# Triple product $p$-adic $L$-functions for finite slope families and a
$p$-adic Gross–Zagier formula

Ting-Han Huang

###### Abstract

In this paper, we generalize two results of H. Darmon and V. Rotger on triple
product $p$-adic $L$-functions attached to Hida families to finite slope
families. We first prove a $p$-adic Gross–Zagier formula, then demonstrate an
application to a special case of the equivariant Birch and Swinnerton-Dyer
conjecture for supersingular elliptic curves.

##  1 Introduction

In [DR14], a $p$-adic Gross–Zagier formula is proved for the triple product
$p$-adic $L$-functions for ordinary families of modular forms. More precisely,
the formula relates

  1. -

the special values of the Garrett–Rankin triple product $p$-adic $L$-function
at classical points lying outside the region of interpolation, to

  2. -

the $p$-adic Abel–Jacobi images of generalized diagonal cycles in the product
of three Kugo–Sato varieties, evaluated at certain differentials.

The fist part of this paper aims to generalize the above result to triple
product $p$-adic $L$-functions for finite slope families of modular forms.

Let $f\in S_{k}(N_{f},\chi_{f})$, $g\in S_{\ell}(N_{g},\chi_{g})$, $h\in
S_{m}(N_{h},\chi_{h})$ be a triple of normalized primitive cuspidal eigenforms
and set $N:=\ell\mathrm{cm}(N_{f},N_{g},N_{h})$. We will assume that
$\chi_{f}\cdot\chi_{g}\cdot\chi_{h}=1$, which implies that $k+\ell+m$ is even.

The triple $(k,\ell,m)$ is said to be balanced if the largest number is
strictly less than the sum of the other two. A triple that is not balanced is
called unbalanced, and the largest number in an unbalanced triple will be
called the dominant weight.

For such a triple $(f,g,h)$, one has the (complex) Garrett–Rankin $L$-function
$L(f,g,h;s)$. It can be viewed as the $L$-function attached to the tensor
product

$V(f,g,h)=V(f)\otimes V(g)\otimes V(h)$

of the (compatible systems of) $p$-adic Galois representations associated with
$f,g$ and $h$ respectively. This $L$-function admits a functional equation
relating its values at $s$ and $k+\ell+m-2-s$. As a consequence, the order of
vanishing of $L(f,g,h;s)$ at the central point $c:=\frac{k+\ell+m-2}{2}$ is
determined by the root number $\varepsilon\in\\{\pm 1\\}$ of the functional
equation. The root number $\varepsilon$ can be further written as a product
$\prod_{v\mid N\infty}\varepsilon_{v}$ of local root numbers
$\varepsilon_{v}\in\\{\pm 1\\}$. We will assume the following assumption
throughout this article as in [DR14].

Assumption H: The local root numbers $\varepsilon_{v}=+1$ for all finite
primes $v\mid N$.

As indicated in [DR14, § 1], this condition is satisfied in a broad collection
of settings. Under Assumption H, $\varepsilon=\varepsilon_{\infty}$ depends
only on the triple of weights $(k,\ell,m)$:

$\varepsilon_{\infty}=\begin{cases}-1\quad\textrm{if }(k,\ell,m)\textrm{ is
balanced;}\\\ +1\quad\textrm{if }(k,\ell,m)\textrm{ is
unbalanced.}\end{cases}$

In particular, the $L$-function necessarily vanishes at the central point $c$
when $(k,\ell,m)$ is balanced. In this situation, instead of studying the
values of $L(f,g,h;s)$, one usually studies the values of its derivatives or
its associated $p$-adic $L$-functions and expects them to encode arithmetic
information.

Choose a prime $p\nmid N$ such that $f,g,h$ are of (sufficiently small) finite
slope at $p$. We may find ($p$-adic) Coleman families $\mathbf{f}$,
$\mathbf{g}$, $\mathbf{h}$ passing through $f,g,h$, defined over certain
weight spaces $\Omega_{f},\Omega_{g},\Omega_{h}$, respectively. In the
language of rigid analytic geometry, the space $\Omega_{\bullet}$ for
$\bullet\in\\{f,g,h\\}$ is a finite rigid analytic cover over a subset of the
weight space
$\mathcal{W}:=\operatorname{\operatorname{Hom}}_{\mathrm{cts}}(\mathbb{Z}_{p}^{\times},\mathbb{C}_{p}^{\times})$
which always contains the integers $\mathbb{Z}$ via the identification
$k\mapsto(\gamma\mapsto\gamma^{k})$.

When $\mathbf{f}$ is of slope $a\in\mathbb{Q}_{\geq 0}$, a point
$x\in\Omega_{f}$ is said to be classical if its image in $\Omega$, denoted by
$\kappa(x)$, belongs to $\mathbb{Z}_{>a+1}$. We will often identify a
classical weight $x$ with $\kappa(x)\in\mathbb{Z}$ by an abuse of notation. We
denote the set of classical points in $\Omega_{f}$ by
$\Omega_{f,\mathrm{cl}}$. For almost all $x\in\Omega_{f,\mathrm{cl}}$, the
specialization of $\mathbf{f}$ at $x\in\Omega_{f,\mathrm{cl}}$, denoted by
$\mathbf{f}_{x}$ (or simply $f_{x}$), is a normalized eigenform of weight
$\kappa(x)$ on $\Gamma_{1}(N,p):=\Gamma_{1}(N)\cap\Gamma_{0}(p)$. For all but
finitely many such $x$, $\mathbf{f}_{x}$ is a (finite slope) $p$-stabilization
of a normalized eigenform of the same weight on $\Gamma_{1}(N)$, denoted by
$f_{x}^{0}$.

When $\mathbf{f}$, $\mathbf{g}$, $\mathbf{h}$ are Hida (ordinary) families,
one can view them as maps

$\mathbf{f}:\Omega_{f}\rightarrow\mathbb{C}_{p}\llbracket q\rrbracket,\
\mathbf{g}:\Omega_{g}\rightarrow\mathbb{C}_{p}\llbracket q\rrbracket,\
\mathbf{h}:\Omega_{h}\rightarrow\mathbb{C}_{p}\llbracket q\rrbracket.$

The triple product $p$-adic $L$-functions for Hida families in [DR14] is
defined as

$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h}):=\frac{(\mathbf{f}^{*},e_{\operatorname{\operatorname{ord}}}(\theta^{\bullet}\mathbf{g}^{[p]}\times\mathbf{h}))}{(\mathbf{f}^{*},\mathbf{f}^{*})},$

where $\mathbf{f}^{*}$ is the Atkin–Lehner involution of $\mathbf{f}$,
$e_{\operatorname{\operatorname{ord}}}$ is Hida’s ordinary projector,
$\theta=q\frac{d}{dq}$ is Serre’s operator,
$\mathbf{g}^{[p]}:=(1-VU)\mathbf{g}$ is the $p$-depletion of $\mathbf{g}$, and
$(\phantom{e},\phantom{e})$ is the Petersson product.

For finite slope families, one needs a different approach. A more geometric
construction of triple product $p$-adic $L$-functions for finite slope
families is developed by F. Andreatta and A. Iovita in [AI21]. Section §2 is
dedicated to recall their construction.

Roughly speaking, one wants to define the triple product $p$-adic $L$-function
as the ratio

$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h}):=\frac{(\mathbf{f}^{*},\nabla^{\bullet}\mathbf{g}^{[p]}\times\mathbf{h})}{(\mathbf{f}^{*},\mathbf{f}^{*})}.$

In order for this expression to make sense, one needs to $p$-adically iterate
the Gauss–Manin connection $\nabla$. When $\mathbf{f}$,
$\mathbf{g},\mathbf{h}$ are Hida families, as we saw in above definition, one
may replace $\nabla$ by $\theta=q\frac{d}{dq}$ and works only on the
$q$-expansions. Then there is a straightforward way to iterate $\theta$ to
$p$-adic powers, provided that the form it acts on lies in the kernel of $U$
(e.g. the $p$-depletion $\mathbf{g}^{[p]}$). The case for $\nabla$ is more
complicated and requires a careful study on the convergence. However, we would
like to refer the detailed computations to[AI21].

The triple product $p$-adic $L$-function
$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$ can be viewed as a
three-variable function

$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h}):\Omega_{f}\times\Omega_{g}\times\Omega_{h}\rightarrow\mathbb{C}_{p},$

whose value at $(x,y,z)\in\Omega_{f}\times\Omega_{g}\times\Omega_{h}$ is

$\frac{(\mathbf{f}_{x}^{*},\nabla^{-t}\mathbf{g}_{y}^{[p]}\times\mathbf{h}_{z})}{(\mathbf{f}_{x}^{*},\mathbf{f}_{x}^{*})},$

where $t\in\Omega$ satisfies $\kappa(x)=\kappa(y)+\kappa(z)-2t$. As one can
observe, the form $\mathbf{f}$ plays a distinct role in $\mathscr{L}^{f}_{p}$.
What is not so obvious is that
$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$ is symmetric, up to a
$\pm$ sign, in the last two components. Similarly, one can define
$\mathscr{L}^{g}_{p}=\mathscr{L}^{g}_{p}(\mathbf{g},\mathbf{f},\mathbf{h})$
and
$\mathscr{L}^{h}_{p}=\mathscr{L}^{h}_{p}(\mathbf{h},\mathbf{f},\mathbf{g})$.

Set $\Sigma=\Omega_{f}\times\Omega_{g}\times\Omega_{h}$ and
$\Sigma_{\operatorname{cl}}=\Omega_{f,\operatorname{cl}}\times\Omega_{g,\operatorname{cl}}\times\Omega_{h,\operatorname{cl}}$.
The set $\Sigma_{\operatorname{cl}}$ can be naturally divided into four
disjoint subsets:

$\displaystyle\Sigma_{\operatorname{\operatorname{bal}}}$
$\displaystyle:=\\{(x,y,z)\in\Sigma_{\operatorname{cl}}\mid(\kappa(x),\kappa(y),\kappa(z))\textrm{
is balanced}\\};$ $\displaystyle\Sigma_{f}$
$\displaystyle:=\\{(x,y,z)\in\Sigma_{\operatorname{cl}}\mid\kappa(x)\geq\kappa(y)+\kappa(z)\\};$
$\displaystyle\Sigma_{g}$
$\displaystyle:=\\{(x,y,z)\in\Sigma_{\operatorname{cl}}\mid\kappa(y)\geq\kappa(x)+\kappa(z)\\};$
$\displaystyle\Sigma_{h}$
$\displaystyle:=\\{(x,y,z)\in\Sigma_{\operatorname{cl}}\mid\kappa(z)\geq\kappa(x)+\kappa(y)\\}.$

The $p$-adic $L$-function $\mathscr{L}^{f}_{p}$ on $\Sigma_{f}$ interpolates
the square roots of central critical values of the classical $L$-functions
$L(f_{x}^{0},g_{y}^{0},h_{z}^{0};s)$, which is a combined result of the Ichino
formula, [DR14, Theorem 4.7] and [AI21, Corollary 5.13].

Points in $\Sigma_{\operatorname{\operatorname{bal}}}$, on the other hand, lie
outside this region of interpolation, and we are interested in the values of
$\mathscr{L}^{f}_{p}$ at these points. By definition, the value of
$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$ at a balanced point
$(x,y,z)$ involves a ‘negative’ power $\nabla^{-t}$, which is originally
defined as a $p$-adic limit. On the other hand, the interpolating property of
$\nabla$ implies the relation $\nabla^{s}\circ\nabla^{-t}=\nabla^{-t+s}$. In
other words, this negative power also serves as a $t$-th anti-derivative with
respect to $\nabla$. Hence, we may expect to express these special values via
a $p$-adic integration theory. A suitable candidate is Coleman $p$-adic
integration or its generalization, Besser’s finite polynomial cohomology.
Furthermore, these $p$-adic integration theories allow us to compute $p$-adic
Abel–Jacobi maps, which will be discussed bellow.

Let $E\rightarrow X=X_{1}(N)$ be the universal elliptic curve over the modular
curve defined over $\mathbb{Q}_{p}$ (or a finite extension $K$ of
$\mathbb{Q}_{p}$). For any $n\geq 0$, we let $W_{n}$ be the $n$-th Kuga–Sato
variety over $X$. It is an $(n+1)$-dimensional variety obtained by
desingularizing the $n$-fold fiber product $E^{n}$ over $X$ (c.f. [BDP13,
Apeendix]).

Now fix a balanced weight
$(x,y,z)\in\Sigma_{\operatorname{\operatorname{bal}}}$. We write
$(x,y,z)=(r_{1}+2,r_{2}+2,r_{3}+2)$, $r:=\frac{1}{2}(r_{1}+r_{2}+r_{3})$, and
set

$W:=W_{r_{1}}\times W_{r_{2}}\times W_{r_{3}}.$

As in [DR14, § 3.1], one can construct a homologically trivial cycle

$\Delta_{x,y,z}\in\operatorname{\operatorname{CH}}^{r+2}(W)_{0}:=\ker(\operatorname{\operatorname{CH}}^{r+2}(W)\xrightarrow{\operatorname{cl}}H^{2r+4}_{\operatorname{\operatorname{dR}}}(W/\mathbb{C}_{p})),$

and study its image under the $p$-adic Abel–Jacobi map

$\operatorname{\operatorname{AJ}}_{p}:\operatorname{\operatorname{CH}}^{r+2}(W)_{0}\rightarrow[\operatorname{\operatorname{Fil}}^{r+2}H^{2r+3}_{\operatorname{\operatorname{dR}}}(W/\mathbb{C}_{p})]^{\vee}.$

Suppose that the family $\mathbf{f}$ is of slope $a$ with $a<x-1$, and assume
that the specializations $\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}$ are
$p$-stabilizations of classical modular forms $f_{x}^{0},g_{y}^{0},h_{z}^{0}$,
respectively. For simplicity, we will further assume that
$f_{x}^{0},g_{y}^{0},h_{z}^{0}$ are newforms of the same level $N$ in the
introduction.

Associated with the triple $(f_{x}^{0},g_{y}^{0},h_{z}^{0})$, we construct an
element

$\displaystyle\eta_{f}^{a}\otimes\omega_{g}\otimes\omega_{h}$
$\displaystyle\in
H^{r_{1}+1}_{\operatorname{\operatorname{dR}}}(W_{r_{1}})\otimes\operatorname{\operatorname{Fil}}^{r_{2}+1}H^{r_{2}+1}_{\operatorname{\operatorname{dR}}}(W_{r_{2}})\otimes\operatorname{\operatorname{Fil}}^{r_{3}+1}H^{r_{3}+1}_{\operatorname{\operatorname{dR}}}(W_{r_{3}})$
$\displaystyle\subset\operatorname{\operatorname{Fil}}^{r+2}(H^{r_{1}+1}_{\operatorname{\operatorname{dR}}}(W_{r_{1}})\otimes
H^{r_{2}+1}_{\operatorname{\operatorname{dR}}}(W_{r_{2}})\otimes
H^{r_{3}+1}_{\operatorname{\operatorname{dR}}}(W_{r_{3}}))$
$\displaystyle\subset\operatorname{\operatorname{Fil}}^{r+2}H^{2r+3}_{\operatorname{\operatorname{dR}}}(W/\mathbb{C}_{p}),$

where the first inclusion is by the balancedness assumption, and the second
one is from the Künneth decomposition of $W=W_{r_{1}}\times W_{r_{2}}\times
W_{r_{3}}$. In particular, $\eta_{f}^{a}\otimes\omega_{g}\otimes\omega_{h}$
lies inside the domain of
$\operatorname{\operatorname{AJ}}_{p}(\Delta_{x,y,z})$.

Before stating the Gross–Zagier formula, we need to introduce several Euler
factors. For any $\phi\in S_{k}(\Gamma_{1}(N),\chi)$, we shall write the Hecke
polynomial as

$x^{2}-a_{p}(\phi)x+\chi(p)p^{k-1}=(x-\alpha_{\phi})(x-\beta_{\phi})$

with
$\operatorname{\operatorname{ord}}_{p}(\alpha_{\phi})\leq\operatorname{\operatorname{ord}}_{p}(\beta_{\phi})$.
For the modular form $f_{x}^{0}$ associated with $\mathbf{f}_{x}$, we also
assume that $a=\operatorname{\operatorname{ord}}_{p}(\alpha_{f_{x}^{0}})$. To
simplify the notations, we will write $\alpha_{f}$ for $\alpha_{f_{x}^{0}}$
and similarly for the modular forms $g$ and $h$ when no confusion might occur.

The first result of this paper is the theorem below.

######  Theorem 1.1.

Given $(x,y,z)\in\Sigma_{\operatorname{\operatorname{bal}}}$. Let
$c:=(x+y+z-2)/2$ and write $x=y+z-2t$ with $t\in\mathbb{Z}_{>0}$. Then

$\mathscr{L}_{p}^{f}(\mathbf{f},\mathbf{g},\mathbf{h})(x,y,z)=(-1)^{t-1}\frac{\mathscr{E}(f_{x}^{0},g_{y}^{0},h_{z}^{0})}{(t-1)!\mathscr{E}_{0}(f_{x}^{0})\mathscr{E}_{1}(f_{x}^{0})}\operatorname{\operatorname{AJ}}_{p}(\Delta_{x,y,z})(\eta_{f}^{a}\otimes\omega_{g}\otimes\omega_{h}),$
(1)

where the Euler factors are given by

$\displaystyle\begin{split}\mathscr{E}_{0}(f_{x}^{0})&:=1-\beta_{f}^{2}\chi_{f}^{-1}(p)p^{1-x},\\\
\mathscr{E}_{1}(f_{x}^{0})&:=1-\beta_{f}^{2}\chi_{f}^{-1}(p)p^{-x},\\\
\mathscr{E}(f_{x}^{0},g_{y}^{0},h_{z}^{0})&:=(1-\beta_{f}\alpha_{g}\alpha_{h}p^{-c})(1-\beta_{f}\alpha_{g}\beta_{h}p^{-c})(1-\beta_{f}\beta_{g}\alpha_{h}p^{-c})(1-\beta_{f}\beta_{g}\beta_{h}p^{-c}).\end{split}$
(2)

After establishing the $p$-adic Gross–Zagier formula, we proceed to generalize
two results in the paper [DR17]. Let $E$ be an elliptic curve defined over
$\mathbb{Q}$ and

$\rho:\operatorname{\operatorname{Gal}}_{\mathbb{Q}}:=\operatorname{\operatorname{Gal}}(\overline{\mathbb{Q}}/\mathbb{Q})\rightarrow\operatorname{\operatorname{Aut}}_{L}(V_{\rho})\cong\operatorname{\operatorname{GL}}_{n}(L)$

be an Artin representation with coefficient in a finite extension $L$ of
$\mathbb{Q}$, which factors through a finite Galois extension $H/\mathbb{Q}$.
Then one has $L(E,\rho,s)$, the Hasse–Weil–Artin $L$-function of $E$ twisted
by $\rho$. Let
$E(H)^{\rho}:=\operatorname{\operatorname{Hom}}_{G_{\mathbb{Q}}}(V_{\rho},E(H)\otimes_{\mathbb{Z}}L)$
be the $\rho$-isotypic component of the Mordell–Weil group. One defines the
analytic and algebraic rank of the twist of $E$ by $\rho$ as

$r_{\mathrm{an}}(E,\rho):=\operatorname{\operatorname{ord}}_{s=1}L(E,\rho,s),\quad
r(E,\rho):=\dim_{L}(E(H))^{\rho}.$

The Galois-equivariant Birch and Swinnerton-Dyer conjecture then predicts that

$r_{\mathrm{an}}(E,\rho)=r(E,\rho).$

The triple product $L$-function appears in the special case
$\rho=\rho_{1}\otimes\rho_{2}$, where

$\rho_{1},\rho_{2}:G_{\mathbb{Q}}\rightarrow\operatorname{\operatorname{GL}}_{2}(L)$

are two-dimensional odd irreducible Artin representation with
$\chi=\det(\rho_{1})=\det(\rho_{2})^{-1}$. As the elliptic curve $E$ is
associated with a cuspidal newform $f\in S_{2}(N_{f})$, and the
representations $\rho_{1},\rho_{2}$ are associated with cuspidal newforms
$g\in S_{1}(N_{g},\chi),h\in S_{1}(N_{h},\chi^{-1})$, respectively,
$L(E,\rho;s)$ is equal to the triple product $L$-function $L(f,g,h;s)$.

Assuming that the modular forms $f,g,h$ are ordinary at a fixed prime $p$, and
let $\mathbf{f},\mathbf{g},\mathbf{h}$ be three Hida families passing through
respectively $f,g,h$, Darmon–Rotger proved the following result.

######  Theorem 1.2 ([DR17, Theorem B]).

If $L(f,g,h;1)=0$ and
$\mathscr{L}_{p}^{g}(\mathbf{g},\mathbf{f},\mathbf{h})(1,2,1)\neq 0$ for some
choice of test vectors, then

$\dim_{L_{p}}\operatorname{\operatorname{Sel}}_{p}(E,\rho)\geq 2$

where $L_{p}$ is the completion of $L$ in $\overline{\mathbb{Q}}_{p}$,
$\operatorname{\operatorname{Sel}}_{p}(E,\rho)$ is the $\rho$-isotypic
component of the Bloch–Kato Selmer group (c.f. §6.1 in loc. cit.). The
statement stays true if we replace
$\mathscr{L}_{p}^{g}(\mathbf{g},\mathbf{f},\mathbf{h})$ by
$\mathscr{L}_{p}^{h}(\mathbf{h},\mathbf{f},\mathbf{g})$.

The technique used in [DR17] was later improved by Bertolini–Seveso–Venerucci
in [BSV22]. We will give an over-simplified illustration of their approach
below.

First, one constructs an element (called the diagonal class)

$\underline{\kappa}=\underline{\kappa}(\mathbf{f},\mathbf{g},\mathbf{h})\in
H^{1}(\mathbb{Q},V(\mathbf{f},\mathbf{g},\mathbf{h}))$

in the cohomology group of a certain big Galois representation
$V(\mathbf{f},\mathbf{g},\mathbf{h})$, such that the image of
$\underline{\kappa}_{p}:=res_{p}(\underline{\kappa})$ under the Perrin-Riou
big logarithm $\mathscr{L}og_{\xi}$ is the triple product $p$-adic
$L$-function $\mathscr{L}_{p}^{\xi}$ for $\xi\in\\{f,g,h\\}$, where $res_{p}$
denotes the restriction to $G_{\mathbb{Q}_{p}}$.

Second, one studies the specialization
$\underline{\kappa}_{p}(\mathbf{f_{2}},\mathbf{g_{1}},\mathbf{h_{1}})$ of the
class $\underline{\kappa}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$ to weight
$(2,1,1)$. It turns out that the complex $L$-function encodes certain
information about this specialization. In particular, one has

######  Theorem 1.3 ([DR14, Theorem 6.4] and [BSV22, Theorem B]).

The class
$\underline{\kappa}_{p}(\mathbf{f_{2}},\mathbf{g_{1}},\mathbf{h_{1}})$ is
crystalline if and only if the complex $L$-function $L(f,g,h;s)$ vanishes at
the central point $s=1$.

######  Remark 1.4.

The above theorem actually holds for specializations at general unbalanced
weights $(x,y,z)$, with the central point being $c=\frac{x+y+z-2}{2}$.

Lastly, one can construct, out of the crystalline class
$\kappa(\mathbf{f_{2}},\mathbf{g_{1}},\mathbf{h_{1}})$, at least two linearly
independent elements in $\operatorname{\operatorname{Sel}}_{p}(E,\rho)$ and
completes the proof.

######  Remark 1.5.

Since $g,h$ are modular forms of weight $1$, they are necessarily ordinary at
$p$, the two $p$-adic $L$-functions
$\mathscr{L}_{p}^{g}(\mathbf{g},\mathbf{f},\mathbf{h})$ and
$\mathscr{L}_{p}^{h}(\mathbf{h},\mathbf{f},\mathbf{g})$ are already defined in
[DR14]. The only remaining obstacle for the finite slope case is about
$\mathscr{L}_{p}^{f}$ and the existence of the class
$\underline{\kappa}(\mathbf{f},\mathbf{g},\mathbf{h})$ that gives rise to
$L_{p}^{f}$, whose proof will rely on our $p$-adic Gross–Zagier formula for
finite slope families.

To summarize, our main result in the second part is the following theorem.

######  Theorem 1.6.

Let $(\mathbf{f},\mathbf{g},\mathbf{h})$ be as in the setting of Theorem 1.1.
Then there is a unique element
$\underline{\kappa}(\mathbf{f},\mathbf{g},\mathbf{h})\in
H^{1}(\mathbb{Q},V(\mathbf{f},\mathbf{g},\mathbf{h}))$ such that

$\mathscr{L}og_{f}(\underline{\kappa}_{p}(\mathbf{f},\mathbf{g},\mathbf{h}))=\mathscr{L}_{p}^{f}(\mathbf{f},\mathbf{g},\mathbf{h}).$

Moreover, when specialized to a unbalanced triple $(x,y,z)\in\Sigma_{f}$, the
class $\underline{\kappa}_{p}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})$
is crystalline if and only if the classical $L$-function
$L(f_{x}^{0},g_{y}^{0},h_{z}^{0};s)$ vanishes at the central point
$c=\frac{x+y+z-2}{2}$.

Put in other words, the second part aims to answer the question raised in
[BSV22, Remark 1.3]; namely, whether their two main results can be generalized
to the finite slope case. Most of the materials are already provided in their
paper, except for various Galois cohomology groups and the big logarithm map
for finite slope families. Once these objects are carefully introduced, the
proof of Theorem 1.6 will follow identically the arguments in [BSV22, § 8 and
§ 9]. For this reason, we will mainly adopt their notations and omit most of
the proofs in §4.

With Theorem 1.6 established, it is not hard to see that one can prove an
analogous result of Theorem 1.2 by the same strategy as in [DR17, § 6].

######  Remark 1.7.

We would like to mention that we do not generalize or improve [DR17, Theorem
A] in any sense, since one can always find a prime $p$ such that the elliptic
curve $E$ and the modular forms $g,h$ are ordinary at $p$.

Acknowledgements. The author would like thank his supervisors Adrian Iovita
and Giovanni Rosso for their constant support. In addition, the author is
grateful to Henri Darmon for his insightful advice on the $p$-adic
Gross–Zagier formula. The work of this article is funded by an NSERC grant and
an FRQNT grant.

##  2 Triple product $p$-adic $L$-functions

In this section, we will recall the definition of the triple product $p$-adic
$L$-function for finite slope families. The main reference is [AI21], but we
will follow closely [AI19] for simpler setups. Before going through the
construction, we would like to briefly explain the idea of their work.

Fix an integer $N\geq 5$ and a prime $p\nmid N$. Let $X=X_{1}(N)$ be the
modular curve defined over $\mathbb{Z}_{p}$ with the universal generalized
elliptic curve $\pi:E\rightarrow X$. Let $Y\subset X$ be the affine modular
curve and $C\subset X$ be the closed subscheme of cusps. Then the two sheaves

$\underline{\omega}:=\pi_{*}(\Omega^{1}_{E/Y})\textrm{ and
}\mathcal{H}:=\mathbb{R}^{1}\pi_{*}(\Omega^{\bullet}_{{E}/Y})$

on $Y$ have canonical extensions to $X$ (c.f. [BDP13, § 1]), which will still
be denoted by the same notations $\underline{\omega}$ and $\mathcal{H}$.

For any $r\in\mathbb{Z}$, $H^{0}(X,\underline{\omega}^{r})$ can be identified
as the space of modular forms of weight $r$. For a strict neighborhood
$\mathcal{U}$ of the ordinary locus in $X$,
$H^{0}(\mathcal{U},\underline{\omega}^{r})$ can be identified as the space of
overconvergent modular forms of weight $r$.

The sheaf $\mathcal{H}$ is equipped with a (increasing) filtration given by
the exact sequence

$0\rightarrow\underline{\omega}\rightarrow\mathcal{H}\rightarrow\underline{\omega}^{-1}\rightarrow
0$

and a Gauss–Manin connection

$\nabla:\mathcal{H}\rightarrow\mathcal{H}\otimes\Omega^{1}_{X}(\log C).$

For any $r\in\mathbb{Z}$, we let
$\mathcal{H}^{r}:=\operatorname{\operatorname{Sym}}^{r}\mathcal{H}$ be its
$r$-th symmetric power. The filtration and the connection on $\mathcal{H}$
then extend naturally to $\mathcal{H}^{r}$. By composing with the
Kodaira–Spencer isomorphism

$\mathrm{KS}:\Omega^{1}_{X}(\log C)\xrightarrow{\sim}\underline{\omega}^{2},$

the Gauss–Manin connection may be viewed as a map from $\mathcal{H}^{r}$ to
$\mathcal{H}^{r+2}$.

The idea is to construct a sheaf $\mathfrak{w}_{k}$ (over a certain strict
neighborhood) which $p$-adically interpolates the collection
$\\{\underline{\omega}^{r}\\}_{r\in\mathbb{Z}}$, such that a $p$-adic family
of overconvergent modular forms may be viewed as a section of
$\mathfrak{w}_{k}$ over some strict neighborhood $\mathcal{U}$ of the ordinary
locus.

There is also a sheaf $\mathbb{W}_{k}$ which $p$-adically interpolates
$(\mathcal{H}^{n},\operatorname{\operatorname{Fil}}_{\bullet},\nabla)_{n\in\mathbb{Z}}$.
In particular, it allows us to take certain $p$-adic powers of the connection
$\nabla$, which is crucial for defining the triple product $p$-adic
$L$-function $\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$ as
explained in the introduction.

Since their construction is highly technical, we prefer not to provide
detailed proofs in this section. Instead, we will only set up notations and
statements that are sufficient for our application.

###  2.1 The modular sheaf $\mathfrak{w}_{k}$ and the de Rham sheaf
$\mathbb{W}_{k}$

Let $N,p$ be as before and $X=X_{1}(N)$ be the modular curve defined over
$\mathbb{Z}_{p}$. We write $\mathfrak{X}$ for the formal completion of $X$
along its special fiber. We denote by $\operatorname{Hdg}$ be the
$\mathcal{O}_{\mathfrak{X}}$-ideal that is locally generated by a lift of the
Hasse invariant.

For any integer $n\geq 2$, we denote by $\mathfrak{X}_{n}$ the formal open
subscheme in the formal admissible blow-up of $\mathfrak{X}$ with respect to
the ideal $\mathcal{J}=(p,\operatorname{Hdg}^{n})$, defined such that
$\mathcal{J}$ is generated by $\operatorname{Hdg}^{n}$ over
$\mathfrak{X}_{n}$. The generalized elliptic curve
${E}\rightarrow\mathfrak{X}_{n}$ has a canonical subgroup of order $p^{r}$,
where this $r$ depends on $n$ (c.f. [AI21, § 3.1]). We will write
$\mathcal{X}_{n}$ for the adic generic fiber of $\mathfrak{X}_{n}$.

######  Definition 2.1.

Let $\Lambda$ be a $p$-adically complete and separated ring, and
$k:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda^{\times}$ a continuous
homomorphism. Then $k$ is called an analytic weight of radius $r\in\mathbb{N}$
if there exists an element $u=u_{k}\in p^{1-r}\Lambda$ such that
$k(t)=\exp(u\log(t))$ for all $t\in 1+p^{r}\mathbb{Z}_{p}$.

######  Definition 2.2.

Let $k:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda^{\times}$ be an analytic
weight. By saying ‘specialized to a classical weight
$\ell\in\mathbb{Z}_{>0}$,’ we mean that we take the composition $\gamma\circ
k$ for some homomorphism $\gamma:\Lambda\rightarrow\mathbb{Z}_{p}$ such that
$\gamma\circ k:\mathbb{Z}_{p}^{\times}\rightarrow\mathbb{Z}_{p}^{\times}$ is
the $\ell$-th power map.

One has the following results (c.f. [AI21, § 3.2 & § 3.3]):

######  Proposition 2.3.

Let $k$ be an analytic weight of radius $r$ as above, and take an integer $n$
such that ${E}\rightarrow\mathfrak{X}_{n}$ admits a canonical subgroup of
order $p^{r}$. Then there are
$\mathcal{O}_{\mathfrak{X}_{n}}\hat{\otimes}_{\mathbb{Z}_{p}}\Lambda$-modules
$\mathfrak{w}_{k}$ and $\mathbb{W}_{k}$ with the following properties:

  1. 1.

$\mathfrak{w}_{k}$ is an invertible
$\mathcal{O}_{\mathfrak{X}_{n}}\hat{\otimes}_{\mathbb{Z}_{p}}\Lambda$-module.
Over the generic fiber $\mathcal{X}_{n}$, the global sections of
$\mathfrak{w}_{k}$ can then be identified as $p$-adic families of
overconvergent modular forms of weight $k$;

  2. 2.

$\mathbb{W}_{k}$ has a natural increasing filtration
$(\operatorname{\operatorname{Fil}}_{n})_{n\geq 0}$ by
$\mathcal{O}_{\mathfrak{X}_{n}}\hat{\otimes}\Lambda$-modules with
$\operatorname{\operatorname{Fil}}_{0}=\mathfrak{w}_{k}$. Moreover, when
specializing to a classical weight $r\geq 1$,
$\operatorname{\operatorname{Fil}}_{r}\mathbb{W}_{r}\cong\mathcal{H}^{r}$ over
$\mathcal{X}_{n}$ and is compatible with the Hodge filtration of
$\mathcal{H}^{r}$.

  3. 3.

There is a Gauss–Manin connection $\nabla_{k}$ (with poles at
$V(\operatorname{Hdg})$) on $\mathbb{W}_{k}$ that satisfies Griffiths
transversality. By composing with the Kodaira–Spencer map, the connection
$\nabla_{k}$ can be viewed as a map from $\mathbb{W}_{k}$ to
$\mathbb{W}_{k+2}$.

#### The $q$-expansions.

As explained in [AI21, § 3.5], we have the $q$-expansion map

$H^{0}(\mathfrak{X}_{n},\mathbb{W}_{k})\rightarrow\Lambda((q))$

obtained by first restricting to the ordinary locus then applying the unit
root splitting. On the other hand, the $q$-expansion map can also be described
by using the Tate curve.

Let $T=\mathrm{Tate}(q^{N})$ be the Tate curve over
$\operatorname{\operatorname{Spf}}(S)$, where $S=\Lambda((q))$. We fix the
canonical basis
$(\omega_{\operatorname{\operatorname{can}}},\eta_{\operatorname{\operatorname{can}}}=\nabla(\theta)(\omega_{\operatorname{\operatorname{can}}}))$
of $\cal H$ on $T$. Let $\mathbb{W}_{k}(q)$ be the module of $\mathbb{W}_{k}$
over the Tate curve, then we have the following local description:
$\mathbb{W}_{k}(q)=S\langle\frac{Y}{1+p^{r}Z}\rangle(1+p^{r}Z)^{k}$ (c.f.
[AI21, § 3.5]). If we set $V_{k,i}:=Y^{i}(1+p^{r}Z)^{k-i}$, then the
filtration of $\mathbb{W}_{k}$ is given by
$\operatorname{\operatorname{Fil}}_{h}\mathbb{W}_{k}(q)=\sum_{i=0}^{h}S\cdot
V_{k,i}$. The $q$-expansion map then corresponds to the map
$\sum_{i}a_{i}(q)V_{k,i}\mapsto a_{0}(q)$.

######  Remark 2.4.

Since $k$ is analytic, the element $k(1+p^{r}Z)$ is well-defined in $S\langle
Z\rangle$ and we sometimes denote it by $(1+p^{r}Z)^{k}$. The description
above comes directly from the explicit construction of vector bundles with
marked sections (c.f. [AI21, § 2]). One should think of $1+p^{r}Z$ (resp. $Y$)
as an element that parametrizes the deformation of
$\omega_{\operatorname{\operatorname{can}}}$ (resp.
$\eta_{\operatorname{\operatorname{can}}})$. In particular, the specialization
of $V_{k,i}$ to a classical weight $\ell\in\mathbb{Z}_{>0}$ can be identified
as the element
$\omega_{\operatorname{\operatorname{can}}}^{\ell-i}\eta_{\operatorname{\operatorname{can}}}^{i}$
of $\mathcal{H}^{\ell}$ on the Tate curve when $i\leq\ell$.

#### The operators $U$ and $V$.

Similar to the classical case, we can define Hecke operators $U$ on
$H^{0}(\mathcal{X}_{n},\mathbb{W}_{k})$ and
$V:H^{0}(\mathfrak{X}_{n},\mathfrak{w}_{k})\rightarrow
H^{0}(\mathfrak{X}_{n+1},\mathfrak{w}_{k})$, whose actions on the
$q$-expansions are the usual ones (c.f. [AI21, § 3.6, § 3.7]). Moreover, $U$
is a compact operator. In particular, one has the following proposition.

######  Proposition 2.5 ([AI21, Cor. 3.26]).

The operator $U$ on $H^{0}(\mathcal{X}_{n},\mathbb{W}_{k})$ admits a Fredholm
determinant $P(T)\in\Lambda\llbracket T\rrbracket$, and for any non-negative
rational number $a$ the space $H^{0}(\mathcal{X}_{n},\mathbb{W}_{k})$ admits a
slope $a$-decomposition

$H^{0}(\mathcal{X}_{n},\mathbb{W}_{k})=H^{0}(\mathcal{X}_{n},\mathbb{W}_{k})^{\leq
a}\oplus H^{0}(\mathcal{X}_{n},\mathbb{W}_{k})^{>a}.$

######  Definition 2.6.

Let $\mathbf{f}\in H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k})$. We define
$\mathbf{f}^{[p]}:=(1-VU)\mathbf{f}$ and call it the $p$-depletion of
$\mathbf{f}$. If
$\mathbf{f}(q)=\sum_{n=0}^{\infty}a_{n}(\mathbf{f})q^{n}V_{k,0}$ is the
$q$-expansion of $\mathbf{f}$, then the $q$-expansion of $\mathbf{f}^{[p]}$ is
$\mathbf{f}^{[p]}(q)=\sum_{n=0,\ p\nmid
n}^{\infty}a_{n}(\mathbf{f})q^{n}V_{k,0}$.

####  $p$-adic iterations of the Gauss–Manin connection.

Another important property of the sheaf $\mathbb{W}_{k}$ is that it is
possible to iterate the Gauss–Manin connection $\nabla$ to certain $p$-adic
powers and can be described explicitly in terms of $q$-expansions.

We first need an assumption on the weights.

######  Assumption 2.7.

Let $k$ and $s$ be two analytic weights for the same ring $\Lambda$ that
satisfy

$k(t)=\chi(t)t^{a}\exp(u_{k}\log(t)),\quad
s(t)=\chi^{\prime}(t)t^{b}\exp(u_{s}\log(t))\quad\forall
t\in\mathbb{Z}_{p}^{\times}$

where

  1. $\bullet$

$a,b\in\mathbb{Z}$;

  2. $\bullet$

$\chi$ and $\chi^{\prime}$ are finite character of $\mathbb{Z}_{p}^{\times}$,
and $\chi$ is even;

  3. $\bullet$

$u_{k}\in p\Lambda$ and $u_{s}\in p^{2}\Lambda$.

Under this assumption, we have the following theorem.

######  Theorem 2.8 ([AI21, Theorem 4.3]).

Suppose $\mathbf{g}\in H^{0}(\mathfrak{X}_{n},\mathbb{W}_{k})^{U=0}$ and let
$\mathbf{g}(q)=\sum_{h}\mathbf{g}_{h}(q)V_{k,h}$ be its $q$-expansion. Then
there are positive integers $b$ and $\gamma$ depending on $k,s,n$ and an
element
$\nabla^{s}(\mathbf{g})\in\mathrm{Hdg}^{-\gamma}H^{0}(\mathfrak{X}_{b},\mathbb{W}_{k+2s})$
such that

$\nabla^{s}(\mathbf{g})(q):=\sum_{h}\sum_{j=0}^{\infty}\binom{u_{s}}{j}\prod_{i=0}^{j-1}(u_{k}+u_{s}-h-1-i)\theta^{s-j}(\mathbf{g}_{h}(q))V_{k+2s,j+h},$

where $\theta=q\frac{d}{dq}$.

######  Remark 2.9.

Let $\gamma:R\rightarrow\mathbb{Z}_{p}$ be a homomorphism such that
$\gamma\circ s:\mathbb{Z}_{p}^{\times}\rightarrow\mathbb{Z}_{p}^{\times}$ is a
positive classical weight $\ell$. Then the specialization of
$\nabla^{s}(\mathbf{g})$ via $\gamma$ is $\nabla^{\ell}(\mathbf{g})$, the
usual $\ell$-th iteration of $\nabla$.

######  Remark 2.10.

The requirement that $u_{k}$ and $u_{s}$ are divisible by powers of $p$ is for
convergence reason. We should mention that an improved version is developed by
Ananyo Kazi in his Ph.D. thesis [Kaz22], which slightly releases the
conditions.

#### The overconvergent projections.

We have seen how to define $\nabla^{s}(\mathbf{g}^{[p]})$ for a suitable
weight $s$. However, it is only a section of $\mathbb{W}_{k+2s}$ instead of
$\mathfrak{w}_{k+2s}$. In order to define the triple product $p$-adic
$L$-function via Petersson product of families of overconvergent modular
forms, we need the overconvergent projection, which sends
$\nabla^{s}(\mathbf{g}^{[p]})$ to a certain section of $\mathfrak{w}_{k+2s}$.

We first consider the following commutative diagram of sheaves on
$\mathcal{X}_{n}$

${0}$${\operatorname{\operatorname{Fil}}_{m}(\mathbb{W}_{k})}$${\mathbb{W}_{k}}$${\mathbb{W}_{k}/\operatorname{\operatorname{Fil}}_{m}(\mathbb{W}_{k})}$${0}$${0}$${\operatorname{\operatorname{Fil}}_{m+1}(\mathbb{W}_{k+2})}$${\mathbb{W}_{k+2}}$${\mathbb{W}_{k+2}/\operatorname{\operatorname{Fil}}_{m+1}(\mathbb{W}_{k+2})}$${0}$$\scriptstyle{\nabla_{k}}$$\scriptstyle{\nabla_{k}}$$\scriptstyle{\nabla_{k}}$

and view the columns as complexes
$\operatorname{\operatorname{Fil}}_{m}^{\bullet}\mathbb{W}_{k}$,
$\mathbb{W}_{k}^{\bullet}$ and
$(\mathbb{W}_{k}/\operatorname{\operatorname{Fil}}^{m})^{\bullet}$
respectively.

Note that the sheaves $\operatorname{\operatorname{Fil}}_{m}\mathbb{W}_{k}$
and $\operatorname{\operatorname{Fil}}_{m+1}\mathbb{W}_{k+2}$ are coherent,
while the rest are not. So the hypercohomology of
$\operatorname{\operatorname{Fil}}_{m}^{\bullet}\mathbb{W}_{k}$ on the Stein
space $\mathcal{X}_{n}$ can be computed by the cohomology of the complex of
global sections. That is, for $i\geq 0$, we have

$H^{i}_{\operatorname{\operatorname{dR}}}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m}^{\bullet}\mathbb{W}_{k}):=\mathbb{H}^{i}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m}^{\bullet}\mathbb{W}_{k})=H^{i}(H^{0}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m}\mathbb{W}_{k})\xrightarrow{\nabla_{k}}H^{0}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m+1}\mathbb{W}_{k+2})).$

######  Theorem 2.11 ([AI21, Lemma 3.32]).

For any $m\in\mathbb{N}$, there is an isomorphism

$H^{\dagger}_{m}:H^{1}_{\operatorname{\operatorname{dR}}}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m}^{\bullet}\mathbb{W}_{k})\otimes_{\Lambda}\Lambda\left[\prod_{i=0}^{m}(u_{k}-i)\right]\cong
H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k+2})\otimes_{\Lambda}\Lambda\left[\prod_{i=0}^{m}(u_{k}-i)\right].$

######  Theorem 2.12 ([AI21, Theorem 3.34]).

Fix a positive slope $a$. Then we have an integer $m_{a}\in\mathbb{N}$ and an
isomorphism

$H^{\dagger,\leq
a}:H^{1}_{\operatorname{\operatorname{dR}}}(\mathcal{X}_{n},\mathbb{W}_{k})^{\leq
a}\otimes_{\Lambda}\Lambda\left[\prod_{i=0}^{m_{a}}(u_{k}-i)\right]\cong
H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k+2})^{\leq
a}\otimes_{\Lambda}\Lambda\left[\prod_{i=0}^{m_{a}}(u_{k}-i)\right]$

where the superscript $\leq a$ means the slope $\leq a$ subspace with respect
to the $U$ operator.

The maps $H^{\dagger}_{m}$ and $H^{\dagger}$ above are referred as the
overconvergent projections.

######  Remark 2.13.

When specialized to a classical weight $\ell\in\mathbb{Z}$, one can see that
the map $H^{\dagger}_{m}$ for $m=\ell$ coincides with the overconvergent
projection defined in [Urb14, § 3.2].

We shall now give the description of the map $H^{\dagger}_{m}$ on the
$q$-expansions, which is an important tool for later computations. Recall that
on $\mathbb{W}_{k}(q)$, we have the identity

$\nabla_{k}(aV_{k,i})=\theta(a)V_{k+2,i}+a(u_{k}-i)V_{k+2,h+1},\quad
a\in\Lambda((q)).$

By definition, the overconvergent projection of a class $[\gamma]$ for
$\gamma\in
H^{0}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m+1}\mathbb{W}_{k+2})$
is the unique element in $H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k+2})$ that
represents the same class modulo
$\nabla(H^{0}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m}\mathbb{W}_{k}))$.
Hence, after shifting the numbers, we have the following result.

######  Proposition 2.14 ([AI21, Prop. 3.37]).

Consider an element $\gamma\in
H^{0}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m}\mathbb{W}_{k})$
with its class $[\gamma]\in
H^{1}_{\operatorname{\operatorname{dR}}}(\mathcal{X}_{n},\operatorname{\operatorname{Fil}}_{m-1}\mathbb{W}_{k-2}^{\bullet})$.
Let $\gamma(q)=\sum_{i=0}^{m}\gamma_{i}(q)V_{k,i}$ be its $q$-expansion. Then
the $q$-expansion of $H^{\dagger}([\gamma])\in
H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k})$ is

$\sum_{i=0}^{m}(-1)^{i}\frac{\theta^{i}\gamma_{i}(q)}{(u_{k}-2-i+1)(u_{k}-2-i+2)\cdots(u_{k}-2)}V_{k,0}.$

###  2.2 Definition of the triple product $p$-adic $L$-function

We are now able to define triple product $p$-adic $L$-functions for finite
slope families. We first recall some results in the complex case. For any
newform $f$ of level $N_{f}$ and nebentypus $\chi_{f}$, we write
$\mathbb{Q}_{f}$ for the number field generated by all Hecke eigenvalues of
$f$. We denote by $\pi_{f}$ the automorphic representation of
$\operatorname{\operatorname{GL}}_{2}(\mathbb{A}_{\mathbb{Q}})$ generated by
$f$. If $N$ is a multiple of $N_{f}$ and $\mathbb{Q}_{f}\subset L$, then we
let $S_{k}(N,L)[\pi_{f}]$ be the $f$-isotypic subspace of $S_{k}(N,L)$.

Now let $f\in S_{k}(N_{f},\chi_{f})$, $g\in S_{\ell}(N_{g},\chi_{g})$, $h\in
S_{m}(N_{h},\chi_{h})$ be a triple of normalized primitive cuspidal
eigenforms. We set $N=\operatorname{\ell\textrm{cm}}(N_{f},N_{g},N_{h})$ and
$\mathbb{Q}_{f,g,h}=\mathbb{Q}_{f}\cdot\mathbb{Q}_{g}\cdot\mathbb{Q}_{h}$. We
assume that $\chi_{f}\cdot\chi_{g}\cdot\chi_{h}=1$ and the triple $(k,\ell,m)$
is unbalanced with $k=\ell+m+2t^{\prime}$ for some
$t^{\prime}\in\mathbb{Z}_{\geq 0}$. In this situation, we have the following
result of M. Harris and S. Kulda [HK91], refined by A. Ichino [Ich08] and T.
C. Watson [Wat02], which is usual known as the Ichino formula.

######  Theorem 2.15.

Let $f,g,h$ be as above, then there exist holomorphic modular forms

$\breve{f}\in S_{k}(N,\mathbb{Q}_{f,g,h})[\pi_{f}],\ \breve{g}\in
S_{k}(N,\mathbb{Q}_{f,g,h})[\pi_{g}],\ \breve{h}\in
S_{k}(N,\mathbb{Q}_{f,g,h})[\pi_{h}]$

and constants $C_{q}\in\mathbb{Q}_{f,g,h}$ depending only on the local
components of $\breve{f},\breve{g},\breve{h}$ for all $q\mid N\infty$ such
that

$\frac{\prod_{q}C_{q}}{\pi^{2k}}L\left(\breve{f},\breve{g},\breve{h};\frac{k+l+m-2}{2}\right)=|I(\breve{f},\breve{g},\breve{h})|^{2}.$

Moreover, there exists a choice of $\breve{f},\breve{g},\breve{h}$ such that
all $C_{q}\neq 0$.

In the above theorem, $L(f,g,h;s)$ is the complex Garrett-Rankin triple
product $L$-function and

$I(\breve{f},\breve{g},\breve{h}):=(\breve{f}^{*},\delta^{t}(\breve{g})\cdot\breve{h})$

where $(\phantom{e},\phantom{e})$ is the Petersson inner product, $\delta$ is
the Maass-Shimura differential operator, and
$\breve{f}^{*}=\breve{f}\otimes\chi_{f}^{-1}$ is the Atkin–Lehner involution
of $\breve{f}$.

#### Triple product $p$-adic $L$-functions for finite slope families.

Let $f,g,h$ be as before and suppose that $f$ is of finite slope $a>0$. We
further assume that and $a<1$ if $k=2$, and $2a<k-1$ if $k\geq 2$ (c.f. Remark
2.19). We fix modular forms $\breve{f}$, $\breve{g}$, $\breve{h}$ as in
Theorem 2.15 such that all the constants $C_{q}$ are non-zero. For simplicity,
we write $K=\mathbb{Q}_{f,g,h}$ and let $\mathcal{O}_{K}$ be the ring of
integers in $K$.

Let $\mathbf{f},\mathbf{g},\mathbf{h}$ be overconvergent families of modular
forms deforming the $p$-stabilizations of $f,g,h$, respectively, and similarly
$\breve{\mathbf{f}},\breve{\mathbf{g}},\breve{\mathbf{h}}$ for
$\breve{f},\breve{g},\breve{h}$. We denote by

$\displaystyle k_{f}:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda_{f},\quad
k_{g}:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda_{g},\quad
k_{h}:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda_{h}$

the analytic weights of these families. After base-change to
$\mathcal{O}_{K}$, we may assume that $\Lambda_{f},\Lambda_{g},\Lambda_{h}$
are $\mathcal{O}_{K}$-algebras. Then there exists some $n\in\mathbb{N}$ such
that we have

$\displaystyle\mathbf{f},\breve{\mathbf{f}}\in
H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k_{f}}),\quad\mathbf{g},\breve{\mathbf{g}}\in
H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k_{g}}),\quad\mathbf{h},\breve{\mathbf{h}}\in
H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k_{h}}).$

As we want to apply $p$-adic powers of $\nabla$ to the $p$-depletion
$\breve{\mathbf{g}}^{[p]}$ later, we need the following assumption on the
weights.

######  Assumption 2.16.

Suppose the weight $k_{f}-k_{g}-k_{h}$ is even, i.e., there is a weight
$\nu:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda_{f}\hat{\otimes}_{\mathcal{O}_{K}}\Lambda_{g}\hat{\otimes}_{\mathcal{O}_{K}}\Lambda_{h}$
such that $k_{f}-k_{g}-k_{h}=2\nu$. Furthermore, $k_{g}$ and $\nu$ satisfy
Assumption 2.7.

Under this assumption, we have

$\nabla_{k_{g}}^{\nu}(\breve{\mathbf{g}}^{[p]})\in
H^{0}(\mathcal{X}_{n^{\prime}},\mathbb{W}_{k_{g}+2\nu})$

for some $n^{\prime}\geq n$. Therefore,
$\nabla_{k_{g}}^{\nu}(\breve{\mathbf{g}}^{[p]})\times\breve{\mathbf{h}}\in
H^{0}(\mathcal{X}_{n^{\prime}},\mathbb{W}_{k_{f}})$ and we may consider its
class in
$H^{1}_{\operatorname{\operatorname{dR}}}(\mathcal{X}_{n^{\prime}},\mathbb{W}_{k_{f}-2})$.
After base change to $\mathfrak{K}_{f}$, we obtain a section in
$H^{1}_{\operatorname{\operatorname{dR}}}(\mathcal{X}_{n^{\prime}},\mathbb{W}_{k_{f}-2})\otimes_{\Lambda_{f}}\mathfrak{K}_{f}$,
where $\mathfrak{K}_{f}$ is obtained from $\Lambda_{f}$ by inverting the
elements $\\{u_{k_{f}}-n\mid n\in\mathbb{N}\\}$ (or one may simply take it to
be $\operatorname{\operatorname{Frac}}(\Lambda_{f})$). We then consider its
overconvergent projection

$H^{\dagger,\leq
a}(\nabla_{k_{g}}^{\nu}(\breve{\mathbf{g}}^{[p]})\times\breve{\mathbf{h}})\in
H^{0}(\mathcal{X}_{n^{\prime}},\mathfrak{w}_{k_{f}})^{\leq
a}\otimes\mathfrak{K}_{f}.$

To define the triple product $L$-function, we consider the Atkin-Lehner
involution $\breve{\mathbf{f}}^{*}=w_{N}(\breve{\mathbf{f}})\in
H^{0}(\mathcal{X}_{n},\mathfrak{w}_{k_{f}})$ (c.f. [AI21, Definition 5.2]). At
a classical point $x\in\Omega_{f,\operatorname{cl}}$, the specialization
$\breve{\mathbf{f}}^{*}_{x}$ is simply the Atkin–Lehner involution of
$\breve{\mathbf{f}}_{x}$.

######  Definition 2.17.

The Garrett-Rankin triple product $p$-adic $L$-function attached to the triple
$(\breve{\mathbf{f}},\breve{\mathbf{g}},$ $\breve{\mathbf{h}})$ of $p$-adic
families of modular forms is

$\mathscr{L}^{f}_{p}(\breve{\mathbf{f}},\breve{\mathbf{g}},\breve{\mathbf{h}}):=\frac{(\breve{\mathbf{f}}^{*},H^{\dagger,\leq
a}(\nabla_{k_{g}}^{\nu}(\breve{\mathbf{g}}^{[p]})\times\breve{\mathbf{h}}))}{(\breve{\mathbf{f}}^{*},\breve{\mathbf{f}}^{*})}\in\mathfrak{K}_{f}\hat{\otimes}\Lambda_{g}\hat{\otimes}\Lambda_{h}$

where $(\phantom{e},\phantom{e})$ is the Petersson product of families defined
in [AI21, § 5.2.1].

######  Proposition 2.18 ([AI21, Corollary 5.13]).

For a unbalanced classical weight $(x,y,z)$ with $x$ dominant, one has the
interpolating formula

$\mathscr{L}^{f}_{p}(\breve{\mathbf{f}},\breve{\mathbf{g}},\breve{\mathbf{h}})(x,y,z)=C\cdot
I(\breve{f}_{x},\breve{g}_{y},\breve{h}_{z})$

for some non-zero constant $C$.

######  Remark 2.19.

As explained in [AI21, § 5.2.1], when specialized to a classical weight $x>2$
with $2a<x-1$ or $x=2$ with $a<1$, the Petersson product interpolates the
classical Petessons inner product on $S_{x}(\Gamma_{1}(N,p))$, up to a
constant multiple.
In fact, given $\mathbf{f}$ of finite slope $a$, what we really require is the
condition $\alpha(\mathbf{f}_{x})^{2}\neq p^{x-1}b(\mathbf{f}_{x})$, where
$\alpha(\mathbf{f}_{x}),b(\mathbf{f}_{x})$ are the eigenvalues of $U_{p}$ and
$\langle p\rangle$ acting on $\mathbf{f}_{x}$, respectively. In other words,
if $f_{x}^{0}$ is the classical modular form on $\Gamma_{1}(N)$ with one of
its $p$-stabilizations being $\mathbf{f}_{x}$, it means that the two roots
$\alpha_{f},\beta_{f}$ of the Hecke polynomial of $f_{x}^{0}$ at $p$ are
different. Equivalently in a fancier language, one wants to restrict to a
subset of the weight space over which the eigencurve giving rise to
$\mathbf{f}$ is étale over it.

######  Remark 2.20.

When $\mathbf{f},\mathbf{g},\mathbf{h}$ are Hida families, one can see that
this definition coincides with the one given in [DR14, § 4.2].

##  3 The $p$-adic Gross–Zagier formula

In this section, we will prove the $p$-adic Gross–Zagier formula. We examine
one side of the equation, i.e., the special values of the $p$-adic
$L$-function in §3.1. Then we recall some computations on the $p$-adic
Abel–Jacobi image of the generalized diagonal cycle in §3.2. In §3.3, we will
prove the formula by comparing the two ingredients.

For simplicity, we will drop the $\breve{\phantom{e}}$ notation throughout
this section.

###  3.1 Special values at balanced classical weights

We first specify at which classical points we want to study the values of
$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$.

Suppose $\mathbf{f}$ is a family of finite slope $a=a_{f}$, and
$\mathbf{g},\mathbf{h}$ are of slopes $a_{g},a_{h}$ respectively. We are
interested in the values of
$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$ at classical weights
$(x,y,z)\in\Sigma_{\operatorname{\operatorname{bal}}}$ such that

  1. i.

The specializations of $\mathbf{f},\mathbf{g},\mathbf{h}$ at $(x,y,z)$ are
$p$-old, i.e., $\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}$ are
$p$-stabilizations of some classical modular forms
$f^{0}_{x},g^{0}_{y},h^{0}_{z}$ on $X_{1}(N)$;

  2. ii.

The triple $(x,y,z)$ is balanced with $x=y+z-2t$ for some
$t\in\mathbb{Z}_{>0}$;

  3. iii.

$x>2a+1$, $y>2a_{g}+1$ and $z>2a_{h}+1$ (c.f. Remark 2.19).

We denote the set consisting of all such weights by
$\Sigma_{\mathbf{f},\mathbf{g},\mathbf{h}}$. To simplify the notations, we
will write $f_{x}:=\mathbf{f}_{x}$, $g_{y}:=\mathbf{g}_{y}$,
$h_{z}:=\mathbf{h}_{z}$ and denote the classical forms
$f^{0}_{x},g^{0}_{y},h^{0}_{z}$ by $f,g,h$. Notice that the new $f,g,h$ are
different from the modular forms (of unbalanced weights $k,\ell,m$) which the
families $\mathbf{f},\mathbf{g},\mathbf{h}$ deform respectively.

As we will proceed our proof by studying the $q$-expansions of various modular
forms, we here recall several definitions. On the Tate curve, one has the
canonical basis
$\\{\omega_{\operatorname{\operatorname{can}}},\eta_{\operatorname{\operatorname{can}}}\\}$
of $\mathcal{H}$ that satisfies

$\nabla(\omega_{\operatorname{\operatorname{can}}})=\eta_{\operatorname{\operatorname{can}}}\otimes\frac{dq}{q},\quad\nabla(\eta_{\operatorname{\operatorname{can}}})=0.$

As mentioned in Remark 2.4, the element $V_{r,i}$ of $\mathbb{W}_{r}(q)$
corresponds to the element
$\omega_{\operatorname{\operatorname{can}}}^{r-i}\eta_{\operatorname{\operatorname{can}}}^{i}$
when $r$ is a positive integer and $i\leq r$. From now on, we will make no
difference between these two notations and use them interchangeably.

We now examine the $q$-expansion of
$H^{\dagger}(\nabla_{k_{g}}^{\nu}(\mathbf{g}^{[p]})\times\mathbf{h})$ at
$(y,z)$. Let

$g_{y}^{[p]}(q)V_{y,0}\textrm{ and }h_{z}(q)V_{z,0}$

be the $q$-expansions of the specializations of $\mathbf{g}^{[p]}$ and
$\mathbf{h}$ at $y$ and $z$ respectively.

Since we assume the triple $(x,y,z)$ to be balanced, $b:=y-t$ is a positive
integer. Apply the formula in Theorem 2.8, we see that

$\displaystyle\nabla^{-t}(g_{y}^{[p]}V_{y,0})$
$\displaystyle=\sum_{j}^{\infty}\binom{-t}{j}\prod_{i=0}^{j-1}(y-t-1-i)\theta^{-t-j}g_{y}^{[p]}(q)V_{y-2t,j}$
$\displaystyle=\sum_{j=0}^{b-1}\binom{-t}{j}\prod_{i=0}^{j-1}(y-t-1-i)\theta^{-t-j}g_{y}^{[p]}(q)V_{y-2t,j}$

is in fact a finite sum. We simplify the above formula as

$\nabla^{-t}(g_{y}^{[p]}V_{y,0})=\sum_{j=0}^{b-1}C_{j}\cdot\theta^{-t-j}g_{y}^{[p]}(q)V_{y-2t,j}$

by letting $C_{j}:=\binom{-t}{j}\prod_{i=0}^{j-1}(y-t-1-i)$.

The $q$-expansion of the product $\nabla^{-t}g_{y}^{[p]}(q)\times h_{z}(q)$
can now be expressed as

$\sum_{j=0}^{b-1}C_{j}\cdot\theta^{-t-j}g_{y}^{[p]}(q)V_{y-2t,j}\times
h_{z}(q)V_{z,0}=\sum_{j=0}^{b-1}C_{j}\cdot\theta^{-t-j}g_{y}^{[p]}(q)\times
h_{z}(q)V_{x,j}.$

We here remark that $b-1\leq x$. Since if $b-1=y-t-1>x$, by using the
balancedness assumption $x+z-y>0$, we have $z>t+1$. But this would imply
$x=y-t+z-t>y-t+1>y-t-1$, which is a contradiction. As a result, one has the
identification

$\sum_{j=0}^{b-1}C_{j}\cdot\theta^{-t-j}g_{y}^{[p]}(q)\times
h_{z}(q)V_{x,j}=\sum_{j=0}^{b-1}C_{j}\cdot\theta^{-t-j}g_{y}^{[p]}(q)\times
h_{z}(q)\omega_{\operatorname{\operatorname{can}}}^{x-j}\eta_{\operatorname{\operatorname{can}}}^{j}.$

Applying the overconvergent projection formula in Prop. 2.14 with

$\gamma_{j}(q)=C_{j}\cdot\theta^{-t-j}g_{y}^{[p]}(q)\times h_{z}(q),$

we get

$\displaystyle H^{\dagger}(\nabla^{-t}(g_{y}^{[p]}V_{y,0})\times
h_{z}V_{z,0})(q)=\sum_{j=0}^{b-1}(-1)^{j}C_{j}\cdot\frac{\theta^{j}(\theta^{-t-j}g_{y}^{[p]}(q)\times
h_{z}(q))}{(x-2-j+1)(x-2-j+2)\cdots(x-2)}V_{x,0}.$

###  3.2 The $p$-adic Abel–Jacobi maps and the generalized diagonal cycles

#### Kuga–Sato varieties and the generalized diagonal cycles.

For any $r\geq 0$, we have the Kuga–Sato variety $W_{r}$, which is the
desingularization (c.f. [BDP13, Appendix]) of the $r$-fold fiber product

$W_{r}^{\prime}:=E\times_{X}E\cdots\times_{X}E.$

Then one may see the parabolic cohomolgy
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r})$ as a subspace
in a correct degree of the de Rham cohomology of $W_{r}$. This is illustrated
in the following lemma.

######  Lemma 3.1 ([BDP13, Lemma 2.2]).

Assume that $r\geq 1$. Then there is a idempotent
$\epsilon_{r}\in\mathbb{Q}[\operatorname{\operatorname{Aut}}(W_{r}/X)]$,
defined in [DR14, §3.1], such that we have

$\epsilon_{r}H^{j}_{\operatorname{\operatorname{dR}}}(W_{r}/K)=\begin{cases}0&j\neq
r+1\\\
H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r})&j=r+1\end{cases}.$

Furthermore, the Hodge filtration on
$\epsilon_{r}H^{r+1}_{\operatorname{\operatorname{dR}}}(W_{r}/K)$ is given by

$\displaystyle\operatorname{\operatorname{Fil}}^{0}=H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r}),$
$\displaystyle\operatorname{\operatorname{Fil}}^{1}=\operatorname{\operatorname{Fil}}^{2}=\cdots=\operatorname{\operatorname{Fil}}^{r+1}=H^{0}(X,\underline{\omega}^{r}\otimes\Omega^{1}_{X}),$
$\displaystyle\operatorname{\operatorname{Fil}}^{r+2}=0.$

######  Remark 3.2.

We here recall that the parabolic cohomology
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r})$ is defined to
be the hypercohomology of the complex

$0\rightarrow\mathcal{H}^{r}\rightarrow\left(\mathcal{H}^{r}\otimes\Omega^{1}_{X}+\nabla(\mathcal{H}^{r})\right)\rightarrow
0.$

As we are mainly interested in cusp forms, the parabolic cohomology has the
advantage that it comes with the exact sequence (c.f. [Sch85, § 2])

$0\rightarrow H^{0}(X,\underline{\omega}^{r}\otimes\Omega^{1}_{X})\rightarrow
H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r})\rightarrow
H^{1}(X,\underline{\omega}^{-r})\rightarrow 0.$

Suppose now we have a triple of balanced weights
$(x,y,z)=(r_{1}+2,r_{2}+2,r_{3}+2)$. We further assume that
$r_{1}>0,r_{2}>0,r_{3}>0$ and $r:=\frac{r_{1}+r_{2}+r_{3}}{2}\in\mathbb{N}$.

Set $W=W_{r_{1}}\times W_{r_{2}}\times W_{r_{3}}$, which is of dimension
$2r+3$. We now briefly recall the definition of the generalized diagonal cycle
$\Delta_{x,y,z}\in\mathrm{CH}^{r+2}(W)$ (c.f. [DR14, Definition 3.3]).

Choose three subsets

$A=\\{a_{1},\ldots,a_{r_{1}}\\},\quad B=\\{b_{1},\ldots,b_{r_{2}}\\},\quad
C=\\{c_{1},\ldots,c_{r_{3}}\\}$

of $\\{1,\ldots,r\\}$ such that $A\cap B\cap C=\emptyset$ and $A\cup B=B\cup
C=A\cup C=\\{1,\ldots,r\\}$. One can see that the balancedness assumption
guarantees the existence of such sets. Then we consider the map

$\displaystyle\varphi_{ABC}:W_{r}$ $\displaystyle\rightarrow
W:=W_{r_{1}}\times W_{r_{2}}\times W_{r_{3}}$ $\displaystyle(x;P_{1},\ldots
P_{r})$
$\displaystyle\mapsto((x;P_{a_{1}},\ldots,P_{a_{r_{1}}}),(x;P_{b_{1}},\ldots
P_{b_{r_{2}}}),(x;P_{c_{1}},\ldots P_{c_{r_{3}}})),$

which is a closed embedding of $W_{r}$ into $W$.

######  Definition 3.3.

The generalized diagonal cycle is defined by

$\Delta_{x,y,z}=(\epsilon_{r_{1}},\epsilon_{r_{2}},\epsilon_{r_{3}})\varphi_{ABC}(W_{r})\in\mathrm{CH}^{r+2}(W).$

######  Remark 3.4.

Here we assume that $x,y,z>2$ for simplicity. The definition of
$\Delta_{x,y,z}$ when some of the weights are equal to $2$ can be found in
[DR14, § 3].

By using the Künneth decomposition on
$H^{2r+4}_{\operatorname{\operatorname{dR}}}(W)$ and examining the image of
the idempotents $\epsilon_{r_{i}}$’s, it follows that the cycle
$\Delta_{x,y,z}$ is homologous to zero. That is,

$\Delta_{x,y,z}\in\mathrm{CH}^{r+2}(W)_{0}:=\ker(\operatorname{cl}:\mathrm{CH}^{r+2}(W)\rightarrow
H^{2r+4}_{\operatorname{\operatorname{dR}}}(W)).$

In particular, one may consider the image of $\Delta_{x,y,z}$ under the
$p$-adic Abel–Jacobi map (c.f. [Nek00])

$\operatorname{\operatorname{AJ}}_{p}:\mathrm{CH}^{r+2}(W)_{0}\rightarrow\operatorname{\operatorname{Fil}}^{r+2}H^{2r+3}_{\operatorname{\operatorname{dR}}}(W)^{\vee}.$

#### Explicit computations of the Abel–Jacobi maps.

We now study the value of
$\operatorname{\operatorname{AJ}}_{p}(\Delta_{x,y,z})$ at a certain element
$\eta\otimes\omega_{2}\otimes\omega_{3}$ in
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})\otimes
H^{0}(X,\underline{\omega}^{r_{2}}\otimes\Omega^{1}_{X})\otimes
H^{0}(X,\underline{\omega}^{r_{3}}\otimes\Omega^{1}_{X}).$

First of all, by the balancedness assumption, we see that

$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})\otimes
H^{0}(X,\underline{\omega}^{r_{2}}\otimes\Omega^{1}_{X})\otimes
H^{0}(X,\underline{\omega}^{r_{3}}\otimes\Omega^{1}_{X})\subset\operatorname{\operatorname{Fil}}^{r+2}H^{2r+3}_{\operatorname{\operatorname{dR}}}(W).$

We now specify the forms $\eta,\omega_{2},\omega_{3}$ we will encounter later.
Let $f,g,h$ be a triple of cusp forms on $X_{1}(N)$ of balanced weights
$(x=r_{1}+2,\ y=r_{2}+2,\ z=r_{3}+2)$ such that $x=y+z-2t$ as before. The cusp
form $f$ then corresponds to a section $\omega_{f}\in
H^{0}(X,\underline{\omega}^{r_{1}}\otimes\Omega^{1}_{X})$ and similar for $g$
and $h$. We also reserve the notation $\mathcal{U}$ for some strict
neighborhood of the ordinary locus in $X$ (which may vary according to the
situation) and view $\omega_{g},\omega_{h}$ as sections over $\mathcal{U}$ via
restriction.

We write the Hecke polynomial of $f$ as

$T^{2}-a_{p}(f)T+\chi_{f}(p)p^{x-1}=(T-\alpha_{f})(T-\beta_{f})$

with $a=\operatorname{\operatorname{ord}}_{p}(\alpha_{f})$, and similarly for
$g,h$. We also write

$\displaystyle P_{g}(T)$
$\displaystyle=1-a_{p}(g)p^{1-y}T+\chi_{g}(p)p^{1-y}T^{2}=(1-\alpha_{g}p^{1-y}T)(1-\beta_{g}p^{1-y}T),$
$\displaystyle P_{h}(T)$
$\displaystyle=1-a_{p}(h)p^{1-z}T+\chi_{h}(p)p^{1-z}T^{2}=(1-\alpha_{h}p^{1-z}T)(1-\beta_{h}p^{1-z}T).$

We will write $\alpha_{g}^{\prime}:=\alpha_{g}p^{1-y}$,
$\beta_{g}^{\prime}:=\beta_{g}p^{1-y}$,
$\alpha_{h}^{\prime}:=\alpha_{h}p^{1-z}$ and
$\beta_{h}^{\prime}:=\beta_{h}p^{1-z}$. The polynomial $P_{g}(T)$ is defined
such that $P_{g}(\phi)$ annihilates the class of $\omega_{g}$, where $\phi$ is
the Frobenius on the cohomology
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{2}})$. In fact,
$P_{g}(\phi)\omega_{g}=\omega_{g^{[p]}}$ as section over $\mathcal{U}$.

Similarly as in [DR14, Cor. 2.13] and the paragraph before it, we have a class
$\overline{\eta_{f}}\in H^{1}(X,\underline{\omega}^{-r_{1}})$ such that for
any cusp form $\omega\in S_{x}(\Gamma_{1}(N))$, we have

$\langle\overline{\eta_{f}},\omega\rangle_{\operatorname{\operatorname{dR}}}=\frac{(f^{*},\omega)_{N}}{(f^{*},f^{*})_{N}}.$

We want to find a suitable lift of $\overline{\eta_{f}}$ in
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r_{1}})$. First
recall that Poincaré pairing on
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r_{1}})$
descends to a perfect pairing

$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r_{1}})[f]/\operatorname{\operatorname{Fil}}^{r_{1}+1}\times
S_{k}(N;K)[{f}^{*}]\rightarrow K.$

Hence, one can viewed $\overline{\eta_{f}}$ as an element in
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})[f]/\operatorname{\operatorname{Fil}}^{r+1}$.
We would also like to remark that on the $f$-isotypic part
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r_{1}})[f]$, the
Frobenius $\phi$ acts with eigenvalues $\alpha_{f}/p^{x-1}=\beta_{f^{*}}$ and
$\beta_{f}/p^{x-1}=\alpha_{f^{*}}$. Notice that
$\alpha_{f^{*}}=\chi_{f}(p)^{-1}\alpha_{f}$ is also of $p$-adic valuation $a$.

Suppose that $a<x-1$, then one has the following decomposition (c.f. [BC09, §
2.4.3])

$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})[f]=\operatorname{\operatorname{Fil}}^{r_{1}+1}H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})[f]\oplus
H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})[f]^{\phi=\alpha_{f^{*}}}.$

This splitting allows us to fix a lift $\eta_{f}=\eta_{f}^{a}$ of
$\overline{\eta_{f}}$ such that the Frobenius $\phi$ acts as multiplication by
$\alpha_{f^{*}}$.

We now turn our attention to the modular forms $g$ and $h$. As $1-VU$
annihilates the cohomology group
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{2}})$, the form
$\omega_{g}^{[p]}$ over $\mathcal{U}$ is $\nabla$-exact. A primitive $G^{[p]}$
over $\mathcal{U}$ can be chosen (uniquely) such that its polynomial
$q$-expansion takes the form

$G^{[p]}(q)=\sum_{i=0}^{r_{2}}(-1)^{i}i!\binom{r_{2}}{i}\theta^{-i-1}g^{[p]}(q)\omega_{\operatorname{\operatorname{can}}}^{r_{2}-i}\eta_{\operatorname{\operatorname{can}}}^{i}.$

The product $G^{[p]}\times\omega_{h}$ can then be viewed as a section of
$\mathcal{H}^{r_{2}}\otimes\mathcal{H}^{r_{3}}\otimes\Omega_{X}^{1}$. By
assumption on the weights $(x,y,z)$, it follows that there is a projection
$\operatorname{pr}_{r_{1}}:\mathcal{H}^{r_{2}}\otimes\mathcal{H}^{r_{3}}\rightarrow\mathcal{H}^{r_{1}}(1-t)$
(c.f. [DR14, Prop. 2.9]). As a result, one has a section
$\operatorname{pr}_{r_{1}}(G^{[p]}\times\omega_{h})$ of
$\mathcal{H}^{r_{1}}(1-t)\otimes\Omega^{1}_{X}$.

######  Remark 3.5.

We here explain the projection $\operatorname{pr}_{r_{1}}$ in detail, for it
is crucial in later computations.

Set $r:=r_{2}+r_{3}-(t-1)$, and let
$\mathcal{H}^{(1)},\ldots,\mathcal{H}^{(r)}$ be $r$-copies of $\mathcal{H}$
and $\mathcal{H}^{\otimes
r}:=\mathcal{H}^{(1)}\otimes\cdots\mathcal{H}^{(r)}$. We then have a natural
embedding of
$\mathcal{H}^{r}:=\operatorname{\operatorname{Sym}}^{r}\mathcal{H}$ into
$\mathcal{H}^{\otimes r}$.

We choose subsets $B\subset\\{1,2,\ldots,r\\}$ of cardinality $r_{2}$ and
$C\subset\\{1,2,\ldots,r\\}$ of cardinality $r_{3}$ such that $B\cup
C=\\{1,2,\ldots,r\\}$ as before. Notice that we automatically have $\\#(B\cap
C)^{c}=r_{1}$. We may fix a simple choice $B=\\{1,2,\ldots,r_{2}\\}$ and
$C=\\{r-r_{3}+1=r_{2}-t+2,\ldots,r\\}$.

We may embed $\mathcal{H}^{r_{2}}$ canonically into $\mathcal{H}^{\otimes
r_{2}}$, then embed it into $\mathcal{H}^{\otimes r}$ via the set $B$, and
embed $\mathcal{H}^{r_{3}}$ into $\mathcal{H}^{\otimes r}$ via $C$ similarly.
In terms of the basis
$\\{\omega_{\operatorname{\operatorname{can}}},\eta_{\operatorname{\operatorname{can}}}\\}$,
the element $1\cdot\omega_{\operatorname{\operatorname{can}}}^{r_{2}}$ of
$\mathcal{H}^{r_{2}}$ is sent to
$\omega_{\operatorname{\operatorname{can}}}^{(r_{2})}=1\cdot\omega_{\operatorname{\operatorname{can}}}\otimes\omega_{\operatorname{\operatorname{can}}}\cdots\otimes\omega_{\operatorname{\operatorname{can}}}$
of $\mathcal{H}^{\otimes r_{2}}$. On the other hand,
$1\cdot\omega_{\operatorname{\operatorname{can}}}^{r_{2}-1}\eta_{\operatorname{\operatorname{can}}}$
is sent to

$\frac{1}{\binom{r_{2}}{1}}\sum_{j=1}^{r_{2}}\omega_{\operatorname{\operatorname{can}}}^{(\hat{j})}\otimes\eta_{\operatorname{\operatorname{can}}}^{(j)}$

where $\eta_{\operatorname{\operatorname{can}}}^{(j)}=1\otimes\cdots
1\otimes\eta_{\operatorname{\operatorname{can}}}\otimes 1\cdots\otimes 1$ with
only one $\eta_{\operatorname{\operatorname{can}}}$ at the $j$-th component,
and
$\omega_{\operatorname{\operatorname{can}}}^{(\hat{j})}=\omega_{\operatorname{\operatorname{can}}}\otimes\cdots\omega_{\operatorname{\operatorname{can}}}\otimes
1\otimes\omega_{\operatorname{\operatorname{can}}}\cdots\otimes\omega_{\operatorname{\operatorname{can}}}$
with only one $1$ at the $j$-th component. One should be able to work out the
general cases explicitly.

Now apply the Poincaré pairing
$\mathcal{H}\times\mathcal{H}\rightarrow\mathcal{O}_{X}(-1)$ component-wise on
the images of $\mathcal{H}^{r_{2}}$ and $\mathcal{H}^{r_{3}}$. Since there are
$(t-1)$-many overlapping components corresponding to $B\cap C$, after
symmetrization, we may identify the resulting sheaf as
$\mathcal{H}^{r_{1}}(1-t)$. We remark that the symmetrization sends, for
example,
$\sum_{j=1}^{r_{2}}\alpha_{j}\omega_{\operatorname{\operatorname{can}}}^{(\hat{j})}\otimes\eta_{\operatorname{\operatorname{can}}}^{(j)}$
to
$(\sum_{j=1}^{r_{2}}\alpha_{j})\cdot\omega_{\operatorname{\operatorname{can}}}^{r_{2}-1}\eta_{\operatorname{\operatorname{can}}}$.

Note that the projection $\operatorname{pr}_{r_{1}}$ generalizes the
decomposition

$\operatorname{\operatorname{Sym}}^{r_{2}}V\otimes\operatorname{\operatorname{Sym}}^{r_{3}}V\cong\bigoplus_{j=0}^{\min\\{r_{2},r_{3}\\}}\operatorname{\operatorname{Sym}}^{r_{2}+r_{3}-2j}V$

for a two dimensional vector space $V$ (or viewed as the standard
representation of $\operatorname{\operatorname{SL}}_{2}$).

Now we are able to describe the Abel–Jacobi image.

######  Theorem 3.6.

Let $\Delta_{x,y,z}$ be the general diagonal cycle defined as above, and view
$\eta_{f}\otimes\omega_{g}\otimes\omega_{h}$ as elements in the de Rham
cohomology
$\operatorname{\operatorname{Fil}}^{r+2}H^{2r+3}_{\operatorname{\operatorname{dR}}}(W)$.
Then we have

$\operatorname{\operatorname{AJ}}_{p}(\Delta_{x,y,z})(\eta_{f}\otimes\omega_{g}\otimes\omega_{h})=\frac{\mathscr{E}_{1}(f)}{\mathscr{E}(f,g,h)}\cdot\langle\eta_{f},e^{\leq
a}\operatorname{pr}_{r_{1}}(G^{[p]}\times\omega_{h})\rangle$

where $\langle\phantom{e},\phantom{e}\rangle$ is the Poincaré pairing between
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})$ and
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}}(1-t))$ and
$e^{\leq a}$ is the projection to the slope $\leq a$ part with respect to the
operator $U$. Note that on the right hand side, we identify $e^{\leq
a}\operatorname{pr}_{r_{1}}(G^{[p]}\times\omega_{h})$ with its class in
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}}(1-t))$.

The terms $\mathscr{E}_{1}(f)$ and $\mathscr{E}(f,g,h)$ are Euler factors
defined by

$\displaystyle\mathscr{E}_{1}(f)$
$\displaystyle=(1-\beta^{2}\alpha_{g}^{\prime}\beta_{g}^{\prime}\alpha_{h}^{\prime}\beta_{h}^{\prime})=(1-\beta_{f}^{2}\chi_{f}^{-1}(p)p^{-x}),$
$\displaystyle\mathscr{E}(f,g,h)$
$\displaystyle=(1-\beta\alpha_{g}^{\prime}\alpha_{h}^{\prime})(1-\beta\alpha_{g}^{\prime}\beta_{h}^{\prime})(1-\beta\beta_{g}^{\prime}\alpha_{h}^{\prime})(1-\beta\beta_{g}^{\prime}\beta_{h}^{\prime})$

where $\beta:=p^{x-1+t-1}/\alpha_{f^{*}}$.

###### Proof.

The proof of this theorem is essentially identical to the proof in [DR14, §
3.4]. The main difference is that instead of the ordinary projection
$e_{\operatorname{\operatorname{ord}}}$, we use the finite slope projection
$e^{\leq a}$.

Notice that the projection $e^{\leq a}$ can be written as a power series in
$U$ without constant term (c.f. Remark 5.12). Consequently, one has the
following analogous result of [DR14, Lemma 2.17]:

$e^{\leq a}(G^{[p]}\cdot V\omega_{h})=0.$

In addition, one needs the following lemma (c.f. [DR14, Proposition 2.11]),
which will be proved in the Appendix.

######  Lemma 3.7.

Suppose $\eta\in H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r})$
is an element for which the Frobenius $\phi$ acts with eigenvalue $\alpha$
such that $v_{p}(\alpha)\leq a$. Then for any $\omega\in
H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r})$, we have

$\langle\eta,\omega\rangle=\langle\eta,e^{\leq a}\omega\rangle$

where the pairing is the Poincaré pairing as in Theorem 3.6.

With these two facts, one then follow the exact same argument in [DR14, § 3.4]
and conclude the proof. ∎

######  Remark 3.8.

In [Bes16], it is showed that the proof of the above theorem in the case
$(x,y,z)=(2,2,2)$ can be greatly simplified by using the explicit cup product
formula for finite polynomial cohomology on $X$. It is expected that the
general case can also be proved in a similar manner if one has a suitable
theory with coefficients. Currently, there is a joint work of me and Ju-Feng
Wu [HW22] aiming to develop a theory of finite polynomial cohomology and
$p$-adic Abel–Jacobi maps with coefficients. Alternatively, it is also proved
in my Ph.D. thesis [Hua24].

###  3.3 Proof of the $p$-adic Gross–Zagier formula

After studying the two ingredients, we are now able to prove the $p$-adic
Gross–Zagier formula. Before doing so, we first recall our notations.

Let $\mathbf{f},\mathbf{g},\mathbf{h}$ be three finite slope families and
$(x,y,x)\in\Sigma_{\mathbf{f},\mathbf{g},\mathbf{h}}$ be a triple of balanced
weights as in Section 3.1. Their specializations $f_{x},g_{y},h_{z}$ are
classical modular forms on $X_{1}(N,p)$ and will be viewed as sections over
the strict neighborhood $\mathcal{U}$ in $X$. The forms $f_{x},g_{y},h_{z}$
are the $p$-stabilizations of modular forms $f,g,h$ on $X_{1}(N)$ and
$f_{x}=(1-\beta_{f}V)f$ by our assumption. We also have the elements
$\eta_{f},\omega_{g},\omega_{h}$ associated with $f,g,h$ in their respective
cohomology groups as defined in Section 3.2.

######  Theorem 3.9.

($p$-adic Gross–Zagier formula) Let notations be as above. We have

$\mathscr{L}_{p}^{f}(\mathbf{f},\mathbf{g},\mathbf{h})(x,y,z)=(-1)^{t-1}\frac{\mathscr{E}(f,g,h)}{(t-1)!\mathscr{E}_{0}(f)\mathscr{E}_{1}(f)}\times\operatorname{\operatorname{AJ}}_{p}(\Delta_{x,y,z})(\eta_{f}\otimes\omega_{g}\otimes\omega_{h}),$

where the Euler factors $\mathscr{E}(f,g,h)$ and $\mathscr{E}_{1}(f)$ are as
Theorem 3.6, and

$\mathscr{E}_{0}(f)=(1-\beta_{f}^{2}\chi_{f}^{-1}(p)p^{1-x}).$

The strategy of the proof is the following: from the two overconvergent forms
$g_{y},h_{z}$, there are two ways to construct an overconvergent form of
weight $x$ (or an element in
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})$).

One is $H^{\dagger}(\nabla^{-t}(g_{y}^{[p]}V_{y,0})\times h_{z}V_{z,0})$,
which appears in the triple product $L$-function. The other is
$H^{\dagger}(\operatorname{pr}_{r_{1}}(G^{[p]}\times\omega_{h}))$, which is
related to the $p$-adic Abel-Jacobi image. Hence, proving the $p$-adic
Gross–Zagier formula is essentially equal to relating the two overconvergent
forms (or cohomological classes) mentioned above.

The crucial observation is the following lemma, which should be considered the
main novelty of this paper.

######  Lemma 3.10.

Let notations be as before, then we have

$(-1)^{t-1}(t-1)!\cdot H^{\dagger}(\nabla^{-t}(g_{y}^{[p]}V_{y,0})\times
h_{z}V_{z,0})=H^{\dagger}(\operatorname{pr}_{r_{1}}(G^{[p]}\times
h_{z}))V_{x,0}.$

###### Proof.

We prove this equality by examining their $q$-expansions.

First, we observe that the two forms $g$ and $g_{y}=(1-\beta_{g}V)g$ has the
same $p$-depletion. So we may replace $g_{y}^{[p]}$ with $g^{[p]}$.

Let $b:=y-t$ be as before, recall that

$H^{\dagger}(\nabla^{-t}(g^{[p]}V_{y,0})\times
hV_{z,0})=\sum_{j=0}^{b-1}(-1)^{j}\binom{-t}{j}\prod_{i=0}^{j-1}(y-t-1-i)\frac{\theta^{j}(\theta^{-t-j}g^{[p]}\times
h)}{(x-2-j+1)\cdots(x-2)}V_{x,0}.$ (3)

On the other hand, recall that we have

$G^{[p]}=\sum_{s=0}^{r_{2}}(-1)^{s}s!\binom{r_{2}}{s}\theta^{-s-1}g^{[p]}(q)\omega_{\operatorname{\operatorname{can}}}^{r_{2}-s}\eta_{\operatorname{\operatorname{can}}}^{s}$

as the primitive of the overconvergent form $g^{[p]}$ with respect to the
Gauss–Manin connection $\nabla$. We also view $h_{z}$ as a section of
$\mathcal{H}^{r_{3}}\otimes\Omega^{1}_{X}$ over $\mathcal{U}$ and write its
$q$-expansion as
$h_{z}(q)\omega_{\operatorname{\operatorname{can}}}^{r_{3}}\cdot\frac{dq}{q}$.

Following the recipe in Remark 3.5, we have the polynomial $q$-expansion

$\operatorname{pr}_{r_{1}}(G^{[p]}\times
h_{z})=\sum_{s=t-1}^{r_{2}}(-1)^{s}s!\binom{r_{2}-(t-1)}{s-(t-1)}\theta^{-s-1}g^{[p]}\times
h_{z}\
\omega_{\operatorname{\operatorname{can}}}^{r_{1}-(s-(t-1))}\eta_{\operatorname{\operatorname{can}}}^{s-(t-1)}\cdot\frac{dq}{q}.$

After a change of variable $\alpha:=s-(t-1)$ and noticing that
$r_{2}-(t-1)=b-1$, it can be written as

$\operatorname{pr}_{r_{1}}(G^{[p]}\times
h_{z})=\sum_{\alpha=0}^{b-1}(-1)^{\alpha+(t-1)}(\alpha+t-1)!\binom{b-1}{\alpha}\theta^{-t-\alpha}g^{[p]}\times
h_{z}\
\omega_{\operatorname{\operatorname{can}}}^{r_{1}-\alpha}\eta_{\operatorname{\operatorname{can}}}^{\alpha}\cdot\frac{dq}{q}.$

The overconvergent projection
$H^{\dagger}(\operatorname{pr}_{r_{1}}(G^{[p]}\times h_{z}))$ then takes the
form

$\sum_{\alpha=0}^{b-1}(-1)^{2\alpha+(t-1)}(\alpha+t-1)!\binom{b-1}{\alpha}\frac{\theta^{\alpha}(\theta^{-t-\alpha}g^{[p]}\times
h_{z})}{(k-2-\alpha+1)\cdots(k-2)}\
\omega_{\operatorname{\operatorname{can}}}^{r_{1}}\cdot\frac{dq}{q}.$ (4)

The lemma is then reduced to comparing the numbers

$(-1)^{j}\binom{-t}{j}\prod_{i=0}^{j-1}(b-1-i)$

and

$(-1)^{t-1}(j+t-1)!\binom{b-1}{j}.$

Observe that

$\displaystyle(-1)^{j}\binom{-t}{j}\prod_{i=0}^{j-1}(b-1-i)$
$\displaystyle=(-1)^{j}\frac{(-t)(-t-1)\cdots(-t-j+1)}{j!}\prod_{i=0}^{j-1}(b-1-i)$
$\displaystyle=(-1)^{2j}\cdot
t(t+1)\cdots(t+j-1)\cdot\frac{1}{j!}\cdot\prod_{i=0}^{j-1}(b-1-i)$
$\displaystyle=t(t+1)\cdots(t+j-1)\frac{1}{j!}\cdot j!\binom{b-1}{j}$
$\displaystyle=t(t+1)\cdots(t+j-1)\cdot\binom{b-1}{j}.$

Hence we have

$(-1)^{t-1}(t-1)!\cdot\left((-1)^{j}\binom{-t}{j}\prod_{i=0}^{j-1}(b-1-i)\right)=(-1)^{t-1}(j+t-1)!\binom{b-1}{j}$

for all $0\leq j\leq b-1$. The lemma now follows. In fact, one sees that the
equality holds even without applying the overconvergent projection. ∎

######  Remark 3.11.

In the ordinary case (c.f. [DR14, Prop 2.9]), the proof is easier as one may
replace the overconvergent projection with the unit root splitting. In other
words, one only needs to prove the equality between the first terms of the two
polynomial $q$-expansions.

######  Corollary 3.12.

With notations as before, we have

$\displaystyle\operatorname{\operatorname{AJ}}(\Delta_{x,y,z})(\eta_{f}\otimes\omega_{g}\otimes\omega_{h})$
$\displaystyle=\frac{\mathscr{E}_{1}(f)}{\mathscr{E}(f,g,h)}\cdot\langle\eta_{f},e^{\leq
a}(\operatorname{pr}_{r_{1}}(G^{[p]}\times h)\rangle$
$\displaystyle=\frac{\mathscr{E}_{1}(f)}{\mathscr{E}(f,g,h)}\cdot\langle\eta_{f},e^{\leq
a}(\operatorname{pr}_{r_{1}}(G^{[p]}\times h_{z})\rangle$
$\displaystyle=(-1)^{t-1}(t-1)!\frac{\mathscr{E}_{1}(f)}{\mathscr{E}(f,g,h)}\cdot\langle\eta_{f},e^{\leq
a}H^{\dagger}(\nabla^{-t}(g^{[p]})\times h_{z})\rangle.$

###### Proof.

It only suffices to prove the second equation. As $U(G^{[p]}\cdot Vh)=0$, and
$e^{\leq a}$ has no constant term as a power series in $U$, $e^{\leq a}$
annihilates $\operatorname{pr}_{r_{1}}(G^{[p]}\times Vh)$. Hence we may
replace $h$ by one of its $p$-stabilization $h_{z}$. The second equality then
follows. ∎

Now back to the triple product $p$-adic $L$-function
$\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})$. By definition, its
value at the point $(x,y,z)$ is

$\frac{(f_{x}^{*},H^{\dagger,\leq a}(\nabla^{-t}(g^{[p]}_{y})\times
h_{z}))_{N,p}}{(f_{x}^{*},f_{x}^{*})_{N,p}}$

where the Petersson product is on $X_{1}(N,p)$.

We need to translate the Petersson product on $X_{1}(N,p)$ back to $X_{1}(N)$.
Recall that $f_{x}$ is the $p$-stabilization of $f$. In other words, as
classes in $H^{1}_{\operatorname{\operatorname{dR}}}(X,\mathcal{H}^{r_{1}})$,
we have $f_{x}=(1-\beta_{f}^{\prime}\phi)f=(1-\beta_{f}V)f$ and
$f^{*}_{x}=(1-\beta_{f^{*}}V)f^{*}$.

We now use the formula in [DR14, Lemma 2.19]. Let $\omega$ be any modular form
on $X_{1}(N,p)$ of weight $x$, we have

$\frac{(f_{x}^{*},\omega)_{N,p}}{(f_{x}^{*},f_{x}^{*})_{N,p}}=\frac{(f_{x}^{*},e_{f_{x}^{*}}\omega)_{N,p}}{(f_{x}^{*},f_{x}^{*})_{N,p}}=\frac{(f^{*},\omega^{0})_{N}}{(f^{*},f^{*})_{N}}=\langle\eta_{f},\omega^{0}\rangle_{\operatorname{\operatorname{dR}}}.$

Here $e_{f_{x}^{*}}$ is the projection to the $f_{x}^{*}$-isotypic component,
and $\omega^{0}$ is a modular form on $X_{1}(N)$ such that

$e_{f*}\omega=(1-\beta_{f^{*}}^{\prime}\phi)\omega^{0}=(1-\beta_{f^{*}}V)\omega^{0}.$

Now we take $\omega$ to be the modular form $H^{\dagger,\leq
a}(\nabla^{-t}(g^{[p]}_{y})\times h_{z})$, which we also view as a class in
$H^{1}_{\operatorname{\operatorname{par}}}(X,\mathcal{H}^{r_{1}})$. Then,

$\displaystyle\langle\eta_{f},\omega\rangle=\langle\eta_{f},e_{f^{*}}\omega\rangle$
$\displaystyle=\langle\eta_{f},(1-\beta_{f^{*}}V)\omega^{0}\rangle$
$\displaystyle=\langle\eta_{f},\omega^{0}\rangle-\beta_{f^{*}}\langle\eta_{f},V\omega^{0}\rangle$
$\displaystyle=\langle\eta_{f},\omega^{0}\rangle-\beta_{f^{*}}\langle\phi^{-1}\eta_{f},\omega^{0}\rangle$
$\displaystyle=(1-\frac{\beta_{f^{*}}}{\alpha_{f^{*}}})\langle\eta_{f},\omega^{0}\rangle$
$\displaystyle=\mathscr{E}_{0}(f)\langle\eta_{f},\omega^{0}\rangle$

where
$\mathscr{E}_{0}(f):=(1-\frac{\beta_{f^{*}}}{\alpha_{f^{*}}})=(1-\frac{\beta_{f}}{\alpha_{f}})=(1-\beta_{f}^{2}\chi_{f}^{-1}(p)p^{1-x})$.
As a result,

$\mathscr{L}_{p}^{f}(\mathbf{f},\mathbf{g},\mathbf{h})(x,y,z)=\frac{1}{\mathscr{E}_{0}(f)}\langle\eta_{f},H^{\dagger,\leq
a}(\nabla^{-t}(g^{[p]}_{y})\times
h_{z})\rangle_{\operatorname{\operatorname{dR}}}.$

This equality, together with Corollary 3.12, then imply

$\mathscr{L}_{p}^{f}(\mathbf{f},\mathbf{g},\mathbf{h})(x,y,z)=(-1)^{t-1}\frac{\mathscr{E}(f,g,h)}{(t-1)!\mathscr{E}_{0}(f)\mathscr{E}_{1}(f)}\times\operatorname{\operatorname{AJ}}(\Delta_{x,y,z})(\eta_{f}\otimes\omega_{g}\otimes\omega_{h}).$

##  4 An application to equivariant BSD conjecture

In this section, we will show an application of the $p$-adic Gross–Zagier
formula for finite slope families.

###  4.1 Coleman families and big Galois representations

Let $\mathcal{W}$ be the weight space defined over $\mathbb{Q}_{p}$ as before.
In this section, we will focus on families defined over wide open disks in
$\mathcal{W}$ as in [LZ16, § 4].

######  Definition 4.1.

Suppose $\mathcal{V}$ is a open disk in $\mathcal{W}$ defined over a finite
extension $L$ over $\mathbb{Q}_{p}$. We denote by $\mathscr{O}_{\mathcal{V}}$
(resp. $\Lambda_{\mathcal{V}}$) the ring of bounded (resp. bounded by $1$)
rigid analytic functions on $\mathcal{V}$. In particular, if
$k_{0}\in\mathbb{Z}$ is a center of $\mathcal{V}$,
$\Lambda_{\mathcal{V}}\cong\mathcal{O}_{L}\llbracket\bm{k}-k_{0}\rrbracket$
for some formal variable $\bm{k}$ and
$\mathscr{O}_{\mathcal{V}}=\Lambda_{\mathcal{V}}[\frac{1}{p}]$. We will always
endow $\Lambda_{\mathcal{V}}$ with the topology induced by its maximal ideal
$m_{\mathcal{V}}$. We let
$\kappa_{\mathcal{V}}:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda_{\mathcal{V}}^{\times}$
be the universal character (shifted by $-2$) on $\mathcal{V}$. In other words,
$\kappa_{\mathcal{V}}(t)=\omega(t)^{k_{0}-2}\langle t\rangle^{\mathbf{k}-2}$
for $t\in\mathbb{Z}_{p}^{\times}$, where $\omega$ is the Teichmüller character
and $\langle t\rangle:=t/\omega(t)\in 1+p\mathbb{Z}_{p}$.

######  Definition 4.2.

Let $\mathcal{V}\subset\mathcal{W}$ be an open disk such that the classical
weights $\mathcal{V}\cap\mathbb{Z}_{\geq 0}$ is dense in $\mathcal{V}$. A
Coleman family $\mathbf{f}$ (of level $N_{f}$ and nebentypus character
$\chi_{f}$) defined over $\mathcal{V}$ of slope $a\in\mathbb{Q}_{\geq 0}$ is a
power series

$\mathbf{f}=\sum_{n\geq
1}a_{n}(\mathbf{f})q^{n}\in\Lambda_{\mathcal{V}}\llbracket q\rrbracket$

such that $a_{1}(\mathbf{f})=1$, $a_{p}(\mathbf{f})$ is invertible in
$\mathscr{O}_{\mathcal{V}}$ with norm $p^{-a}$, and for all but finitely many
classical weights $k\in\mathcal{V}$, the specialization
$\mathbf{f}_{k}\in\mathcal{O}_{L}\llbracket q\rrbracket$ is the $q$-expansion
of a classical normalized eigenform of weight $k$, level $\Gamma_{1}(N_{f},p)$
and nebentypus $\chi_{f}$.

We will often write $\mathcal{V}_{f}=\mathcal{V}$ and similarly
$\mathscr{O}_{f}=\mathscr{O}_{\mathcal{V}_{f}}$,
$\Lambda_{f}=\Lambda_{\mathcal{V}_{f}}$, $\kappa_{f}=\kappa_{\mathcal{V}_{f}}$
to specify their dependence to the family $\mathbf{f}$.

One now wants to construct a $p$-adic family of Galois representations
attached to a finite slope family. It turns out that this can be achieved by
considering certain big étale sheaf on modular curves (c.f. [AIS15] and
[AS08]). We shall briefly recall several notations and results without any
proofs, mainly following the setups in [BSV22, § 4].

As we will consider both families and their specializations to classical
weights, we use the terminology ‘big’ to refer to anything defined for
families and ‘small’ for the specialized case.

#### Locally analytic functions and distributions.

######  Definition 4.3.

For any integer $m\geq 0$, we write
$LA_{m}(\mathbb{Z}_{p},\Lambda_{\mathcal{V}})$ for the space of functions
$\mathbb{Z}_{p}\rightarrow\Lambda_{\mathcal{V}}$ that are analytic on every
open disk $z+p^{m}\mathbb{Z}_{p}$ of $\mathbb{Z}_{p}$.

We define $T:=\mathbb{Z}_{p}^{\times}\times\mathbb{Z}_{p}$ and
$T^{\prime}=p\mathbb{Z}_{p}\times\mathbb{Z}_{p}^{\times}$. One has right
actions of $M_{2\times 2}(\mathbb{Z}_{p})$ on $T$ and $T^{\prime}$. An easy
calculation shows that $T$ is preserved by

$\Sigma_{0}(p)=\begin{pmatrix}\mathbb{Z}_{p}^{\times}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&\mathbb{Z}_{p}\end{pmatrix}$

and $T^{\prime}$ is preserved by

$\Sigma^{\prime}_{0}(p)=\begin{pmatrix}\mathbb{Z}_{p}&\mathbb{Z}_{p}\\\
p\mathbb{Z}_{p}&\mathbb{Z}_{p}^{\times}\end{pmatrix}.$

In particular, they are both preserved by scalar multiplication by
$\mathbb{Z}_{p}^{\times}$ and the subgroup
$\Gamma_{0}(p\mathbb{Z}_{p}):=\Sigma_{0}(p)\cap\Sigma^{\prime}_{0}(p)$.

######  Definition 4.4.

For any $m\geq 0$, we let $A_{\mathcal{V},m}(T)$ be the space of functions
$f:T\rightarrow\Lambda_{\mathcal{V}}$ such that

$f(\gamma\cdot t)=\kappa_{\mathcal{V}}(\gamma)f(t)$

for all $\gamma\in\mathbb{Z}_{p}^{\times}$ and $f(1,z)\in
LA_{m}(\mathbb{Z}_{p},\Lambda_{\mathcal{V}})$. This space is a
$\Lambda_{\mathcal{V}}$-module and we endow it with the topology induced by
$m_{\mathcal{V}}$. Similarly, one defines $A_{\mathcal{V},m}(T^{\prime})$ to
be the space of functions $f:T^{\prime}\rightarrow\Lambda_{\mathcal{V}}$ with
$f(\gamma\cdot t)=\kappa_{\mathcal{V}}(\gamma)f(t)$ for all
$\gamma\in\mathbb{Z}_{p}^{\times}$ and $f(pz,1)\in
LA_{m}(\mathbb{Z}_{p},\Lambda_{\mathcal{V}})$.

######  Definition 4.5.

We define

$D_{\mathcal{V},m}(T):=\operatorname{\operatorname{Hom}}_{\Lambda_{\mathcal{V}}}(A_{\mathcal{V},m}(T),\Lambda_{\mathcal{V}})$

and $D_{\mathcal{V},m}(T^{\prime})$ in a similar manner. We endow
$D_{\mathcal{V},m}(T)$ with the weak-$\ast$ topology, which is the weakest
topology such that the evaluation-at-$f$ maps are continuous for all $f\in
A_{\mathcal{V},m}(T)$.

######  Remark 4.6.

As we will frequently give statements that work both for $T$ and T′, we will
use the convention $T^{\cdot}$ as in [BSV22] to denote either $T$ or
$T^{\prime}$.

One can also define the small versions $A_{k,m}(T^{\cdot})$ and
$D_{k,m}(T^{\cdot})$, where the pair
$(\Lambda_{\mathcal{V}},\kappa_{\mathcal{V}})$ is replaced by
$(\mathcal{O}_{L},k)$ with $k\in\mathcal{V}\cap\mathbb{Z}$. Then one has
specialization maps (c.f. [BSV22, § 4.1.3])

$A_{\mathcal{V},m}(T^{\cdot})\xrightarrow{\rho_{k}}A_{k,m}(T^{\cdot})\textrm{
and }D_{\mathcal{V},m}(T^{\cdot})\xrightarrow{\rho_{k}}D_{k,m}(T^{\cdot})$

induced by quotient by the maximal ideal $\mathfrak{m}_{k}$ corresponding to
the point $k$.

######  Remark 4.7 ([BSV22, § 4]).

The objects $A_{\mathcal{V},m}(T^{\cdot})$ and $D_{\mathcal{V},m}(T^{\cdot})$
as well as the small versions have natural actions of
$\Gamma=\Gamma_{1}(N,p)$. In particular, there are notions of Hecke operators
$T_{\ell}^{\cdot},U^{\cdot}=U_{p}^{\cdot}$ and Atkin–Lehner involution $w_{p}$
on $H^{1}(\Gamma,A_{\mathcal{V},m}(T^{\cdot}))$ and
$H^{1}(\Gamma,D_{\mathcal{V},m}(T^{\cdot}))$.

Later we will introduce big étale sheaves whose construction is based on
locally analytic functions and distributions. For this purpose, we need the
following lemma.

######  Lemma 4.8 ([LZ16, Lemma 4.2.7] ).

We may write $D_{\mathcal{V},m}(T^{\cdot})$ as an inverse limit

$D_{\mathcal{V},m}(T^{\cdot})=\varprojlim_{n\geq
0}D_{\mathcal{V},m}(T^{\cdot})/\operatorname{\operatorname{Fil}}^{n}D_{\mathcal{V},m}(T^{\cdot})$

where $\operatorname{\operatorname{Fil}}^{n}D_{\mathcal{V},m}(T^{\cdot})$ is a
decreasing filtration preserved by $\Sigma^{\cdot}_{0}(p)$ and each quotient
is finite of $p$-power order.

#### Big étale sheaves.

Let $Y=Y_{1}(N,p)$ be the affine modular curve defined over
$\mathbb{Z}[1/Np]$. Fix a geometric point
$\eta:\operatorname{\operatorname{Spec}}(\overline{\mathbb{Q}})\rightarrow Y$
and denote by $\mathcal{G}$ the étale fundamental group
$\pi_{1}^{\mathrm{\acute{e}t}}(Y,\eta)$, which admits a natural map
$\mathcal{G}\rightarrow\Gamma_{0}(p\mathbb{Z}_{p})$.

Let $S_{f}(Y_{\mathrm{\acute{e}t}})$ be the category of locally constant
constructible sheaves on $Y_{\acute{e}t}$ with finite stalk of $p$-power order
at $\eta$. For any topological group $G$, we denote by $M_{f}(G)$ the category
of finite sets of $p$-power order with continuous $G$-actions.

Following [BSV22, § 4.2], one sees that taking stalks at $\eta$ defines an
equivalence of categories between $S_{f}(Y_{\mathrm{\acute{e}t}})$ and
$M_{f}(\mathcal{G})$. We define $M_{\mathrm{cts}}(G)$ be the category of
$G$-modules $M$ of the form $M=\cup_{i\in I}M_{i}$ with $M_{i}\in M_{f}(G)$
for all $i\in I$, and $M(G)$ be the category of inverse systems of objects in
$M_{\mathrm{cts}}(G)$. We also define similarly
$S_{\mathrm{cts}}(Y_{\mathrm{\acute{e}t}})$ and $S(Y_{\mathrm{\acute{e}t}})$.
If $G$ is either $\mathcal{G}$ or $\Gamma_{0}(p\mathbb{Z}_{p})$, one has a
functor

$\cdot^{\mathrm{\acute{e}t}}:M(G)\rightarrow S(Y_{\mathrm{\acute{e}t}})$

extending the previous equivalence. For any
$\mathscr{F}=(\mathscr{F}_{i})_{i}\in S(Y_{\mathrm{\acute{e}t}})$, we will
mainly consider the following cohomology group

$H^{i}_{\mathrm{\acute{e}t}}(Y,\mathscr{F})=\varprojlim_{i}H^{i}_{\mathrm{\acute{e}t}}(Y,\mathscr{F}_{i}).$

We now apply the above construction to the modules, viewed as element in
$M(\Gamma_{0}(p\mathbb{Z}_{p}))$,

$\displaystyle A_{\mathcal{V},m}(T^{\cdot})$
$\displaystyle=\varprojlim_{n}A_{\mathcal{V},m}(T^{\cdot})/m_{\mathcal{V}}^{n},$
$\displaystyle D_{\mathcal{V},m}(T^{\cdot})$
$\displaystyle=\varprojlim_{n}D_{\mathcal{V},m}(T^{\cdot})/\operatorname{\operatorname{Fil}}^{n}.$

As a consequence, we have objects
$A_{\mathcal{V},m}(T^{\cdot})^{\operatorname{\mathrm{\acute{e}t}}}$ and
$D_{\mathcal{V},m}(T^{\cdot})^{\operatorname{\mathrm{\acute{e}t}}}$ in
$S(Y_{\operatorname{\mathrm{\acute{e}t}}})$. We will often drop the
superscript $\operatorname{\mathrm{\acute{e}t}}$ when no confusion might be
made.

######  Remark 4.9.

We conclude this subsection with a list of facts about these big étale sheaves
(c.f. [BSV22, § 4.2]).

  1. 1.

One has isomorphisms

$\displaystyle
H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},A_{\mathcal{V},m}(T^{\cdot})^{\operatorname{\mathrm{\acute{e}t}}})$
$\displaystyle\cong H^{1}(\Gamma,A_{\mathcal{V},m}(T^{\cdot})),$
$\displaystyle
H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},D_{\mathcal{V},m}(T^{\cdot})^{\operatorname{\mathrm{\acute{e}t}}})$
$\displaystyle\cong H^{1}(\Gamma,D_{\mathcal{V},m}(T^{\cdot})),$
$\displaystyle
H^{1}_{\operatorname{\mathrm{\acute{e}t}},c}(Y_{\bar{\mathbb{Q}}},D_{\mathcal{V},m}(T^{\cdot})^{\operatorname{\mathrm{\acute{e}t}}})$
$\displaystyle\cong H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\cdot})).$

where we refer the definition of
$H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\cdot}))$ to [BSV22, (77)]. We remark
that to prove these isomorphisms, one needs the fact that
$A_{\mathcal{V},m}(T^{\cdot})$ and $D_{\mathcal{V},m}(T^{\cdot})$ are
profinite.

  2. 2.

For any continuous character
$\chi:\mathbb{Z}_{p}^{\times}\rightarrow\Lambda_{\mathcal{V}}^{\times}$, one
can define the twisted modules $D_{\mathcal{V},m}(T^{\cdot})(\chi)$ and
$A_{\mathcal{V},m}(T^{\cdot})(\chi)$. Their resulting étale sheaves will be
denoted by
$D_{\mathcal{V},m}(T^{\cdot})(\chi)^{\operatorname{\mathrm{\acute{e}t}}}$ and
$A_{\mathcal{V},m}(T^{\cdot})(\chi)^{\operatorname{\mathrm{\acute{e}t}}}$.
Then one has the following identifications of $G_{\mathbb{Q}}$-modules

$\displaystyle
H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},D_{\mathcal{V},m}(T^{\cdot})(\chi)^{\operatorname{\mathrm{\acute{e}t}}})$
$\displaystyle=H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},D_{\mathcal{V},m}(T^{\cdot})^{\operatorname{\mathrm{\acute{e}t}}})(\chi_{\mathbb{Q}}^{\prime}),$
$\displaystyle
H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},A_{\mathcal{V},m}(T^{\cdot})(\chi)^{\operatorname{\mathrm{\acute{e}t}}})$
$\displaystyle=H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},A_{\mathcal{V},m}(T^{\cdot})^{\operatorname{\mathrm{\acute{e}t}}})(\chi_{\mathbb{Q}}^{\prime})$

where
$\chi_{\mathbb{Q}}^{\prime}:=\chi\circ\chi_{\mathrm{cyc}}^{-1}:G_{\mathbb{Q}}\rightarrow\Lambda_{\mathcal{V}}^{\times}$.
Similar statements also hold for the compactly supported versions.

  3. 3.

The Hecke operator $U^{\cdot}$ on the $G_{\mathbb{Q}}$-modules

$H^{1}(\Gamma,A_{\mathcal{V},m}(T^{\cdot})),H^{1}(\Gamma,D_{\mathcal{V},m}(T^{\cdot})),H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\cdot}))$

admits slope $\leq h$ decompositions for all $h\in\mathbb{Q}_{\geq 0}$ (with
possibly shrinking $\mathcal{V}$).

  4. 4.

The evaluation map
$A_{\mathcal{V},m}(T^{\cdot})\otimes_{\Lambda_{\mathcal{V}}}D_{\mathcal{V},m}(T^{\cdot})\rightarrow\Lambda_{\mathcal{V}}$
and the trace map induce a cup-product

$H^{1}(\Gamma,A_{\mathcal{V},m}(T^{\cdot}))\otimes
H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\cdot}))\rightarrow
H^{2}_{c}(\Gamma,\Lambda_{\mathcal{V}})\cong\Lambda_{\mathcal{V}},$

under which the action of $U^{\cdot}$ on
$H^{1}(\Gamma,A_{\mathcal{V},m}(T^{\cdot}))$ is adjoint to the one on
$H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\cdot}))$. For $h\in\mathbb{Q}_{\geq
0}$ and possibly shrinking $\mathcal{V}$, we have a morphism of
$\mathscr{O}_{\mathcal{V}}=\Lambda_{\mathcal{V}}[1/p]$-modules

$\xi^{\cdot}_{\mathcal{V},m}:H^{1}(\Gamma,A_{\mathcal{V},m}(T^{\cdot}))^{\leq
h}\rightarrow\operatorname{\operatorname{Hom}}_{\mathscr{O}_{\mathcal{V}}}(H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\cdot}))^{\leq
h},\mathscr{O}_{\mathcal{V}}).$

  5. 5.

On the other hand, the determinant map $\det:T\times
T^{\prime}\rightarrow\mathbb{Z}_{p}^{\times}$ induces a cup-product pairing

$\det:H^{1}(\Gamma,D_{\mathcal{V},m}(T))\otimes_{\Lambda_{\mathcal{V}}}H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\prime}))\rightarrow\Lambda_{\mathcal{V}}$

with $U$ and $U^{\prime}$ adjoint to each other. In particular, for every
$h\in\mathbb{Q}_{\geq 0}$, we have isomorphisms

$\zeta^{\prime}_{\mathcal{V},m}:\operatorname{\operatorname{Hom}}_{\mathscr{O}_{\mathcal{V}}}(H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T^{\prime})^{\leq
h},\mathscr{O}_{\mathcal{V}})\cong
H^{1}(\Gamma,H^{1}(\Gamma,D_{\mathcal{V},m})(T))^{\leq h}$

and similarly

$\zeta_{\mathcal{V},m}:\operatorname{\operatorname{Hom}}_{\mathscr{O}_{\mathcal{V}}}(H^{1}_{c}(\Gamma,D_{\mathcal{V},m}(T)^{\leq
h},\mathscr{O}_{\mathcal{V}})\cong
H^{1}(\Gamma,H^{1}(\Gamma,D_{\mathcal{V},m})(T^{\prime}))^{\leq h}.$

  6. 6.

For $h\in\mathbb{Q}_{\geq 0}$ and $\mathcal{V}$ sufficiently small, the
composition of $\zeta_{\mathcal{V},m}$ and $\xi_{\mathcal{V},m}$ gives a
morphism of $G_{\mathbb{Q}}$-modules

$s_{\mathcal{V},h}:H^{1}(\Gamma,A_{\mathcal{V},m}(T))^{\leq
h}(\kappa_{\mathcal{V}})\rightarrow
H^{1}(\Gamma,D_{\mathcal{V},m}(T^{\prime}))^{\leq h},$ (5)

where by an abuse of notation we let
$\kappa_{\mathcal{V}}:G_{\mathbb{Q}}\rightarrow\Lambda_{\mathcal{V}}^{\times}$
be the composition of $\chi_{\mathrm{cyc}}$ and the universal character
$\kappa_{\mathcal{V}}$. Moreover, for every integer
$k=r+2\in\mathcal{V}\cap\mathbb{Z}$ with $h<k-1$, the specialization maps
induce the following commutative diagram

${H^{1}(\Gamma,A_{\mathcal{V},m}(T))^{\leq
h}(\kappa_{\mathcal{V}})}$${H^{1}(\Gamma,D_{\mathcal{V},m}(T^{\prime}))^{\leq
h}}$${H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},\mathscr{S}_{r})_{L}^{\leq^{\prime}h}(r)}$${H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},\mathscr{L}_{r})_{L}^{\leq^{\prime}h}}$$\scriptstyle{\rho_{k}}$$\scriptstyle{s_{\mathcal{V},h}}$$\scriptstyle{\rho_{k}}$$\scriptstyle{s_{r}}$

where $\leq^{\prime}h$ means the slope $\leq h$ part for the operator
$U^{\prime}$. The sheaves $\mathscr{S}_{r}$ and $\mathscr{L}_{r}$ are the
small étale sheaves defined in [BSV22, § 2.3] and $s_{r}$ is the natural
duality between the two cohomology groups. Roughly speaking, the spaces
$H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},\mathscr{S}_{r})_{L}$
and
$H^{1}_{\operatorname{\mathrm{\acute{e}t}}}(Y_{\bar{\mathbb{Q}}},\mathscr{L}_{r})_{L}$
are where the small Galois representations attached to modular forms of weight
$k$ (with coefficients in $L$) live in. Similarly, one has an isomorphism

$s^{\prime}_{\mathcal{V},h}:H^{1}(\Gamma,A_{\mathcal{V},m}(T^{\prime}))^{\leq
h}(\kappa_{\mathcal{V}})\rightarrow H^{1}(\Gamma,D_{\mathcal{V},m}(T))^{\leq
h}$ (6)

with an analogous commutative diagram.

  7. 7.

Suppose that $\mathbf{f}$ is a Coleman family of slope $\leq a$ defined over
an open disk $\mathcal{V}_{f}$. For simplicity, we will write
$A_{f,m}(T^{\cdot})$ for $A_{\mathcal{V}_{f},m}(T^{\cdot})$ and
$D_{f,m}(T^{\cdot})$ for $D_{\mathcal{V}_{f},m}(T^{\cdot})$. Then, for some
$m$ and possibly shrinking $\mathcal{V}_{f}$, one defines the big Galois
representation attached to $\mathbf{f}$,

$H^{1}(\Gamma,D_{f,m}(T^{\prime}))^{\leq h}(1)\twoheadrightarrow
V(\mathbf{f})$

to be the maximal $\mathscr{O}_{f}$-quotient on which $T^{\prime}_{\ell}$,
$U^{\prime}$ and $\langle d\rangle^{*}$ acts respectively as multiplication by
$a_{\ell}(\mathbf{f})$, $a_{p}(\mathbf{f})$ and $\chi_{f}(d)$ for all
$\ell\nmid NP$ and $d\in(\mathbb{Z}/N_{f}\mathbb{Z})^{\times}$. Similarly, one
define

$V^{*}(\mathbf{f})\hookrightarrow H^{1}_{c}(\Gamma,D_{f,m}(T))^{\leq
h}(-\kappa_{f})$

to be the maximal $\mathscr{O}_{f}$-submodule by using the Hecke operators
$T_{\ell}$, $U$ and $\langle d\rangle$ instead.

These two $\mathscr{O}_{f}$-modules are free and in fact are direct summands
of $H^{1}(\Gamma,D_{f,m}(T^{\prime}))^{\leq h}(1)$ and
$H^{1}_{c}(\Gamma,D_{f,m}(T))^{\leq h}(-\kappa_{f})$ respectively. In
particular, there are natural maps

$\operatorname{pr}_{f}:H^{1}(\Gamma,D_{f,m}(T^{\prime}))^{\leq
h}(1)\twoheadrightarrow V(\mathbf{f})\ \textrm{ and
}\operatorname{pr}_{f}^{*}:H^{1}_{c}(\Gamma,D_{f,m}(T))^{\leq
h}(-\kappa_{f})\twoheadrightarrow V(\mathbf{f})^{*}.$

If $\mathbf{f}$ is a Hida family, then there is a natural short exact sequence
of $\mathscr{O}_{f}[G_{\mathbb{Q}_{p}}]$-modules (c.f. [BSV22, (99)])

$0\rightarrow V(\mathbf{f})^{+}\rightarrow V(\mathbf{f})\rightarrow
V(\mathbf{f})^{-}\rightarrow 0$

with explicit descriptions for $V(\mathbf{f})^{\pm}$. However, in the finite
slope case, this exact sequence does not exist, at least not in the category
of Galois representations. To obtain an analogous result, one needs to
consider the category of $(\operatorname{\varphi,\Gamma})$-modules, which is
the main subject in next subsection.

###  4.2 On $(\varphi,\Gamma)$-modules and triangulations

This subsection aims to recall several facts about $(\varphi,\Gamma)$-modules
and develop necessary setups for defining the big logarithm maps for finite
slope families.

A brief reference on $(\varphi,\Gamma)$-modules and explicit computations of
the logarithm maps can be found in [Ber03]. For the definitions of various
Selmer groups for $(\varphi,\Gamma)$-modules, we follow the notations in
[Ben11, § 1]. Since we are interested in the case of modular forms, we will
also recall several useful results in [LZ16, § 6].

Let $\mathscr{R}$ be the Robba ring of $\mathbb{Q}_{p}$. It is the ring
consists of formal Laurent series over $\mathbb{Q}_{p}$ in one variable $\pi$
which converges in some annulus of the form
$\\{p^{-\epsilon}<|z|_{p}<1\\}\in\mathbb{A}^{1,\operatorname{rig}}$ for some
$\epsilon\in\mathbb{Q}$. There are an action of Frobenius $\varphi$ and an
action of
$\Gamma:=\operatorname{\operatorname{Gal}}(\mathbb{Q}_{p}(\mu_{p^{\infty}})/\mathbb{Q}_{p})\cong\mathbb{Z}_{p}^{\times}$
on $\mathscr{R}$, defined by

$\displaystyle\tau(\pi)$
$\displaystyle=(1+\pi)^{\chi_{\mathrm{cyc}}(\tau)}-1,\quad\tau\in\Gamma,$
$\displaystyle\varphi(\pi)$ $\displaystyle=(1+\pi)^{p}-1,$

where $\chi_{\mathrm{cyc}}$ is the cyclotomic character. There is also a left
inverse $\psi$ of $\varphi$, whose definition we would like to refer to
[Ber03, § I.2].

######  Definition 4.10.

By $p$-adic representation of $G_{\mathbb{Q}_{p}}$, we always means a finite
dimensional $\mathbb{Q}_{p}$-vector space with continuous linear action of
$G_{\mathbb{Q}_{p}}$. A $(\varphi,\Gamma)$-module $M$ over $\mathscr{R}$ is a
free $\mathscr{R}$-module of finite rank with $\mathscr{R}$-semilinear actions
of $\varphi$ and $\Gamma$.

Given a reduced affinoid algebra $A$ over $\mathbb{Q}_{p}$, let
$\mathscr{R}_{A}:=\mathscr{R}\hat{\otimes}_{\mathbb{Q}_{p}}A$. By an
$A$-representation of $G_{\mathbb{Q}_{p}}$, we means a finitely generated
locally free $A$-module $M$ with a continuous $A$-linear action of
$G_{\mathbb{Q}_{p}}$. One can also define $(\varphi,\Gamma)$-modules over
$\mathscr{R}_{A}$ similarly.

As is well known, there is a functor
$\mathbf{D}^{\dagger}_{\operatorname{rig}}$ from the category of $p$-adic
representations of $G_{\mathbb{Q}_{p}}$ to the category of
$(\varphi,\Gamma)$-modules over $\mathscr{R}$. A similar functor also exists
from $A$-representations to $(\varphi,\Gamma)$-modules over $\mathscr{R}_{A}$,
which will still be denoted by $\mathbf{D}^{\dagger}_{\operatorname{rig}}$
(c.f. [Nak14]).

######  Proposition 4.11 ([Ben11, § 1]).

Let notations be as before, then we have the following results.

  1. 1.

The functor $\mathbf{D}^{\dagger}_{\operatorname{rig}}$ establishes an
equivalence between the category of $p$-adic representations of
$G_{\mathbb{Q}_{p}}$ and the category of $(\varphi,\Gamma)$-modules over
$\mathscr{R}$ of slope $0$.

  2. 2.

Let $V$ be a $p$-adic representation and $D_{\bullet}$ be the functors of
Fontaine for
$\bullet\in\\{\operatorname{\operatorname{dR}},\operatorname{\operatorname{cris}},\mathrm{st}\\}$.
Then there are functors $\mathcal{D}_{\bullet}$ on the category of
$(\varphi,\Gamma)$-modules over $\mathscr{R}$ such that

$D_{\bullet}(V)=\mathcal{D}_{\bullet}(\mathbf{D}^{\dagger}_{\operatorname{rig}}(V))$

for
$\bullet\in\\{\operatorname{\operatorname{dR}},\operatorname{\operatorname{cris}},\mathrm{st}\\}$.
In particular, one can define the notations of de Rham, crystalline, and
semistable $(\varphi,\Gamma)$-modules over $\mathscr{R}$.

######  Definition 4.12.

For any continuous character
$\delta:\mathbb{Q}_{p}^{\times}\rightarrow\mathscr{R}_{A}^{\times}$, we let
$\mathscr{R}_{A}\\{\delta\\}$ be the module $\mathscr{R}_{A}\cdot e_{\delta}$,
with $\varphi(e_{\delta})=\delta(p)e_{\delta}$ and
$\gamma(e_{\delta})=\delta(\gamma)e_{\delta}$ for $\gamma\in\Gamma$. The
object $\mathscr{R}_{A}\\{\delta\\}$ serves as an important example for
$(\varphi,\Gamma)$-modules of rank $1$ over $\mathscr{R}_{A}$. If we want to
separate the actions of $\varphi$ and $\Gamma$, we will use the notation
$\mathscr{R}_{A}\\{\delta\\}=\mathscr{R}_{A}(a)[b]$ where
$a:=\delta(p)\in\mathscr{R}_{A}$ and
$b:=\delta|_{\Gamma}:\Gamma\rightarrow\mathscr{R}_{A}$.

######  Definition 4.13.

Fix a generator $\gamma$ of $\Gamma$. For a $(\varphi,\Gamma)$-module $M$ over
$\mathscr{R}_{A}$, we define the cohomology groups $H^{i}(\mathbb{Q}_{p},M)$
to be the cohomology of the following complex

$M\xrightarrow[]{(\varphi-1,\gamma-1)}M\oplus
M\xrightarrow[]{(\gamma-1,-(\varphi-1))}M.$

######  Proposition 4.14 ([Ber02, Theorem 0.2] and [KPX14, Proposition
2.3.7]).

Let $V$ be either a $p$-adic representation or an $A$-representation. Then we
have

$H^{i}(\mathbb{Q}_{p},\mathbf{D}^{\dagger}_{\operatorname{rig}}(V))=H^{i}(\mathbb{Q}_{p},V)$

for all $i$, where $H^{i}(\mathbb{Q}_{p},V)$ is the continuous cohomology.

######  Theorem 4.15 ([Nak14]).

If $A$ is a finite extension of $\mathbb{Q}_{p}$, and $M$ is a de Rham
$(\varphi,\Gamma)$-module over $\mathscr{R}_{A}$. Then there are Bloch–Kato
exponential and dual-exponential maps

$\exp:\mathcal{D}_{\operatorname{\operatorname{dR}}}(M)\rightarrow
H^{1}(\mathbb{Q}_{p},M),\quad\exp^{*}:H^{1}(\mathbb{Q}_{p},M)\rightarrow\mathcal{D}_{\operatorname{\operatorname{dR}}}(M),$

which are compatible with the usual definitions when
$M=\operatorname{\mathbf{D}^{\dagger}_{\operatorname{rig}}}(V)$ for a de Rham
representation $V$.

Given a $(\varphi,\Gamma)$-module $M$, similarly to the case of Galois
representations, one may interpret $H^{1}(\mathbb{Q}_{p},M)$ as isomorphism
classes of extensions of $\mathscr{R}$ by $M$. By using the functor
$\mathcal{D}_{\operatorname{\operatorname{cris}}}$, one can define
$H^{1}_{f}(\mathbb{Q}_{p},M)$ to be the subset of crystalline extensions (c.f.
[Ben11, § 1.4.1]). Its relation with the finite Bloch–Kato Selmer groups of
$p$-adic representations is illustrated in the following proposition.

######  Proposition 4.16 ([Ben11, Proposition 1.4.2]).

Suppose $V$ is a $p$-adic representation. Then

$H^{1}_{f}(\mathbb{Q}_{p},V)\cong
H^{1}_{f}(\mathbb{Q}_{p},\mathbf{D}^{\dagger}_{\operatorname{rig}}(V)).$

######  Remark 4.17.

In fact, one can also define $H^{1}_{g}$ and $H^{1}_{e}$ (c.f. [Nak09,
Definition 2.4]). They also agree with the usual Selmer groups $H^{1}_{g}$ and
$H^{1}_{e}$ for $p$-adic representations.

Let $\mathbf{f}$ be a primitive Coleman family of finite slope $a$, defined
over a wide open disk $\mathcal{V}_{f}\subset\mathcal{W}$, with nebentypus
character $\chi_{f}$ as in §4.1.

######  Theorem 4.18 ([LZ16, Theorem 6.3.2], [Liu15] ).

Let $V(\mathbf{f})$ be the big Galois representation of $G_{\mathbb{Q}_{p}}$
over $\mathscr{O}_{f}$ associated with $\mathbf{f}$. Set
$M(\mathbf{f}):=\mathbf{D}^{\dagger}_{\operatorname{rig}}(V(\mathbf{f}))$ and
$\mathscr{R}_{f}:=\mathscr{R}_{\mathscr{O}_{f}}$. Then one has a triangulation
of $M(\mathbf{f})$

$0\rightarrow F^{+}M(\mathbf{f})\rightarrow M(\mathbf{f})\rightarrow
F^{-}M(\mathbf{f})\rightarrow 0,$

which satisfies the following properties:

  1. 1.

$F^{-}M(\mathbf{f})\cong\mathscr{R}_{f}(a_{p}(\mathbf{f}))$, i.e., $\varphi$
acts as multiplication by $a_{p}(\mathbf{f})$;

  2. 2.

$F^{+}M(\mathbf{f})\cong\mathscr{R}_{f}(\chi_{f}(p)-a_{p}(\mathbf{f}))[1+\kappa_{f}]$.

A similar result hold for the dual module
$M^{*}(\mathbf{f}):=\mathbf{D}^{\dagger}_{\operatorname{rig}}(V^{*}(\mathbf{f}))$.
Notice that we follow the additive notation of characters as in [BSV22].

###  4.3 The Balanced Selmer group

Let $\mathbf{f},\mathbf{g},\mathbf{h}$ be three Coleman families with tame
characters satisfying the relation $\chi_{f}\cdot\chi_{g}\cdot\chi_{h}=1$, and
let $a_{f},a_{g},a_{h}$ be their respective slopes. Suppose
$\mathcal{V}_{f},\mathcal{V}_{g},\mathcal{V}_{h}$ are defined over some finite
extension $L$ of $\mathbb{Q}_{p}$. We fix centers $k_{0},\ell_{0},m_{0}$ and
uniformizers $\mathbf{k}-k_{0},\bm{\ell}-\ell_{0},\mathbf{m}-m_{0}$ for
$\mathcal{V}_{f},\mathcal{V}_{g},\mathcal{V}_{h}$ respectively. We set
$\mathscr{O}_{fgh}:=\mathscr{O}_{f}\hat{\otimes}_{L}\mathscr{O}_{g}\hat{\otimes}_{L}\mathscr{O}_{h}$
and $\mathscr{R}_{fgh}:=\mathscr{R}_{\mathscr{O}_{fgh}}$.

Consider the character
$\Xi:G_{\mathbb{Q}_{p}}\rightarrow\mathscr{O}_{fgh}^{\times}$ defined by

$\displaystyle\Xi(g):$
$\displaystyle=(\omega^{(4-k_{0}-l_{0}-m_{0})/2}\cdot\langle\phantom{e}\rangle^{(4-\mathbf{k}-\bm{\ell}-\mathbf{m})/2})\circ\chi_{\mathrm{cyc}}(g)$
$\displaystyle=\frac{1}{2}(-\kappa_{f}-\kappa_{g}-\kappa_{h}-2)\circ\chi_{\mathrm{cyc}}(g).$

We set

$V(\mathbf{f},\mathbf{g},\mathbf{h}):=V(\mathbf{f})\hat{\otimes}_{L}V(\mathbf{g})\hat{\otimes}_{L}V(\mathbf{h})\otimes_{\mathscr{O}_{fgh}}\Xi,$

which is Kummer self-dual, and set
$M(\mathbf{f},\mathbf{g},\mathbf{h}):=\operatorname{\mathbf{D}^{\dagger}_{\operatorname{rig}}}(V(\mathbf{f},\mathbf{g},\mathbf{h}))$.
One defines a filtration on $M(\mathbf{f})$ by
$F^{0}M(\mathbf{f})=M(\mathbf{f})$, $F^{1}M(\mathbf{f})=F^{+}M(\mathbf{f})$,
$F^{2}M(\mathbf{f})=0$, and similarly for $M(\mathbf{g}),M(\mathbf{h})$. These
filtrations induce a filtration on $M(\mathbf{f},\mathbf{g},\mathbf{h})$,
defined by

$F^{n}M(\mathbf{f},\mathbf{g},\mathbf{h}):=\left[\sum_{p+q+r=n}F^{p}M(\mathbf{f})\hat{\otimes}_{L}F^{q}M(\mathbf{g})\hat{\otimes}_{L}F^{r}M(\mathbf{h})\right]\\{\Xi\\}$

where by $\\{\Xi\\}$ we mean the image of $\Xi$ under the functor
$\operatorname{\mathbf{D}^{\dagger}_{\operatorname{rig}}}$ (c.f. [Ben11, §
1.5]).

######  Definition 4.19.

Analogous to [BSV22, § 7.2], we define the balanced Selmer group to be

$H^{1}_{\mathrm{bal}}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h})):=H^{1}(\mathbb{Q}_{p},F^{2}M(\mathbf{f},\mathbf{g},\mathbf{h}))\subset
H^{1}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h})).$ (7)

######  Remark 4.20.

We remark that by examining the $(\varphi,\Gamma)$-module structure of
$\textrm{gr}^{0}M(\mathbf{f},\mathbf{g},\mathbf{h})$, the natural map
$F^{2}M(\mathbf{f},\mathbf{g},\mathbf{h})\hookrightarrow
M(\mathbf{f},\mathbf{g},\mathbf{h})$ induces an injection on $H^{1}$ (c.f.
[BSV22, (148)]).

One important property of this balanced Selmer group is the following lemma.

######  Lemma 4.21 (c.f. [BSV22, Lamma 7.2]).

If $(x,y,z)\in\Sigma_{\mathrm{bal}}$ is a balanced triple of classical
weights, then

$H^{1}_{\mathrm{bal}}(\mathbb{Q}_{p},M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))=H^{1}_{f}(\mathbb{Q}_{p},M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))=H^{1}_{f}(\mathbb{Q}_{p},V(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))$
(8)

where the last one is the Bloch–Kato finite Selmer group. As a consequence,
one has the Bloch–Kato exponential map

$\exp_{p}:D_{\operatorname{\operatorname{dR}}}(V(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))/\operatorname{\operatorname{Fil}}^{0}\cong
H^{1}_{\mathrm{bal}}(\mathbb{Q}_{p},M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))$

and we denote its inverse by $\log_{p}$.

###### Proof.

The proof is essentially identical to that of [BSV22, Lamma 7.2], which boils
down to compute the Hodge–Tate weights, or the actions of $\Gamma$ in the case
of $(\varphi,\Gamma)$-modules. In particular, one has

$\operatorname{\operatorname{Fil}}^{0}\mathcal{D}_{\operatorname{\operatorname{dR}}}(F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))=0\textrm{
and
}\operatorname{\operatorname{Fil}}^{0}\mathcal{D}_{\operatorname{\operatorname{dR}}}(M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})/F^{2})=\mathcal{D}_{\operatorname{\operatorname{dR}}}(M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})).$

The second identity implies that
$H^{1}_{e}(\mathbb{Q}_{p},M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})/F^{2})=0$.
Notice that $F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})$ is Kummer
dual to $M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})/F^{2}$. So one has

$H^{1}(\mathbb{Q}_{p},F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))=H^{1}_{g}(\mathbb{Q}_{p},F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})).$

By [BSV22, Lemma 3.5],
$H^{1}_{g}(\mathbb{Q}_{p},F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))=H^{1}_{f}(\mathbb{Q}_{p},F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))$.
One then deduces that
$H^{1}_{\operatorname{\operatorname{bal}}}(\mathbb{Q}_{p},F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))\subset
H^{1}_{f}(\mathbb{Q}_{p},F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))$.
For the other direction, it follows similarly by using the fact that
$\operatorname{\operatorname{Fil}}^{0}\mathcal{D}_{\operatorname{\operatorname{dR}}}(F^{2}M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))=0$.
∎

If we set

$M(\mathbf{f},\mathbf{g},\mathbf{h})_{f}:=F^{-}M(\mathbf{f})\hat{\otimes}F^{+}M(\mathbf{g})\hat{\otimes}F^{+}M(\mathbf{h})\\{\Xi\\}$

and similarly for $M(\mathbf{f},\mathbf{g},\mathbf{h})_{g}$ and
$M(\mathbf{f},\mathbf{g},\mathbf{h})_{h}$, then the graded piece $F^{2}/F^{3}$
decomposes as

$F^{2}/F^{3}=M(\mathbf{f},\mathbf{g},\mathbf{h})_{f}\oplus
M(\mathbf{f},\mathbf{g},\mathbf{h})_{g}\oplus
M(\mathbf{f},\mathbf{g},\mathbf{h})_{h}.$ (9)

As a result, there are natural projections
$F^{2}M(\mathbf{f},\mathbf{g},\mathbf{h})\rightarrow
M(\mathbf{f},\mathbf{g},\mathbf{h})_{\xi}$ for $\xi\in\\{f,g,h\\}$.

###  4.4 The three-variable big logarithm

Write
$V_{\operatorname{\operatorname{dR}}}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}):=D_{\operatorname{\operatorname{dR}}}(V(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))$
and similarly for its Kummer dual
$V_{\operatorname{\operatorname{dR}}}^{*}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})$.

If $w=(x,y,z)\in\Sigma_{\mathrm{bal}}$, one can construct an element
$\eta^{a}_{\mathbf{f}_{x}}\otimes\omega_{\mathbf{g}_{y}}\otimes\omega_{\mathbf{h}_{z}}\in\operatorname{\operatorname{Fil}}^{0}V_{\operatorname{\operatorname{dR}}}^{*}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})$
similarly to the construction in §3.2 (c.f. [BSV22, § 3.1]). The Bloch–Kato
logarithm

$\log_{p}:H^{1}_{\mathrm{bal}}(\mathbb{Q}_{p},M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))\xrightarrow{\sim}D_{\operatorname{\operatorname{dR}}}(V(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))\cong[\operatorname{\operatorname{Fil}}^{0}V_{\operatorname{\operatorname{dR}}}^{*}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})]^{\vee}$

then enables us to define
$\log_{p,f}:H^{1}_{\mathrm{bal}}(\mathbb{Q}_{p},M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))\rightarrow
L$ as the composition of $\log_{p}$ and evaluation at
$\eta^{a}_{\mathbf{f}_{x}}\otimes\omega_{\mathbf{g}_{y}}\otimes\omega_{\mathbf{h}_{z}}$.

On the other hand, if $w\in\Sigma_{f}$, one has the Bloch–Kato dual
exponential

$\exp_{p}^{*}:H^{1}(\mathbb{Q}_{p},M(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}))\rightarrow\operatorname{\operatorname{Fil}}^{0}V_{\operatorname{\operatorname{dR}}}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z}).$

We then let $\exp_{p,f}^{*}$ be the composition of $\exp_{p}^{*}$ with
evaluation at
$\eta^{a}_{\mathbf{f}_{x}}\otimes\omega_{\mathbf{g}_{y}}\otimes\omega_{\mathbf{h}_{z}}$.
Notice that here one needs to identify
$\operatorname{\operatorname{Fil}}^{0}V_{\operatorname{\operatorname{dR}}}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})$
as a subspace of
$[V^{*}_{\operatorname{\operatorname{dR}}}(\mathbf{f}_{x},\mathbf{g}_{y},\mathbf{h}_{z})]^{\vee}$
via Poincaré duality.

Before stating the proposition about the three variable big logarithm, we
would like to recall some notations. Let
$w=(x,y,z)\in\Sigma_{\operatorname{cl}}$, we write
$\alpha_{f_{x}}=a_{p}(\mathbf{f})_{x}$ and
$\beta_{f_{x}}=\chi_{f}(p)p^{x-1}/\alpha_{f_{x}}$ and similarly for $g$ and
$h$. We also set $c_{w}:=(x+y+z-2)/2$ to be the point of symmetry.

######  Proposition 4.22 ([BSV22, Proposition 7.3]).

There is a unique $\mathscr{O}_{fgh}$-module morphism

$\mathscr{L}_{f}=\mathscr{L}og_{f}(\mathbf{f},\mathbf{g},\mathbf{h}):H^{1}_{\operatorname{\operatorname{bal}}}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h}))\rightarrow\mathscr{O}_{fgh}$

such that, for all $w=(x,y,z)\in\Sigma_{\operatorname{cl}}$ with
$\alpha_{f_{x}}\beta_{g_{y}}\beta_{h_{z}}\neq p^{c_{w}}$ and $Z\in
H^{1}_{\mathrm{bal}}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h}))$,

$\mathscr{L}_{f}(Z)_{w}=(p-1)\alpha_{f_{x}}\cdot\frac{\left(1-\frac{\beta_{f_{x}}\alpha_{g_{y}}\alpha_{h_{z}}}{p^{c_{w}}}\right)}{\left(1-\frac{\alpha_{f_{x}}\beta_{g_{y}}\beta_{h_{z}}}{p^{c_{w}}}\right)}\cdot\begin{cases}\frac{(-1)^{c_{w}-x}}{(c_{w}-x)!}\log_{p,f}(Z_{w})&\textrm{
if }w\in\Sigma_{\operatorname{\operatorname{bal}}}\\\
(x-c_{w}-1)!\exp_{p,f}^{*}(Z_{w})&\textrm{ if }w\in\Sigma_{f}\end{cases},$
(10)

where the subscript $w$ means the specialization at $w$. Moreover,
$\mathscr{L}_{f}$ factors through the natural map

$H^{1}_{\operatorname{\operatorname{bal}}}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h}))\rightarrow
H^{1}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h})_{f}).$

Analogous results also hold for $g$ and $h$.

###### Proof.

The construction of the big logarithm is a consequence based on the work of
Coleman–Perrin-Riou [Col79, Per94], Ochiai [Och03], and Loeffler–Zerbes [LZ14,
Theorem 4.15]. Here we only explain the factor

$(p-1)\alpha_{f_{x}}\cdot{\left(1-\frac{\beta_{f_{x}}\alpha_{g_{y}}\alpha_{h_{z}}}{p^{c_{w}}}\right)}{\left(1-\frac{\alpha_{f_{x}}\beta_{g_{y}}\beta_{h_{z}}}{p^{c_{w}}}\right)}^{-1}.$

It suffices to understand the $(\varphi,\Gamma)$-module structure of
$M(\mathbf{f},\mathbf{g},\mathbf{h})_{f}$. To be more precise, one need to
compute the $\varphi$-eigenvalue of $M(\mathbf{f},\mathbf{g},\mathbf{h})_{f}$
when specialized to $w=(x,y,z)\in\Sigma_{\operatorname{\operatorname{bal}}}$.
According to [BSV22, § 7.1], one would expect this eigenvalue to be

$\frac{\chi_{g}(p)\chi_{h}(p)\alpha_{f_{x}}}{\alpha_{g_{y}}\alpha_{h_{z}}}=\frac{p^{k-1}}{\beta_{f_{x}}\alpha_{g_{y}}\alpha_{h_{z}}}$

up to a twist of $p^{j}$ for some $j=j(w)\in\mathbb{Z}$.

First, one sees from the description in Theorem 4.18 that the term

$\frac{\chi_{g}(p)\chi_{h}(p)\alpha_{f_{x}}}{\alpha_{g_{y}}\alpha_{h_{z}}}$

appears. One now calculates the Hodge–Tate weight of
$(M(\mathbf{f},\mathbf{g},\mathbf{h})_{f})_{w}$, which is
$j:=\frac{x-y-z}{2}$. Then by [Ben11, Proposition 1.5.8], the
$\varphi$-eigenvalue on $(M(\mathbf{f},\mathbf{g},\mathbf{h})_{f})_{w}$ is

$A:=\frac{\chi_{g}(p)\chi_{h}(p)\alpha_{f_{x}}}{\alpha_{g_{y}}\alpha_{h_{z}}}\cdot
p^{-j}=\frac{p^{k-1}}{\beta_{f_{x}}\alpha_{g_{y}}\alpha_{h_{z}}}\cdot p^{-j}.$
(11)

Notice that $k-1-j=c_{w}$. Hence one gets the Euler factor

$(1-A^{-1})(1-p^{-1}A)^{-1}={\left(1-\frac{\beta_{f_{x}}\alpha_{g_{y}}\alpha_{h_{z}}}{p^{c_{w}}}\right)}{\left(1-\frac{\alpha_{f_{x}}\beta_{g_{y}}\beta_{h_{z}}}{p^{c_{w}}}\right)}^{-1}.$

On the other hand, it follows from Eichler–Shimura relation in families (c.f.
[LZ16, Theorem 6.4.1 & Corollary 6.4.3]) that one can also $p$-adically
interpolate the class
$\eta^{a}_{\mathbf{f}_{x}},\omega_{\mathbf{g}_{y}},\omega_{\mathbf{h}_{z}}$.
Here we follows the normalization in [BSV22, (132)]. Namely, the big class
$\eta^{a}_{\mathbf{f}}$ is such that for $x\in\Sigma_{f}$,

$(\eta^{a}_{\mathbf{f}})_{x}=(p-1)\alpha_{f_{x}}\eta^{a}_{\mathbf{f}_{x}}.$

For $\omega_{\mathbf{g}}$ and $\omega_{\mathbf{h}}$, they interpolate
respectively $\omega_{\mathbf{g}_{y}},\omega_{\mathbf{h}_{z}}$ without extra
multiples. This explains the factor $(p-1)\alpha_{f_{x}}$ in (10). ∎

###  4.5 The diagonal class $\underline{\kappa}$

We now recall the definition of the big diagonal class $\underline{\kappa}$,
which interpolates the ($p$-stabilization of) the classes given by
$\Delta_{x,y,z}$’s in the étale cohomology groups. If one examines the
construction in [BSV22, § 8.1] carefully, one sees that their argument is also
valid for the finite slope case, with only few modifications needed.

Let $\mathbf{f},\mathbf{g},\mathbf{h}$ be as before. We fix a integer $m$
large enough and set $A_{\xi}^{\cdot}=A_{\mathcal{V}_{\xi},m}(T^{\cdot})$ and
$D_{\xi}^{\cdot}=D_{\mathcal{V}_{\xi},m}(T^{\cdot})$ for $\xi\in\\{f,g,h\\}$.

Set $(T\times T)_{0}:=\\{(t_{1},t_{2})\in T\times
T\mid\det(t_{1},t_{2})\in\mathbb{Z}_{p}^{\times}\\}$ and let $(T\times T)^{0}$
be the complement of $(T\times T)_{0}$ in $T\times T$. Both $(T\times T)_{0}$
and $(T\times T)^{0}$ are compact open in $T\times T$ and are preserved by
$\Gamma_{0}(p\mathbb{Z}_{p})$. The decomposition $T\times T=(T\times
T)_{0}\sqcup(T\times T)^{0}$ induces a decomposition of
$\Gamma_{0}(p\mathbb{Z}_{p})$-modules

$A_{g}\hat{\otimes}A_{h}=(A_{g}\hat{\otimes}A_{h})_{0}\oplus(A_{g}\hat{\otimes}A_{h})^{0}$

where $(A_{g}\hat{\otimes}A_{h})_{0}$ consists of locally analytic functions
supported on $(T\times T)_{0}$ and similarly for
$(A_{g}\hat{\otimes}A_{h})^{0}$. Let
$\Lambda_{fgh}:=\Lambda_{f}\hat{\otimes}_{\mathcal{O}_{L}}\Lambda_{g}\hat{\otimes}_{\mathcal{O}_{L}}\Lambda_{h}$
and define the following characters from $\mathbb{Z}_{p}^{\times}$ to
$\Lambda_{fgh}^{\times}$

$\displaystyle\kappa_{f}^{*}(u)$
$\displaystyle:=\omega(u)^{(\ell_{0}+m_{0}-k_{0}-2)/2}\cdot\langle
u\rangle^{(\bm{\ell}+\bm{m}-\bm{k}-2)/2},$ $\displaystyle\kappa_{fgh}^{*}(u)$
$\displaystyle:=\omega(u)^{(k_{0}+\ell_{0}+m_{0}-6)/2}\cdot\langle
u\rangle^{(\bm{k}+\bm{\ell}+\bm{m}-6)/2}.$

One also define $\kappa_{g}^{*},\kappa_{h}^{*}$ similarly so that
$\kappa_{fgh}^{*}=\kappa_{f}^{*}+\kappa_{g}^{*}+\kappa_{h}^{*}$.

Consider now the function

$\bm{Det}:T^{\prime}\times T\times T\rightarrow\Lambda_{fgh}^{\times}$

which is identically zero on $T^{\prime}\times(T\times T)^{0}$ and on an
element $(t_{1},t_{2},t_{3})\in T^{\prime}\times(T\times T)_{0}$ takes the
value

$\bm{Det}(t_{1},t_{2},t_{3})=\det(t_{1},t_{2})^{\kappa_{h}^{*}}\cdot\det(t_{1},t_{3})^{\kappa_{g}^{*}}\cdot\det(t_{2},t_{3})^{\kappa_{f}^{*}}.$

A direct computation shows that $\bm{Det}$ belongs to
$A^{\prime}_{f}\hat{\otimes}A_{g}\hat{\otimes}A_{h}(-\kappa_{fgh}^{*})$. By
Remark 4.9, is corresponds to a class

$\bm{Det}^{\operatorname{\mathrm{\acute{e}t}}}\in
H^{0}_{\operatorname{\mathrm{\acute{e}t}}}(Y,A^{\prime}_{f}\otimes
A_{g}\otimes A_{h}(-\kappa_{fgh}^{*}))$ (12)

where we use the same notation $A^{\cdot}_{\xi}$ for its corresponding étale
sheave.

Let $d:Y\rightarrow Y^{3}$ be the diagonal embedding. Define
$\underline{\kappa}^{\circ\circ}:=\bm{\mathrm{AJ}}_{\operatorname{\mathrm{\acute{e}t}}}(\bm{Det}^{\operatorname{\mathrm{\acute{e}t}}})$,
where $\bm{\mathrm{AJ}}_{\operatorname{\mathrm{\acute{e}t}}}$ denotes the
following composition of maps:

$\displaystyle\begin{split}H^{0}_{\operatorname{\mathrm{\acute{e}t}}}(Y,A^{\prime}_{f}\otimes
A_{g}\otimes&A_{h}(-\kappa_{fgh}^{*}))\xrightarrow{d_{*}}H^{4}_{\operatorname{\mathrm{\acute{e}t}}}(Y^{3},A^{\prime}_{f}\boxtimes
A_{g}\boxtimes
A_{h}(-\kappa_{fgh}^{*})\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p}(2))\\\
\xrightarrow{\mathrm{HS}}&H^{1}(\mathbb{Q},H^{3}_{\operatorname{\mathrm{\acute{e}t}}}(Y^{3}_{\bar{\mathbb{Q}}},A^{\prime}_{f}\boxtimes
A_{g}\boxtimes A_{h})(2+\kappa_{fgh}^{*}))\\\
\xrightarrow{K}&H^{1}(\mathbb{Q},H^{1}(\Gamma,A^{\prime}_{f})\hat{\otimes}_{L}H^{1}(\Gamma,A_{g})\hat{\otimes}_{L}H^{1}(\Gamma,A_{h})(2+\kappa_{fgh}^{*}))\\\
\xrightarrow{w_{p}\otimes\operatorname{\operatorname{id}}\otimes\operatorname{\operatorname{id}}}&H^{1}(\mathbb{Q},H^{1}(\Gamma,A_{f})\hat{\otimes}_{L}H^{1}(\Gamma,A_{g})\hat{\otimes}_{L}H^{1}(\Gamma,A_{h})(2+\kappa_{fgh}^{*}))\\\
\xrightarrow{s_{fgh}}&H^{1}(\mathbb{Q},H^{1}(\Gamma,D^{\prime}_{f})^{\leq{a_{f}}}\hat{\otimes}_{L}H^{1}(\Gamma,D^{\prime}_{g})^{\leq{a_{g}}}\hat{\otimes}_{L}H^{1}(\Gamma,D^{\prime}_{h})^{\leq{a_{h}}}(2-\kappa_{fgh}^{*}))\\\
\xrightarrow{\operatorname{pr}_{fgh}}&H^{(}\mathbb{Q},V(\mathbf{f})\hat{\otimes}_{L}V(\mathbf{g})\hat{\otimes}V(\mathbf{h})(-1-\kappa_{fgh}^{*}))=H^{1}(\mathbb{Q},V(\mathbf{f},\mathbf{g},\mathbf{h})).\end{split}$
(13)

Here $\kappa_{fgh}^{*}:G_{\mathbb{Q}}\rightarrow\Lambda_{fgh}^{\times}$
denotes, by an abuse of notation, the composition of the $p$-adic cyclotomic
character with $\kappa_{fgh}^{*}$. The map $d_{*}$ is the push-forward map.
The map $\mathrm{HS}$ is the map arising from the Hochschild–Serre spectral
sequence (since
$H^{4}_{\operatorname{\mathrm{\acute{e}t}}}(Y^{3}_{\bar{\mathbb{Q}}},\cdot\ )$
vanishes) and $2.$ of Remark 4.9. The map $K$ comes from the Künneth
decomposition and the isomorphism in $1.$ of Remark 4.9. The meaning of the
map
$w_{p}\otimes\operatorname{\operatorname{id}}\otimes\operatorname{\operatorname{id}}$
is clear, with $w_{p}:H^{1}(\Gamma,A^{\prime}_{f})\rightarrow
H^{1}(\Gamma,A_{f})$ being the Atkin–Lehner involution. The map $s_{fgh}$ is
induced by the tensor product of the $G_{\mathbb{Q}}$-morphisms

$H^{1}(\Gamma,A_{\xi})\twoheadrightarrow H^{1}(\Gamma,A_{\xi})^{\leq
a_{\xi}}\xrightarrow{s_{\xi}}H^{1}(\Gamma,D^{\prime}_{\xi})^{\leq
a_{\xi}}(-\kappa_{\xi})$

for $\xi\in\\{f,g,h\\}$. The last map $\operatorname{pr}_{fgh}$ is the tensor
product of the maps $\operatorname{pr}_{\xi}$ in $7$. of Remark 4.9.

We now define

$\underline{\kappa}^{\circ}(\mathbf{f},\mathbf{g},\mathbf{h}):\frac{1}{a_{p}(\mathbf{f})}\cdot\underline{\kappa}^{\circ\circ}\in
H^{1}(\mathbb{Q},V(\mathbf{f},\mathbf{g},\mathbf{h})).$

and let
$\underline{\kappa}^{\circ}_{p}:=res_{p}(\underline{\kappa}^{\circ})\in
H^{1}(\mathbb{Q}_{p},V(\mathbf{f},\mathbf{g},\mathbf{h}))$ be its restriction
to $G_{\mathbb{Q}_{p}}$. Finally, we let

$\underline{\kappa}_{p}\in
H^{1}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h}))$

be the corresponding element of $\underline{\kappa}^{\circ}_{p}$.

######  Theorem 4.23 ([BSV22, Theorem A]).

The class $\underline{\kappa}_{p}$ belongs to
$H^{1}_{\operatorname{\operatorname{bal}}}(\mathbb{Q}_{p},M(\mathbf{f},\mathbf{g},\mathbf{h}))$
and

$\mathscr{L}og_{f}(\underline{\kappa}_{p})=\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h}).$

###### Proof.

The proof is the same as in [BSV22, § 8]. Through §8.2 to §8.4 in loc. cit.,
one attempts to describe the specialization of
$\mathscr{L}og_{f}(\underline{\kappa}_{p})$ at a balanced triple
$w=(x,y,z)\in\Sigma_{\operatorname{\operatorname{bal}}}$ in terms of
$\log_{p}(\kappa(f_{x}^{0},g_{y}^{0},h_{z}^{0}))$, where
$\kappa(f_{x}^{0},g_{y}^{0},h_{z}^{0})$ is the small diagonal class defined in
[BSV22, § 3]. In the end, one gets an equality

$\mathscr{L}og_{f}(\underline{\kappa}_{p})_{w}=(-1)^{t-1}\frac{\mathscr{E}(f_{x}^{0},g_{y}^{0},h_{z}^{0})}{(t-1)!\mathscr{E}_{0}(f_{x}^{0})\mathscr{E}_{1}(f_{x}^{0})}\cdot\log_{p}(\kappa(f_{x}^{0},g_{y}^{0},h_{z}^{0}))(\eta^{a}_{f_{x}^{0}}\otimes\omega_{g_{y}^{0}}\otimes\omega_{h_{z}^{0}}).$

On the other hand, the Bloch–Kato logarithm $\log_{p}$ can be interpreted as
the $p$-adic Abel–Jacobi map (c.f. [Bes12, § 1.6.1] and [Bes00, Proposition
9.11]). In particular,

$\log_{p}(\kappa(f_{x}^{0},g_{y}^{0},h_{z}^{0}))(\eta^{a}_{f_{x}^{0}}\otimes\omega_{g_{y}^{0}}\otimes\omega_{h_{z}^{0}})=\operatorname{\operatorname{AJ}}_{p}(\Delta_{x,y,z})(\eta^{a}_{f_{x}^{0}}\otimes\omega_{g_{y}^{0}}\otimes\omega_{h_{z}^{0}}).$

Now, our $p$-adic Gross–Zagier formula tells us that

$\mathscr{L}og_{f}(\underline{\kappa}_{p})_{w}=\mathscr{L}^{f}_{p}(\mathbf{f},\mathbf{g},\mathbf{h})(w)$

for all $w\in\Sigma_{\operatorname{\operatorname{bal}}}$ with the condition
$\alpha_{f_{x}}\beta_{g_{y}}\beta_{h_{z}}\neq p^{c_{w}}$. As such a subset of
weights is dense in the weight space $\Sigma$, one obtains the desired
identity. ∎

We now shift our attention to the unbalanced region $\Sigma_{f}$. For
simplicity, we will only deal with $w=(x,y,z)\in\Sigma_{f}$ that is not
exceptional in the sense of [BSV22, § 1.2]. Following the same argument as in
[BSV22, § 9.1], we have the following corollary.

######  Corollary 4.24 ([BSV22, Theorem B]).

Let $w=(x,y,z)\in\Sigma_{f}$ be a unbalanced triple that is not exceptional.
Then the class $\underline{\kappa}_{p}(x,y,z)$ is crystalline (i.e., belongs
to $H^{1}_{f}$) if and only if the complex $L$-function
$L(f_{x}^{0},g_{y}^{0},h_{z}^{0};s)$ vanishes at $s=\frac{x+y+z-2}{2}$.

######  Remark 4.25.

In the case where $w$ is exceptional, the arguments in [BSV22, § 9.2 to §9.4]
can also be applied to the finite slope case.

With Corollary 4.24, one can also establish an analogue of Theorem 1.2. The
proof will be identical to one given in [DR17, § 6.4]. So we would like leave
the verification to the readers.

##  5 Appendix. A lemma on pairings of $p$-adic Banach spaces

In this appendix, we will give the proof of Lemma 3.7.

The goal is to study the interaction of the finite slope projector $e^{\leq
a}$ with the Poincaré pariring or the Petersson product. To be more precise,
we wish to establish an analogous result of [DR14, Prop 2.3] and [DR14, Prop
2.11]. A combined statement as well as its proof will be recalled below.

Let $K$ be a finite extension of $\mathbb{Q}_{p}$ and $X_{K}=X_{1}(N)_{K}$ be
the modular curve base-changed to $K$.

######  Lemma 5.1.

Let $\eta\in
H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})^{\mathrm{u-r}}$
be a class in the unit root part, i.e., the Frobenius $\phi$ acts on $\eta$ as
multiplying by a $p$-adic unit. Let $g$ be a nearly overconvergent modular
form of weight $k=r+2$ on $\Gamma_{1}(N)$ with vanishing residues at all the
supersigular annuli, which induces a class $\omega=\omega_{g}$ in
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})$. Then

$\langle\eta,\omega\rangle=\langle\eta,e_{\operatorname{\operatorname{ord}}}\omega\rangle$

where $\langle\phantom{e},\phantom{e}\rangle$ is the Poincaré pairing on
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})$.

###### Proof.

([Bes16, [Prop. 3.2]) First, recall that we the operators $U$ and $V$ are
inverse to each other on the cohomology. In addition, the Frobenius morphism
$\phi$ satisfies the relations
$\langle\phi\eta,\phi\omega\rangle=p^{k-1}\langle\eta,\omega\rangle$ and
$\phi=p^{k-1}V=p^{k-1}U^{-1}$ on the cohomology. Let $\beta$ be the eigenvalue
of $\eta$ for $\phi$, which is a $p$-adic unit by assumption. Then,

$\displaystyle\langle\eta,U\omega\rangle$
$\displaystyle=p^{1-k}\langle\phi\eta,\phi U\omega\rangle$
$\displaystyle=\langle\phi\eta,p^{1-k}\phi U\omega\rangle$
$\displaystyle=\langle\phi\eta,\omega\rangle$
$\displaystyle=\beta\langle\eta,\omega\rangle$

As a result, we have

$\langle\eta,e_{\operatorname{\operatorname{ord}}}\omega\rangle=\langle\eta,\lim_{n\rightarrow\infty}U^{n!}\omega\rangle=\lim_{n\rightarrow\infty}\beta^{n!}\langle\eta,\omega\rangle=\langle\eta,\omega\rangle$

as desired. ∎

######  Remark 5.2.

A more conceptual proof is outlined in [DR14]. First, one uses the relation
$\langle\phi\eta,\phi\omega\rangle=p^{k-1}\langle\eta,\omega\rangle$ to deduce
that the Poincaré pairing descends to a well-defined pairing

$\langle\phantom{e},\phantom{e}\rangle:H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})^{\mathrm{u-r}}\times
H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})^{\phi,k-1}\rightarrow
K(1-r)$

where the subscript $\phi,k-1$ denotes the slope $=k-1$ subspace for $\phi$.
Then one can identify $S^{\operatorname{\operatorname{ord}}}(N)$, the space of
ordinary overconvergent modular forms, with
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})^{\phi,k-1}$.

The main result in this section is the following lemma:

######  Lemma 5.3.

(General form) Let $M$ be the $p$-adic Banach space over a $p$-adic field $K$
and $u$ be a compact operator on $M$. Suppose there is a bilinear pairing
$\langle\phantom{e},\phantom{e}\rangle:M\times M\rightarrow K$ and another
operator $\phi$ on $M$ that satisfies

$\langle m_{1},um_{2}\rangle=\langle\phi m_{1},m_{2}\rangle$

for any $m_{1},m_{2}\in M$.

Let $\eta\in M$ be an eigenvector for $\phi$ with eigenvalue $\alpha$, whose
$p$-adic valuation is $a\in\mathbb{Q}_{\geq 0}$. Let $e^{\leq a}$ be the
projector onto the slope $\leq a$ subspace of $M$ for the operator $u$. Then
we have

$\langle\eta,m\rangle=\langle\eta,e^{\leq a}m\rangle$

for all $m\in M$.

In our application, it takes the following form:

######  Lemma 5.4.

Suppose $\eta\in
H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})$ is an
element for which the Frobenius $\phi$ acts with eigenvalue $\alpha$ such that
$v_{p}(\alpha)\leq a$. Then for any $\omega\in
H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})$, we have

$\langle\eta,\omega\rangle=\langle\eta,e^{\leq a}\omega\rangle$

where $e^{\leq a}$ is the projector to the slope $\leq a$ part for the $U$
operator and the pairing is the Poincaré pairing.

######  Remark 5.5.

As we will eventually restrict to classical forms on $X_{1}(N)$, we are in
fact dealing with the finite dimensional vector space
$H^{1}_{\operatorname{\operatorname{par}}}(X_{K},\mathcal{H}^{r})$. Hence the
projector $e^{\leq a}$ can be expressed as a polynomial in $U$ instead of a
convergent power series. So we do not really need to deal with (infinite
dimensional) $p$-adic Banach spaces and compact operators. However, we wish to
provide a proof that may be applied to more general cases.

We will skip the the basic definitions of $p$-adic Banach spaces and compact
operators. Some standard references are [Ser62], [Col97] and [Buz07].

Let $K$ be a complete non-archimedean valuation field, $A$ be its ring of
valuation, $\mathfrak{m}$ be its maximal ideal, and $k=A/\mathfrak{m}$ be the
residue field. Suppose $E$ is a $p$-adic Banach space over $K$, we denote by
$\mathcal{L}(E,E)$ (resp. $\mathcal{C}(E,E)$) the space of all linear
operators (resp. compact operators) on $E$.

######  Proposition 5.6.

For any $u\in\mathcal{C}(E,E)$, we can define the Fredholm determinant
$\det(1-Tu)\in K\llbracket T\rrbracket$ that satisfies the following
properties:

  1. i.

$\det(1-Tu)$ is an entire function of $T$ with values in $\mathcal{L}(E,E)$.

  2. ii.

If $u_{i}\rightarrow u$ with $u\in\mathcal{C}(E,E)$. Then
$\det(1-Tu_{i})\rightarrow\det(1-Tu)$ coefficient-wise.

  3. iii.

If $u$ is of finite rank, then $\det(1-Tu)$ coincides with the polynomial
defined in the classical sense (c.f. §5. of [Ser62]).

######  Definition 5.7.

The Fredholm resolvant $R(T,u)$ is defined to be

$R(T,u)=\frac{\det(1-Tu)}{(1-Tu)}=\sum_{m=0}^{\infty}v_{m}T^{m}$

where $v_{m}\in\mathcal{L}(E,E)$ are polynomials in $u$.

######  Proposition 5.8 ([Ser62, Proposition 10]).

As a function with value in $\mathcal{L}(E,E)$, $R(T,u)$ is an entire function
of $T$.

From now on, we will fix a compact operator $u\in\mathcal{C}(E,E)$ and simply
write $P(T)=P_{u}(T)$ for $\det(1-Tu)$.

######  Proposition 5.9.

Suppose $\lambda\in K$. Then $1-\lambda u$ is invertible in $\mathcal{L}(E,E)$
if and only if $P(\lambda)\neq 0$.

######  Definition 5.10.

For a function $F(T)=\sum b_{m}T^{m}$, define
$\Delta^{s}f=\sum\binom{m+s}{m}b_{m+s}T^{m}$ for any $s\in\mathbb{N}$. Then a
zero $\lambda\in K$ of $f$ is said to have order $h\in\mathbb{N}$ if
$\Delta^{s}(\lambda)=0$ for all $s<h$ and $\Delta^{h}f(\lambda)\neq 0$. Note
that if the characteristic of $K$ is $0$, we have
$\Delta^{s}=\frac{1}{s!}\frac{d^{s}}{dT^{s}}$.

One important result of compact operators on $p$-adic Banach spaces is the
following proposition.

######  Proposition 5.11.

(Riesz theory) If $\lambda\in K$ is a zero of $P(T)=\det(1-Tu)$ of order $h$.
Then the space $E$ can be decomposed uniquely as a direct sum of $u$-invariant
closed subspaces

$E=N(\lambda)\oplus F(\lambda)$

such that

  1. i.

$(1-\lambda u)$ is invertible on $F(\lambda)$

  2. ii.

$(1-\lambda u)$ is nilpotent on $N(\lambda)$. More precisely, $(1-\lambda