak{P}_{i}}\colon{g_{x_{i}\mathfrak{P}_{i}^{-1}\mathfrak{c}}}_{*}\mathfrak{w}^{G,x_{i}\mathfrak{P}_{i}^{-1}\mathfrak{c}}_{k,\alpha,I}\to{g_{\mathfrak{c}}}_{*}\mathfrak{w}^{G,\mathfrak{c}}_{k,\alpha,I}$
as
$\mathbf{V}_{\mathfrak{P}_{i}}=(x_{i}/\varpi_{i})^{v_{\text{un}}}V_{\mathfrak{P}_{i}}$.

######  Proposition 6.17.

The operators $\mathbf{U}_{\mathfrak{P}_{i}}$ and
$\mathbf{V}_{\mathfrak{P}_{i}}$ are independent of the choice of $x_{i}$, and
commute with the action of $\text{\emph{Princ}}(L)^{+,(p)}$. Hence they induce
endomorphisms $\mathbf{U}_{\mathfrak{P}_{i}}$ and
$\mathbf{V}_{\mathfrak{P}_{i}}$ of $\mathfrak{w}^{G}_{k,\alpha,I}$ preserving
cuspforms. Moreover,
$\mathbf{U}_{\mathfrak{P}_{i}}\circ\mathbf{V}_{\mathfrak{P}_{i}}=\text{id}$,
and $\prod_{i=1}^{h}\mathbf{U}_{\mathfrak{P}_{i}}=U$, and
$\prod_{i=1}^{h}\mathbf{V}_{\mathfrak{P}_{i}}=V$.

###### Proof.

Left to the reader. ∎

###  6.4. Overconvergent projection

In this section we will define the overconvergent projection in families upon
studying the cohomology of the complex of
$\mathcal{O}_{\bar{\mathcal{M}}_{r,\alpha,I}}$ sheaves obtained by the
connection $\nabla$ which is described as follows. In particular in this
section all sheaves are considered over the analytic adic spaces. Hence to
simplify notation we will still write $\mathbb{W}_{k,\alpha,I}$ but it is to
be understood that this is a sheaf over $\bar{\mathcal{M}}_{r,\alpha,I}$.
Consider the complex

$\mathbb{W}_{k,\alpha,I}\xrightarrow{\nabla}\mathbb{W}_{k,\alpha,I}\hat{\otimes}\Omega^{1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}\cdots\xrightarrow{\nabla}\mathbb{W}_{k,\alpha,I}\hat{\otimes}\Omega^{g}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}.$
(13)

Denote by $\mathbb{W}^{\bullet}_{k,\alpha,I}$ the complex obtained by
tensoring the above complex by
$\mathcal{O}_{\bar{\mathcal{M}}_{r,\alpha,I}}(-D)$ (and the universal
derivation on it) where we recall $D$ is the boundary divisor. By Griffiths’
transversality we obtain a complex corresponding to the filtration on
$\mathbb{W}_{k,\alpha,I}$ as follows.

$\mathrm{Fil}_{n}\mathbb{W}_{k,\alpha,I}\xrightarrow{\nabla}\mathrm{Fil}_{n+1}\mathbb{W}_{k,\alpha,I}\hat{\otimes}\Omega^{1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}\cdots\xrightarrow{\nabla}\mathrm{Fil}_{n+g}\mathbb{W}_{k,\alpha,I}\hat{\otimes}\Omega^{g}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}.$
(14)

Denote by $\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I}$ the complex
obtained by tensoring the above complex with
$\mathcal{O}_{\bar{\mathcal{M}}_{r,\alpha,I}}(-D)$. By taking quotient of the
first complex by the second we obtain a third complex
$(\mathbb{W}_{k,\alpha,I}/\mathrm{Fil}_{n}\mathbb{W}_{k,\alpha,I})^{\bullet}$
which sits in a short exact sequence of complexes on
$\bar{\mathcal{M}}_{r,\alpha,I}$ that gives a long exact sequence of
hypercohomology groups.

$\displaystyle\begin{split}0&\to
H^{0}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I})\to
H^{0}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathbb{W}^{\bullet}_{k,\alpha,I})\to
H^{0}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},(\mathbb{W}_{k,\alpha,I}/\mathrm{Fil}_{n}\mathbb{W}_{k,\alpha,I})^{\bullet})\\\
&\to
H^{1}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I})\to
H^{1}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathbb{W}^{\bullet}_{k,\alpha,I})\to\cdots\end{split}$
(15)

######  Lemma 6.18.

The cohomology of the de Rham complex of coherent sheaves
$\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I}$ can be computed using
global sections.

###### Proof.

We recall that if
$f\colon\bar{\mathcal{M}}_{r,\alpha,I}\to\mathcal{M}^{*}_{r,\alpha,I}$ is the
projection to the minimal compactification, which is an affinoid adic space,
$R^{i}f_{*}\mathfrak{w}_{k,\alpha,I}(-D)=0$. We note that in order to prove
the lemma, it will be enough to prove that each sheaf in the complex
$\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I}$ is acyclic for the direct
image functor $f_{*}$. Because then an injective resolution of
$\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I}$ will give an acyclic
resolution of $f_{*}\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I}$, and
since $\Gamma(\mathcal{M}^{*}_{r,\alpha,I},\cdot)$ is exact, the lemma will
follow.

We first show that $\mathrm{Fil}_{n}\mathbb{W}_{k,\alpha,I}(-D)$ is acyclic
for $f_{*}$. By Lemma 5.2, the sheaf
$\mathrm{Fil}_{n}\mathbb{W}_{k,\alpha,I}(-D)$ is equipped with a finite
filtration such that the graded pieces are finite direct sums of sheaves of
cuspforms. Thus the graded pieces of the filtration are acyclic for $f_{*}$ by
what we just recalled above. Then a simple spectral sequence argument proves
that $\mathrm{Fil}_{n}\mathbb{W}_{k,\alpha,I}(-D)$ is $f_{*}$-acyclic.
Moreover, using the Kodaira-Spencer isomorphism, this same proof shows that
$\mathrm{Fil}_{n+i}\mathbb{W}_{k,\alpha,I}(-D)\hat{\otimes}{\Omega}^{i}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}$
is also $f_{*}$-acyclic for all $0\leq i\leq g$. ∎

######  Lemma 6.19.

Let $t_{L}=\sum\sigma$ be the generator of the parallel weights. There exists
an exact sequence

$0\to
H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathfrak{w}_{k+2t_{L}}(-D))\xrightarrow{i}H^{g}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I})\to\operatorname{coker}{i}\to
0$

where $i$ is $U$-equivariant and $\operatorname{coker}{i}$ is killed by
$\prod_{\sigma}\prod_{i=0}^{n+g-1}(u_{\sigma}-i)$.

###### Proof.

We first note that

$H^{g}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k})=\frac{H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n+g}\mathbb{W}_{k+2t_{L}}(-D))}{\nabla
H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n+g-1}\mathbb{W}_{k}(-D)\hat{\otimes}\Omega^{g-1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}})}$

The $U$-equivariant inclusion of
$H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathfrak{w}_{k+2t_{L}}(-D))$ inside
$H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n+g}\mathbb{W}_{k+2t_{L}}(-D))$
induces a map
$H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathfrak{w}_{k+2t_{L}}(-D))\to
H^{g}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k})$
which is an inclusion as can be seen from the local description below. We are
left to understand the cokernel of the inclusion, and in particular to show
that the cokernel is killed by
$\prod_{\sigma}\prod_{i=0}^{n+g-1}(u_{\sigma}-i)$.

The proof relies on the local description of the connection (11). Choosing a
numbering $\sigma\colon\\{1,\dots,g\\}\simeq\Sigma$, we first write the sheaf
$\mathrm{Fil}_{n+g-1}\mathbb{W}_{k}\hat{\otimes}\Omega^{g-1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}$
on local coordinates as

$\mathrm{Fil}_{n+g-1}\mathbb{W}_{k}\hat{\otimes}\Omega^{g-1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}=\bigoplus_{i}\mathfrak{w}_{k}^{\leq
n+g-1}[V_{1},\dots,V_{g}]\mathrm{d}\hat{X_{i}}.$

Here $\mathrm{d}\hat{X_{i}}$ corresponds via Kodaira-Spencer to a generator of
$\omega_{\mathcal{A}}^{2(t_{L}-\sigma_{i})}$ and the superscript $\leq n+g-1$
denotes we take the polynomials in $V_{j}$’s of degree at most $n+g-1$.

The map
$\nabla\colon\mathrm{Fil}_{n+g-1}\mathbb{W}_{k}(-D)\hat{\otimes}\Omega^{g-1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}\to\mathrm{Fil}_{n+g}\mathbb{W}_{k+2t_{L}}(-D)$
can be described as the twist by
$\mathcal{O}_{\bar{\mathcal{M}}_{r,\alpha,I}}(-D)$ of a map

$\bigoplus_{i}\mathfrak{w}_{k}^{\leq
n+g-1}[V_{1},\dots,V_{g}]\mathrm{d}\hat{X_{i}}\xrightarrow{\nabla}\mathfrak{w}_{k+2t_{L}}^{\leq
n+g}[V_{1},\dots,V_{g}].$

that can be described using formula (11). In particular, the image of $\nabla$
consists of polynomials in $V_{i}$ of positive total degree and hence the map

$\mathfrak{w}_{k+2t_{l}}(-D)\to\frac{\mathrm{Fil}_{n+g}\mathbb{W}_{k+2t_{L}}(-D)}{\nabla\mathrm{Fil}_{n+g-1}\mathbb{W}_{k}(-D)\hat{\otimes}\Omega^{g-1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}}$

is injective. Thus taking global sections we get
$H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathfrak{w}_{k+2t_{L}}(-D))\xhookrightarrow{}H^{g}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k})$.

We prove the claim about the annihilator of $\operatorname{coker}{i}$ by
induction on $n$, the base case being $n=1-g$. For $n=1-g$ we have
$\mathrm{Fil}_{1}\mathbb{W}_{k+2t_{L}}/(\nabla\mathfrak{w}_{k}+\mathfrak{w}_{k+2t_{L}})\simeq\oplus_{i}\mathfrak{w}_{k+2t_{L}}V_{i}/u_{i}\mathfrak{w}_{k+2t_{L}}V_{i}$.
This proves the base case. We have a diagram as follows with exact rows.

${0}$${\mathrm{Fil}_{n+g-2}\mathbb{W}_{k}\hat{\otimes}\Omega^{g-1}}$${\mathrm{Fil}_{n+g-1}\mathbb{W}_{k}\hat{\otimes}\Omega^{g-1}}$${\mathrm{Gr}_{n+g-1}\mathbb{W}_{k}\hat{\otimes}\Omega^{g-1}}$${0}$${0}$${\mathrm{Fil}_{n+g-1}\mathbb{W}_{k+2t_{L}}}$${\mathrm{Fil}_{n+g}\mathbb{W}_{k+2t_{L}}}$${\mathrm{Gr}_{n+g}\mathbb{W}_{k+2t_{L}}}$${0}$$\scriptstyle{\nabla}$$\scriptstyle{\nabla}$$\scriptstyle{\nabla}$

To complete the induction, we need to understand the connection on the graded
pieces. Letting $k(1+\beta_{n}Z)$ be a local generator of $\mathfrak{w}_{k}$,
the map $\nabla$ on the graded pieces can be described as follows.

$\displaystyle\oplus\mathfrak{w}_{k}^{n+g-1}[V_{1},\dots,V_{g}]\mathrm{d}\hat{X}_{i}$
$\displaystyle\xrightarrow{\nabla}\mathfrak{w}_{k+2t_{L}}^{n+g}[V_{1},\dots,V_{g}]$
$\displaystyle k(1+\beta_{n}Z)\prod_{i}V_{i}^{n_{i}}\mathrm{d}\hat{X}_{j}$
$\displaystyle\mapsto(u_{j}-n_{j})(k+2t_{L})(1+\beta_{n}Z)V_{j}\prod_{i}V_{i}^{n_{i}}$

This shows that the cokernel of
$\nabla\colon\mathrm{Gr}_{n+g-1}\mathbb{W}_{k}\hat{\otimes}\Omega^{g-1}_{\bar{\mathcal{M}}_{r,\alpha,I}/\Lambda_{\alpha,I}}\xrightarrow{}\mathrm{Gr}_{n+g}\mathbb{W}_{k+2t_{L}}$
is annihilated by $\prod_{\sigma}\prod_{i=0}^{n+g-1}(u_{\sigma}-i)$. The lemma
then follows by applying Snake lemma to the diagram above and by the induction
hypothesis. ∎

######  Lemma 6.20.

For $\delta\in\mathbb{Q}_{\geq 0}$, there exists $n\geq 0$ such that the map

$H^{i}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k,\alpha,I})^{\leq\delta}\to
H^{i}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathbb{W}^{\bullet}_{k,\alpha,I})^{\leq\delta}$

is an isomorphism.

###### Proof.

Arguing as in Proposition 6.11, the sheaf
$H^{i}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},(\mathbb{W}_{k}/\mathrm{Fil}_{n}\mathbb{W}_{k})^{\bullet})$
admits slope decomposition locally on the weight space. Moreover, by the same
proposition
$H^{i}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},(\mathbb{W}_{k}/\mathrm{Fil}_{n}\mathbb{W}_{k})^{\bullet})^{\leq\delta}=0$
for large enough $n$. The lemma then follows from the long exact sequence
(15). ∎

######  Definition 6.21.

For $\delta\geq 0$, let $n$ be as in the above lemma. Let
$\lambda=\prod_{\sigma}\prod_{i=0}^{n+g-1}(u_{\sigma}-i)$. For the finite
slope $\delta\geq 0$, define the overconvergent projection in families to be
the map induced by the isomorphisms as follows.

$\displaystyle H^{\dagger}\colon
H^{g}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathbb{W}_{k}^{\bullet})^{\leq\delta}\otimes\Lambda_{\alpha,I}[\lambda^{-1}]$
$\displaystyle\xrightarrow{\sim}H^{g}_{\textrm{dR}}(\bar{\mathcal{M}}_{r,\alpha,I},\mathrm{Fil}_{n}^{\bullet}\mathbb{W}_{k})^{\leq\delta}\otimes\Lambda_{\alpha,I}[\lambda^{-1}]$
$\displaystyle\xrightarrow{\sim}H^{0}(\bar{\mathcal{M}}_{r,\alpha,I},\mathfrak{w}_{k+2t_{L}}(-D))^{\leq\delta}\otimes\Lambda_{\alpha,I}[\lambda^{-1}]$

##  7\. Twisted triple product $p$-adic $L$-function

In this section we finally construct the twisted triple product $p$-adic
$L$-function associated to finite slope families of Hilbert modular forms. We
note that there has recently been a construction of these $p$-adic
$L$-functions associated to Hida families due to I. Blanco-Chacón and M.
Fornea [BF20]. We follow their article to describe the generalities of the
setting.

Let $L/F$ be a quadratic extension of totally real number fields, with
$[F\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]=d$. Assume $p$ is
unramified in $L$. Let $\Sigma_{L}$ and $\Sigma_{F}$ be their respective set
of embeddings into a common finite extension $K$ of $\mathbb{Q}_{p}$ where
both $L$ and $F$ are split. Let
$G_{L}=\operatorname{Res}_{L/\mathbb{Q}}\mathbf{GL}_{2,L}$ and
$G_{F}=\operatorname{Res}_{F/\mathbb{Q}}\mathbf{GL}_{2,F}$. Let
$(v,n)\in\mathbb{Z}[\Sigma_{L}]\times\mathbb{Z}$ and
$(w,m)\in\mathbb{Z}[\Sigma_{F}]\times\mathbb{Z}$ be two classical weights.

Consider primitive eigenforms $g\in S^{G_{L}}({N_{g}},(v,n),\bar{\mathbb{Q}})$
and $f\in S^{G_{F}}({N_{f}},(w,m),\bar{\mathbb{Q}})$ generating irreducible
cuspidal automorphic representations $\pi,\sigma$ of $G_{L}(\mathbb{A})$ and
$G_{F}(\mathbb{A})$ respectively. Here
$S^{G_{L}}({N_{g}},(v,n),\bar{\mathbb{Q}})$ denotes the space of arithmetic
Hilbert cuspforms of level $N_{g}$ and weight $(v,n)$ with $\bar{\mathbb{Q}}$
coefficients, and similarly for $S^{G_{F}}({N_{f}},(w,m),\bar{\mathbb{Q}})$.
Denote by $\pi^{u},\sigma^{u}$ their unitarizations and define a
representation of $\mathbf{GL}_{2}(\mathbb{A}_{L\times F})$ by
$\Pi=\pi^{u}\otimes\sigma^{u}$. Let $\rho\colon\mathrm{Gal}(\bar{F}/F)\to
S_{3}$ be the homomorphism mapping the absolute Galois group of $F$ to the
symmetric group over 3 elements associated to the étale cubic algebra $L\times
F/F$. Let $G_{L\times F}=\operatorname{Res}_{L\times
F/F}\mathbf{GL}_{2,L\times F}$. The $L$-group ${}^{L}(G_{L\times F})$ is given
by the semi-direct product of $\hat{G}\rtimes\mathrm{Gal}(\bar{F}/F)$, where
$\mathrm{Gal}(\bar{F}/F)$ acts on $\hat{G}=\mathbf{GL}_{2}(\mathbb{C})^{\times
3}$ through $\rho$. Assume the central character of $\Pi$ is trivial when
restricted to $\mathbb{A}_{F}^{\times}$.

######  Definition 7.1.

The twisted triple product $L$-function associated with the unitary
automorphic representation $\Pi$ is given by the Euler product

$L(s,\Pi,r)=\prod_{v}L_{v}(s,\Pi_{v},r)^{-1}$

where $\Pi_{v}$ is the local representation at the place $v$ of $F$ appearing
in the restricted tensor product decomposition
$\Pi=\otimes^{\prime}_{v}\Pi_{v}$, and representation $r$ gives the action of
the $L$-group of $G_{L\times F}$ on
$\mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}$ which restricts to
the natural $8$-dimensional representation of $\hat{G}$ and for which
$\mathrm{Gal}(\bar{F}/F)$ acts via $\rho$ permuting the vectors.

Let $D_{/F}$ be a quarternion algebra. We denote by $\Pi^{D}$ the irreducible
unitary cuspidal automorphic representation of $D^{\times}(\mathbb{A}_{L\times
F})$ associated with $\Pi$ by the Jacquet-Langlands correspondence when it
exists. For a vector $\phi\in\Pi^{D}$ one defines its period integral as

$I^{D}(\phi)=\int_{[D^{\times}(\mathbb{A}_{F})]}\phi(x)\mathrm{d}x$

where
$[D^{\times}(\mathbb{A}_{F})]=\mathbb{A}_{F}^{\times}D^{\times}(F)\backslash
D^{\times}(\mathbb{A}_{F})$. For simplicity of notation we write $I(\phi)$ to
denote the period integral when $D=M_{2}(F)$.

######  Theorem 7.2.

Let $\eta\colon\mathbb{A}_{F}^{\times}\to\mathbb{C}^{\times}$ be the quadratic
character attached to $L/F$ by class field theory. Then the following are
equivalent:

  1. (1)

The central $L$-value $L(\frac{1}{2},\Pi,r)$ does not vanish, and for every
place $v$ of $F$ the local $\epsilon$-factor satisfies
$\epsilon_{v}(\frac{1}{2},\Pi_{v},r)\eta_{v}(-1)=1$.

  2. (2)

There exists a vector $\phi\in\Pi$, called a test vector, whose period
integral $I(\phi)$ does not vanish.

###### Proof.

This relies on Ichino’s formula [Ich08, Theorem 1.1]. For a complete proof see
[BF20, Theorem 3.2]. ∎

######  Definition 7.3.

Let $(v,n)\in\mathbb{Z}[\Sigma_{L}]\times\mathbb{Z}$ and
$(w,m)\in\mathbb{Z}[\Sigma_{F}]\times\mathbb{Z}$ be two classical weights. We
say the pair $(v,n)$ and $(w,m)$ are $F$-dominated if there exists
$r\in\mathbb{N}[\Sigma_{L}]$ such that $w=(v+r)_{|F}$ and
$mt_{F}={nt_{L}}_{|F}$ (or equivalently, $m=2n$).

######  Remark 7.4.

In their article [BF20], the authors give sufficient criterion for the
equivalent conditions of Theorem 7.2 to hold. If the weights $(v,n)$ and
$(w,m)$ of $g$ and $f$ respectively are $F$-dominated, then the local
$\epsilon$-factor at the archimedean places satisfy the hypothesis of the
theorem. Moreover the same is true of the finite places if we assume that
$\mathrm{N}_{L/F}(N_{g})\cdot d_{L/F}$ and $N_{f}$ are coprime, and that every
finite prime dividing $N_{f}$ splits in $L$.

######  Assumption 7.5.

Assume that the equivalent conditions of Theorem 7.2 are satisfied. Also
assume that the weights $(v,n),(w,m)$ of $g,f$ respectively are $F$-dominated.

Let $\mathfrak{A}$ be an ideal in $\mathcal{O}_{F}$. Denote by
$K_{11}(\mathfrak{A})$ the following open compact group.

$K_{11}(\mathfrak{A})=\left\\{\begin{pmatrix}a&b\\\
c&d\end{pmatrix}\in\mathbf{GL}_{2}(\hat{\mathcal{O}}_{F})\,|\,a\equiv d\equiv
1,c\equiv 0\text{ mod }N\right\\}.$

Let $\mathfrak{J}$ be the element

${\begin{pmatrix}-1&0\\\
0&1\end{pmatrix}}^{\Sigma_{F}}\in\mathbf{GL}_{2}(\mathbb{R})^{\Sigma_{F}}.$

For any $h\in\sigma^{u}$, we define $h^{\mathfrak{J}}\in\sigma^{u}$ to be the
vector obtained by right translation $h^{\mathfrak{J}}(x)=h(x\mathfrak{J})$.

Let $\delta$ be the Shimura-Maass differential operator [BF20, 1956] on nearly
holomorphic cuspforms.

The natural inclusion
$\mathbf{GL}_{2}(\mathbb{A}_{F})\to\mathbf{GL}_{2}(\mathbb{A}_{L})$ defines by
composition a _diagonal restriction_ map

$\zeta^{*}\colon
S^{G_{L}}(K_{11}(\mathfrak{A}\mathcal{O}_{L}),(v,n),\mathbb{C})\to
S^{G_{F}}(K_{11}(\mathfrak{A}),(v_{|F},2n),\mathbb{C}).$

######  Definition 7.6.

Let $\zeta\colon M(\mu_{\mathfrak{A}},\mathfrak{c})\to
M(\mu_{\mathfrak{A}\mathcal{O}_{L}},\mathfrak{c}\mathcal{O}_{L}\mathfrak{d}^{-1}_{L/F})$
be the map given on points by sending an abelian variety $A\mapsto
A\otimes_{\mathcal{O}_{F}}\mathcal{O}_{L}$ [BF20, §4.1.2]. Here
$\mathfrak{d}_{L/F}$ is the different ideal of the extension $L/F$.

Then as shown in loc. cit. $\zeta$ induces the map on Shimura varieties given
by the inclusion
$\mathbf{GL}_{2}(\mathbb{A}_{F})\to\mathbf{GL}_{2}(\mathbb{A}_{L})$ as above.

######  Lemma 7.7.

For $A/\textrm{Spf}\hskip 1.99997pt{R}$ parametrized by a point of
$\mathfrak{IG}_{n,r,I}$, the trace map
$\text{Tr}_{L/F}\colon\mathrm{H}^{\sharp}_{A}\otimes_{\mathcal{O}_{F}}\mathcal{O}_{L}\to\mathrm{H}^{\sharp}_{A}$
preserves marked sections and the kernel of the marked splitting.

###### Proof.

The first statement follows from the functoriality of the
$\operatorname{dlog}$ map since
$H^{\vee}_{n}(A\otimes_{\mathcal{O}_{F}}\mathcal{O}_{L})=H_{n}^{\vee}(A)\otimes_{\mathcal{O}_{F}}\mathcal{O}_{L}$.
The second statement follows from the fact that the kernel of the splitting is
the kernel of Verschiebung, and the trace commutes with Verschiebung. ∎

######  Lemma 7.8.

Let $r\in\mathbb{N}[\Sigma_{L}]$ be such that $w=(v+r)_{|F}$. Then there is an
$\mathcal{O}_{F}$ ideal $\mathfrak{A}$ supported on a subset of the prime
factors of $N_{f}\cdot\mathrm{N}_{L/F}(N_{g})\cdot d_{L/F}$ such that a test
vector $\phi$ as in the statement of Theorem 7.2 can be chosen to be of the
form $\phi=(\delta^{r}(\breve{g}))^{u}\otimes(\breve{f}^{\mathfrak{J}})^{u}$
for $\breve{g}\in
S^{G_{L}}(K_{11}(\mathfrak{A}\mathcal{O}_{L}),(v,n),\bar{\mathbb{Q}})$ and
$\breve{f}\in S^{G_{F}}(K_{11}(\mathfrak{A}),(w,m),\bar{\mathbb{Q}})$. The
cuspforms $\breve{g},\breve{f}$ are eigenforms for all Hecke operators outside
$N_{f}\cdot\mathrm{N}_{L/F}(N_{g})\cdot d_{L/F}$ with the same eigenvalues of
$g$ and $f$ respectively. Moreover, in this case the period integral can be
expressed as

$I(\phi)=\int_{[\mathbf{GL}_{2}(\mathbb{A}_{F})]}(\delta^{r}\breve{g})^{u}\otimes(\breve{f}^{\mathfrak{J}})^{u}\mathrm{d}x=\langle\zeta^{*}(\delta^{r}\breve{g}),\breve{f}^{*}\rangle$

where $\breve{f}^{*}$ is the cuspform in
$S^{G_{F}}(K_{11}(\mathfrak{A}),(w,m),\bar{\mathbb{Q}})$ whose adelic
$q$-expansion coefficients are complex conjugates of those of $\breve{f}$,
i.e. $\breve{f}^{*}=c\cdot\breve{f}|w_{N}$, where $w_{N}$ is the Atkin-Lehner
operator, and $c\neq 0$ is a constant.

###### Proof.

[BF20, Lemma 3.4]. ∎

######  Proposition 7.9.

The value $L^{\text{alg}}(g,f)\mathrel{\mathop{\mathchar
58\relax}}=\frac{\langle\zeta^{*}(\delta^{r}\breve{g}),\breve{f}^{*}\rangle}{\langle
f^{*},f^{*}\rangle}$ is algebraic. We have

$L\left(\frac{1}{2},\Pi,r\right)=(\ast)\cdot|L^{\text{alg}}(g,f)|^{2}$

where $(\ast)$ is a constant which can be ensured to be non-zero by suitably
choosing the test vectors $\breve{g},\breve{f}$.

###### Proof.

The first statement is [BF20, Proposition 3.5]. The second statement follows
from the proof of Lemma 7.8 using Ichino’s formula. ∎

######  Assumption 7.10.

Let $g,f$ be as above with weights $(v,n)$ and $(w,m)$ respectively that are
$F$-dominated. Let $K$ be a finite extension of $\mathbb{Q}_{p}$ containing
all the Hecke eigenvalues of $g$ and $f$, and such that $L$ is split in $K$.
Assume $f$ has finite slope $\leq a$ at a rational prime $p$ not dividing
$N_{g}N_{f}$, and unramified in $L$. Let $\breve{g},\breve{f}$ be as in Lemma
7.8, which by [Miy71, Proposition 6] are $K$-linear combinations of the
oldforms coming from $g$ and $f$ respectively. Let $\omega_{g},\omega_{f}$ be
overconvergent families of weight $(v_{g},n_{g})$ and $(w_{f},m_{f})$
deforming $g$ and $f$. Let $\breve{\omega}_{g},\breve{\omega}_{f}$ be the
overconvergent families of level $K_{11}(\mathfrak{A}\mathcal{O}_{L})$ and
$K_{11}(\mathfrak{A})$ respectively, obtained from $\omega_{g}$ and
$\omega_{f}$ using the same linear combinations of oldforms as with
$\breve{g}$ and $\breve{f}$. Therefore $\breve{\omega}_{g}$ and
$\breve{\omega}_{f}$ interpolate $\breve{g}$ and $\breve{f}$. Let
$\Lambda_{g}$ and $\Lambda_{f}$ be the $p$-adically complete $K$-algebras such
that $(v_{g},n_{g})\in\mathcal{W}^{G_{L}}(\Lambda_{g})$ and
$(w_{f},m_{f})\in\mathcal{W}^{G_{F}}(\Lambda_{f})$. Let
$k_{g}=2v_{g}+n_{g}t_{L}$, and $k_{f}=2w_{f}+m_{f}t_{F}$. We assume that there
exists a continuous character
$r\colon(\mathcal{O}_{L}\otimes\mathbb{Z}_{p})^{\times}\to\Lambda_{g}^{\times}$
such that the following diagram commutes.

${{\mathrm{Spm}(\Lambda_{g}\hat{\otimes}_{K}\Lambda_{f})}}$${{\mathrm{Spm}(\Lambda_{g})}}$${{\mathcal{W}^{G_{L}}}}$${{\mathrm{Spm}(\Lambda_{f})}}$${{\mathcal{W}^{G_{F}}}}$$\scriptstyle{(w_{f},m_{f})}$$\scriptstyle{(v_{g}+r,n_{g})}$$\scriptstyle{(v,n)\mapsto(v_{|F},2n)}$

Assume that $r=r_{0}\cdot r_{\textrm{an}}$ with
$r_{0}\in\mathbb{N}[\Sigma_{L}]$ and $r_{\textrm{an}}$ is an analytic weight
satisfying the same conditions as satisfied by the character $s$ in
Proposition 5.18. Assume that $k_{g}$ satisfies the same conditions as
satisfied by $k$ in Proposition 5.18, i.e. the analytic part of $k_{g}$ is
given by
$k^{\text{an}}_{g}(x)=\prod_{\sigma\in\Sigma}\exp(u_{\sigma}\log(\sigma(x)))$,
for all $x\in(\mathcal{O}_{L}\otimes\mathbb{Z}_{p})^{\times}$ and for
$u_{\sigma}\in\Lambda_{g}$ for all $\sigma$. For $\tau\in\Sigma_{F}$, let
$v_{\tau}=u_{\sigma}+u_{\bar{\sigma}}$, where $\sigma,\bar{\sigma}$ are the
two extensions of $\tau$ to $L$. Assume that the central character of
$\zeta^{*}\omega_{g}\cdot\omega_{f}$ is trivial.

Let $\mathcal{P}$ be the set of primes lying above $p$ in $L$. Denote by
$\breve{\omega}_{g}^{[\mathcal{P}]}$ the depletion of $\breve{\omega}_{g}$
with respect to all the primes in $\mathcal{P}$. Under the above assumptions,
$\nabla^{r}(\breve{\omega}_{g}^{[\mathcal{P}]})=\nabla^{r_{\textrm{an}}}(\nabla^{r_{0}}(\breve{\omega}_{g}^{[\mathcal{P}]}))$
makes sense, and is a section of $\mathbb{W}^{G_{L}}_{k_{g}+2r,p,I}(-D)$ in
the sense of Definition 4.59.

Let $\Lambda_{g,f}=\Lambda_{g}\hat{\otimes}_{K}\Lambda_{f}$. Consider the
sheaves
$\mathbb{W}_{k_{g}+2r,p,I}^{G_{L}}(-D)\otimes_{\Lambda_{g}}\Lambda_{g,f}$, and
$\mathbb{W}_{k_{f},p,I}^{G_{F}}(-D)\otimes_{\Lambda_{f}}\Lambda_{g,f}$. By
Lemma 7.7 and functoriality of VBMS, the trace map induces a map
$\zeta^{*}\colon\mathbb{W}_{k_{g}+2r,p,I}^{G_{L}}(-D)\otimes_{\Lambda_{g}}\Lambda_{g,f}\to\mathbb{W}_{k_{f},p,I}^{G_{F}}(-D)\otimes_{\Lambda_{f}}\Lambda_{g,f}$,
which is nothing but the diagonal restriction.

For each choice of representative $\mathfrak{c}_{i}$ of
$\mathrm{Cl}^{+}(\mathfrak{A})$, the corresponding component
$\zeta^{*}\nabla^{r}(\breve{\omega}_{g}^{[\mathcal{P}]})_{i}$ can be viewed as
a class in
$H^{d}_{\textrm{dR}}(\bar{\mathcal{M}}_{\ell,p,I}^{\mathfrak{c}_{i}}\otimes\Lambda_{g,f},\mathbb{W}_{k_{f}-2t_{F}})$
for some suitable $\ell$ determined by Proposition 5.18, and $I\subset[0,1]$.
Here we recall that $d=[F\mathrel{\mathop{\mathchar 58\relax}}\mathbb{Q}]$.
Let $\zeta^{*}\nabla^{r}(\breve{\omega}_{g}^{[\mathcal{P}]})^{\leq a}_{i}$ be
its projection onto the slope “$\leq a$” part
$H^{d}_{\textrm{dR}}(\bar{\mathcal{M}}_{\ell,p,I}^{\mathfrak{c}_{i}}\otimes\Lambda_{g,f},\mathbb{W}_{k_{f}-2t_{F}})^{\leq
a}$ of the de Rham cohomology group, possible by the spectral theory of the
$U$ operator [AIP18, Appendice B]. Then inverting
$\lambda=\prod_{\tau\in\Sigma_{F}}\prod_{i=0}^{N}({v}_{\tau}-i)$ for $N\gg 0$,
it can be realized in the space
$H^{0}(\bar{\mathcal{M}}_{\ell,p,I}^{\mathfrak{c}_{i}}\otimes\Lambda_{g,f}[\lambda^{-1}],\mathfrak{w}_{k_{f}})^{\leq
a}$ of slope $\leq a$ using the overconvergent projection $H^{\dagger}$ of
Definition 6.21. Since all of these operations commute with the action of
$\Gamma$ and $\text{Princ}(L)^{+,(p)}$, the sections
$H^{\dagger}(\zeta^{*}\nabla^{r}(\breve{\omega}_{g}^{[\mathcal{P}]})^{\leq
a}_{i})$ together define an arithmetic overconvergent Hilbert modular form.
Let us call this form $H^{\dagger,\leq
a}(\zeta^{*}\nabla^{r}\breve{\omega}_{g}^{[\mathcal{P}]})$, which is a section
of the $\mathrm{Spm}(\Lambda_{g,f}[\lambda^{-1}])$ sheaf
$\mathfrak{w}_{k_{f},p,I}^{G_{F}}(-D)^{\leq a}$ in the sense of Definition
4.57.

Let $\mathrm{Spm}(A)$ be a connected affinoid of
$\mathrm{Spm}(\Lambda_{g,f}[\lambda^{-1}])$. Consider the $A$-Banach module

$M=H^{0}(\mathrm{Spm}(A),\mathfrak{w}_{k_{f},p,I}^{G_{F}}(-D)^{\leq a}).$

######  Definition 7.11.

Define the Hecke algebra $\mathcal{H}$ acting on $M$ to be the subring of
$\operatorname{End}_{A}{M}$ generated by the operators $T_{\ell},S_{\ell}$ for
$(\ell,p\mathfrak{A})=1$, $U_{\ell}$ for $\ell|\mathfrak{A}$ a prime, and
$\mathbf{U}_{\mathfrak{P}}$ for every prime $\mathfrak{P}|p$ in $F$.

We note that for any element $f\in M$, the adelic $q$-expansion coefficient
$c_{p}(y,f)$ makes sense for any
$y\in\hat{\mathcal{O}}_{F}F_{\infty,+}^{\times}$. This is because these
coefficients only depend on the $q$-expansions of $f$ at appropriate cusps as
discussed in §2.2.3. Therefore, one can define a Hecke equivariant map

$\displaystyle M$ $\displaystyle\to\mathcal{H}^{\vee}$ $\displaystyle f$
$\displaystyle\mapsto[T\mapsto c_{p}(1,f|T)].$

The open affinoid $\mathrm{Spm}(\mathcal{H})$ of the eigenvariety is finite
and generically étale over the weight space $\mathrm{Spm}(A)$. Therefore the
trace map $\mathrm{Tr}\colon\mathcal{H}\to\mathcal{H}^{\vee}$ is an
isomorphism over the total ring of fractions $\mathbb{K}$ of $A$. Using this
isomorphism we can define the Petersson inner product on $M$ as follows.

######  Definition 7.12.

Define the Petersson inner product as the pairing

$\langle\cdot,\cdot\rangle\colon M\times
M\to\mathcal{H}^{\vee}\otimes_{A}\mathbb{K}\times\mathcal{H}^{\vee}\otimes_{A}\mathbb{K}\xrightarrow{\mathrm{Tr}}\mathbb{K}.$

######  Proposition 7.13.

Let $x\in\mathrm{Spm}(A)$ be a classical weight such that $M_{x}$ consists
entirely of classical forms of slope less than or equal to $a$, and the fibre
$\mathcal{H}_{x}$ if finite étale over $k(x)$. Then the Hecke equivariant map
$M_{x}\to\mathcal{H}^{\vee}_{x}$ is an isomorphism, and the inner product
$\langle\cdot,\cdot\rangle_{x}$ coincides upto a scalar with the classical
Petersson inner product after base change $k(x)\to\mathbb{C}$.

###### Proof.

Under the assumption of étaleness, the fibre $\mathcal{H}_{x}$ equals the
Hecke algebra of $M_{x}$, and the first claim then follows from [Hid91,
Theorem 2.2]. For the second claim, we notice that the trace pairing on
$\mathcal{H}^{\vee}_{x}$ is perfect. Hence $\langle\cdot,\cdot\rangle_{x}$ is
a Hecke equivariant perfect pairing on $M_{x}$. The result follows since the
classical Petersson inner product is also a Hecke equivariant perfect pairing
on $M_{x}\otimes_{k(x)}\mathbb{C}$. ∎

######  Definition 7.14.

Let $\omega_{f}^{*}=\omega_{f}|w_{\mathfrak{A}}$ where $w_{\mathfrak{A}}$ is
the Atkin-Lehner operator. The $p$-adic twisted triple product $L$-function
associated to $g$ and $f$ satisfying the assumptions above is defined to be

$\mathscr{L}^{g}_{p}(\breve{\omega}_{g},\breve{\omega}_{f})=\frac{\langle
H^{\dagger,\leq
a}(\zeta^{*}\nabla^{r}\breve{\omega}_{g}^{[\mathcal{P}]}),{\breve{\omega}_{f}^{*}}\rangle}{\langle{\omega_{f}^{*}},{\omega_{f}^{*}}\rangle}.$

######  Corollary 7.15.

For a pair of classical weights
$\left((x,c),(y,2c)\right)\in\mathrm{Spm}(\Lambda_{g,f}[\lambda^{-1}])$
factoring through $\mathrm{Spm}(A)$ that are $F$-dominated, i.e. there exists
$t\in\mathbb{N}[\Sigma_{L}]$, such that $(x+t)_{|F}=y$, letting
$g_{x},\breve{g}_{x}$ and $f_{y},\breve{f}_{y}$ be the specializations of
${\omega}_{g},\breve{\omega}_{g}$ and $\omega_{f},\breve{\omega}_{f}$ at
$(x,c)$ and $(y,2c)$ respectively, we have

$\mathscr{L}_{p}^{g}(\breve{\omega}_{g},\breve{\omega}_{f})\left((x,c),(y,2c)\right)=\frac{\langle
H^{\dagger,\leq
a}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]}),{\breve{f}_{y}^{*}}\rangle}{\langle{f_{y}^{*}},{f_{y}^{*}}\rangle}.$

###### Proof.

Follows from construction since the projection to slope “$\leq a$” space, the
overconvergent projection, and the diagonal restriction all commute with the
specialization map. ∎

Next we compare our $p$-adic $L$-function with the definition given by Fornea
and Blanco-Chacon [BF20, Definition 3.8].

So let us now assume that $g,f$ are ordinary, and the families
$\omega_{g},\omega_{f}$ are Hida families interpolating ordinary
$p$-stabilizations $g^{(p)}$ and $f^{(p)}$ respectively. The families
$\breve{\omega}_{g},\breve{\omega}_{f}$ are $\Lambda_{g,f}$-adic families
interpolating $\breve{g}^{(p)}$ and $\breve{f}^{(p)}$ as in loc. cit.

######  Proposition 7.16.

Fix a pair of classical weights $((x,c),(y,2c))\in\mathrm{Spm}(A)$ such that
there exists $t\in\mathbb{Z}[\Sigma_{L}]$ such that $y=(x+t)_{|F}$, and such
that $(x,c)$ and $(y,2c)$ satisfy the condition of Proposition 7.13. Then
letting $g_{x},\breve{g}_{x}$ and $f_{y},\breve{f}_{y}$ be the specializations
as above, we have

$\mathscr{L}^{g}_{p}(\breve{\omega}_{g},\breve{\omega}_{f})((x,c),(y,2c))=\frac{\langle
e_{n.o}\zeta^{*}(\theta^{t}\breve{g}^{[\mathcal{P}]}_{x}),\breve{f}_{y}^{*(p)}\rangle}{\langle
f_{y}^{*(p)},f_{y}^{*(p)}\rangle}.$

Here $e_{n.o}$ is the ordinary projector $e_{n.o}=\lim_{n\to\infty}U^{n!}$. In
particular, it coincides with Definition 3.8 in _[BF20]_ at these points.

###### Proof.

We first note that since the Petersson product is Hecke equivariant, the
special $L$-value
$C=\mathscr{L}^{g}_{p}(\breve{\omega}_{g},\breve{\omega}_{f})((x,c),(y,2c))$
only depends on the projection of $H^{\dagger,\leq
a}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})$ to the
$\breve{f}_{y}^{*(p)}$ isotypic component. In particular, since
$\breve{f}_{y}^{*(p)}$ is ordinary,

$C=\frac{\langle e_{n.o}H^{\dagger,\leq
0}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]}),{\breve{f}_{y}^{*(p)}}\rangle}{\langle{f_{y}^{*(p)}},{f_{y}^{*(p)}}\rangle}.$

We note that since $\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]}$ is already
realized at a finite step of the filtration on $\mathbb{W}_{k_{f}}$,
$H^{\dagger}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})$ makes sense,
and
$e_{n.o}H^{\dagger}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})=e_{n.o}H^{\dagger,\leq
0}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})$. Now by the proof of
Lemma 6.19,
$\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]}=H^{\dagger}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})+\nabla(h)$
for some
$h\in\mathrm{Fil}_{n+d-1}\mathbb{W}_{k_{f}}(-D)\hat{\otimes}\Omega^{g-1}_{\bar{\mathcal{M}}_{r,\alpha,I}}$.
Let $\psi_{\textrm{Frob}}$ be the unit root splitting over the ordinary locus.
Then, since by Lemma 7.7, the trace commutes with Verschiebung, implying the
diagonal restriction commutes with unit root splitting, we have

$\displaystyle
H^{\dagger}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})=\psi_{\textrm{Frob}}(H^{\dagger}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]}))$
$\displaystyle=\psi_{\textrm{Frob}}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})-\psi_{\textrm{Frob}}(\nabla
h)$
$\displaystyle=\zeta^{*}\theta^{t}\breve{g}_{x}^{[\mathcal{P}]}-\psi_{\textrm{Frob}}(\nabla
h).$

Now $\psi_{\textrm{Frob}}\nabla=\sum_{\tau\in\Sigma_{F}}\theta_{\tau}$. Since
$e_{n.o}\theta_{\tau}=0$ for all $\tau\in\Sigma_{F}$, we have

$e_{n.o}H^{\dagger,\leq
0}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]})=\zeta^{*}\theta^{t}\breve{g}_{x}^{[\mathcal{P}]}.$

This proves the claim. ∎

###  7.1. Interpolation formula

We now prove an interpolation formula relating special values of this $p$-adic
$L$-function to classical $L$-values.

Assume $(x,c)$ and $(y,2c)$ are a pair of $F$-dominated weights as before such
that $y=(x+t)_{|F}$ for some $t\in\mathbb{N}[\Sigma_{L}]$, and such that
$(x,c)$ and $(y,2c)$ satisfy the condition of Proposition 7.13. Assume further
that the specializations $g_{x}\mathrel{\mathop{\mathchar
58\relax}}={(\omega_{g})}_{x}$ and $f_{y}\mathrel{\mathop{\mathchar
58\relax}}={(\omega_{f})}_{y}$, which are classical by assumption, are
eigenforms for all the Hecke operators $T_{\ell}$, where
$\ell\nmid\mathfrak{A}$. Note in particular $\ell$ can be $p$, and we assume
$f_{y}$ has slope $\leq a$. Moreover, assume that the Hecke polynomials for
$T_{\mathfrak{p}}$ for any prime $\mathfrak{p}|p$ associated to both $g_{x}$
and $f_{y}$ are separable.

To relate the value $\frac{\langle H^{\dagger,\leq
a}(\zeta^{*}\nabla^{t}\breve{g}_{x}^{[\mathcal{P}]}),\breve{f}_{y}^{*}\rangle}{\langle
f_{y}^{*},f_{y}^{*}}$ to the algebraic part of the classical $L$-value
$\frac{\langle\zeta^{*}(\delta^{t}\breve{g}_{x}),\breve{f}_{y}^{*}\rangle}{\langle\breve{f}_{y}^{*},\breve{f}_{y}^{*}\rangle}$,
we will split the computation into two cases, based on whether a prime
$\mathfrak{p}|p$ in $F$ splits in $L$ or stays inert. We will use this
distinction to simplify the computation of Euler factors while passing from
the $[\mathcal{P}]$-depletion of $\breve{g}$ to $\breve{g}$.

We begin with some preparatory lemmas that we will repeatedly use in the
computation.

######  Lemma 7.17.

Let $\mathfrak{p}|p$ be a prime in $F$.

  1. (1)

$\mathbf{U}_{\mathfrak{p}}\nabla^{t}=p^{t_{\mathfrak{p}}}\nabla^{t}\mathbf{U}_{\mathfrak{p}}$,
where $t_{\mathfrak{p}}=\sum_{\sigma|\mathfrak{p}}t_{\sigma}$.

  2. (2)

$\mathbf{U}_{\mathfrak{p}}H^{\dagger}=H^{\dagger}\mathbf{U}_{\mathfrak{p}}$
over any filtered piece of $\mathbb{W}_{k}$.

  3. (3)

$\mathbf{U}_{\mathfrak{p}}\zeta^{*}=\zeta^{*}\mathbf{U}_{\mathfrak{p}}$.

###### Proof.

  1. (1)

We note that upto multiplication by a constant normalizing factor
$p^{x_{\mathfrak{p}}}$ which we can ignore,
$\mathbf{U}_{\mathfrak{p}}V_{\tau}=pV_{\tau}$ if $\tau|\mathfrak{p}$, and
$V_{\tau}$ otherwise. By [Hid91, §7G], on adelic $q$-expansions
$c_{p}(y,f|\theta^{r})=y_{p}^{r}c_{p}(y,f)$. The claim then follows from the
formula $c_{p}(y,f|\mathbf{U}_{\mathfrak{p}})=c_{p}(y\varpi_{\mathfrak{p}},f)$
and equation (12), where $\varpi_{\mathfrak{p}}$ is the idele with the
uniformizer $p$ in the $\mathfrak{p}$ place and $1$ elsewhere.

  2. (2)

This again follows from equation (12), noting that $H^{\dagger}\nabla=0$. In
fact one can work out a formula like [AI21, Proposition 3.37] for the
overconvergent projection, and check the identity.

  3. (3)

The identity is clear on $V_{\tau}$ for any $\tau$. We thus only need to check
on adelic $q$-expansions of modular forms. For any
$y\in\mathbb{A}_{F,+}^{\times}$, using Theorem 2.8, we have $c_{p}(\xi
y\mathrm{d}_{F},\zeta^{*}f)=\sum_{\mathrm{Tr}_{L/F}(\Upsilon)=\xi}c_{p}(\Upsilon
y\mathrm{d}_{L},f)$ for any $\xi\in F^{\times}_{+}$. The claim follows again
by looking at the effect of $\mathbf{U}_{\mathfrak{p}}$ on adelic
$q$-expansion.

∎

_Inert case_ : Let $\mathfrak{p}|p$ be a prime in $F$ that stays inert in $L$.
Let $\alpha_{g},\beta_{g}$ be the two roots of the Hecke polynomial of
$T_{\mathfrak{p}}$ associated to $g_{x}$, which by assumption are not equal.
Let $\breve{g}^{(\mathfrak{p})}_{\bullet}$ be the $\mathfrak{p}$-stabilization
of $\breve{g}_{x}$ with eigenvalue $\bullet$ for
$\bullet=\alpha_{g},\beta_{g}$. Remark that
$\breve{g}^{[\mathfrak{p}]}_{x}=\breve{g}^{(\mathfrak{p})[\mathfrak{p}]}_{\bullet}$
for each of the $\mathfrak{p}$-stabilizations. This simply follows from
$(1-\mathbf{V}_{\mathfrak{p}}\mathbf{U}_{\mathfrak{p}})(1-(\bullet)\mathbf{V}_{\mathfrak{p}})=1-\mathbf{V}_{\mathfrak{p}}\mathbf{U}_{\mathfrak{p}}$.
Finally, note that
$\breve{g}_{x}=\frac{\alpha_{g}\breve{g}^{(\mathfrak{p})}_{\alpha}-\beta_{g}\breve{g}^{(\mathfrak{p})}_{\beta}}{\alpha_{g}-\beta_{g}}$.

######  Lemma 7.18.

$H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p}]}_{x})=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p})[\mathfrak{p}]}_{\bullet})=\left(1-(\bullet)p^{t_{\mathfrak{p}}}\mathbf{U}_{\mathfrak{p}}^{-1}\right)H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p})}_{\bullet}).$

###### Proof.

We have

$H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p})[\mathfrak{p}]}_{\bullet})=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p})}_{\bullet})-(\bullet)H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\mathbf{V}_{\mathfrak{p}}\breve{g}^{(\mathfrak{p})}_{\bullet}).$

By Lemma 7.17,

$\displaystyle H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p})}_{\bullet})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\mathbf{U}_{\mathfrak{p}}\mathbf{V}_{\mathfrak{p}}\breve{g}^{(\mathfrak{p})}_{\bullet})$
$\displaystyle=p^{-t_{\mathfrak{p}}}\mathbf{U}_{\mathfrak{p}}H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\mathbf{V}_{\mathfrak{p}}\breve{g}^{(\mathfrak{p})}_{\bullet}).$

The claim follows. ∎

######  Lemma 7.19.

Let
$\mathscr{E}_{\mathfrak{p}}(g_{x},T)=(1-p^{t_{\mathfrak{p}}}\alpha_{g}T^{-1})(1-p^{t_{\mathfrak{p}}}\beta_{g}T^{-1})$.
Then,

$H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}_{x})=\frac{1}{\mathscr{E}_{\mathfrak{p}}(g_{x},\mathbf{U}_{\mathfrak{p}})}H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p}]}_{x}).$

###### Proof.

Follows from a simple computation using

$H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}_{x})=\frac{\alpha_{g}H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p})}_{\alpha})-\beta_{g}H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p})}_{\beta})}{\alpha_{g}-\beta_{g}}.$

∎

######  Proposition 7.20.

Let $E(x,y)=(1-xy^{-1})$. Let $\alpha_{f}^{*},\beta_{f}^{*}$ be the two roots
of the Hecke polynomial of $T_{\mathfrak{p}}$ associated to $f_{y}^{*}$, which
are in particular conjugates of the $T_{\mathfrak{p}}$ roots
$\alpha_{f},\beta_{f}$ associated to $f_{y}$. Then

$\langle H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p}]}_{x}),\breve{f}^{*}_{y}\rangle=\left[\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\alpha_{f}^{*})}{E(\beta_{f},\alpha_{f})}+\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\beta_{f}^{*})}{E(\alpha_{f},\beta_{f})}\right]\langle
H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}_{x}),\breve{f}^{*}_{y}\rangle.$

###### Proof.

We have
$\breve{f}^{*}_{y}=(\alpha_{f}^{*}-\beta_{f}^{*})^{-1}(\alpha_{f}^{*}\breve{f}^{*(\mathfrak{p})}_{\alpha}-\beta_{f}^{*}\breve{f}^{*(\mathfrak{p})}_{\beta})$
for the two $\mathfrak{p}$-stabilizations
$\breve{f}^{*(\mathfrak{p})}_{\alpha}$ and
$\breve{f}^{*(\mathfrak{p})}_{\beta}$ of $\breve{f}^{*}_{y}$. The formula then
follows by observing that on the
$\breve{f}^{*(\mathfrak{p})}_{\star}$-component, $\mathbf{U}_{\mathfrak{p}}$
acts via $\star$, for $\star=\alpha_{f}^{*},\beta_{f}^{*}$. Notice that since
the Petersson product is conjugate linear in the second variable, we have
$E(\beta_{f},\alpha_{f})$ (and similarly $E(\alpha_{f},\beta_{f})$) instead of
the conjugate terms. ∎

_Split case_ : Let $\mathfrak{p}|p$ be a prime in $F$ that splits as
$\mathfrak{p}\mathcal{O}_{L}=\mathfrak{p}_{1}\mathfrak{p}_{2}$ in $L$. For
$i=1,2$, let $\breve{g}^{(\mathfrak{p}_{i})}_{\bullet}$ be the
$\mathfrak{p}_{i}$-stabilization of $\breve{g}_{x}$ with eigenvalue
$\bullet=\alpha_{i},\beta_{i}$.

######  Lemma 7.21.

For $\mathfrak{i}\neq\mathfrak{j}$, we have
$\mathbf{U}_{\mathfrak{p}}\nabla^{t}\mathbf{V}_{\mathfrak{p_{i}}}\breve{g}^{[\mathfrak{p_{j}}]}=0$
for any $\breve{g}$. In particular, it is true for $\breve{g}_{x}$ and any of
its stabilizations.

###### Proof.

By Lemma 7.17, we have

$\mathbf{U}_{\mathfrak{p}}\nabla^{t}\mathbf{V}_{\mathfrak{p_{i}}}\breve{g}^{[\mathfrak{p_{j}}]}=p^{t_{\mathfrak{p}}}\nabla^{t}\mathbf{U}_{\mathfrak{p}}\mathbf{V}_{\mathfrak{p_{i}}}\breve{g}^{[\mathfrak{p_{j}}]}=p^{t_{\mathfrak{p}}}\nabla^{t}\mathbf{U}_{\mathfrak{p_{j}}}\breve{g}^{[\mathfrak{p_{j}}]}=0.$

∎

######  Lemma 7.22.

For any $\mathfrak{p_{i}}$-stabilization
$\breve{g}^{(\mathfrak{p_{i}})}_{\bullet}$, we have $H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{j}}]}_{\bullet})=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x})$
for $\mathfrak{i}\neq\mathfrak{j}$. Hence, by taking the appropriate linear
combination, we have $H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{j}}]}_{x})=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x})$
for any $\mathfrak{j}=1,2$.

###### Proof.

We have the following.

$\displaystyle H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}(1-\mathbf{V}_{\mathfrak{p_{i}}}\mathbf{U}_{\mathfrak{p_{i}}}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{j}}]}_{\bullet})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{j}}]}_{\bullet})-(\bullet)H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\mathbf{V}_{\mathfrak{p_{i}}}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{j}}]}_{\bullet})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{j}}]}_{\bullet})-(\bullet)H^{\dagger,\leq
a}\mathbf{U}_{\mathfrak{p}}^{-1}\zeta^{*}(\mathbf{U}_{\mathfrak{p}}\nabla^{t}\mathbf{V}_{\mathfrak{p_{i}}}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{j}}]}_{\bullet})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{j}}]}_{\bullet}).$

Here the last equality follows from Lemma 7.21. ∎

######  Lemma 7.23.

We have

$H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x})=\frac{\prod\limits_{\bullet,\star\in\\{\alpha,\beta\\}}1-p^{t_{\mathfrak{p}}}\bullet_{i}\star_{j}\mathbf{U}_{\mathfrak{p}}^{-1}}{1-p^{2t_{\mathfrak{p}}}\alpha_{i}\alpha_{j}\beta_{i}\beta_{j}\mathbf{U}_{\mathfrak{p}}^{-2}}=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}_{x}).$

###### Proof.

We have the following chain of identities.

$\displaystyle H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{i}}]}_{x})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})[\mathfrak{p_{i}}]}_{\bullet})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet})-(\bullet)H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\mathbf{V}_{\mathfrak{p_{i}}}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet})$
$\displaystyle=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet})-(\bullet)p^{t_{\mathfrak{p}}}\mathbf{U}_{\mathfrak{p}}^{-1}H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\mathbf{U}_{\mathfrak{p_{j}}}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet})$
$\displaystyle=\left[1-(\bullet)p^{t_{\mathfrak{p}}}(\alpha_{j}+\beta_{j})\mathbf{U}_{\mathfrak{p}}^{-1}\right]H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet})+(\bullet)p^{t_{\mathfrak{p}}}\alpha_{j}\beta_{j}\mathbf{U}_{\mathfrak{p}}^{-1}H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\mathbf{V}_{\mathfrak{p_{j}}}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet})$
$\displaystyle=\left[1-(\bullet)p^{t_{\mathfrak{p}}}(\alpha_{j}+\beta_{j})\mathbf{U}_{\mathfrak{p}}^{-1}+(\bullet)^{2}p^{2t_{\mathfrak{p}}}\alpha_{j}\beta_{j}\mathbf{U}_{\mathfrak{p}}^{-2}\right]H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet})$
$\displaystyle=(1-(\bullet)p^{t_{\mathfrak{p}}}\alpha_{j}\mathbf{U}_{\mathfrak{p}}^{-1})(1-(\bullet)p^{t_{\mathfrak{p}}}\beta_{j}\mathbf{U}_{\mathfrak{p}}^{-1})H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{(\mathfrak{p_{i}})}_{\bullet}).$

Now by Lemma 7.22, since $H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{i}}]}_{\bullet})=H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x})$,
we have

$\displaystyle H^{\dagger,\leq a}\zeta^{*}(\nabla^{t}\breve{g}_{x})$
$\displaystyle=\frac{1}{\alpha_{i}-\beta_{i}}\left[\frac{\alpha_{i}}{(1-p^{t_{\mathfrak{p}}}\alpha_{i}\alpha_{j}\mathbf{U}_{\mathfrak{p}}^{-1})(1-p^{t_{\mathfrak{p}}}\alpha_{i}\beta_{j}\mathbf{U}_{\mathfrak{p}}^{-1})}-\frac{\beta_{i}}{(1-p^{t_{\mathfrak{p}}}\beta_{i}\alpha_{j}\mathbf{U}_{\mathfrak{p}}^{-1})(1-p^{t_{\mathfrak{p}}}\beta_{i}\beta_{j}\mathbf{U}_{\mathfrak{p}}^{-1})}\right]$
$\displaystyle\quad\quad\quad\quad\quad\quad\quad\times H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x})$
$\displaystyle=\frac{1-p^{2t_{\mathfrak{p}}}\alpha_{i}\alpha_{j}\beta_{i}\beta_{j}\mathbf{U}_{\mathfrak{p}}^{-2}}{\prod\limits_{\bullet,\star\in\\{\alpha,\beta\\}}1-p^{t_{\mathfrak{p}}}\bullet_{i}\star_{j}\mathbf{U}_{\mathfrak{p}}^{-1}}H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x}).$

∎

######  Proposition 7.24.

Let
$\mathscr{E}_{\mathfrak{p}}(g_{x},T)=\prod\limits_{\bullet,\star\in\\{\alpha,\beta\\}}(1-p^{t_{\mathfrak{p}}}\bullet_{i}\star_{j}T^{-1})$
for a split prime $\mathfrak{p}$. Let
$\mathscr{E}_{0,\mathfrak{p}}(g_{x},T)=1-p^{2t_{\mathfrak{p}}}\alpha_{i}\alpha_{j}\beta_{i}\beta_{j}T^{-2}$
With assumptions as in Proposition 7.20, we have

$\langle H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}^{[\mathfrak{p_{1}},\mathfrak{p_{2}}]}_{x}),\breve{f}^{*}_{y}\rangle=\left[\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\alpha_{f}^{*})}{E(\beta_{f},\alpha_{f})\mathscr{E}_{0,\mathfrak{p}}(g_{x},\alpha_{f}^{*})}+\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\beta_{f}^{*})}{E(\alpha_{f},\beta_{f})\mathscr{E}_{0,\mathfrak{p}}(g_{x},\beta_{f}^{*})}\right]\langle
H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}_{x}),\breve{f}^{*}_{y}\rangle.$

###### Proof.

This is similar to the proof of Proposition 7.20. Write
$\breve{f}^{*}_{y}=(\alpha_{f}^{*}-\beta_{f}^{*})^{-1}(\alpha_{f}^{*}\breve{f}^{*(\mathfrak{p})}_{\alpha}-\beta_{f}^{*}\breve{f}^{*(\mathfrak{p})}_{\beta})$
and apply Lemma 7.23 to each of the $\mathfrak{p}$-stabilizations. ∎

######  Lemma 7.25.

We have $\langle H^{\dagger,\leq
a}\zeta^{*}(\nabla^{t}\breve{g}_{x}),\breve{f}^{*}_{y}\rangle/\langle
f^{*}_{y},f^{*}_{y}\rangle=\langle\zeta^{*}(\delta^{t}\breve{g}_{x}),\breve{f}^{*}_{y}\rangle/\langle
f^{*}_{y},f^{*}_{y}\rangle$ upon base change to $\mathbb{C}$.

###### Proof.

By [DR14, Lemma 2.5], we have
$\langle\zeta^{*}(\delta^{t}\breve{g}_{x}),\breve{f}^{*}_{y}\rangle=\langle\Pi^{\text{hol}}\zeta^{*}(\delta^{t}\breve{g}_{x}),\breve{f}^{*}_{y}\rangle$
where $\Pi^{\text{hol}}$ is the holomorphic projection of the nearly
holomorphic modular form $\zeta^{*}(\delta^{t}\breve{g}_{x})$. But the
holomorphic projection is by definition the same thing as the overconvergent
projection of $\zeta^{*}(\nabla^{t}\breve{g}_{x})$ [DR14, Proposition 2.8].
The Petersson inner product cuts out the slope $\leq a$ part since
$\breve{f}^{*}_{y}$ has slope $\leq a$. The lemma then follows from
Proposition 7.13, which shows that the pairing defined algebraically is a
scalar multiple of the Petersson inner product. ∎

We can now collect all the results above to state the following theorem.

######  Theorem 7.26.

With all the assumptions as above, the value of the $p$-adic $L$-function is
related to the classical central $L$-value in the following manner.

$\displaystyle\begin{aligned}
&\mathscr{L}^{g}_{p}(\breve{\omega}_{g},\breve{\omega}_{f})((x,c),(y,2c))\\\
&=\left(\prod_{\mathfrak{p}\text{
inert}}\left[\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\alpha_{f}^{*})}{E(\beta_{f},\alpha_{f})}+\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\beta_{f}^{*})}{E(\alpha_{f},\beta_{f})}\right]\prod_{\mathfrak{p}\text{
split}}\left[\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\alpha_{f}^{*})}{E(\beta_{f},\alpha_{f})\mathscr{E}_{0,\mathfrak{p}}(g_{x},\alpha_{f}^{*})}+\frac{\mathscr{E}_{\mathfrak{p}}(g_{x},\beta_{f}^{*})}{E(\alpha_{f},\beta_{f})\mathscr{E}_{0,\mathfrak{p}}(g_{x},\beta_{f}^{*})}\right]\right)\end{aligned}$
$\displaystyle\times\frac{\langle\zeta^{*}(\delta^{t}\breve{g}_{x}),\breve{f}^{*}_{y}\rangle}{\langle
f^{*}_{y},f^{*}_{y}\rangle}.$

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