# A $p$-adic Shimura-Maass operator on Mumford curves

Matteo Longo  Dipartimento di Matematica, Università di Padova, Via Trieste
63, 35121 Padova, Italy<EMAIL_ADDRESS>
###### Abstract.

We study a $p$-adic Shimura-Maass operator in the context of Mumford curves
defined by C. Franc in [Fra11]. We prove that this operator arises from a
splitting of the Hodge filtration, thus answering a question in [Fra11]. We
also study the relation of this operator with generalized Heegner cycles, in
the spirit of [BDP13], [HB15], [Kri18] and [AI19].

###### Contents

  1. 1 Introduction
  2. 2 Algebraic de Rham cohomology of Shimura curves
  3. 3 Special values of $L$-series
  4. 4 The Shimura-Maass operator on the $p$-adic upper half plane
  5. 5 The $p$-adic Shimura-Maass operator on Shimura curves
  6. 6 The Coleman primitive
  7. 7 The generalised Kuga-Sato motive
  8. 8 The Abel-Jacobi map
  9. 9 Generalized Heegner cycles

##  1\. Introduction

The main purpose of this paper is to study in the context of Mumford curves a
$p$-adic variant of the Shimura-Maass operator, and relate it to generalized
Heegner cycles.

The real analytic Shimura-Maass operator is defined by the formula

(1) $\delta_{k}\left(f(z)\right)=\frac{1}{2\pi
i}\left(\frac{\partial}{\partial z}+\frac{k}{z-\bar{z}}\right)f(z)$

where $z$ is a variable in the complex upper half plane $\mathcal{H}$, $f(z)$
is a real analytic modular form of weight $k$, and $z\mapsto\bar{z}$ denotes
the complex conjugation; here $\delta_{k}\left(f(z)\right)$ is a real analytic
modular form of weight $k+2$. The relevance of this operator arises in studyin
algebraicity properties of Eisenstein series and $L$-functions: see Shimura
[Shi75], Hida [Hid93, Chapter 10]. One of the main results in [Shi75] is the
following. Let

$\delta_{k}^{r}=\delta_{k+2(r-1)}\circ\delta_{k+2(r-2)}\circ\dots\circ\delta_{k}$

for any $r\geq 1$, and let $K$ be an imaginary quadratic field. Then there
exists $\Omega_{K}\in\mathbb{C}^{\times}$ such that for every CM point $z\in
K\cap\mathcal{H}$, every congruence subgroup
$\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Z})$, every integer $k\geq 0$,
$r\geq 1$, and every modular form of weigh $k$ and level $\Gamma$ with
algebraic Fourier coefficients we have

(2) $\frac{\delta_{k}^{r}(f)(z)}{\Omega_{K}^{(k+2r}}\in\bar{\mathbb{Q}}.$

Katz described in [Kat68] the Shimura-Maass operator in more abstract terms by
means of the Gauss-Manin connection (see also [KO68]). More precisely. let
$N\geq 1$ be an integer, $X_{1}(N)$ the modular form of level $\Gamma_{1}(N)$
over $\mathbb{Q}$, and let $\pi:\mathcal{E}\rightarrow X_{1}(N)$ be the
universal elliptic curve. Consider the relative de Rham cohomology sheaf

$\mathcal{L}_{1}=\mathbb{R}^{1}\pi_{*}\left(0\rightarrow\mathcal{O}_{\mathcal{E}}\rightarrow\Omega^{1}_{\mathcal{E}/X_{1}(N)}\right)$

on $X_{1}(N)$, and define $\mathcal{L}_{r}=\mathrm{Sym}^{r}(\mathcal{L}_{1})$.
Let
$\underline{\omega}=\pi_{*}\left(\Omega^{1}_{\mathcal{E}/X_{1}(N)}\right)$.
The sheaf $\underline{\omega}$ is invertible and we have the Hodge filtration

(3)
$0\rightarrow\underline{\omega}\rightarrow\mathcal{L}_{1}\rightarrow\underline{\omega}^{-1}\rightarrow
0.$

Once we are given a splitting $\Psi$ of the Hodge filtration (3), one may
define by means of the Gauss-Manin connection and the Kodaira-Spencer map an
operator
$\Theta^{(r)}_{\Psi}\colon\underline{\omega}^{r}\rightarrow\underline{\omega}^{r+2}$,
for any even integer $r\geq 2$. In particular, considering the associated real
analytic sheaves, which we denote by a superscript ra, the Hodge exact
sequence admits a splitting

(4)
$\Psi_{\infty}:\mathcal{L}_{1}^{\mathrm{ra}}\simeq\underline{\omega}_{\mathrm{ra}}\oplus\underline{\bar{\omega}}_{\mathrm{ra}},$

where $\underline{\bar{\omega}}_{\mathrm{ra}}$ is obtained from
$\underline{\omega}$ by applying the complex conjugation. The Shimura-Maass
operator can then be described as the map $\Theta^{(r)}_{\Psi_{\infty}}$ on
real analytic differentials by appealing to the general procedure alluded to
above applied to the real analytic splitting (4). For details on this
construction, the reader is referred to [Kat78, §1.8] and [BDP13, §1.2]; for
the case of Siegel modular forms, see [Har81, §4] while for the case of
Shimura curves see [HB15, §3], [Mor11, §2].

As hinted from the above discussion, Katz description of the Shimura-Maass
operator rests on the fact that the real analytic Hodge sequences (3) splits.
In [Kat78, §1.11], Katz introduces a $p$-adic analogue of this splitting.
Suppose that $p\nmid N$ is a prime number, and let $X_{1}^{\mathrm{ord}}(N)$
denote the ordinary locus of the modular curve, viewed as a rigid analytic
scheme over $\mathbb{Q}_{p}$. Let $\mathcal{F}^{\mathrm{rig}}$ be the rigid
analytic sheaf associated with a sheaf $\mathcal{F}$ on $X_{1}(N)$. Then
$\mathcal{L}_{1}^{\mathrm{rig}}$ splits over $X_{1}^{\mathrm{ord}}(N)$ as the
direct sum

$\Psi_{p}:\mathcal{L}_{1}^{\mathrm{rig}}\simeq\underline{\omega}^{\mathrm{rig}}\oplus\mathcal{L}_{1}^{\mathrm{Frob}}$

(where $\mathcal{L}_{1}^{\mathrm{Frob}}$ has the property that the Frobenius
endomorphism acts on this sheaf invertibly). This allows to define a
differential operator $\Theta_{\Psi_{p}}^{(r)}$, which can be seen as a
$p$-adic analogue of the Shimura-Maass operator; this operator can be also
described in terms of Atkin-Serre derivative. At CM points the splittings
$\Psi_{\infty}$ and $\Psi_{p}$ coincide, and therefore one deduces by
comparison rationality results for the values of $\Theta_{\Psi_{p}}^{(r)}$ at
CM points from intro rationality. For details, see [BDP13, Proposition 1,12].
The $p$-adic Shimura-Maass operator is then used in [Kat78] and [BDP13] to
construct $p$-adic $L$-functions and study their properties.

We now fix an integer $N$, a prime $p\nmid N$, and a quadratic imaginary field
in which $p$ is inert. In this context, Kritz introduced in [Kri18] for
modular forms of level $N$ a new $p$-adic Shimura-Maass operator by using
perfectoid techniques, and define $p$-adic $L$-functions by means of this
operator, thus removing the crucial assumption that $p$ is split in $K$, but
keeping the assumption that $p$ is a prime of good reduction for the modular
curve. Moreover, Andreatta-Iovita [AI19] introduced still an other $p$-adic
Shimura-Maass operator in [AI19], and obtained results analogue to [BDP13],
thus extending their work to the non-split case.

On the other hand, Franc in his thesis [Fra11] proposed still an other
$p$-adic Shimura-Maass operator for primes $p$ which are inert in $K$, in the
following context. Let $N\geq 1$ be an integer, $K/\mathbb{Q}$ a quadratic
imaginary field, $p\nmid N$ a prime number which is inert in $K$, and let
$Np=N^{+}\cdot N^{-}p$ be a factorization of $Np$ into coprime integers such
that $N^{+}$ is divisible only by primes which are split in $K$, and $N^{-}p$
is a square-free product of an even number of primes factors which are inert
in $K$. Let $\mathcal{B}$ be the indefinite quaternion of discriminant
$N^{-}$, $\mathcal{R}$ an Eichler order of $X$ of level $N^{+}$, and
$\mathcal{C}$ the Shimura curve attached to $(\mathcal{B},\mathcal{R})$. The
rigid analytic curve $X^{\mathrm{rig}}$ over $\mathbb{Q}_{p}$ is then a
Mumford curve, namely $X^{\mathrm{ord}}(\mathbb{C}_{p})$ is isomorphic to the
rigid analytic quotient of the $p$-adic upper half plane
$\mathcal{H}_{p}(\mathbb{Q}_{p})=\mathbb{C}_{p}-\mathbb{Q}_{p}$ by an
arithmetic subgroup $\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Q}_{p})$.
Franc defines in this context a $p$-adic Shimura-Maass operator $\delta_{p,k}$
by mimicking the definition (1) and formally replacing the variable
$z\in\mathcal{H}$ with the $p$-adic variable
$z\in\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})=\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}-\mathbb{Q}_{p}$,
and replacing the complex conjugation with the Frobenius map (here
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$ is the completion of the maximal
unramified extension of $\mathbb{Q}_{p}$). Following the arguments of [Shi75],
Franc proves statement analogue to (2) (see [Fra11, Theorem 5.1.5]).

In [Fra11, §6.1.3], Franc asks for a construction of his $p$-adic Shimura-
Maass operator by means of a (non-rigid analytic) splitting $\Psi_{p}$ of the
Hodge filtration, similar to what happens over $X_{1}(N)$ (in the real
analytic case [Shi75]) and $X_{1}^{\mathrm{ord}}(N)$ (in the $p$-adic rigid
analytic case [Kat78]). The first result of this paper is to provide such a
splitting $\Psi_{p}$, and define the associated $p$-adic Shimura-Maass
operator. In particular, we show that our splitting $\Psi_{p}$ coincides at CM
points with the Hodge splitting $\Psi_{\infty}$, and therefore, as in [Kat78],
we reprove the main results of [Fra11] by the comparison of the two Shimura-
Mass operators. We also derive a relation between our $p$-adic Shimura-Maass
operator and generalized Heegner cycles in the context of Mumford curves,
which can be vied as an analogue of [BDP13, Proposition 3.24]. In the
remaining part of the introduction we describe more precisely the results of
this paper.

Instead of the curve $X$ attached to the Eichler order $\mathcal{R}$, we
follow [HB15] and consider a covering $\mathcal{C}\rightarrow X$ where
$\mathcal{C}$ is a geometrically connected curve defined over $\mathbb{Q}$
corresponding to a $\Gamma_{1}(N^{+})$-level structure subgroup of
$\hat{\mathcal{R}}^{\times}$, where
$\hat{\mathcal{R}}=\mathcal{R}\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}$ is the
profinite completion of $\mathcal{R}$. The advantage of using $\mathcal{C}$ is
that $\mathcal{C}$ is the solution of a moduli problem, and we have a
universal false elliptic curve $\pi:\mathcal{A}\rightarrow\mathcal{C}$ (see
§2.2). Following [Has95], [Mor11], [HB15], we define a quaternionic projector
$e$, acting on the relative de Rham cohomology of
$\pi:\mathcal{A}\rightarrow\mathcal{C}$, and define the sheaf

$\mathcal{L}_{1}=e\cdot\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{A}/\mathcal{C})$

and the line bundle

$\underline{\omega}=e\cdot\pi_{*}\left(\Omega_{\mathcal{A}/\mathcal{C}}^{1}\right).$

We have a corresponding Hodge filtration

$0\longrightarrow\underline{\omega}\longrightarrow\mathcal{L}_{1}\longrightarrow\underline{\omega}^{-1}\longrightarrow
0.$

The rigid analytic curve $\mathcal{C}^{\mathrm{rig}}$ associated with
$\mathcal{C}$ admits a $p$-adic uniformization

$\mathcal{C}^{\mathrm{rig}}(\mathbb{C}_{p})\simeq\Gamma\backslash\mathcal{H}_{p}(\mathbb{C}_{p})$

for a suitable subgroup
$\Gamma\subseteq\operatorname{SL}_{2}(\mathbb{Q}_{p})$. Modular forms on
$\mathcal{C}^{\mathrm{rig}}$ are then $\Gamma$-invariant sections of
$\mathcal{H}_{p}$, and therefore, to define a $p$-adic Shimura-Maass operator
on $\mathcal{C}$ one is naturally led to consider the analogue problem for
$\mathcal{H}_{p}$.

Let $\mathcal{C}^{0}$ denote the $\mathbb{C}_{p}$-vector space of continuous
(for the standard $p$-adic topology on both spaces) $\mathbb{C}_{p}$-valued
functions on ${\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})}$, and let
$\mathcal{A}$ denote the $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector space of
rigid analytic global sections of
$\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$. We have a map of
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector spaces
$r:\mathcal{A}\rightarrow\mathcal{C}^{0}$ and, following [Fra11], we denote
$\mathcal{A}^{*}$ the image of the morphism of $\mathcal{A}$-algebras
$\mathcal{A}[X,Y]\rightarrow\mathcal{C}^{0}$ defined by sending $X$ to the
function $z\mapsto 1/(z-\sigma(z))$ and $Y$ to the function
$z\mapsto\sigma(z)$, where
$\sigma:\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}\rightarrow\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$
is the Frobenius automorphism (note that the function $z\mapsto z-\sigma(z)$
is invertible on $\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$).
Denote $\hat{\mathcal{H}}_{p}$ the formal $\mathbb{Z}_{p}$-scheme whose
generic fiber is $\mathcal{H}_{p}$, let $\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$
be its base change to $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$ and let
$\mathcal{G}\rightarrow\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$ be the universal
$\operatorname{SFD}$-module. Denote
$\underline{\omega}_{\mathcal{G}}=e_{\mathcal{G}}^{*}(\Omega^{1}_{\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})$,
where
$e_{\mathcal{G}}:\hat{\mathcal{H}}_{p}^{\mathrm{unr}}\rightarrow\mathcal{G}$
is the zero-section, and let $\mathcal{L}ie_{\mathcal{G}^{\vee}}$ be the Lie
algebra of the Cartier dual $\mathcal{G}^{\vee}$ of $\mathcal{G}$. Then
$\underline{\omega}_{\mathcal{G}}$ and $\mathcal{L}ie_{\mathcal{G}^{\vee}}$
are locally free $\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}$-modules,
dual to each other and we have the Hodge-Tate exact sequence of
$\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}$-modules

$0\longrightarrow\underline{\omega}_{\mathcal{G}}\longrightarrow\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})\longrightarrow\mathcal{L}ie_{\mathcal{G}^{\vee}}\longrightarrow
0.$

Set
$\mathcal{L}_{\mathcal{G}}^{0}=e\cdot\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}_{p}^{\mathrm{unr}}})$
and $\omega_{\mathcal{G}}^{0}=e\cdot\underline{\omega}_{\mathcal{G}}$. Define
$\Lambda_{\mathcal{G}}=H^{0}(\mathcal{H}_{p}^{\mathrm{unr}},\mathcal{L}_{\mathcal{G}}^{0})$,
$\Lambda_{\mathcal{G}}^{\ast}=\Lambda_{\mathcal{G}}\otimes_{\mathcal{A}}\mathcal{A}^{\ast}$,
$w_{\mathcal{G}}=H^{0}(\mathcal{H}_{p}^{\mathrm{unr}},\underline{\omega}_{\mathcal{G}}^{0})$,
${w}_{\mathcal{G}}^{\ast}=w_{\mathcal{G}}\otimes_{\mathcal{A}}\mathcal{A}^{\ast}$.
We have then an injective map of $\mathcal{A}^{*}$-algebras

(5)
${w}_{\mathcal{G}}^{\ast}\mbox{\;$\lhook\joinrel\longrightarrow$\;}\Lambda_{\mathcal{G}}^{\ast}.$

######  Theorem 1.1.

The injection (5) of $\mathcal{A}^{*}$-algebras admits a canonical splitting
$\Psi_{p}^{*}:\Lambda_{\mathcal{G}}^{\ast}\rightarrow{w}_{\mathcal{G}}^{\ast}$.

This is the main result of this paper, Theorem 4.7. We may then attach to
$\Psi_{p}^{*}$ a $p$-adic Shimura-Maass operator $\Theta_{p}^{(r)}$. We have
the following two corollaries.

######  Corollary 1.2.

The $p$-adic Shimura-Maass operator $\delta_{p,k}$ defined by Franc in [Fra11]
coincides with the $p$-adic Shimura-Maass operator $\Theta_{p}^{(r)}$ defined
by means of the splitting in Theorem 1.1.

The main tool which is used to prove Theorem 1.1 and Corollary 1.2 is
Drinfel’d interpretation of $\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$ as moduli
space of special formal modules with quaternionic multiplication; following
[Tei89], we call these objects $\operatorname{SFD}$-modules. We study the
relative de Rham cohomology of the universal $\operatorname{SFD}$-module
$\mathcal{G}\rightarrow\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$ by means of
techniques from [Tei89], [Fal97] and [IS03]. The upshot of our analysis is an
explicit description of the Gauss-Manin connection and the Kodaira-Spencer
isomorphism for $\mathcal{G}\rightarrow\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$,
once we apply to the relevant sheaves the projector $e$. This detailed study
is contained in Section 4, which we believe is of independent interest and is
the technical hart of the paper.

For the next corollary, define as in the real analytic case

$\delta_{p,k}^{r}=\delta_{p,k+2(r-1)}\circ\delta_{p,k+2(r-2)}\circ\dots\circ\delta_{k}.$

Moreover, we fix an embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$, and we say that
$\xi\in\bar{\mathbb{Q}}_{p}$ belongs to $\bar{\mathbb{Q}}$ if $\xi$ belongs to
the image of this embedding.

######  Corollary 1.3.

Let $f$ be a modular form on $\mathcal{C}$. Then there exists
$t_{p}\in\mathbb{C}_{p}^{\times}$, independent of $f$, such that for every CM
point $z\in K\cap\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$ we have
$\frac{\delta_{p,k}^{r}(f)(z)}{t_{p}^{j}}\in\bar{\mathbb{Q}}$.

As remarked above, this is the main result of Franc thesis [Fra11], which he
proves via an explicit approach following Shimura. Instead, we derive this
result in Theorem 5.4 from a comparison between the values at CM points of our
$p$-adic Shimura-Maass operator $\Theta_{p}^{(r)}$ and the real analytic
Shimura-Maass operator $\Theta_{\infty}^{(r)}$.

We explain now the connection with generalized Heegner cycles. These cycles
were introduced in [BDP13] with the aim of studying certain anticyclotomic
$p$-adic $L$-functions. Generalized Heegner cycles have been also studied in
the context of Shimura curves with good reduction at $p$ by [HB15], and in the
context of Mumford curves in [Mas12], [LP19]. In this paper we introduce still
an other variant of Generalized Heegner cycles. Fix a false elliptic curve
$A_{0}$ with CM by $\mathcal{O}_{K}$. To any isogeny $\varphi:A_{0}\rightarrow
A$, where $A$ is a false elliptic curve, we construct a cycle
$\Upsilon_{\varphi}$ in the Chow group
$\operatorname{CH}^{m}(\mathcal{A}\times A_{0})$ of the Chow motive
$\mathcal{A}\times A_{0}$, where $m=n/2$ with $n=k-2$. The work of Brooks
[HB15] gives us a projector $\epsilon$ in the ring of correspondences of
$X_{m}=\mathcal{A}^{m}\times A_{0}^{m}$, which defines the motive
$\mathcal{D}=(X^{m},\epsilon)$. The generalised Heegner cycle
$\Delta_{\varphi}$ in the image of $\Upsilon_{\varphi}$ in
$\operatorname{CH}^{m}(\mathcal{D})$ via this projector. Let $M_{k}(\Gamma)$
be the $\mathbb{C}_{p}$-vector space of rigid analytic quaternionic modular
forms of weight $k$ and level $\Gamma$; elements of $M_{k}(\Gamma)$ are
functions from $\mathcal{H}_{p}(\mathbb{C}_{p})=\mathbb{C}_{p}-\mathbb{Q}_{p}$
to $\mathbb{C}_{p}$ which transform under the action of $\Gamma$ by the
automorphic factor of weight $k$. We construct a $p$-adic Abel-Jacobi map

$\operatorname{AJ}_{p}:\operatorname{CH}^{m}(\mathcal{D})\longrightarrow\left(M_{k}(\Gamma)\otimes\mathrm{Sym}^{m}eH^{1}_{\mathrm{dR}}(A_{0})\right)^{\vee}$

where ∨ denotes $\mathbb{C}_{p}$-linear dual. If follows from our work that
$\mathcal{L}_{\mathcal{G}}^{0}=e\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})$
is equipped with two canonical sections $\omega_{\mathrm{can}}$ and
$\eta_{\mathrm{can}}$, such that $\omega_{\mathrm{can}}$ is a generator of the
invertible sheaf $\omega_{\mathcal{G}}^{0}$. Let $\omega_{f}\in
w_{\mathcal{G}}^{0}$ be the $\Gamma$-invariant differential form associated
with $f\in M_{k}(\Gamma)$, and let $F_{f}$ its Coleman primitive satisfying
$\nabla(F_{f})=\omega_{f}$, where $\nabla$ is the Gauss-Manin connection.
Denote $\langle,\rangle$ is the Poincaré pairing on
$\mathrm{Sym}^{n}eH^{1}_{\mathrm{dR}}(A_{z})$, where $A_{z}$ is the fiber of
$\mathcal{A}$ at $z$. Define the function

$H(z)=\langle F_{f}(z),\omega_{\mathrm{can}}^{n}(z)\rangle.$

######  Theorem 1.4.

Let $\varphi:A_{0}\rightarrow A$ be an isogeny and $z_{A}$ the fiber of $A$ of
$\mathcal{A}\rightarrow\mathcal{C}$. Then for each integer $j=n/2,\dots,n$ we
have

$\delta_{p,k}^{n-j}(H_{n})(z_{A})=\operatorname{AJ}_{p}(\Delta_{\varphi})(\omega_{f}\otimes\omega^{j}_{\mathrm{can}}\eta^{n-j}_{\mathrm{can}}).$

Theorem 1.4 relates the Shimura-Maass operator with generalised Heegner
cycles, and corresponds to Corollary 9.8.

We finally make a remark on $p$-adic $L$-functions. It would be interesting to
use our $p$-adic Shimura-Maass operator to construct $p$-adic $L$-functions
interpolating special values of the complex $L$-function of $f$ twisted by
Hecke characters as in [BDP13], [HB15], [Kri18], [AI19]. We would like to come
back to this problem in a future work.

##  2\. Algebraic de Rham cohomology of Shimura curves

Throughout this section, let $k\geq 2$ be an even integer and $N\geq 1$ an
integer. Fix an imaginary quadratic field $K/\mathbb{Q}$ of discriminant
$D_{K}$ prime to $N$ and factor $N=N^{+}\cdot N^{-}$ by requiring that all
primes dividing $N^{+}$ (respectively $N^{-}$) split in $K$ (respectively, are
inert in $K$). Assume that $N^{-}$ is a square-free of an odd number of
primes, and let $p\nmid N$ be a prime number which is inert in $K$ (thus
$N^{-}p$ is a square-free of an even number of primes). Let $f\in
S_{k}(\Gamma_{0}(Np))$ be a weight $k$ newform of level $\Gamma_{0}(Np)$. Fix
also embeddings $\bar{\mathbb{Q}}\hookrightarrow\mathbb{C}$ and
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{\ell}$ for each prime number
$\ell$.

###  2.1. Quaternion algebras

Fix a indefinite quaternion algebra $\mathcal{B}/\mathbb{Q}$ of discriminant
$N^{-}p$, a maximal order $\mathcal{R}_{\mathrm{max}}\subseteq\mathcal{B}$ and
an Eichler order $\mathcal{R}\subseteq\mathcal{R}_{\mathrm{max}}$ of level
$N^{+}$. Fix isormorphisms
$\iota_{\ell}:\mathcal{B}_{\ell}=\mathcal{B}\otimes_{\mathbb{Q}}\mathbb{Q}_{\ell}\simeq\operatorname{M}_{2}(\mathbb{Q}_{\ell})$
for each prime $\ell\nmid N^{-}p$ such that
$\iota_{\ell}(\mathcal{R}_{\mathrm{max}}\otimes_{\mathbb{Z}}\mathbb{Z}_{\ell})$
is the subgroup $\operatorname{M}_{2}(\mathbb{Z}_{\ell})$ and moreover for
each prime $\ell\mid N^{+}$ we require that
$\iota_{\ell}(\mathcal{R}\otimes_{\mathbb{Z}}\mathbb{Z}_{\ell})$ is the
subgroup consisting of matrices which are upper triangular modulo $N^{+}$.
Define $V_{1}(N^{+})$ to be the subgroup of
$\hat{\mathcal{R}}^{\times}_{\mathrm{max}}$ to consisting of elements $x$ such
that $\iota_{\ell}(x)\equiv\bigl{(}\begin{smallmatrix}*&*\\\
0&1\end{smallmatrix}\bigr{)}\pmod{N^{+}}$ (as a general notation, for any
$\mathbb{Z}$-algebra $A$, let $\hat{A}=A\otimes_{\mathbb{Z}}\hat{\mathbb{Z}}$,
where $\hat{\mathbb{Z}}$ is the pro finite completion of $\mathbb{Z}$).

We need to fix a convenient basis for the $\mathbb{Q}$-algebra $\mathcal{B}$,
called _Hashimoto model_. Denote $M=\mathbb{Q}(\sqrt{p_{o}})$ the splitting
field of the quadratic polynomial $X^{2}-p_{0}$, where $p_{0}$ if an auxiliary
prime number fixed as in [Mor11, §1.1] and [HB15, §2.1], such that:

  1. (1)

for all primes $\ell$ we have $(p_{0},pN^{-})_{\ell}=-1$ if and only if
$\ell\mid pN^{-}$, where $(a,b)_{\ell}$ denotes the Hilbert symbol,

  2. (2)

all primes $\ell\mid N^{+}$ are split in the real quadratic field
$M=\mathbb{Q}(\sqrt{p_{0}})$, where $\sqrt{p_{0}}$ is a square root of $p_{0}$
in $\bar{\mathbb{Q}}$.

The choice of $p_{0}$ fixes a $\mathbb{Q}$-basis of $\mathcal{B}$ as in
[Has95, §2] given by $\\{\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}\\}$ with
$\mathbf{i}^{2}=-pN^{-}$, $\mathbf{j}^{2}=p_{0}$,
$\mathbf{k}=\mathbf{ij}=-\mathbf{ji}$, and $\mathbf{1}$ the unit of
$\mathcal{B}$; of course, if $x\in\mathbb{Q}$ we will often just write $x$ for
$x\cdot\mathbf{1}$.

###  2.2. Moduli problem

A _false elliptic curve_ $A$ over a scheme $S$ is a an abelian scheme
$A\rightarrow S$ of relative dimension $2$ equipped with an embedding
$\iota_{A}:\mathcal{R}_{\mathrm{max}}\hookrightarrow\operatorname{End}_{S}(A)$.
An _isogeny of false elliptic curves_ is an isogeny which commutes with the
action of $\mathcal{R}_{\mathrm{max}}$. A _full level $N^{+}$-structure_ on
$A$ is an isomorphism of group schemes
$\alpha_{A}:A[N^{+}]\simeq(\mathcal{R}_{\mathrm{max}}\otimes_{\mathbb{Z}}(\mathbb{Z}/N^{+}\mathbb{Z}))_{S}$,
where for any group $G$ we denote $G_{S}$ the constant group scheme $G$ over
$S$. Note that $\hat{\mathcal{R}}^{\times}_{\mathrm{max}}$ acts from the right
on $A[N^{+}]$ and
$\mathcal{R}_{\mathrm{max}}\otimes_{\mathbb{Z}}(\mathbb{Z}/N^{+}\mathbb{Z})$
through the canonical projection
$\hat{\mathcal{R}}_{\mathrm{max}}^{\times}\rightarrow\prod_{\ell\mid{N^{+}}}(\mathcal{R}_{\mathrm{max}}\otimes_{\mathbb{Z}}\mathbb{Z}_{\ell})^{\times}$.
A _level structure of $V_{1}(N^{+})$-type_ is an equivalence class of full
level $N^{+}$-structures under the (right) action of $V_{1}(N^{+})$.

The moduli problem which associates to any $\mathbb{Z}[1/Np]$-scheme $S$ the
set of isomorphism classes of false elliptic curves equipped with a
$V_{1}(N^{+})$-level structure is representable by a smooth proper scheme
$\mathcal{C}$ defined over $\operatorname{Spec}(\mathbb{Z}[1/Np])$ ([HB15,
Theorem 2.2]). Let $\pi:\mathcal{A}\twoheadrightarrow\mathcal{C}$ be the
universal false elliptic curve. For any $\mathbb{Z}[1/Np]$-algebra $R$, let
$\pi_{R}:\mathcal{A}_{R}\twoheadrightarrow\mathcal{C}_{R}$ be the base change
of $\pi$ to $R$.

###  2.3. Algebraic de Rham cohomology

We review some preliminaries on the algebraic de Rham cohomology of Shimura
curves, including the Gauss-Manin connection and the Kodaira-Spencer map,
referring for details to [KO68], and especially to [Mor11, §2.1] for the case
under consideration of Shimura curves.

We first recall some general notation. For any morphism of schemes
$\phi:X\rightarrow S$, denote $(\Omega^{\bullet}_{X/S},d^{\bullet}_{X/S})$, or
simply $\Omega^{\bullet}_{X/S}$ understanding the differentials
$d^{\bullet}_{X/S}$, the complex of sheaves of relative differential forms for
the morphism $\phi$. For a sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-modules
over a scheme $X$, we denote $\mathcal{F}^{\vee}$ its $\mathcal{O}_{X}$-linear
dual and, for a positive integer $k$, we put $\mathcal{F}^{\otimes
k}=\mathcal{F}\otimes_{\mathcal{O}_{X}}\dots\otimes_{\mathcal{O}_{X}}\mathcal{F}$
($k$ factors). If $\mathcal{F}$ is invertible, we denote $\mathcal{F}^{-1}$
its inverse, and in this case for $k$ a negative integer,
$\mathcal{F}^{\otimes{k}}$ denotes $(\mathcal{F}^{-1})^{\otimes{k}}$ as usual.

Fix a field $F$ of characteristic zero. The relative de Rham cohomology bundle
for the morphism
$\mathcal{A}_{F}\overset{\pi_{F}}{\rightarrow}\mathcal{C}_{F}$ is defined by

$\mathcal{H}^{q}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})=\mathbb{R}^{q}\pi_{F*}\left(\Omega^{\bullet}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\right).$

We first recall the construction of the Gauss-Manin connection. We have a
canonical short exact sequence of locally free sheaves

(6)
$0\longrightarrow\pi_{F}^{*}\left(\Omega^{1}_{\mathcal{C}_{F}/F}\right)\longrightarrow\Omega^{1}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\longrightarrow\Omega^{1}_{\mathcal{A}_{F}/F}\longrightarrow
0$

(the exactness is because $\pi_{F}$ is smooth). This exact sequence induces
maps

$\Omega^{\bullet-i}_{\mathcal{A}_{F}/F}\otimes_{\mathcal{O}_{\mathcal{A}_{F}}}\pi_{F}^{*}\left(\Omega^{i}_{\mathcal{C}_{F}/F}\right)\overset{\psi^{i}_{\mathcal{A}_{F}/F}}{\longrightarrow}\Omega^{\bullet}_{\mathcal{A}_{F}/F}$

for each integer $i$, defining a filtration
$F^{i}\Omega^{\bullet}_{\mathcal{A}_{F}/F}=\mathrm{Im}(\psi^{i}_{\mathcal{A}_{F}/F})$
on $\Omega^{\bullet}_{\mathcal{A}_{F}/F}$ with associated graded objects

$\mathrm{gr}^{i}\left(\Omega^{\bullet}_{\mathcal{A}_{F}/F}\right)=\Omega^{\bullet-i}_{\mathcal{A}_{F}/F}\otimes_{\mathcal{O}_{\mathcal{A}_{F}}}\pi_{F}^{*}\left(\Omega^{i}_{\mathcal{C}_{F}/F}\right).$

Let $E_{i}^{p,q}$ denote the spectral sequence associated with this
filtration. The $E_{1}^{p,q}$ terms are then given by
$E_{1}^{p,q}=\mathbb{R}^{p+q}\pi_{F*}\left(\mathrm{gr}^{p}\left(\Omega^{\bullet}_{\mathcal{A}_{F}/F}\right)\right)$.
Since $\Omega^{p}_{\mathcal{C}_{F}/F}$ is locally free, and the differentials
in the complex
$\pi_{F}^{*}\left(\Omega_{\mathcal{C}_{F}/F}^{p}\right)\otimes_{\mathcal{O}_{\mathcal{A}_{F}}}\Omega^{\bullet}_{\mathcal{A}_{F}/\mathcal{C}_{F}}$
are $\pi^{-1}(\mathcal{O}_{\mathcal{C}_{F}})$-linear, one can show that

$E_{1}^{p,q}\simeq\Omega^{p}_{\mathcal{C}_{F}/F}\otimes_{\mathcal{O}_{\mathcal{C}_{F}}}\mathcal{H}^{q}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})$

([KO68, (7)]). The Gauss-Manin connection

$\nabla:\mathcal{H}^{i}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})\longrightarrow\Omega^{1}_{\mathcal{C}_{F}/F}\otimes_{\mathcal{O}_{\mathcal{C}_{F}}}\mathcal{H}^{i}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})$

is then defined as the differential $d_{1}^{0,i}:E_{1}^{0,i}\rightarrow
E_{1}^{1,i}$ in this spectral sequence.

We now recall various descriptions of the Kodaira-Spencer map. It is defined
to be the boundary map

$\mathrm{KS}_{\mathcal{A}_{F}/\mathcal{C}_{F}}:\pi_{F*}\left(\Omega^{1}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\right)\longrightarrow\mathbb{R}^{1}\pi_{F*}\left(\pi_{F}^{*}\left(\Omega^{1}_{\mathcal{C}_{F}/F}\right)\right)$

in the long exact sequence of derived functors obtained from (6). It can also
be reconstructed from the Gauss-Manin connection as follows. Let
$\pi_{F}^{\vee}:\mathcal{A}_{F}^{\vee}\rightarrow\mathcal{C}_{F}$ denote the
dual abelian variety. By a result of Buzzard ([Buz97, Section1], see also
[HB15, page 4183]), it is known that the abelian surface $\mathcal{A}_{F}$ is
equipped with a canonical principal polarization
$\iota_{\mathcal{A}_{F}}:\mathcal{A}_{F}\simeq\mathcal{A}_{F}^{\vee}$ over
$\mathcal{C}_{F}$, which we use to identify $\mathcal{A}_{F}^{\vee}$ and
$\mathcal{A}_{F}$ in the following without explicit mentioning it; we recall
that this polarization is characterized by the fact that the associated Rosati
involution in $\operatorname{End}_{\mathcal{C}_{F}}(\mathcal{A}_{F})$
restricted to the image of $\mathcal{R}_{\mathrm{max}}$ via the map
$\mathcal{R}_{\mathrm{max}}\hookrightarrow\operatorname{End}_{\mathcal{C}_{F}}(\mathcal{A}_{F})$
coincides with the involution $x\mapsto x^{\dagger}$ of
$\mathcal{R}_{\mathrm{max}}$, defined by $x^{\dagger}=i^{-1}\bar{b}i$ (as
usual, if $x=a+bi+cj+dk$, then $\bar{x}=a-bi-cj-dk$). Using the principal
polarization and the isomorphism between
$\mathbb{R}^{1}\pi_{F*}(\Omega^{1}_{\mathcal{A}_{F}/F})$ and the tangent
bundle of $\mathcal{A}^{\vee}_{F}$, the Hodge exact sequence can be written as

(7)
$0\longrightarrow\pi_{F*}\left(\Omega^{1}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\right)\longrightarrow\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})\longrightarrow\left(\pi_{F*}\left(\Omega^{1}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\right)\right)^{\vee}\longrightarrow
0$

(_cf._ [Mor11, (2.2)], [HB15, §2.6]). The Kodaira-Spencer map can be defined
using the Gauss-Manin connection as the composition

$\begin{split}\mathrm{KS}_{\mathcal{A}_{F}/\mathcal{C}_{F}}:\pi_{F*}\left(\Omega^{1}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\right)\overset{\eqref{Hodge}}{\mbox{\;$\lhook\joinrel\longrightarrow$\;}}&\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})\overset{\nabla}{\longrightarrow}\mathcal{H}^{i}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})\otimes_{\mathcal{O}_{\mathcal{C}_{F}}}\Omega^{1}_{\mathcal{C}_{F}/F}\overset{\eqref{Hodge}}{\mbox{\;$\relbar\joinrel\twoheadrightarrow$\;}}\\\
&\overset{\eqref{Hodge}}{\mbox{\;$\relbar\joinrel\twoheadrightarrow$\;}}\left(\pi_{F*}\left(\Omega^{1}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\right)\right)^{\vee}\otimes_{\mathcal{O}_{\mathcal{C}_{F}}}\Omega^{1}_{\mathcal{C}_{F}/F}\end{split}$

in which the first and the last map come from the Hodge exact sequence (7).
Therefore the Kodaira-Spencer map can also be seen as a map of
$\mathcal{O}_{\mathcal{C}_{F}}$-modules, denoted again with the same symbol,

$\mathrm{KS}_{\mathcal{A}_{F}/\mathcal{C}_{F}}:\pi_{F*}\left(\Omega^{1}_{\mathcal{A}_{F}/\mathcal{C}_{F}}\right)^{\otimes
2}\longrightarrow\Omega^{1}_{\mathcal{C}_{F}/F}.$

###  2.4. Idempotents and line bundles

Let

$e=\frac{1}{2}\left(\mathbf{1}\otimes
1+\frac{1}{p_{0}}\mathbf{j}\otimes\sqrt{p_{0}}\right)\in\mathcal{R}_{M}=\mathcal{R}_{\mathrm{max}}\otimes_{\mathbb{Z}}\mathcal{O}_{M}[1/(2p_{0})]$

be the idempotent in [Mor11, (1.10)], [HB15, §2.1], where $\mathcal{O}_{M}$ is
the ring of integers of $M$. We have an isomorphism
$\iota_{M}:B\otimes_{\mathbb{Q}}{M}\simeq\operatorname{M}_{2}(M)$.

Suppose we have an embedding $M\hookrightarrow F$, allowing us to identify $M$
with a subfield of $F$; in the cases we are interested in, either
$F\subseteq\bar{\mathbb{Q}}$ (and then we require that $F$ contains $M$), or
$F=\mathbb{C}$ (and then we view $M\hookrightarrow\mathbb{C}$ via the fixed
embedding $\bar{\mathbb{Q}}\hookrightarrow\mathbb{C}$) or
$F\subseteq\bar{\mathbb{Q}}_{p}$ (and then we require that $F$ contains the
image of $M$ via the fixed embedding
$\bar{\mathbb{Q}}\hookrightarrow\bar{\mathbb{Q}}_{p}$).

Since $M$ is contained in $F$, we have an action of $\mathcal{R}_{M}$ on the
sheaves $\pi_{F*}\left(\Omega_{\mathcal{A}_{F}/\mathcal{C}_{F}}^{1}\right)$
and $\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F})$, and we
may therefore define the invertible sheaf of
$\mathcal{O}_{\mathcal{C}_{F}}$-modules

(8)
$\underline{\omega}_{F}=e\cdot\pi_{F*}\left(\Omega_{\mathcal{A}_{F}/\mathcal{C}_{F}}^{1}\right)$

and the sheaf of $\mathcal{O}_{\mathcal{C}_{F}}$-modules

(9)
$\mathcal{L}_{F}=e\cdot\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{F}/\mathcal{C}_{F}).$

Using that $e$ is fixed by the Rosati involution, the Hodge exact sequence (7)
becomes

$0\longrightarrow\underline{\omega}_{F}\longrightarrow\mathcal{L}_{F}\longrightarrow\underline{\omega}_{F}^{-1}\longrightarrow
0$

(see [HB15, §2.6] for details). For any integer $n\geq 1$, define

(10) $\mathcal{L}_{F,n}=\operatorname{Sym}^{n}(\mathcal{L}_{F}).$

The Gauss-Manin connection is compatible with the quaternionic action ([Mor11,
Proposition 2.2]). Therefore, restricting to $\mathcal{L}_{F,1}$ and using the
Leibniz rule (see for example [HB15, §3.2]), the Gauss-Manin connection
defines a connection

$\nabla_{n}:\mathcal{L}_{F,n}\longrightarrow\mathcal{L}_{F,n}\otimes\Omega^{1}_{\mathcal{C}_{F}/F}.$

By [Mor11, Theorem 2.5], restricting the Kodaira-Spencer map to
$\underline{\omega}_{F}^{\otimes{2}}$ gives an isomorphism

$\mathrm{KS}_{F}:\underline{\omega}_{F}^{\otimes{2}}\simeq\Omega^{1}_{\mathcal{C}_{F}/F}.$

We may then define a map
$\tilde{\nabla}_{n}:\mathcal{L}_{F,n}\rightarrow\mathcal{L}_{F,n+2}$ by the
composition

(11)
$\textstyle{\tilde{\nabla}_{n}:\mathcal{L}_{F,n}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla_{n}}$$\textstyle{\mathcal{L}_{F,n}\otimes_{\mathcal{O}_{\mathcal{C}_{L}}}\Omega^{1}_{\mathcal{C}_{F}/F}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\mathrm{id}\otimes{\mathrm{KS}_{F}^{-1}}}$$\textstyle{\mathcal{L}_{F,n}\otimes_{\mathcal{O}_{\mathcal{C}_{L}}}\underline{\omega}_{F}^{\otimes{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{F,n}\otimes_{\mathcal{O}_{\mathcal{C}_{L}}}\mathcal{L}_{F,2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{F,n+2}}$

where the last map is the product map in the symmetric algebras.

###  2.5. Algebraic modular forms

For any $F$-algebra $R$, we define the $R$-algebra

$S_{k}^{\mathrm{alg}}\left(V_{1}(N^{+}),R)\right)=H^{0}(\mathcal{C}_{R},\underline{\omega}_{R}^{\otimes{k}})$

of _algebraic modular forms_ of weight $k$ and level $V_{1}(N^{+})$ over $R$.
One can show ([HB15, §3.1]) that the $R$-algebra
$S_{k}^{\mathrm{alg}}(V_{1}(N^{+}),R)$ can be alternatively described in
modular terms. Let $R^{\prime}$ be an $R$ algebra. A _test triple_ over
$R^{\prime}$ is a triplet $(A^{\prime},t^{\prime},\omega^{\prime})$ consisting
of a false elliptic curve $A^{\prime}/R^{\prime}$, a $V_{1}(N^{+})$-level
structure $t^{\prime}$ and a global section $\omega^{\prime}$ of
$\underline{\omega}_{A^{\prime}/R^{\prime}}$. An isomorphism of test triples
$(A^{\prime},t^{\prime},\omega^{\prime})$ and
$(A^{\prime\prime},t^{\prime\prime},\omega^{\prime\prime})$ is an isomorphism
of false elliptic curves $\phi:A^{\prime}\rightarrow A^{\prime\prime}$ such
that $\phi(t^{\prime})=t^{\prime\prime}$ and
$\phi^{*}(\omega^{\prime\prime})=\omega^{\prime}$. A _test pair_ over
$R^{\prime}$ is a pair $(A^{\prime},t^{\prime})$ obtained from a test triplet
by forgetting the datum of the global section. Then one can identify global
sections of $\underline{\omega}_{R}^{\otimes{k}}$ with:

  1. (1)

A rule $F$ which assigns, to each $R$-algebra $R^{\prime}$ and each
isomorphism class of test triplets $(A^{\prime},t^{\prime},\omega^{\prime})$
over $R^{\prime}$, an element $F(A^{\prime},t^{\prime},\omega^{\prime})\in
R^{\prime}$, subject to the _base change axiom_ (for all maps of $R$-algebras
$\phi:R^{\prime}\rightarrow R^{\prime\prime}$, we have
$F(A^{\prime},t^{\prime},\phi^{*}(\omega^{\prime\prime}))=F(A^{\prime\prime},\phi(t^{\prime}),\omega^{\prime\prime})$,
where $A^{\prime}$ is the base change of $A^{\prime\prime}$ via $\phi$) and
_the weight $k$ condition_
($F(A^{\prime},t^{\prime},\lambda\omega^{\prime})=\lambda^{-k}F(A^{\prime},t^{\prime},\omega^{\prime})$
for any $\lambda\in(R^{\prime})^{\times}$) ([HB15, Definition 3.2]).

  2. (2)

A rule $F$ which assigns to each $R$-algebra $R^{\prime}$ and each isomorphism
class of test pairs $(A^{\prime},t^{\prime})$ over $R^{\prime}$, a translation
invariant section
$F(A^{\prime},t^{\prime})\in\underline{\omega}^{\otimes{k}}_{A^{\prime}/R^{\prime}}$
subject to the _base change axiom_ (for all maps of $R$-algebras
$\phi:R^{\prime}\rightarrow R^{\prime\prime}$, we have
$F(A^{\prime},t^{\prime})=\phi^{*}(F(A^{\prime\prime},\phi(t^{\prime}))$,
where $A^{\prime}$ is the base change of $A^{\prime\prime}$ via $\phi$)
([HB15, Definition 3.3])).

Let us make the relations between these definition more explicit ([HB15, page
4193]). Given a global section $f\in
H^{0}(\mathcal{C}_{R},\underline{\omega}^{\otimes{k}}_{R})$, we get a function
as in (2) above associating to each test pair $(A^{\prime},t^{\prime})$ over
$R^{\prime}$ the point
$x_{(A^{\prime},t^{\prime})}\in\mathcal{C}_{R}(R^{\prime})$, and taking the
value of $f$ at $x_{(A^{\prime},t^{\prime})}$; if $F$ is as in (2), we get a
function $G$ on test triples $(A^{\prime},t^{\prime},\omega^{\prime})$ over
$R^{\prime}$ as in (1) by the formula
$F(A^{\prime},t^{\prime})=G(A^{\prime},t^{\prime},\omega^{\prime})\omega^{\otimes{k}}$
where $\omega\in\underline{\omega}_{A^{\prime}/R^{\prime}}$ is the choice of
any translation invariant global section.

##  3\. Special values of $L$-series

In this section we review the work of Brooks [HB15] expressing special values
of certain $L$-functions of modular forms in terms of CM-values of the
Shimura-Maass operator applied to the modular form in question.

###  3.1. The real analytic Shimura-Maass operator

We denote
$(X,\mathcal{O}_{X})\leadsto(X^{\mathrm{an}},\mathcal{O}_{X}^{\mathrm{an}})$
the analytification functor which takes a scheme of finite type over
$\mathbb{C}$ to its associated complex analytic space ([Ser56, §2]). For each
sheaf $\mathcal{F}$ of $\mathcal{O}_{X}$-modules on $X$, we also denote
$\mathcal{F}^{\mathrm{an}}$ the analytification of $\mathcal{F}$, and for each
morphism $\varphi:\mathcal{F}\rightarrow\mathcal{G}$ of
$\mathcal{O}_{X}$-modules, we let
$\varphi^{\mathrm{an}}:\mathcal{F}^{\mathrm{an}}\rightarrow\mathcal{G}^{\mathrm{an}}$
the corresponding morphism of analytic sheaves ([Ser56, §3]). If
$(X,\mathcal{O}_{X})$ is an analytic space, we denote
$\mathcal{O}_{X}^{\mathrm{r\text{-}an}}$ the ring of real analytic functions
on $X$; this is a sheaf of $\mathcal{O}_{X}$-modules, and for any sheaf
$\mathcal{F}$ of $\mathcal{O}_{X}$-modules, we let
$\mathcal{F}^{\mathrm{r\text{-}an}}=\mathcal{F}\otimes_{\mathcal{O}_{X}}\mathcal{O}_{X}^{\mathrm{r\text{-}an}}$;
when $\mathcal{F}=\mathcal{F}^{\mathrm{an}}$, we simplify the notation by
writing $\mathcal{F}^{\mathrm{r\text{-}an}}$ instead of
$(\mathcal{F}^{\mathrm{an}})^{\mathrm{r\text{-}an}}$.

Since $\mathcal{C}_{\mathbb{C}}$ is proper and smooth over $\mathbb{C}$, the
analytification functor $\mathcal{F}\leadsto\mathcal{F}^{\mathrm{an}}$ induces
an equivalence of categories between the category of coherent sheaves
$\mathcal{C}_{\mathbb{C}}$ and the category of analytic coherent sheaves of
$\mathcal{O}_{\mathcal{C}_{\mathbb{C}}}^{\mathrm{an}}$-modules ([Ser56, Th. 2,
3]). Also, the analytic sheaf obtained from the sheaf of algebraic de Rham
cohomology
$\mathcal{H}^{i}_{\mathrm{dR}}(\mathcal{A}_{\mathbb{C}}/\mathcal{C}_{\mathbb{C}})$
coincides with the derived functor
$\mathbb{R}^{1}\pi_{\mathbb{C}*}^{\mathrm{an}}\left(\Omega^{1}_{\mathcal{A}_{\mathbb{C}}^{\mathrm{an}}/\mathcal{C}_{\mathbb{C}}^{\mathrm{an}}}\right)$
in the category of analytic sheaves over
$\mathcal{C}_{\mathbb{C}}^{\mathrm{an}}$ ([Ser56, Theorem 1]).

Hodge theory gives a splitting

$\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{\mathbb{C}}/\mathcal{C}_{\mathbb{C}})^{\mathrm{r\text{-}an}}\longrightarrow\left(\pi_{\mathbb{C}*}\left(\Omega^{1}_{\mathcal{A}_{\mathbb{C}}/\mathcal{C}_{\mathbb{C}}}\right)\right)^{\mathrm{r\text{-}an}}$

of the corresponding Hodge exact sequence of real analytic sheaves obtained
from (7). Since this splitting is the identity on the image of
$\left(\pi_{\mathbb{C}*}\left(\Omega^{1}_{\mathcal{A}_{\mathbb{C}}/\mathcal{C}_{\mathbb{C}}}\right)\right)^{\mathrm{r\text{-}an}}$
in
$\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{A}_{\mathbb{C}}/\mathcal{C}_{\mathbb{C}})^{\mathrm{r\text{-}an}}$,
it gives rise to a map
$\Psi_{\infty}:\mathcal{L}_{\mathbb{C},1}^{\mathrm{r\text{-}an}}\rightarrow\underline{\omega}_{\mathbb{C}}^{\mathrm{r\text{-}an}}$
(_cf._ [Mor11, Proposition 2.8]). We may then consider the induced maps
$\Psi_{\infty,n}:\mathcal{L}_{\mathbb{C},n}^{\mathrm{r\text{-}an}}\rightarrow(\underline{\omega}_{\mathbb{C}}^{\otimes{n}})^{\mathrm{r\text{-}an}}$
for any integer $n\geq 1$. Further, the map $\tilde{\nabla}_{n}$ gives rise to
a map
$\tilde{\nabla}_{n}^{\mathrm{r\text{-}an}}:\mathcal{L}_{\mathbb{C},n}^{\mathrm{r\text{-}an}}\rightarrow\mathcal{L}_{\mathbb{C},n+2}^{\mathrm{r\text{-}an}}$
of real analytic sheaves. The composition

$\textstyle{\Theta_{\infty,n}:(\underline{\omega}_{\mathbb{C}}^{\otimes{n}})^{\mathrm{r\text{-}an}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\mathbb{C},n}^{\mathrm{r\text{-}an}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{{\tilde{\nabla}}_{n}^{\mathrm{r\text{-}an}}}$$\textstyle{\mathcal{L}_{\mathbb{C},n+2}^{\mathrm{r\text{-}an}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{\infty,n}}$$\textstyle{(\underline{\omega}_{\mathbb{C}}^{\otimes{n+2}})^{\mathrm{r\text{-}an}}}$

is the real-analytic Shimura-Maas operator.

The effect of $\Theta_{\infty,n}$ on modular forms is described in [HB15,
Proposition 3.4] and [Mor11, Proposition 2.9]. Denote
$\Gamma=\Gamma_{1}(N^{+})$ the subgroup of $\mathcal{B}^{\times}\cap
V_{1}(N^{+})$ consisting of elements of norm equal to $1$. Fix an isomorphism
$\mathcal{B}\otimes_{\mathbb{Q}}\mathbb{R}\simeq\operatorname{M}_{2}(\mathbb{R})$
and denote $\Gamma_{\infty}$ the image of $\Gamma$ in
$\operatorname{GL}_{2}(\mathbb{R})$. Let $S_{k}(\Gamma_{\infty})$ denote the
$\mathbb{C}$-vector space of _holomorphic modular forms_ of weight $k$ and
level $\Gamma_{\infty}$ consisting of those holomorphic functions on
$\mathcal{H}_{\infty}$, the complex upper half plane, such that
$f(\gamma(z))=j(\gamma,z)^{k}f(z)$ for all $\gamma\in\Gamma_{\infty}$; here
$\Gamma_{\infty}$ acts on $\mathcal{H}_{\infty}$ by fractional linear
transformations via the map
$\mathcal{B}\hookrightarrow\mathcal{B}\otimes_{\mathbb{Q}}\mathbb{R}\simeq\operatorname{M}_{2}(\mathbb{R})$.
We have (_cf_. [HB15, §2.7])

$S_{k}(\Gamma_{\infty})\simeq
H^{0}\left(\mathcal{C}_{\mathbb{C}}^{\mathrm{an}},(\underline{\omega}_{\mathbb{C}}^{\otimes{k}})^{\mathrm{an}}\right).$

Define the space $S_{k}^{\mathrm{r\text{-}an}}(\Gamma_{\infty})$ of real
analytic modular forms of level $\Gamma_{\infty}$ and weight $k$ to be the
$\mathbb{C}$-vector space of real analytic functions
$f:\mathcal{H}_{\infty}\rightarrow\mathbb{C}$ such that
$f(\gamma(z))=j(\gamma,z)^{k}f(z)$ for all $\gamma\in\Gamma_{\infty}$. One
then has

(12) $S_{k}^{\mathrm{r\text{-}an}}(\Gamma_{\infty})\simeq
H^{0}\left(\mathcal{C}_{\mathbb{C}}^{\mathrm{an}},(\underline{\omega}_{\mathbb{C}}^{\otimes{k}})^{\mathrm{r\text{-}an}}\right).$

The operator $\Theta_{\infty,k}$ gives then rise to a map
$\delta_{\infty,k}:S_{k}(\Gamma_{\infty})\rightarrow
S_{k+2}^{\mathrm{r\text{-}an}}(\Gamma_{\infty})$ and we have

$\delta_{\infty,k}\left(f(z)\right)=\frac{1}{2\pi
i}\left(\frac{d}{dz}+\frac{k}{z+\bar{z}}\right)f(z).$

###  3.2. CM points and triples

Fix an embedding $\bar{\mathbb{Q}}\hookrightarrow\mathbb{C}$. For any
embedding $\varphi:K\hookrightarrow\mathcal{B}$ there exists a unique
$\tau\in\mathcal{H}$ such that
$\iota_{\infty}(\varphi(K^{\times}))(\tau)=\tau$. The additive map
$K\hookrightarrow\mathbb{C}$ defined by $\alpha\mapsto
j(\iota_{\infty}(\varphi(\alpha)),\tau)$ gives an embedding
$K\hookrightarrow\mathbb{C}$; we say that $\varphi$ is _normalized_ if
$\alpha\mapsto j(\iota_{\infty}(\varphi(\alpha)),\tau)$ is the identity (with
respect to our fixed embedding $\bar{\mathbb{Q}}\hookrightarrow\mathbb{C}$).

We say that $\tau\in\mathcal{H}$ is a _CM point_ if there exists an embedding
$\varphi:K\hookrightarrow\mathcal{B}$ which has $\tau$ as fixed point as
above, and that a CM point $\tau$ is _normalized_ if $\varphi$ is normalized.
Finally, we say that a CM point $\tau\in\mathcal{H}$ is a _Heegner point_ if
$\varphi(\mathcal{O}_{K})\subseteq\mathcal{R}$ ([HB15, §2.4 and page 4188]).

Fix a CM point $\tau$ corresponding to an embedding
$\varphi:K\hookrightarrow\mathcal{B}$. Let $\mathfrak{a}$ be an integral ideal
of $\mathcal{O}_{K}$, and define the $\mathcal{R}_{\mathrm{max}}$-ideal
$\mathfrak{a}_{\mathcal{B}}=\mathcal{R}_{\mathrm{max}}\cdot\varphi(\mathfrak{a})$.
This ideal is principal, generated by an element
$\alpha=\alpha_{\mathfrak{a}}\in\mathcal{B}$. Right multiplication by $\alpha$
gives an isogeny $A_{\tau}\rightarrow A_{\alpha^{-1}\tau}$, whose kernel is
$A_{\tau}[\mathfrak{a}]$. Let $\Gamma_{\mathrm{max}}$ be the subgroup of
$\mathcal{R}_{\mathrm{max}}^{\times}$ consisting of elements of norm equal to
$1$. The image of $\alpha\tau$ by the canonical projection map
$\rho_{\mathrm{max}}:\mathcal{H}\rightarrow\Gamma_{\mathrm{max}}\backslash\mathcal{H}$
does not depend on the choice of the representative $\alpha$, and therefore
one may write $A_{\mathfrak{a}\star\tau}$ for the corresponding abelian
surface. Shimura’s reciprocity law states that $\rho_{\mathrm{max}}(\tau)$ is
defined over the Hilbert class field $H$ of $K$, and that
$\rho_{\mathrm{max}}(\tau)^{(\mathfrak{a}^{-1},H/K)}=\rho_{\mathrm{max}}(\mathfrak{a}\star\tau)$,
where $(\mathfrak{a}^{-1},H/K)$ denotes the Artin symbol.

Fix a primitive $N^{+}$-root of unity $\zeta$. Fix a normalized Heegner point
$\tau$, and fix a point $P_{\tau}\in A_{\tau}[N^{+}]$ of exact order $N^{+}$
such that $e\cdot P=P$. Let $(A_{\tau},P_{\tau})$ denote the point on
$\mathcal{C}(F)$ corresponding to the level structure
$\mu_{N^{+}}\times\mu_{N^{+}}\simeq\mathbb{Z}/N^{+}\mathbb{Z}\times\mathbb{Z}/N^{+}\mathbb{Z}\rightarrow
A_{\tau}[N^{+}]$ which takes
$(1,0)\in\mathbb{Z}/N^{+}\mathbb{Z}\times\mathbb{Z}/N^{+}\mathbb{Z}$ to
$P_{\tau}$. A _CM triple_ is an isomorphism class of triples
$(A_{\tau},P_{\tau},\omega_{\tau})$ with $(A_{\tau},P_{\tau})$ as above and
$\omega_{\tau}\in e\cdot\Omega_{A_{\tau}/F}$ non vanishing.

There is an action of $\mathrm{Cl}(\mathcal{O}_{K})$ on the set of CM triples,
given by

$\mathfrak{a}\star(A_{\tau},P_{\tau},\pi^{*}(\omega))=(A_{\tau}/A_{\tau}[\mathfrak{a}],\pi(P_{\tau}),\omega)$

where $\pi:A_{\tau}\twoheadrightarrow A_{\tau}/A_{\tau}[\mathfrak{a}]$ is the
canonical projection.

###  3.3. Special value formulas

Fix a CM triple $(A,P,\omega)=(A_{\tau},P_{\tau},\omega_{\tau})$ with $\omega$
defined over $H$, the Hilbert class field of $K$; recall that $A$ is also
defined over $H$, while in general $P$ is only defined over a field $L$ as in
§2.2.

The complex structure $J_{\tau}$ on $\operatorname{M}_{2}(\mathbb{R})$ defines
a differential form $\omega_{\mathbb{C}}=J_{\tau}^{*}(2\pi idz_{1})$, and let
$\Omega_{\infty}\in\mathbb{C}$ be define by
$\omega=\Omega_{\infty}\cdot\omega_{\mathbb{C}}$; clearly, different choices
of $\omega$ correspond to changing $\Omega$ by a multiple in $H$.

We now let $f$ be a modular form of weight $k$, level
$\Gamma_{1}(N^{+})\cap\Gamma_{0}(N^{-})$, and character $\varepsilon_{f}$, and
let $f^{\mathrm{JL}}$ be the modular form on the Shimura curve
$\mathcal{C}_{\mathbb{C}}$ associated with $f$ by the Jacquet-Langlands
correspondence. We can normalise the choice of $f^{\mathrm{JL}}$ so that the
ration $\langle f,f\rangle/\langle f^{\mathrm{JL}},f^{\mathrm{JL}}\rangle$
belongs to $K$ ([HB15, §2.7 and page 4232]).

Let $\Sigma^{(2)}$ be the set of Hecke characters $\chi$ of $K$ of infinite
type $(\ell_{1},\ell_{2})$ with $\ell_{1}\geq k$ and $\ell_{2}\leq 0$. We say
that $\chi\in\Sigma^{(2)}$ is _central critical_ if $\ell_{1}+\ell_{2}=k$, so
that the infinite type of $\chi$ is $(k+j,-j)$ for some integer $j\geq 0$.
Denote $\Sigma_{\mathrm{cc}}^{(s)}$ the subset of $\Sigma^{(2)}$ consisting of
central critical characters.

For each positive integer $j$, let
$\delta_{\infty,k}^{j}:S_{k}^{\mathrm{r\text{-}an}}(\Gamma_{\infty})\rightarrow
D_{k+2j}^{\mathrm{r\text{-}an}}(\Gamma_{\infty})$ denote the $j$-th iterate of
the Shimura-Mass operator defined by

$\delta_{\infty,k}^{j}=\delta_{\infty,k+2(j-1)}\circ\dots\circ\delta_{\infty,k+2}\circ\delta_{\infty,k}.$

For any Hecke character, one may consider the $L$-function $L(f,\chi^{-1},s)$,
and for $\chi\in\Sigma^{(2)}$ central critical define the algebraic part
$L_{\mathrm{alg}}(f,\chi^{-1})$ of its special value at $s=0$ as in [HB15,
Proposition 8.7]. By [HB15, Proposition 8.7], if
$\chi\in\Sigma^{(2)}_{\mathrm{cc}}$ then
$L_{\mathrm{alg}}(f,\chi^{-1})\in\bar{\mathbb{Q}}$, and we have

$L_{\mathrm{alg}}(f,\chi^{-1})=\left(\sum_{\mathfrak{a}\in\mathrm{Cl}(\mathcal{O}_{K})}\chi_{j}^{-1}(\mathfrak{a})\cdot\delta_{\infty,k}^{j}(f^{\mathrm{JL}})(\mathfrak{a}\star(A,t,\omega))\right)^{2}$

where $\chi_{j}=\chi\cdot\mathrm{nr}^{-j}$ and $\mathrm{nr}$ is the norm map
on ideals of $\mathcal{O}_{K}$. In this formula we view the real analytic
modular $\delta_{\infty,k}(f^{\mathrm{JL}})$ as a function on test triplets,
as in [HB15, Proposition 8.5] via (12) (see also the discussion in [BDP13,
page 1094] in the $\operatorname{GL}_{2}$ case).

For each ideal class $\mathfrak{a}$ in $\mathrm{Cl}(\mathcal{O}_{K})$, let
$\alpha_{\mathfrak{a}}$ be the corresponding element in $\mathcal{B}$, as in
§3.2. Then using the dictionary between real analytic forms as functions on
$\mathcal{H}$ or functions on test triples, and recalling that $A=A_{\tau}$
for a normalized Heegner point $\tau$, we have

$L_{\mathrm{alg}}(f,\chi^{-1})=\left(\Omega_{\infty}^{k+2j}\cdot\sum_{\mathfrak{a}\in\mathrm{Cl}(\mathcal{O}_{K})}\chi_{j}^{-1}(\mathfrak{a})\cdot\delta_{\infty,k}^{j}(f^{\mathrm{JL}})(\alpha_{\mathfrak{a}}\cdot\tau)\right)^{2}.$

In this formula we view $\delta_{\infty,k}^{j}(f^{\mathrm{JL}})$ as a function
on $\mathcal{H}$.

##  4\. The Shimura-Maass operator on the $p$-adic upper half plane

In this Section we define a $p$-adic Shimura-Maass operator in the context of
Drinfel’d upper half plane. These results will be used in the next section to
define a $p$-adic Shimura-Maass operator on Shimura curves, whose values at CM
points will be compared with their complex analogue. As in the complex case,
we will see that this operator plays a special role in defining $p$-adic
$L$-functions.

Let $\hat{\mathcal{H}}_{p}$ denote Drinfel’d $p$-adic upper half plane; this
is a $\mathbb{Z}_{p}$-formal scheme, and we denote $\mathcal{H}_{p}$ its
generic fiber, which is a $\mathbb{Q}_{p}$-rigid space ([BC91, Chapitre I]).

###  4.1. Drinfel’d Theorem

Denote $D$ the unique division quaternion algebra over $\mathbb{Q}_{p}$, and
let $\mathcal{O}_{D}$ be its maximal order. The field $\mathbb{Q}_{p^{2}}$ can
be embedded in $D$, and in the following we will see it as a maximal
commutative subfield of $D$ without explicitly mentioning it. Let $\sigma$
denote the absolute Frobenius automorphism of
$\operatorname{Gal}(\mathbb{Q}_{p}^{\mathrm{unr}}/\mathbb{Q}_{p})$. If fix an
element $\Pi\in\mathcal{O}_{D}$ such that $\Pi^{2}=p$ and $\Pi x=\sigma(x)\Pi$
for $x\in\mathbb{Q}_{p^{2}}$, then $D=\mathbb{Q}_{p^{2}}[\Pi]$. We will denote
$x\mapsto\bar{x}$ the restriction of $\sigma$ to
$\operatorname{Gal}(\mathbb{Q}_{p^{2}}/\mathbb{Q}_{p})$.

For any $\mathbb{Z}_{p}$-algebra $B$, a _formal $\mathcal{O}_{D}$-module over
$B$_ is a commutative $2$-dimensional formal group $G$ over $B$ equipped with
an embedding $\iota_{G}:\mathcal{O}_{D}\hookrightarrow\operatorname{End}(G)$.
A formal $\mathcal{O}_{D}$-module is said to be _special_ if for each
geometric point $P$ of $\operatorname{Spec}(B/pB)$, the representation of
$\mathcal{O}_{D}/\Pi\mathcal{O}_{D}$ on the tangent space
$\operatorname{Lie}(G_{P})$ of $G_{P}=G\times k_{P}$ is the sum of two
distinct characters of $\mathcal{O}_{D}/\Pi\mathcal{O}_{D}$, where $k_{P}$ is
the residue field of $P$; see [Tei89, Definition 1] for more details on this
definition. By an _$\operatorname{SFD}$ -module over $B$_, we mean a special
formal $\mathcal{O}_{D}$-module over $B$. If $G$ is a
$\operatorname{SFD}$-module over $B\in\mathrm{Nilp}$, we denote
$\mathbb{M}(G)$ the (covariant) Cartier-Dieudonné module of $G$ ([BC91,
Chapitre II, §1]); we also denote $\mathbb{F}_{G}$ and $\mathbb{V}_{G}$ (or
simply $\mathbb{F}$ and $\mathbb{V}$ when there is no confusion) the Frobenius
and Verschiebung endomorphisms of $\mathbb{M}(G)$. If $B$ is
$\mathbb{Z}_{p^{2}}$-algebra, and $G$ a formal $\mathcal{O}_{D}$-module, then
we may define

$\operatorname{Lie}^{0}(G)=\\{m\in\operatorname{Lie}(G):\iota_{G}(a)=am,a\in\mathbb{Z}_{p^{2}}\\},$
$\operatorname{Lie}^{1}(G)=\\{m\in\operatorname{Lie}(G):\iota_{G}(a)=\bar{a}m,a\in\mathbb{Z}_{p^{2}}\\}$

and, since $G$ is special, both $\operatorname{Lie}^{0}(G)$ and
$\operatorname{Lie}^{1}(G)$ are free $B$-modules of rank $1$, (recall that
$\bar{x}=\sigma(x)$, so $x\mapsto\bar{x}$ is the non-trivial automorphism of
$\operatorname{Gal}(\mathbb{Q}_{p^{2}}/\mathbb{Q}_{p})$). Moreover,
$\mathbb{M}(G)$ is also equipped with a graduation
$\mathbb{M}(G)=\mathbb{M}^{0}(G)\oplus\mathbb{M}^{1}(G)$ where

$\mathbb{M}^{0}(G)=\\{m\in\mathbb{M}(G):\iota_{G}(a)=am,a\in\mathbb{Z}_{p^{2}}\\},$
$\mathbb{M}^{1}(G)=\\{m\in\mathbb{M}(G):\iota_{G}(a)=\bar{a}m,a\in\mathbb{Z}_{p^{2}}\\}.$

Fix a $\operatorname{SFD}$-module $\Phi=G\times G$ over
$\bar{\mathbb{F}}_{p}$, where $G$ is the reduction modulo $p$ of a Lubin-Tate
formal group $\hat{\mathcal{E}}$ of height $2$ over
$\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$, the completion of the valuation ring of
the maximal unramifed extension $\mathbb{Z}_{p}^{\mathrm{unr}}$ of
$\mathbb{Z}_{p}$; so $\hat{\mathcal{E}}$ is the formal group of a
supersingular elliptic curve $\mathcal{E}$ over
$\mathbb{Z}^{\mathrm{unr}}_{p}$ (see [Tei89, Definition 9 and Remark 27]). The
Dieudonné module $\mathbb{M}(\Phi)$ of $\Phi$ is the
$\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}[\mathbb{F},\mathbb{V}]$-module with
$\mathbb{V}$-basis $g^{0}$ and $g^{1}$, satisfying the relations
$\mathbb{F}(g^{0})=\mathbb{V}(g^{0})$ and
$\mathbb{F}(g^{1})=\mathbb{V}(g^{1})$. The quaternionic order
$\mathcal{O}_{D}$ acts via the rules $\Pi(g^{0})=\mathbb{V}(g^{1})$,
$\Pi(g^{1})=\mathbb{V}(g^{0})$ and $a(g^{0})=ag^{0}$, $a(g^{1})=\bar{a}g^{1}$
for $a\in\mathbb{Z}_{p^{2}}$ (viewed inside $\mathcal{O}_{D}$ by the fixed
embedding $\mathbb{Q}_{p^{2}}\hookrightarrow D$), where $a\mapsto\bar{a}$ is
the non-trivial automorphism of
$\operatorname{Gal}(\mathbb{Q}_{p^{2}}/\mathbb{Q}_{p})$. By [Tei89, Corollary
30], $\eta^{0}(\Phi)$ is generated over $\mathbb{Z}_{p}$ by $[g^{0},0]$ and
$[\mathbb{V}(g^{1}),0]$, and $\eta^{1}(\Phi)$ is generated over
$\mathbb{Z}_{p}$ by $[g^{1},0]$ and $[\mathbb{V}(g^{0}),0]$.

Let $\mathrm{Nilp}$ denote the category of $\mathbb{Z}_{p}$-algebras in which
$p$ is nilpotent. Denote $\mathrm{SFD}$ the functor on $\mathrm{Nilp}$ which
associates to each $B\in\mathrm{Nilp}$ the set $\operatorname{SFD}(B)$ of
isomorphism classes of triples $(\psi,G,\rho)$ where

  1. (1)

$\psi:\bar{\mathbb{F}}_{p}\rightarrow B/pB$ is an homomorphism,

  2. (2)

$G$ is a $\mathrm{SFD}$-module over $B$ of height $4$,

  3. (3)

$\rho:\psi_{*}\Phi\rightarrow G_{B/pB}=G\otimes_{B}{B/pB}$ is a quasi-isogeny
of height $0$, called _rigidification_.

See [Tei89, page 663] or [BC91, Chapitre II (8.3)] for more details on the
definition of the functor $\operatorname{SFD}$.

Drinfel’d shows in [Dri76] that the functor $\operatorname{SFD}$ is
represented by the $\mathbb{Z}_{p}$-formal scheme

$\hat{\mathcal{H}}_{p}^{\mathrm{unr}}=\hat{\mathcal{H}}_{p}\hat{\otimes}_{\mathbb{Z}_{p}}\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$

(see [Tei89, Theorem 28], [BC91, Chapitre II (8.4)]). Note that
$\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$, considered as
$\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$-formal scheme, represents the
restriction $\overline{\operatorname{SFD}}$ of $\operatorname{SFD}$ to the
category $\overline{\mathrm{Nilp}}$ of
$\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$-algebras in which $p$ is nilpotent
(_cf._ [BC91, Chapitre II, §8]). Unless otherwise stated, we will see
$\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$ as a
$\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$-formal scheme.

For later use, we review some of the steps involved in the proof of Drinfel’d
Theorem. The crucial step is the interpretation of the $\mathbb{Z}_{p}$-formal
scheme $\hat{\mathcal{H}}_{p}$ as the solution of a moduli problem. For
$B\in\mathrm{Nilp}$, a _compatible data on $S=\operatorname{Spf}(B)$_ consists
of a quadruplet $(\eta,T,u,\rho)$ where

  1. (1)

$\eta=\eta^{0}\oplus\eta^{1}$ is a sheaf of flat
$\mathbb{Z}/2\mathbb{Z}$-graded $\mathbb{Z}_{p}[\Pi]$-modules on $S$,

  2. (2)

$T=T^{0}\oplus T^{1}$ is a $\mathbb{Z}/2\mathbb{Z}$-graded sheaf of
$\mathcal{O}_{S}[\Pi]$-modules with $T^{i}$ invertible,

  3. (3)

$u:\eta\rightarrow T$ is a homogeneous degree zero map such that $u\otimes
1:\eta\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{S}\rightarrow T$ is surjective,

  4. (4)

$\rho:(\mathbb{Q}_{p}^{2})_{S}\rightarrow\eta_{0}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$
is a $\mathbb{Q}_{p}$-linear isomorphism,

which satisfy natural compatibilities, denoted $\mathbf{(C1)}$,
$\mathbf{(C2)}$, $\mathbf{(C3)}$ in [Tei89, page 652], to which we refer for
details. The first step in Drinfel’d work is to show that the
$\mathbb{Z}_{p}$-formal sheme $\hat{\mathcal{H}}_{p}$ represents the functor
which associates to each $B\in\mathrm{Nilp}$ the set of admissible quadruplets
over $B$. To each compatible data $\mathcal{D}=(\eta,T,u,\rho)$ on $S$ one
associates a $S$-valued point $\Psi:S\rightarrow\hat{\mathcal{H}}_{p}$ of
$\hat{\mathcal{H}}_{p}$, as explained in [Tei89, pages 652-655]. The second
step to prove the representability of $\overline{\operatorname{SFD}}$ is to
associate with any $B\in\overline{\mathrm{Nilp}}$ and
$X=(\psi,G,\rho)\in\operatorname{SFD}(B)$ a quadruplet
$(\eta_{X},T_{X},u_{X},\rho_{X})$ which corresponds to an
$S=\operatorname{Spf}(B)$-valued point on
$\hat{\mathcal{H}}_{p}\hat{\otimes}_{\mathbb{Z}_{p}}\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$.
If $X=(\psi,G,\rho)\in\operatorname{SFD}(B)$ is given as above, the quadruplet
$(\eta_{X},T_{X},u_{X},\rho_{X}$) can be explicitly constructed as follows:

  * •

$T_{X}=T(G)=\mathbb{M}(G)/\mathbb{V}\mathbb{M}(G)$ the tangent space to $G$ at
the origin, equipped with its graduation defined previously;

  * •

Define $\mathbb{N}(G)=\mathbb{M}(G)\times\mathbb{M}(G)/\sim$ where
$(\mathbb{V}(x),0)\sim(0,\Pi(x))$; we denote $[x,y]$ the class in $\mathbb{N}$
represented by the pair $(x,y)$. Let
$\lambda:\mathbb{N}(G)\rightarrow\mathbb{M}(G)$ be the map
$\lambda([x,y])=\Pi(x)-\mathbb{V}(y)$. There is a map
$\mathbb{N}(G)\rightarrow\mathbb{M}(G)/\mathbb{V}\mathbb{M}(G)$ induced by the
projection onto the first component. Further, one easily shows that there
exists a unique map $L:\mathbb{M}(G)\rightarrow\mathbb{N}(G)$ satisfying the
relation $\lambda\circ L=\mathbb{F}$. Let
$\phi:\mathbb{N}(G)\rightarrow\mathbb{N}(G)$ be defined by
$\phi([x,y])=L(x)+[y,0]$. Then
$\eta_{X}=\eta(G)=\mathbb{N}^{\phi=\mathrm{Id}}$. The graduation of
$\mathbb{M}(G)$ defines a graduation $\eta(G)=\eta^{0}(G)\oplus\eta^{1}(G)$.

  * •

$u_{X}=u(G)$ is induced by the projection map
$\mathbb{N}(G)\rightarrow\mathbb{M}(G)/\mathbb{V}\mathbb{M}(G)$.

  * •

Fix an isomorphism $\eta^{0}(\Phi)\simeq\mathbb{Z}_{p}\oplus\mathbb{Z}_{p}$.
The quasi-isogeny $\rho$ induces a map

$\rho_{X}=\rho(G):\mathbb{Q}_{p}\oplus\mathbb{Q}_{p}\simeq\eta^{0}(\psi_{*}(\Phi))\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\rightarrow\eta^{0}(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}.$

We finally discuss rigid analytic parameters ([Tei89]). With an abuse of
notation, let $\operatorname{SFD}$ be the functor from the category
$\mathrm{pro\text{-}Nilp}$ of projective limits of objects in $\mathrm{Nilp}$
associated with $\operatorname{SFD}$. In [Tei89, Def. 10], Teitelbaum
introduces a function

(13)
$z_{0}:\operatorname{SFD}(\hat{\mathbb{Z}}_{p}^{\mathrm{unr}})\longrightarrow\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$

such that the map $X=(\psi,G,\rho)\mapsto(z_{0}(X),\psi)$ gives a bijection
between $\operatorname{SFD}(\hat{\mathbb{Z}}_{p}^{\mathrm{unr}})$ and
$(\hat{\mathcal{H}}_{p}\hat{\otimes}_{\mathbb{Z}_{p}}\hat{\mathbb{Z}}_{p}^{\mathrm{unr}})(\hat{\mathbb{Z}}_{p}^{\mathrm{unr}})$,
which we identify with the set
$\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})\times\operatorname{Hom}(\hat{\mathbb{Z}}_{p}^{\mathrm{unr}},\hat{\mathbb{Z}}_{p}^{\mathrm{unr}})$.
We call the map $X\mapsto z_{0}(X)$ a _rigid analytic parameter on
$\operatorname{SFD}$_. If we let $\overline{\mathrm{pro\text{-}Nilp}}$ the
category of projective limits of objects in $\overline{\mathrm{Nilp}}$, and we
still denote $\overline{\operatorname{SFD}}$ the restriction of
$\operatorname{SFD}$ to $\overline{\mathrm{pro\text{-}Nilp}}$, this implies
that the map $X=(\psi,G,\rho)\mapsto z_{0}(X)$ gives a bijection between
$\overline{\operatorname{SFD}}(\hat{\mathbb{Z}}_{p}^{\mathrm{unr}})$ and
$\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$. By [Tei89, Thm. 45],
for each $z\in\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$, there
exists triple $X=(\psi,G,\rho)$ in
$\overline{\operatorname{SFD}}(\hat{\mathbb{Z}}_{p}^{\mathrm{unr}})$ such that
$z_{0}(X)=z$.

###  4.2. Filtered $\phi$-modules

Let $F$ be an unramified field extension of $\mathbb{Q}_{p}$. For an integer
$a$, a _$\sigma^{a}$ -isocrystal $\mathbb{E}$ over $F$_ is a pair
$\mathbb{E}=(V,\phi)$ consisting of a finite dimensional $F$-vector space $V$
with a $\sigma^{a}$-linear isomorphism $\phi$ (_i.e._ the isomorphism
$\phi:V\rightarrow V$ satisfies the relation $\phi(xv)=\sigma(x)^{a}\cdot v$
for $x\in F$ and $v\in V$; see [Zin84, Chapter VI, §1]). If $a=1$,
$\sigma$-isocrystals are also called _$F$ -isocrystals_ (here $F$ stands for
Frobenius, do not confuse with our fixed $p$-adic field $F$) of _$\phi$
-modules_, in which case the $\sigma$-linear isomorphism $\phi$ is called
_Frobenius_ (in the following we will use both terminologies of $\phi$-modules
and $F$-isocrystals).

A _filtered $F$-isocrystal_, or a _filtered $\phi$-module_ is a $\phi$-module
$(V,\phi)$ equipped with an exhaustive and separate filtration $F^{\bullet}V$.

If $G$ is a $p$-divisible formal group over $\bar{\mathbb{F}}_{p}$, one can
define its first crystalline cohomology cohomology group as in [Gro74],
[BO78], [BBM82, Définition 2.5.7], in terms of the _crystalline Dieudonné
functor_ (among many other references, see for example [Ill76], [CL98], [dJ98]
for self-contained expositions). In the following we will denote
$H^{1}_{\mathrm{cris}}(G)$ the global sections of the crystalline Dieudonné
functor (defined as in [BBM82, Théorème 4.2.8.1]) tensored over
$\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$ with
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$. By construction,
$H^{1}_{\mathrm{cris}}(G)$ is then an $F$-isocrystal. Moreover, the canonical
isomorphism between $H^{1}_{\mathrm{cris}}(G)$ and the first de Rham
cohomology group $H^{1}_{\mathrm{dR}}(G)$ of $G$ equips
$H^{1}_{\mathrm{cris}}(G)$ with a canonical filtration (arising from the Hodge
filtration in the de Rham cohomology), making $H^{1}_{\mathrm{dR}}(G)$ a
filtered $F$-isocrystal; see [Oda69].

Let $G$ be a $\operatorname{SFD}$-module over $\bar{\mathbb{F}}_{p}$. Then the
$F$-isocrystal $H^{1}_{\mathrm{cris}}(G)$ is a four-dimensional
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector space, equipped with its
$\sigma$-linear Frobenius $\phi_{\mathrm{cris}}(G)$. It is also equipped with
a $D$-module structure
$j_{G}:D\hookrightarrow\operatorname{End}_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\left(H^{1}_{\mathrm{cris}}(G)\right)$
which commutes with $\phi_{\mathrm{cris}}(G)$, and a $\mathbb{Q}_{p}$-algebra
embedding
$i_{G}:\operatorname{M}_{2}(\mathbb{Q}_{p})\hookrightarrow\operatorname{End}_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\left(H^{1}_{\mathrm{cris}}(G)\right)$
induced by the isomorphism
$\operatorname{End}_{\mathcal{O}_{D}}(G)\simeq\operatorname{M}_{2}(\mathbb{Q}_{p})$,
which commutes with the $D$-action. Define
$\phi^{\prime}_{\mathrm{cris}}(G)=j_{G}(\Pi)^{-1}\phi_{\mathrm{cris}}(G)$ and
put

$V_{\mathrm{cris}}(G)=H^{1}_{\mathrm{cris}}(G)^{\phi^{\prime}_{\mathrm{cris}}(G)=\mathrm{Id}}.$

Denote $\phi_{V_{\mathrm{cris}}(G)}=j_{G}(\Pi)_{|{V_{\mathrm{cris}}(G)}}$ the
restriction of $j_{G}(\Pi)$ to ${V_{\mathrm{cris}}(G)}$. Moreover, denote

$(\eta^{i}(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}=\operatorname{Hom}_{\mathbb{Q}_{p}}(\eta^{i}(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p},\mathbb{Q}_{p})$

the $\mathbb{Q}_{p}$-linear dual of
$\eta^{i}(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$.

The following lemma is crucial in what follows, and identifies
$V_{\mathrm{cris}}(G)$ with
$(\eta^{i}(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}$, from which one
deduces a complete description of the filtered $F$-isocrystal
$H^{1}_{\mathrm{cris}}(G)$. It appears in a slightly different version in the
proof of [IS03, Lemma 5.10]. Since we did not find an reference for this fact
in the form we need it, we add a complete proof.

######  Lemma 4.1.

There is a canonical isomorphism
$V_{\mathrm{cris}}(G)\simeq\left(\eta(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\right)^{\vee}$
of $\mathbb{Q}_{p}$-vector spaces. Moreover,
$H^{1}_{\mathrm{cris}}(G)=V_{\mathrm{cris}}(G)\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$,
where the right hand side is equipped with the structure of
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector space given by
$x\cdot(v\otimes\alpha)=v\otimes(\sigma(x)\alpha)$ for $v\in
V_{\mathrm{cris}}(G)$, $x,\alpha\in\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$.
Finally, under this isomorphism the Frobenius $\phi_{\mathrm{cris}}(G)$
corresponds to $\phi_{V_{\mathrm{cris}}(G)}\otimes\sigma$.

###### Proof.

The $F$-isocrystal $H^{1}_{\mathrm{cris}}(G)$ is canonically isomorphic to the
contravariant Dieudonné module of $G$ with $p$ inverted, and with
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-action twisted by the Frobenius
automorphism $\sigma$ of $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$, equipped with
the canonical Frobenius of the contravariant Dieudonnè module (see [BBM82,
4.2.14]). More precisely, denote
$\mathbb{D}(G)=\operatorname{Hom}_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}(\mathbb{M}(G)[1/p],\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$
the $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear dual of the covariant
Dieudonné module $\mathbb{M}(G)$ of $G$ with $p$ inverted, and let
$\mathbb{D}(G)^{\sigma}=\mathbb{D}(G)\otimes_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}},\sigma}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$,
where the tensor product is taken with respect to the Frobenius endomorphism
$\sigma$ of $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$. Then as
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector spaces, we have
$H^{1}_{\mathrm{cris}}(G)\simeq\mathbb{D}(G)^{\sigma}$. Under this isomorphism
the Frobenius $\phi_{\mathrm{cris}}(G)$ is given by the map
$\varphi\mapsto\sigma\circ\varphi\circ\mathbb{V}_{G}$ for
$\varphi\in\mathbb{D}(G)$.

Now, by [BC91, Lemme (5.12)], we have an isomorphism of
$\sigma^{-1}$-isocrystals

(14)
$\left(\mathbb{M}^{i}(G)[1/p],\mathbb{V}_{G}\Pi^{-1}\right)\simeq\left(\eta^{i}(G)\otimes_{{\mathbb{Z}}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}},\sigma^{-1}\right)$

for each index $i=0,1$ (where the action of $\sigma^{-1}$ on
$\eta^{i}(G)\otimes_{{\mathbb{Z}}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$ is
on the second factor only). We may therefore compute $V_{\mathrm{cris}}(G)$ in
terms of the isocrystal
$\left(\eta^{i}(G)\otimes_{{\mathbb{Z}}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}},\sigma^{-1}\right)$.

As above, define
$\mathbb{D}^{i}(G)=\operatorname{Hom}_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}(\mathbb{M}^{i}(G)[1/p],\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$
($\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear dual) and let
$\mathbb{D}^{i}(G)^{\sigma}$ denote the base change
$\mathbb{D}^{i}(G)\otimes_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}},\sigma}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$
via $\sigma$. Since $\mathbb{M}(G)=\mathbb{M}^{0}(G)\oplus\mathbb{M}^{1}(G)$,
we have
$\mathbb{D}(G)^{\sigma}=\mathbb{D}^{0}(G)^{\sigma}\oplus\mathbb{D}^{1}(G)^{\sigma}$,
and we may write any element $\varphi\in\mathbb{D}(G)^{\sigma}$ as a pair
$(\varphi_{0},\varphi_{1})$ with $\varphi_{i}\in\mathbb{D}^{i}(G)^{\sigma}$,
$i=0,1$. By definition, an element
$\varphi=(\varphi_{0},\varphi_{1})\in\mathbb{D}(G)^{\sigma}$ belongs to
$V_{\mathrm{cris}}(G)$ if and only if $\varphi_{i}(V_{G}\Pi^{-1}(m_{i}))$ is
equal to $\sigma^{-1}(\varphi_{i}(m_{i}))$ for all
$m_{i}\in\mathbb{M}^{i}(G)[1/p]$, and for all $i=0,1$. Using (14), identify
$\varphi_{i}$ with a $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear homomorphism
$\varphi_{i}:\eta^{i}(G)\otimes_{\mathbb{Z}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}\rightarrow\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$,
denoted with a slight abuse of notation with the same symbol; then the above
equation describing $V_{\mathrm{cris}}(G)$ becomes
$\varphi_{i}(n\otimes\sigma^{-1}(x))=\sigma^{-1}\varphi_{i}(n\otimes x)$ for
all $n\in\eta^{i}(G)$ and all $x\in\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$, or
equivalently, since $\varphi_{i}$ is
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear, $\varphi_{i}(n\otimes
1)=\sigma^{-1}\varphi_{i}(n\otimes 1)$ for all $n\in\eta^{i}(G)$, and we
conclude that $\varphi_{i}(n\otimes 1)\in\mathbb{Q}_{p}$ for all
$n\in\eta^{i}(G)$. So $\varphi_{i}$ is the
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear extension of a
$\mathbb{Q}_{p}$-linear homomorphism
$\eta^{i}(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\rightarrow\mathbb{Q}_{p}$.
Since $\eta(G)=\eta^{0}(G)\oplus\eta^{1}(G)$, we then conclude that
$V_{\mathrm{cris}}(G)\simeq\left(\eta(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}\right)^{\vee}$
as $\mathbb{Q}_{p}$-vector spaces. If $n_{1},\dots,n_{4}$ is a
$\mathbb{Q}_{p}$-basis of $\eta(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$,
then $dn_{1},\dots,dn_{4}$ defined by $dn_{i}(n_{j})=\delta_{i,j}$ (as usual,
$\delta_{i,j}=1$ if $i=j$ and $0$ otherwise) is a basis of
$(\eta(G)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}$ and, by
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear extension, also of
$(\eta(G)\otimes_{\mathbb{Z}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})^{\vee}$.
If we now base change the $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector space
$(\eta(G)\otimes_{\mathbb{Z}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})^{\vee}$
via $\sigma$, we see that $dn_{1},\dots,dn_{4}$ is still a
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-basis, and we have $(x\cdot
dn_{i})(n_{j})=\sigma(x)\delta_{i,j}$ for all
$x\in\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$. Using the above description of
$H^{1}_{\mathrm{cris}}(G)$ in terms of $\mathbb{D}(G)^{\sigma}$, and the
description of $V_{\mathrm{cris}}(G)$ in terms of $\eta(G)$, we have an
isomorphism of $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector spaces,

$H^{1}_{\mathrm{cris}}(G)\simeq\left(V_{\mathrm{cris}}(G)\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}\right)^{\sigma},$

where the upper index $\sigma$ on the right hand side means that the structure
of $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-vector space is twisted by $\sigma$
as explained above. Moreover, the $\sigma^{-1}$-linear isomorphism
$V_{G}\Pi^{-1}$ of $\mathbb{M}(G)[1/p]$ corresponds to the
$\sigma^{-1}$-linear isomorphism $\sigma^{-1}$ of
$\eta(G)\otimes_{\mathbb{Z}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$ (acting
on the second component only), and therefore the isomorphism
$\varphi\mapsto\sigma\circ\varphi\circ V_{G}$ of $\mathbb{M}(G)[1/p]^{\vee}$
($\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear dual) corresponds to the
isomorphism $\Pi\otimes\sigma$ of
$\left((\eta(G)\otimes_{\mathbb{Z}_{p}}{\mathbb{Q}}_{p})^{\vee}\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}\right)^{\sigma}$
given by $dn_{i}\otimes x\mapsto(dn_{i}\circ\Pi)\otimes\sigma(x)$ where
$(dn_{i}\circ\Pi)(n)=n_{i}(\Pi n)$, which corresponds to
$\phi_{V_{\mathrm{cris}}(G)}\otimes\sigma$ by definition of
$\phi_{V_{\mathrm{cris}}(G)}$. ∎

###  4.3. Filtered convergent $F$-isocrystals on
$\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$

To describe the relative de Rham cohomology of the $p$-adic upper half plane,
we first need some preliminaries on the notion of filtered convergent
$F$-isocrystals introduced in [IS03].

We first recall some preliminaries. Let
$F\subseteq\mathbb{Q}_{p}^{\mathrm{unr}}$ be an unramified extension of
$\mathbb{Q}_{p}$, with valuation ring $\mathcal{O}_{F}$. If
$(X,\mathcal{O}_{X})$ is a $p$-adic $\mathcal{O}_{F}$-formal scheme, we denote
$(X^{\mathrm{rig}},\mathcal{O}_{X}^{\mathrm{rig}})$ the associated $F$-rigid
analytic space (or its generic fiber), and if $\mathcal{F}$ is a sheaf of
$\mathcal{O}_{X}$-modules, we denote $\mathcal{F}^{\mathrm{rig}}$ its
associated sheaf of $\mathcal{O}_{X}^{\mathrm{rig}}$-modules ([Bos14, §7.4],
[BG98, §1]). We say that $X$ is a _$p$ -adic $\mathcal{O}_{F}$-formal scheme_
if $X$ is a $\mathcal{O}_{F}$-formal scheme which is locally of finite type.
We will always assume in the following that $X$ is analytically smooth, so
that $X^{\mathrm{rig}}$ is smooth.

An _enlargement_ of $X$ is a pair $(T,z_{T})$ consisting of a flat $p$-adic
$\mathcal{O}_{F}$-formal scheme $T$ and a morphism of $\mathcal{O}_{F}$-formal
schemes $z_{T}:T_{0}\rightarrow X$ where for each $\mathcal{O}_{F}$-formal
scheme $T$ we denote $T_{0}$ the reduced closed subscheme of the closed
subscheme $T_{1}$ of $T$ defined by the PD ideal $p\mathcal{O}_{T}$.

A _convergent isocrystal on $X$_ (_cf._ [IS03, Definition 3.1]) is a rule
$\mathcal{E}$ which assigns to each enlargement $(T,z_{T})$ of $X$ a coherent
$\mathcal{O}_{T}\otimes_{\mathcal{O}_{F}}F$-module $\mathcal{E}_{T}$ such that
for any morphism $g:T^{\prime}\rightarrow T$ of $\mathcal{O}_{F}$-formal
schemes with $g_{0}:T_{0}^{\prime}\rightarrow T_{0}$ satisfying
$z_{T^{\prime}}=z_{T}\circ g_{0}$ (where $g_{0}$ is induced from $g$), there
is an isomorphism of
$\mathcal{O}_{T^{\prime}}\otimes_{\mathcal{O}_{F}}F$-modules
$\theta_{g}:g^{*}(\mathcal{E}_{T})\simeq\mathcal{E}_{T^{\prime}}$ satisfying
the cocycle condition. The $\mathcal{O}_{T}\otimes_{\mathcal{O}_{F}}F$-module
$\mathcal{E}_{T}$ also seen as rigid analytic
$\mathcal{O}_{T}^{\mathrm{rig}}$-module on the $F$-rigid analytic space
$T^{\mathrm{rig}}$ ([Ogu84, Remark (1.5)]); we distinguish the notation and
write $\mathcal{E}_{T}^{\mathrm{rig}}$ to emphasise this viewpoint. If
$\mathcal{E}$ is a convergent isocrystal over $X$, for each enlargement
$(T,z_{T})$ which is analytically smooth over $\mathcal{O}_{F}$ we have an
integrable connection

(15)
$\nabla_{T}^{\mathrm{rig}}:\mathcal{E}_{T}^{\mathrm{rig}}\longrightarrow\mathcal{E}_{T}^{\mathrm{rig}}\otimes_{\mathcal{O}_{T}^{\mathrm{rig}}}\Omega^{1}_{X^{\mathrm{rig}}}.$

A _convergent $F$-isocrystal on $X$_ (_cf._ [IS03, Definition 3.2]) is a
convergent isocrystal $\mathcal{E}$ on $X$ equipped with an isomorphism of
convergent isocrystals $\phi_{\mathcal{E}}:F^{*}\mathcal{E}\simeq\mathcal{E}$,
where $F$ is the absolute Frobenius of $X_{0}$.

A _filtered convergent $F$-isocrystal on $X$_ (_cf._ [IS03, Definition 3.3])
is a $F$-isocrystal $(\mathcal{E},\phi_{\mathcal{E}})$ such that
$\mathcal{E}^{\mathrm{rig}}_{X}$ is equipped with an exhaustive and separated
decreasing filtration $F^{\bullet}\mathcal{E}^{\mathrm{rig}}_{X}$ of coherent
$\mathcal{O}_{X}^{\mathrm{rig}}$-submodules such that
$\nabla_{X}^{\mathrm{rig}}(\mathcal{F}^{i}\mathcal{E}^{\mathrm{rig}}_{X})$ is
contained in
$\mathcal{F}^{i-1}\mathcal{E}^{\mathrm{rig}}_{X}\otimes_{\mathcal{O}_{X}^{\mathrm{rig}}}\Omega^{1}_{X^{\mathrm{rig}}}$
for all $i$.

We present two explicit examples of filtered convergent $F$-isocrystals. A
third example will be discussed in §4.4.

######  Example 4.2.

The first example (_cf._ [IS03, Example 3.4(a)]) is the identity object of the
additive tensor category of filtered isocrystals on $X$. This is the
convergent isocrystal $\mathcal{E}(\mathcal{O}_{X})$ on $X$ given by the rule
$(T,z_{T})\leadsto\mathcal{O}_{T}\otimes_{\mathcal{O}_{F}}F$ equipped with the
canonical Frobenius and the filtration given by
$F^{i}\mathcal{O}_{X}^{\mathrm{rig}}=\mathcal{O}_{X}^{\mathrm{rig}}$ for
$i\leq 0$ and $F^{i}\mathcal{O}_{X}^{\mathrm{rig}}=0$ for $i>0$ (in _loc.
cit._ this filtered convergent $F$-isocrystal is simply denoted
$\mathcal{O}_{X}$).

######  Example 4.3.

Our second example (_cf._ [IS03, pages 345-346]) is the filtered convergent
$F$-isocrystal $\mathcal{E}(V)$ attached to a representation
$\rho:\operatorname{GL}_{2}\times\operatorname{GL}_{2}\rightarrow\operatorname{GL}(V)$,
where $V$ is a finite dimensional $\mathbb{Q}_{p}$-rational representation,
and $\operatorname{GL}_{2}$ is the algebraic group of invertible matrix over
$\mathbb{Q}_{p}$. First, for a given such representation
$\rho:\operatorname{GL}_{2}\times\operatorname{GL}_{2}\rightarrow\operatorname{GL}(V)$,
let $\rho_{1}$ and $\rho_{2}$ denote the restrictions of $\rho$ to the first
and second $\operatorname{GL}_{2}$-factor, respectively. As convergent
isocrystal,
$\mathcal{E}(V)=V\otimes_{\mathbb{Q}_{p}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})$.
The Frobenius, making it a convergent $F$-isocrystal, is defined by
$\phi_{V}\otimes\phi_{\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})}$,
where $\phi_{V}=\rho_{2}\left(\bigl{(}\begin{smallmatrix}0&p\\\
1&0\end{smallmatrix}\bigr{)}\right)$ (so, the Frobenius depends on $\rho_{2}$
only ). To define the filtration, making it a filtered convergent
$F$-isocrystal, we first recall some preliminaries. First, the filtration only
depends on $\rho_{1}:\operatorname{GL}_{2}\rightarrow\operatorname{GL}(V)$,
and therefore it is enough to define the filtration attached to a given
representation $\rho:\operatorname{GL}_{2}\rightarrow\operatorname{GL}(V)$.
For this, let $\mathcal{P}_{n}$ be the $\mathbb{Q}_{p}$-vector space of
polynomials in one variable $X$ of degree at most $n$, equipped with a right
action of $\operatorname{GL}_{2}$ by $P(X)\cdot
A=(cX+d)^{n}P\left(\frac{aX+b}{cX+d}\right)$ for
$A=\bigl{(}\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\bigr{)}$ and
$P(X)\in\mathcal{P}_{n}$. Then put $V_{n}=\mathcal{P}_{n}^{\vee}$
($\mathbb{Q}_{p}$-linear dual), equipped with the left action of
$\operatorname{GL}_{2}$ by

$(A\cdot\varphi)(P(x))=\varphi(P(X)\cdot A).$

Recall that any representation
$\rho:\operatorname{GL}_{2}\rightarrow\operatorname{GL}(V)$ can be written as
a direct sum of a sum of representations of the form
$V_{1}^{\otimes{m}}\otimes(V_{1}^{\vee})^{\otimes{n}}$, where $n$, $m$ are
non-negative integers. To define the filtration on $\mathcal{E}(V)$ it is then
enough to defined it for $V=V_{1}$. We have a map
$\mathrm{ev}_{X-z}:V_{1}\otimes_{\mathbb{Q}_{p}}\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}\rightarrow\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$
of sheaves on $\mathcal{H}_{p}^{\mathrm{unr}}$ defined by

$\varphi\otimes{f}\longmapsto[z\mapsto(\varphi(X)-z\varphi(1))f(z)].$

We put $F^{0}\mathcal{E}(V_{1})=\mathcal{E}(V_{1})$,
$F^{1}\mathcal{E}(V_{1})=\ker(\mathrm{ev}_{X-z})$ and
$F^{2}\mathcal{E}(V_{1})=0$. This defines the filtered convergent
$F$-isocrystal $\mathcal{E}(V_{1})$ attached to $V_{1}$, and therefore for any
representation
$\rho:\operatorname{GL}_{2}\times\operatorname{GL}_{2}\rightarrow\operatorname{GL}(V)$
we obtain a filtered convergent $F$-isocrystal $\mathcal{E}(V)$.

###  4.4. The filtered convergent $F$-isocrystal of the universal
$\operatorname{SFD}$-module

The third example of filtered convergent $F$-isocrystal arises from relative
de Rham cohomology of the universal $\operatorname{SFD}$-module. Since it is
more articulated that the previous ones, we prefer to keep it in a separate
subsection. We follow [Fal97], [IS03].

Let $(\lambda_{\mathcal{G}},\mathcal{G},\rho_{\mathcal{G}})$ be the universal
triple, arising from the representability of the functor $\operatorname{SFD}$
by $\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$; denote
$\lambda:\mathcal{G}\rightarrow\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$ be
universal map. Let $\mathcal{G}^{\vee}$ be the Cartier dual of $\mathcal{G}$
([Fon77, Chapitre III, §5]), which is equipped with a canonical map
$\lambda^{\vee}:\mathcal{G}^{\vee}\rightarrow\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$.
In this setting one may define a convergent $F$-isocrystal

$\mathcal{E}(\mathcal{G})=\mathbb{R}^{1}\lambda_{*}(\mathcal{O}_{\mathcal{G}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}})$

interpolating crystalline cohomology sheaves ([Ogu84, Theorems (3.1), (3.7)]):
for each enlargement $(T,z_{T})$, the value $\mathcal{E(G)}_{T}$ of
$\mathcal{E(G)}$ at $T$ is defined to be the crystalline cohomology sheaf of
coherent $\mathcal{O}_{T}\otimes_{\mathcal{O}_{F}}F$-modules
$\mathbb{R}^{q}f_{T_{1}*}\mathcal{O}_{\mathcal{G}_{T_{1}/T}}\otimes_{\mathcal{O}_{F}}F$.
The notation adopted here is standard, following [Ogu84, §3]:
$f_{T_{1}}:\mathcal{G}\times_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}T_{1}\rightarrow
T_{1}$ is the canonical projection where we use
$z_{T}:T_{1}\rightarrow\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$ to form the fiber
product $\mathcal{G}\times_{Z}T_{1}$, and
$\mathbb{R}^{q}f_{T_{1}*}\mathcal{O}_{\mathcal{G}_{T_{1}/T}}$ is the
cristalline cohomology sheaf on the formal scheme $T$ (note that
$T_{1}\hookrightarrow T$ is defined by the PD ideal $p\mathcal{O}_{T}$ and
$f_{T_{1}}$ is smooth and proper); see [BO83], [BO78]. Since $T$ is
noetherian, these are coherent sheaves of
$\mathcal{O}_{T}\otimes_{\mathcal{O}_{F}}F$-modules, and therefore
$\mathcal{E(G)}^{\mathrm{rig}}_{T}$ are coherent
$\mathcal{O}_{T}^{\mathrm{rig}}$-modules.

The coherent
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}=\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}^{\mathrm{rig}}$-module
$\mathcal{E(G)}^{\mathrm{rig}}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}$ is
equipped as in (15) with a connection
$\nabla_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}^{\mathrm{rig}}$. The coherent
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$-module
$\mathcal{E(G)}^{\mathrm{rig}}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}$ is
canonically isomorphic to the relative rigid de Rham cohomology sheaf

$\mathcal{H}_{\mathrm{dR}}^{1,\mathrm{rig}}(\mathcal{G})=\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}^{\mathrm{rig}}/\mathcal{H}_{p}^{\mathrm{unr}})=\mathbb{R}^{1}\lambda_{*}^{\mathrm{rig}}\left(\Omega^{\bullet}_{\mathcal{G}^{\mathrm{rig}}/\mathcal{H}_{p}^{\mathrm{unr}}}\right).$

Moreover, the connection
$\nabla_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}^{\mathrm{rig}}$ corresponds to
the Gauss-Manin connection

$\nabla_{\mathcal{G}}^{\mathrm{rig}}:\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\longrightarrow\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\otimes_{\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}}\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})$

whose construction in this context follows [KO68], and is the analogue of the
construction we outlined in §2.3; see [IS03, Example 3.4(c)], [Ogu84, Theorem
(3.10)]. The Hodge filtration on the de Rham cohomology
$\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})$ makes then
$\mathcal{E(G)}$ a filtered convergent $F$-isocrystal.

The filtration on $\mathcal{E(G)}$ arising from the Hodge filtration on the de
Rham cohomology can be described more explicitly. Denote
$\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})$
the dual of the Lie algebra of the universal vectorial extension of
$\mathcal{G}$, equipped with its structure of convergent $F$-isocrystal
([Mes72, Chapter IV, §2], [MM74, §§1,9,11)]). By [BBM82, §3.3], we have an
isomorphism of convergent $F$-isocrystals

$\mathcal{E(G)}\simeq\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}}).$

The Hodge-Tate filtration on
$\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\simeq\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})^{\mathrm{rig}}_{\mathcal{H}_{p}^{\mathrm{unr}}}$
can be described in explicit terms as follows. Denote
$\underline{\omega}_{\mathcal{G}}=e_{\mathcal{G}}^{*}(\Omega^{1}_{\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})$,
where
$e_{\mathcal{G}}:\hat{\mathcal{H}}_{p}^{\mathrm{unr}}\rightarrow\mathcal{G}$
is the zero-section, and let $\mathcal{L}ie_{\mathcal{G}^{\vee}}$ be the Lie
algebra of the Cartier dual $\mathcal{G}^{\vee}$ of $\mathcal{G}$. Then
$\underline{\omega}_{\mathcal{G}}$ and $\mathcal{L}_{\mathcal{G}^{\vee}}$ are
locally free $\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}$-modules,
dual to each other ([BBM82, §3.3]). We have the Hodge-Tate exact sequence of
$\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}$-modules

$0\longrightarrow\underline{\omega}_{\mathcal{G}}\longrightarrow\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})\longrightarrow\mathcal{L}ie_{\mathcal{G}^{\vee}}\longrightarrow
0$

(see [BBM82, Corollaire 3.3.5], and recall that $\operatorname{SFD}$-modules
are in particular $p$-divisible group schemes, by [BC91, Ch. II, §7]). Since
the functor $\mathcal{F}\leadsto\mathcal{F}^{\mathrm{rig}}$ is exact by [BG98,
Prop. 1.4], we obtain an exact sequence

(16)
$0\longrightarrow\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}}\longrightarrow\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\longrightarrow\mathcal{L}ie_{\mathcal{G}^{\vee}}^{\mathrm{rig}}\longrightarrow
0.$

The Hodge-Tate filtration (16) coincides with the Hodge filtration on the de
Rham cohomology groups.

One the one hand, the exact sequence (16) defines the filtration on the
convergent $F$-isocrystal
$\mathcal{E(G)}\simeq\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})$.
On the other hand, recall that
$V_{\mathrm{cris}}(\Phi)\otimes_{\mathbb{Q}_{p}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})$
has a structure of filtered convergent $F$-isocrystal on
$\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$, with Frobenius given by
$\phi_{V_{\mathrm{cris}}(\Phi)}\otimes\phi_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}$,
and filtration induced as described before by the restriction
$i_{|\operatorname{GL}_{2}(\mathbb{Q}_{p})}:\operatorname{GL}_{2}(\mathbb{Q}_{p})\rightarrow\operatorname{GL}(V)$
of $i$ to $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ (Example 4.3). A result of
Faltings [Fal97, §5] (see also the discussion in [IS03, Lemma 5.10]) shows
that, as filtered convergent $F$-isocrystals, we have

(17)
$\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})\simeq
V_{\mathrm{cris}}(\Phi)\otimes_{\mathbb{Q}_{p}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}).$

The isomorphism of filtered convergent $F$-isocrystals (17) can be
reformulated as follows ([IS03, Lemma 5.10]). Let
$\rho:\operatorname{GL}_{2}\times\operatorname{GL}_{2}\rightarrow\operatorname{GL}(\operatorname{M}_{2})$
be the representation defined by $\rho_{1}(A)(B)=AB$ and
$\rho_{2}(A)B=B\bar{A}$ where if $A=\bigl{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}$ then $\bar{A}=\bigl{(}\begin{smallmatrix}d&-b\\\
-c&a\end{smallmatrix}\bigr{)}$. Note that $\mathcal{E}(\operatorname{M}_{2})$
is pure of weight $1$. The isomorphism (17) can be rewritten in a more compact
way as the isomorphism of filtered convergent $F$-isocrystals

(18)
$\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p}^{\mathrm{unr}})\simeq\mathcal{E}(\operatorname{M}_{2}).$

The Kodaira-Spencer map in this setting is the composition

$\begin{split}\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}:\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}}\overset{\eqref{Hodge
SFD}}{\mbox{\;$\lhook\joinrel\longrightarrow$\;}}&\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\overset{\nabla_{\mathcal{G}}^{\mathrm{rig}}}{\longrightarrow}\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\otimes_{\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}}\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\overset{\eqref{Hodge
SFD}}{\longrightarrow}\mathcal{L}ie_{\mathcal{G}^{\vee}}^{\mathrm{rig}}\otimes_{\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}}\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\end{split}$

in which the first and the last map come from the Hodge exact sequence (16).
Recalling the duality between $\underline{\omega}_{\mathcal{G}}$ and
$\mathcal{L}ie_{\mathcal{G}}$, we therefore obtain, as in the algebraic case,
a symmetric map of $\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$-modules,
again denoted again with the same symbol,
$\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}:\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}}\otimes\underline{\omega}_{\mathcal{G^{\vee}}}^{\mathrm{rig}}\rightarrow\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}$
(tensor product over ${\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}}$). By
fixing a formal polarization
$\iota_{\mathcal{G}}:\mathcal{G}\simeq\mathcal{G}^{\vee}$ of $\mathcal{G}$
([BC91, Chapitre III, Lemma 4.4]), we obtain isomorphism
$\underline{\omega}_{\mathcal{G}}\simeq\omega_{\mathcal{G^{\vee}}}$ of
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$-modules, and the Kodaira-
Spencer map takes the form

$\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}:(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}})^{\otimes{2}}\longrightarrow\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}$

where the tensor product is again over
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$.

The Kodaira-Spencer map can also also be expressed in a different form, (_cf._
[Har81], [Mor11] in the complex setting). Denote
$\mathcal{H}_{1,\mathrm{dR}}(\mathcal{G}/\mathcal{H}_{p}^{\mathrm{unr}})$ the
universal vectorial extension of $\mathcal{G}$, which is equipped with a
structure of convergent filtered $F$-isocrystal as before; see [Fal97, §5].
Put
$\mathcal{H}_{1,\mathrm{dR}}^{\mathrm{rig}}(\mathcal{G})=\mathcal{H}_{1,\mathrm{dR}}(\mathcal{G}/\mathcal{H}_{p}^{\mathrm{unr}})_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}^{\mathrm{rig}}$.
By definition, the universal vectorial extensions of $\mathcal{G}$ and
$\mathcal{G}^{\vee}$ are dual to each other, since each extension of one each
gives rise to an extension of the other by duality. We therefore obtain a
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{rig}}}$-bilinear skew-symmetric map

$\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\times\mathcal{H}_{1,\mathrm{dR}}^{\mathrm{rig}}(\mathcal{G})\longrightarrow\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}.$

The principal polarization
$\iota_{\mathcal{G}}:\mathcal{G}\simeq\mathcal{G}^{\vee}$ identifies
canonically $\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})$ and
$\mathcal{H}_{1,\mathrm{dR}}^{\mathrm{rig}}(\mathcal{G})$, and we therefore
obtain a pairing

$\langle,\rangle_{\mathrm{dR}}^{\mathrm{rig}}:\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\times\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\longrightarrow\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$

satisfying $\langle dx,y\rangle_{\mathcal{G}}^{\mathrm{rig}}=\langle
x,d^{\dagger}y\rangle_{\mathcal{G}}^{\mathrm{rig}}$ for all $x,y$ sections in
$\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})$ and all $d\in D$
(because $\iota_{\mathcal{G}}(dy)=d^{\dagger}\iota_{\mathcal{G}}(y)$) which we
call _rigid polarization pairing_. We may therefore construct a map

$\rho:\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\longrightarrow\left(\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\right)^{\vee}\longrightarrow(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}})^{\vee}$

where the first map takes a section $s$ to the map defined for a section $t$
by $t\mapsto\langle s,t\rangle_{\mathrm{dR}}^{\mathrm{rig}}$ and the second
map is induced by duality from the inclusion
$\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}}\hookrightarrow\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})$.
Fix now a section $s\in
H^{0}(U,(\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}})^{\vee})$
over some affinoid $U$. Then we may compose the maps to get

(19)
$\begin{split}\rho_{s}:H^{0}\left(U,\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}}\right)\longrightarrow&H^{0}\left(U,\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\right)\overset{\nabla_{\mathcal{G}}^{\mathrm{rig}}}{\longrightarrow}H^{0}\left(U,\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\otimes\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\right)\overset{1\otimes
s}{\longrightarrow}\\\ &\overset{1\otimes
s}{\longrightarrow}H^{0}\left(U,\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})\right)\overset{\rho}{\longrightarrow}H^{0}\left(U,(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}})^{\vee}\right).\end{split}$

The association $s\mapsto\rho_{s}$ defines then a map of sheaves

$(\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}})^{\vee}:(\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}/\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}})^{\vee}\longrightarrow\operatorname{Hom}_{\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}}\left(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}},(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}})^{\vee}\right).$

By construction, the dual of this map is the Kodaira-Spencer map, under the
canonical identification between
$\operatorname{Hom}(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}},(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}})^{\vee})^{\vee}$
and $(\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}})^{\otimes 2}$.

###  4.5. Universal rigid data

The aim of this subsection is to use the results of [Tei89] to better describe
the Hodge filtration (16). For this, we need to recall the universal rigid
data introduced in [Tei89].

Let $V_{0}$ and $V_{1}$ be constant sheaves of one-dimensional
$\mathbb{Q}_{p}$-vector spaces on the $\mathbb{Q}_{p}$-rigid analytic space
$\mathcal{H}_{p}$ with basis $t_{0}$ and $t_{1}$ respectively. Define two
invertible sheaves $T_{0}^{\mathrm{univ}}$ and $T_{1}^{\mathrm{univ}}$ on
$\mathcal{H}_{p}$ by
$T_{i}^{\mathrm{univ}}=\mathcal{O}_{\mathcal{H}_{p}}\otimes V_{i}$ for
$i=0,1$, where $\mathcal{O}_{{\mathcal{H}_{p}}}$ is the structural sheaf of
rigid analytic functions on $\mathcal{H}_{p}$. Define
$T^{\mathrm{univ}}=T_{0}^{\mathrm{univ}}\oplus T_{1}^{\mathrm{univ}}$. For
$i=0,1$, let $\eta_{i}^{\mathrm{univ}}$ be the constant sheaf of two-
dimensional $\mathbb{Q}_{p}$-vector spaces on $\mathcal{H}_{p}$ with basis
$e_{i,0}$ and $e_{i,1}$. One fixes

(20)
$\eta^{\mathrm{univ}}_{i}=\eta^{i}(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$

as in [Tei89, page 664]. Define
$u_{0}^{\mathrm{univ}}:\eta_{0}^{\mathrm{univ}}\rightarrow
T_{0}^{\mathrm{univ}}$ by $u_{0}^{\mathrm{univ}}(e_{0,0})=zt_{0}$ and
$u_{0}^{\mathrm{univ}}(e_{0,1})=t_{0}$, and
$u_{1}^{\mathrm{univ}}:\eta_{1}^{\mathrm{univ}}\rightarrow
T_{1}^{\mathrm{univ}}$ by $u_{1}^{\mathrm{univ}}(e_{1,0})=(p/z)t_{1}$ and
$u_{1}^{\mathrm{univ}}(e_{1,1})=t_{1}$, where $z$ denotes the standard
coordinate function on $\mathcal{H}_{p}$. Define
$\eta^{\mathrm{univ}}=\eta_{0}^{\mathrm{univ}}\oplus\eta_{1}^{\mathrm{univ}}$
and similarly define $u^{\mathrm{univ}}=u_{0}^{\mathrm{univ}}\oplus
u_{1}^{\mathrm{univ}}$. We write
$\rho^{\mathrm{univ}}:(\mathbb{Q}_{p})_{\mathcal{H}_{p}}^{2}\simeq\eta_{0}^{\mathrm{univ}}$
for the isomorphism determined by the choice of the basis
$\\{e_{0,0},e_{0,1}\\}$. For $\gamma=\bigl{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\in\operatorname{M}_{2}(\mathbb{Q}_{p})$ and
$i=0,1$, define endomorphisms $\phi_{i}^{(\gamma)}$ in
$\operatorname{End}_{\mathcal{O}_{\mathcal{H}_{p}}}(T_{i}^{\mathrm{univ}})$ by

(21) $\phi_{0}^{(\gamma)}\left(f(z)\otimes
t_{0}\right)=(cz+d)f(\gamma(z))\otimes t_{0},\text{ and
}\phi_{1}^{(\gamma)}\left(f(z)\otimes t_{1}\right)=(a+b/z)f(\gamma(z))\otimes
t_{1}$

for any $f\in\mathcal{O}_{\mathcal{H}_{p}}(U)$, and any affinoid
$U\subseteq\mathcal{H}_{p}$. Define an action of
$\operatorname{SL}_{2}(\mathbb{Q}_{p})$ on $\eta_{i}^{\mathrm{univ}}$ for
$i=0,1$ in such ta way that $u_{i}^{\mathrm{univ}}$ is equivariant with
respect to these actions, namely, for
$\gamma=\bigl{(}\begin{smallmatrix}a&b\\\ c&d\end{smallmatrix}\bigr{)}$ in
$\operatorname{GL}_{2}(\mathbb{Q}_{p})$, put
$\gamma^{*}\binom{x_{0,0}}{x_{0,1}}=\bigl{(}\begin{smallmatrix}a&b\\\
c&d\end{smallmatrix}\bigr{)}\binom{x_{0,0}}{x_{0,1}}$ and
$\gamma^{*}\binom{x_{1,0}}{x_{1,1}}=\bigl{(}\begin{smallmatrix}a&b/p\\\
pc&d\end{smallmatrix}\bigr{)}\binom{x_{1,0}}{x_{1,1}}$. Let
$\mathbb{Z}_{p}[\Pi]$ act on $T^{\mathrm{univ}}$ by $\Pi t_{0}=(p/z)t_{1}$ and
$\Pi t_{1}=zt_{0}$. We let $\mathbb{Z}_{p}[\Pi]$ act on $\eta^{\mathrm{univ}}$
in such a way that $u^{\mathrm{univ}}$ commutes with this action. We call the
quadruplet

${\mathcal{D}}^{\mathrm{univ}}=(\eta^{\mathrm{univ}},T^{\mathrm{univ}},u^{\mathrm{univ}},\rho^{\mathrm{univ}})$

the _universal rigid data_.

Passing to the associated normed sheaves ([Tei89, Definition 6]), we obtain
from ${\mathcal{D}}^{\mathrm{univ}}$ a quadruplet
$\hat{\mathcal{D}}^{\mathrm{univ}}=(\hat{\eta}^{\mathrm{univ}},\hat{T}^{\mathrm{univ}},\hat{u}^{\mathrm{univ}},\hat{\rho}^{\mathrm{univ}})$
on $\hat{\mathcal{H}}_{p}$, corresponding to a $\hat{\mathcal{H}}_{p}$-valued
point, which is universal in the following sense: for each $B\in\mathrm{Nilp}$
and each $\Psi:S=\operatorname{Spec}(B)\rightarrow\hat{\mathcal{H}}_{p}$
corresponding to a quadruplet $(\eta,T,u,\rho)$, we have

(22)
$(\eta,T,u,\rho)=(\Psi^{-1}\hat{\eta}^{\mathrm{univ}},\Psi^{*}\hat{T}^{\mathrm{univ}},\Psi^{-1}\hat{u}^{\mathrm{univ}},\Psi^{-1}\hat{\rho}^{\mathrm{univ}}).$

See [Tei89, Cor. 18 and Thm. 19] for more precise and complete statements. We
call $\hat{\mathcal{D}}^{\mathrm{univ}}$ the _universal formal data_ , and we
denote the quadruplet on the RHS of (22) by
$\hat{\mathcal{D}}^{\mathrm{univ}}_{\Psi}$ to simplify the notation.

The universal $\operatorname{SFD}$-module $\mathcal{G}$ over
$\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$ can be recovered from a universal rigid
data $\mathcal{D}^{\mathrm{univ}}$. Pulling back via the projection
$\pi_{\hat{\mathcal{H}}_{p}}:\hat{\mathcal{H}}_{p}^{\mathrm{unr}}\rightarrow\hat{\mathcal{H}}_{p}$,
we obtain a quadruplet

$\hat{\mathcal{D}}^{\mathrm{unr}}=(\hat{\eta}^{\mathrm{unr}},\hat{T}^{\mathrm{unr}},\hat{u}^{\mathrm{unr}},\hat{\rho}^{\mathrm{unr}})=(\pi^{-1}_{\hat{\mathcal{H}}_{p}}\hat{\eta}^{\mathrm{univ}},\pi_{\hat{\mathcal{H}}_{p}}^{*}\hat{T}^{\mathrm{univ}},\pi^{-1}_{\hat{\mathcal{H}}_{p}}\hat{u}^{\mathrm{univ}},\pi^{-1}_{\hat{\mathcal{H}}_{p}}\hat{\rho}^{\mathrm{univ}})$

on $\hat{\mathcal{H}}_{p}^{\mathrm{unr}}$. Comparing (22) with the universal
property satisfied by $\mathcal{G}$, we see that the quadruplet
$(\eta_{\mathcal{G}},T_{\mathcal{G}},u_{\mathcal{G}},\rho_{\mathcal{G}})$
associated to $\mathcal{G}$ coincides with the quadruplet
$\hat{\mathcal{D}}^{\mathrm{unr}}$. In particular, the associated quadruplet
$(\eta_{\mathcal{G}}^{\mathrm{rig}},T_{\mathcal{G}}^{\mathrm{rig}},u_{\mathcal{G}}^{\mathrm{rig}},\rho_{\mathcal{G}}^{\mathrm{rig}})$
on the rigid $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-rigid analytic space
$\mathcal{G}^{\mathrm{rig}}$ is the quadruplet

$\mathcal{D}^{\mathrm{unr}}=(\eta^{\mathrm{unr}},{T}^{\mathrm{unr}},{u}^{\mathrm{unr}},{\rho}^{\mathrm{unr}})=(\pi^{-1}_{\mathcal{H}_{p}^{\mathrm{unr}}}\eta^{\mathrm{univ}},\pi_{\mathcal{H}_{p}^{\mathrm{unr}}}^{*}{T}^{\mathrm{univ}},\pi^{-1}_{\mathcal{H}_{p}^{\mathrm{unr}}}{u}^{\mathrm{univ}},\pi^{-1}_{\mathcal{H}_{p}^{\mathrm{unr}}}\rho^{\mathrm{univ}})$

obtained from the quadruplet $\mathcal{D}^{\mathrm{univ}}$, where
$\pi_{\mathcal{H}_{p}^{\mathrm{unr}}}:\mathcal{H}_{p}^{\mathrm{unr}}\rightarrow\mathcal{H}_{p}$
is the canonical projection.

Let $(T^{\mathrm{unr}})^{\vee}$ denote the
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$-dual of $T^{\mathrm{unr}}$,
and, as above, denote
$(\eta^{\mathrm{unr}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}$ the
$\mathbb{Q}_{p}$-linear dual of
$\eta^{\mathrm{unr}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p}$. From the
surjective map
$u^{\mathrm{unr}}:\eta^{\mathrm{unr}}\otimes_{\mathbb{Z}_{p}}\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}\twoheadrightarrow
T^{\mathrm{unr}}$ induced by $u^{\mathrm{univ}}$ we obtain an injective map

$\tau:(T^{\mathrm{unr}})^{\vee}\mbox{\;$\lhook\joinrel\longrightarrow$\;}(\eta^{\mathrm{unr}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}\otimes_{\mathbb{Q}_{p}}\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$

######  Proposition 4.4.

We have canonical isomorphisms

$(T^{\mathrm{unr}})^{\vee}\simeq\omega_{\mathcal{G}}^{\mathrm{rig}}$
$(\eta^{\mathrm{unr}}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}\otimes_{\mathbb{Q}_{p}}\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}\simeq\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G}).$

under which the map $\tau$ corresponds to the canonical map in (16).

###### Proof.

The first statement follows from the canonical isomorphism between
$T_{\mathcal{G}}=\mathcal{L}ie_{\mathcal{G}}$ and $T^{\mathrm{unr}}$, while
the second follows from Proposition 4.1 combined with (17). For the statement
about $\tau$, note that for each $\operatorname{SFD}$-module $G$ over
$\bar{\mathbb{F}}_{p}$, the map $u_{G}$ corresponds under the identification
between $\eta(G)\otimes_{\mathbb{Z}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$
and
$\mathbb{M}(G)\otimes_{\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$
to the canonical projection
$\mathbb{M}(G)/\mathbb{V}_{G}\mathbb{M}(G)\rightarrow T_{G}$, where $T_{G}$ is
the tangent space of $G$ at the origin. ∎

###  4.6. The action of the idempotent $e$

Fix an isomorphism $\mathbb{Q}_{p}(\sqrt{p_{0}})\simeq\mathbb{Q}_{p^{2}}$. By
means of this isomorphism, and the fixed embedding
$\mathbb{Q}_{p^{2}}\hookrightarrow D$, we may identify elements
$a+b\sqrt{p_{0}}$ in $\mathbb{Q}_{p}(\sqrt{p_{0}})$ (where
$a,b\in\mathbb{Q}_{p}$) with elements of $D$ in what follows without
explicitly mentioning it.

######  Lemma 4.5.

$e\cdot\left(\eta(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p^{2}}\right)=\eta^{0}(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p^{2}}$
and
$e\cdot\left(T(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p^{2}}\right)=T^{0}(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Z}_{p^{2}}$.

###### Proof.

The action of $\mathcal{O}_{D}$ on $\eta(\Phi)$ is induced by duality from the
action on $\mathbb{M}(\Phi)$, so any element
$a\in\mathbb{Z}_{p^{2}}\hookrightarrow\mathcal{O}_{D}$ acts on
$\eta^{0}(\Phi)$ by multiplication by $a$ and on $\eta^{1}(\Phi)$ by
multiplication by $\bar{a}$. On the other hand, the action of $1\otimes a$ on
$\eta(\Phi)\otimes_{\mathbb{Q}_{p}}\mathbb{Q}_{p^{2}}$ is given by
multiplication by $a$. An immediate calculation shows then that the action of
$e$ is just the projection $\eta(\Phi)\rightarrow\eta^{0}(\Phi)$. The argument
for $T(\Phi)$ is similar. ∎

Write
$\eta_{0}^{\mathrm{unr}}=\pi^{-1}_{\mathcal{H}_{p}^{\mathrm{unr}}}\eta^{\mathrm{univ}}_{0}$,
$T_{0}^{\mathrm{unr}}=\pi_{\mathcal{H}_{p}^{\mathrm{unr}}}^{*}{T}_{0}^{\mathrm{univ}}$,
$u_{0}^{\mathrm{unr}}=\pi^{-1}_{\mathcal{H}_{p}^{\mathrm{unr}}}{u}_{0}^{\mathrm{univ}}$.

######  Proposition 4.6.

$e\cdot\eta^{\mathrm{unr}}=\eta_{0}^{\mathrm{unr}}$ and $e\cdot
T^{\mathrm{unr}}=T_{0}^{\mathrm{unr}}$.

###### Proof.

This is clear from Lemma 4.5 and (20). ∎

For $i=0,1$, the sheaf $T_{0}^{\mathrm{unr}}$ is a free
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$-module of rank $1$, so it is
invertible; denote $(T_{0}^{\mathrm{unr}})^{\vee}$ its
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$-dual. Taking duals we get a map
$du_{0}:(T^{\mathrm{unr}}_{0})^{\vee}\rightarrow(\eta_{0}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}\otimes_{\mathbb{Q}_{p}}\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$
(where the RHS denotes $\mathbb{Q}_{p}$-duals as above). We set up the
following notation:

  * •

$\underline{\omega}_{\mathcal{G}}^{0}=e\cdot\underline{\omega}_{\mathcal{G}}^{\mathrm{rig}}$;

  * •

$\mathcal{L}^{0}_{\mathcal{G}}=e\cdot\mathcal{H}^{1,\mathrm{rig}}_{\mathrm{dR}}(\mathcal{G})$.

Applying the idempotent $e$ and using Propositions 4.4 and 4.6 we then obtain
a diagram with exact rows in which the vertical arrows are isomorphisms:

(23)
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{(T^{\mathrm{unr}}_{0})^{\vee}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{du_{0}}$$\scriptstyle{\simeq}$$\textstyle{(\eta_{0}\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}\otimes_{\mathbb{Q}_{p}}\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\simeq}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\underline{\omega}_{\mathcal{G}}^{0}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{L}_{\mathcal{G}}^{0}}$

###  4.7. Differential calculus on the $p$-adic upper half plane

We now set up the following notation. Recall that the map $u_{0}$ takes
$x_{0,0}e_{0,0}+x_{0,1}e_{0,1}$ to $(zx_{0,0}+x_{0,1})\otimes t_{0}$;
dualizing, $du_{0}$ can be described in coordinates by the map which takes the
canonical generator $t_{0}$ of the
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$-module
$(T^{\mathrm{unr}}_{0})^{\vee}$ (satisfying the relation $dt_{0}(t_{0})=1$) to
the map $x_{0,0}e_{0,0}+x_{0,1}e_{0,1}\mapsto zx_{0,0}+x_{0,1}$. If we denote
$de_{0,i}$ the dual basis of $e_{0,i}$ (satisfying the condition
$de_{0,i}(e_{0,j})=\delta_{i,j}$), we may write this map as
$zde_{0,0}+de_{0,1}$. To simplify the notation, we put from now on
$\tau=t_{0}$, $d\tau=dt_{0}$, $x=e_{0,0}$, $y=e_{0,1}$, $dx=de_{0,0}$ and
$dy=de_{0,1}$, so that the above map reads simply as

$d\tau=zdx+dy.$

Let
$\mathcal{C}=\mathcal{C}^{0}(\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}),\mathbb{C}_{p})$
denote the $\mathbb{C}_{p}$-vector space of continuous (for the standard
$p$-adic topology on both spaces) $\mathbb{C}_{p}$-valued functions on
${\mathcal{H}_{p}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})}$. Denote
$\mathcal{A}=H^{0}(\mathcal{H}_{p}^{\mathrm{unr}},\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}})$
the $\mathbb{Q}_{p}^{\mathrm{unr}}$-vector space of global sections of
$\mathcal{O}_{\mathcal{H}_{p}^{\mathrm{unr}}}$. Each $f\in\mathcal{A}$ is, in
particular, continuous on $\mathcal{H}_{p}^{\mathrm{unr}}$ for the standard
$p$-adic topology of $\mathbb{Q}_{p}^{\mathrm{unr}}-\mathbb{Q}_{p}$, and
therefore restriction induces a map of $\mathbb{Q}_{p}^{\mathrm{unr}}$-vector
spaces $r:\mathcal{A}\rightarrow\mathcal{C}$. Denote $\mathcal{A}^{*}$ the
image of the morphism of $\mathcal{A}$-algebras
$\mathcal{A}[X,Y]\rightarrow\mathcal{C}$ defined by sending $X$ to the
function $z\mapsto 1/(z-\sigma(z))$ and $Y$ to the function
$z\mapsto\sigma(z)$ (note that the function $z\mapsto z-\sigma(z)$ is
invertible on
$\mathcal{H}_{p}^{\mathrm{unr}}(\hat{\mathbb{Q}}_{p}^{\mathrm{unr}})$). To
simplify the notation, we put from now on

$z^{\ast}=\sigma(z).$

Set up the following notation (here $n\geq 1$ is an integer)

  * •

$\Lambda_{\mathcal{G}}=H^{0}(\mathcal{H}_{p}^{\mathrm{unr}},\mathcal{L}_{\mathcal{G}}^{0})$
and $\Lambda_{\mathcal{G},n}=\Lambda_{\mathcal{G}}^{\otimes{n}}$,

  * •

$\Lambda_{\mathcal{G}}^{\ast}=\Lambda_{\mathcal{G}}\otimes_{\mathcal{A}}\mathcal{A}^{\ast}$
and
$\Lambda_{\mathcal{G},n}^{\ast}=\left(\Lambda_{\mathcal{G},n}^{\ast}\right)^{\otimes{n}}$,

  * •

$w_{\mathcal{G}}=H^{0}(\mathcal{H}_{p}^{\mathrm{unr}},\underline{\omega}_{\mathcal{G}}^{0})$
and $w_{\mathcal{G},n}=w_{\mathcal{G}}^{\otimes{n}}$,

  * •

${w}_{\mathcal{G}}^{\ast}=w_{\mathcal{G}}\otimes_{\mathcal{A}}\mathcal{A}^{\ast}$
and
$w_{\mathcal{G},n}^{\ast}=\left(w_{\mathcal{G},n}^{\ast}\right)^{\otimes{n}}$.

The $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-algebra $\mathcal{A}$ is equipped
with the standard derivation $\frac{d}{dz}$ on power series. The
$\mathcal{A}$-module
$\Omega^{1}_{\mathcal{A}}=H^{0}(\mathcal{H}_{p}^{\mathrm{unr}},\Omega^{1}_{\mathcal{H}_{p}^{\mathrm{unr}}})$
is then one dimensional and generated by $dz$ satisfying
$dz\left(\frac{d}{dz}\right)=1$. We extend differential operator
$\frac{d}{dz}$ to a differential operator
$\frac{\partial}{\partial{z}}:\mathcal{A}^{\ast}\rightarrow\mathcal{A}^{\ast}$
by $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linearity using the product formula
and setting $\frac{\partial}{\partial{z}}(z^{\ast})=0$ and
$\frac{\partial}{\partial{z}}\left(\frac{1}{z-z^{\ast}}\right)=\frac{-1}{(z-z^{\ast})^{2}}$.
Similarly, we define a differential operator
$\frac{\partial}{\partial{z^{\ast}}}:\mathcal{A}^{\ast}\rightarrow\mathcal{A}^{\ast}$
setting $\frac{\partial}{\partial{z^{\ast}}}(z)=0$,
$\frac{\partial}{\partial{z^{\ast}}}(z^{\ast})=1$ and
$\frac{\partial}{\partial{z}}\left(\frac{1}{z-z^{\ast}}\right)=\frac{1}{(z-z^{\ast})^{2}}$.
Define $\Omega^{1}_{\mathcal{A}^{\ast}}$ to be the
$\mathcal{A}^{\ast}$-subalgebra of the algebra of derivations generated by
$dz$ and $dz^{\ast}$ satisfying the usual rules
$dz\left(\frac{\partial}{\partial z}\right)=1$,
$dz\left(\frac{\partial}{\partial z^{\ast}}\right)=0$,
$dz^{\ast}\left(\frac{\partial}{\partial z^{\ast}}\right)=0$,
$dz^{\ast}\left(\frac{\partial}{\partial z^{\ast}}\right)=1$.

###  4.8. Splitting of the rigid analytic Hodge filtration

Recall the notation fixed before for the differential form $d\tau=zdx+dy$.
Define

$d\tau^{\ast}=z^{\ast}dx+dy.$

Then $d\tau^{\ast}$ belongs to $w_{\mathcal{G}}^{\ast}$. Taking global
sections, restricting to $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$, and extending
linearly with $\mathcal{A}^{\ast}$ we obtain a short exact sequence of
$\mathcal{A}^{\ast}$-algebras

(24)
$0\longrightarrow{w}_{\mathcal{G}}^{\ast}\longrightarrow\Lambda_{\mathcal{G}}^{\ast}.$

######  Theorem 4.7.

The exact sequence (24) admits a canonical splitting
$\Psi_{p}:\Lambda_{\mathcal{G}}^{\ast}\rightarrow{w}_{\mathcal{G}}^{\ast}$.

###### Proof.

We have

$dx=\frac{d\tau-d\tau^{\ast}}{z-z^{\ast}},\qquad
dy=\frac{zd\tau^{\ast}-z^{\ast}d\tau}{z-z^{\ast}}.$

We may therefore write any differential form $\omega=f(z)dx+g(z)dy$ with
$f,g\in\mathcal{A}^{*}$ as

$\omega=\left(\frac{f(z)-g(z)z^{\ast}}{z-z^{\ast}}\right)d\tau+d\tau^{\ast}\left(\frac{zg(z)-f(z)}{z-z^{\ast}}\right).$

One then defines the sough-for splitting sending
$\omega\mapsto\left(\frac{f(z)-g(z)z^{\ast}}{z-z^{\ast}}\right)d\tau$. ∎

###  4.9. The $p$-adic Shimura-Maass operator

Taking global section, the Gauss-Manin connection gives rise to a map
$\nabla^{\mathrm{rig}}_{\mathcal{G}}:\Lambda_{\mathcal{G}}\rightarrow\Lambda_{\mathcal{G}}\otimes\Omega^{1}_{\mathcal{A}}$.
We extend $\nabla_{\mathcal{G}}^{\mathrm{rig}}$ to a map
$\nabla_{\mathcal{G}}^{\ast}:\Lambda_{\mathcal{G}}^{\ast}\rightarrow\Lambda_{\mathcal{G}}^{\ast}\otimes\Omega^{1}_{\mathcal{A}^{\ast}}$
as follows. First define
$\nabla_{\mathcal{G}}^{1,0}:\Lambda_{\mathcal{G}}^{\ast}\rightarrow\Lambda_{\mathcal{G}}^{\ast}\otimes\Omega^{1}_{\mathcal{A}^{\ast}}$
to be the derivation satisfying the rules
$\nabla^{1,0}_{\mathcal{G}}(d\tau)=dx\otimes dz$,
$\nabla^{1,0}_{\mathcal{G}}(d\tau^{\ast})=0$,
$\nabla^{1,0}_{\mathcal{G}}(z)=1$, $\nabla^{1,0}_{\mathcal{G}}(z^{\ast})=0$.
Define similarly the derivation
$\nabla_{\mathcal{G}}^{0,1}:\Lambda_{\mathcal{G}}^{\ast}\rightarrow\Lambda_{\mathcal{G}}^{\ast}\otimes\Omega^{1}_{\mathcal{A}^{\ast}}$
by the rules $\nabla^{0,1}_{\mathcal{G}}(d\tau)=0$,
$\nabla^{0,1}_{\mathcal{G}}(d\tau^{\ast})=dx\otimes dz^{\ast}$,
$\nabla^{0,1}_{\mathcal{G}}(z)=0$,
$\nabla^{0,1}_{\mathcal{G}}(\bar{z})=dz^{\ast}$. We finally define

$\nabla^{\ast}_{\mathcal{G}}=\nabla^{1,0}_{\mathcal{G}}+\nabla^{0,1}_{\mathcal{G}}:\Lambda_{\mathcal{G}}^{\ast}\longrightarrow\Lambda_{\mathcal{G}}^{\ast}\otimes\Omega^{1}_{\mathcal{A}^{\ast}}.$

Taking global sections, the Kodaira-Spencer map gives rise to a map
$\mathrm{KS}_{\mathcal{G}}:w_{\mathcal{G}}^{\otimes{2}}\rightarrow\Omega^{1}_{\mathcal{A}}$,
which we extend $\mathcal{A}^{\ast}$-linearly to a map

$\mathrm{KS}_{\mathcal{G}}^{\ast}:(w_{\mathcal{G}}^{\ast})^{\otimes{2}}\longrightarrow\Omega^{1}_{\mathcal{A}^{\ast}}.$

Note that

(25)
$\nabla_{\mathcal{G}}^{\mathrm{rig}}(d\tau)=\nabla_{\mathcal{G}}^{\ast}(d\tau)=\frac{d\tau-d{\tau}^{\ast}}{z-z^{\ast}}\otimes
dz.$

and, since $\nabla^{\ast}_{\mathcal{G}}(z^{\ast})=0$, we have

(26) $\nabla^{\ast}_{\mathcal{G}}(d\tau^{\ast})=0.$

In particular, if $f(z)\otimes d\tau\in w_{\mathcal{G}}^{\ast}$ we have

(27) $\nabla_{\mathcal{G}}^{\mathrm{rig}}\left(f(z)\otimes
d\tau\right)=\left(\frac{\partial}{\partial z}f(z)\otimes
d\tau+f(z)\otimes\frac{d\tau-d\tau^{*}}{z-z^{\ast}}\right)\otimes dz.$

Taking global sections, we can form the pairing
$\langle,\rangle_{\mathcal{G}}^{\mathrm{rig}}:\Lambda_{\mathcal{G}}\otimes_{\mathcal{A}}\Lambda_{\mathcal{G}}\rightarrow\mathcal{A}$.
Extending linearly by $\mathcal{A}^{\ast}$, we obtain a new pairing

$\langle,\rangle_{\mathcal{G}}^{\ast}:\Lambda_{\mathcal{G}}^{\ast}\otimes_{\mathcal{A}^{\ast}}\Lambda_{\mathcal{G}}^{\ast}\longrightarrow\mathcal{A}^{\ast}.$

Using the description of the Kodaira-Spencer map in the end of §4.4, we see
that

$\langle
d\tau,\nabla_{\mathcal{G}}^{\mathrm{rig}}(d\tau)\rangle_{\mathcal{G}}^{\mathrm{rig}}=\langle
d\tau,\nabla_{\mathcal{G}}^{\ast}(d\tau)\rangle_{\mathcal{G}}^{\ast}=\frac{-\langle
d\tau,d\tau^{\ast}\rangle_{\mathcal{G}}^{\ast}}{z-z^{\ast}}dz=-\langle
dx,dy\rangle_{\mathcal{G}}^{\ast}dz$

where for the second equality we use (25), while the last equality easily from
the equality $\langle
zdx+dy,z^{\ast}dx+dy\rangle_{\mathcal{G}}^{\ast}=(z-z^{\ast})\langle
dx,dy\rangle_{\mathcal{G}}^{\ast}$. Therefore

$\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}(d\tau\otimes d\tau)=-\langle
dx,dy\rangle_{\mathcal{G}}^{\mathrm{rig}}dz.$

So, to compute $\mathrm{KS}_{\mathcal{G}}^{\ast}(d\tau\otimes
d\tau)=\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}(d\tau\otimes d\tau)$ we are
reduced to compute $\langle dx,dy\rangle_{\mathcal{G}}^{\mathrm{rig}}$. For
this, we switch to de Rham homology and follow the computations in [Mor11],
[HB15].

To begin with, let $W$ denote the order $\mathcal{O}_{D}$ viewed as free left
$\mathcal{O}_{D}$-module of rank 1; then
$W\simeq\mathcal{R}_{\mathrm{max}}\otimes_{\mathbb{Z}}\mathbb{Z}_{p}$. By
[BC91, Ch. III, Lemma 1.9], the collection of bilinear skew-symmetric maps
$\psi:W\times W\rightarrow\mathbb{Z}_{p}$ which satisfy
$\psi(dx,y)=\psi(x,d^{\dagger}y)$ (for all $x,y\in W$ and
$d\in\mathcal{O}_{D}$) is a free $\mathbb{Z}_{p}$-module of rank $1$, and
every generator $\psi_{0}$ of this $\mathbb{Z}_{p}$-module is a perfect
duality on $W$; the pairing

$\psi_{0}(x,y)=\frac{\mathrm{tr}(\mathbf{i}y^{\dagger}x)}{p}$

is such a generator, which we fix once and for all (recall the notation
introduced in §2.1 and §2.3 for $\mathbf{i}$ and $d^{\dagger}$).

Recall that $H^{1}_{\mathrm{cris}}(\Phi)$ is a free
$D\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-module of rank
$1$ (_cf._ [IS03, page 354]); the structure of
$D\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-module is
induced from the $D$-module structure of
$(\eta(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}$ via the
isomorphisms $(\eta(\Phi)\otimes_{\mathbb{Z}_{p}}\mathbb{Q}_{p})^{\vee}\simeq
V_{\mathrm{cris}}(\Phi)$ and $H^{1}_{\mathrm{cris}}(\Phi)\simeq
V_{\mathrm{cris}}(\Phi)\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$
in Lemma 4.1. We have then from Lemma 4.1 then canonical isomorphisms of
convergent $F$-isocrystals:

(28)
$\begin{split}\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G}/\hat{\mathcal{H}}_{p})&\simeq
H^{1}_{\mathrm{cris}}(\Phi)\otimes_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})\\\
&\simeq\left(D\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}\right)\otimes_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})\\\
&\simeq\left(D\otimes_{\mathbb{Q}_{p}}{\mathbb{Q}}_{p^{2}}\right)\otimes_{{\mathbb{Q}}_{p^{2}}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})\\\
&\simeq\left(D\otimes_{\mathbb{Q}_{p}}\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}\right)\otimes_{\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}})\\\
&\simeq\operatorname{M}_{2}(\mathbb{Q}_{p^{2}})\otimes_{{\mathbb{Q}}_{p^{2}}}\mathcal{E}(\mathcal{O}_{\hat{\mathcal{H}}_{p}^{\mathrm{unr}}}).\end{split}$

Let $\psi_{0}$ denote the $\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear
extension of $\psi_{0}$; under the isomorphism (28), $\psi_{0}$ defines a
pairing $H^{1}_{\mathrm{cris}}(\Phi)\times
H^{1}_{\mathrm{cris}}(\Phi)\rightarrow\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$
still denoted by $\psi_{0}$. If we still denote
$\langle,\rangle_{\mathcal{G}}^{\mathrm{rig}}$ the restriction of
$\langle,\rangle_{\mathcal{G}}^{\mathrm{rig}}$ to
$H^{1}_{\mathrm{cris}}(\Phi)$, it follows from the unicity of $\psi_{0}$ up to
constant that there exists an element $t_{p}\in\mathbb{C}_{p}^{\times}$ such
that

(29)
$\langle,\rangle_{\mathcal{G},W}^{\mathrm{rig}}=\frac{1}{t_{p}}\cdot\psi_{0}.$

Moreover, under the isomorphism (28), the element $d\tau=zdx+dy$ of
$\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G})$ corresponds to the element
$e_{1}\otimes{z}+e_{2}\otimes 1$, where
$e_{1}=\bigl{(}\begin{smallmatrix}1&0\\\ 0&1\end{smallmatrix}\bigr{)}$,
$e_{2}=\bigl{(}\begin{smallmatrix}0&1\\\ 0&0\end{smallmatrix}\bigr{)}$,
$e_{3}=\bigl{(}\begin{smallmatrix}0&0\\\ 1&0\end{smallmatrix}\bigr{)}$,
$e_{4}=\bigl{(}\begin{smallmatrix}0&0\\\ 0&1\end{smallmatrix}\bigr{)}$ is the
standard basis of $\operatorname{M}_{2}(\mathbb{Q}_{p^{2}})$. We therefore
obtain the sought-for recipe to compute the Kodaira-Spencer image of
$d\tau\otimes d\tau$ in terms of $\psi_{0}$:

(30) $\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}(d\tau\otimes
d\tau)=\frac{1}{t_{p}}\cdot\psi_{0}(e_{1}\otimes z,e_{2}\otimes 1).$

######  Remark 4.8.

The number $t_{p}$ may be viewed as the $p$-adic analogue of the complex
period $2\pi i$, relating de Rham cohomology with homology ([Mor11, (2.7)],
[HB15, p. 4197]). This explains why we prefer to keep $t_{p}$ at the
denominator in (29).

We now make more explicit the equations (25) and (30) using Hashimoto basis.
For this part, we follow closely the nice calculations in [Mor11, Prop. 2.3],
to which the reader is referred to for details. Recall the Haschimoto basis
$\\{\mathbf{1},\mathbf{i},\mathbf{j},\mathbf{k}\\}$ in §2.1. As in [Has95,
(2)], define $\epsilon_{1}=\mathbf{1}$,
$\epsilon_{2}=(\mathbf{1}+\mathbf{j})/2$,
$\epsilon_{3}=(\mathbf{i}+\mathbf{i}\mathbf{j})/2$,
$\epsilon_{4}=(apN^{-}\mathbf{j}+\mathbf{i}\mathbf{j})/{p_{0}}$ and use these
elements to define a symplectic basis of $W$ with respect to the pairing
$\psi_{0}$ as in [Has95, (5)] by
$\eta_{1}=\epsilon_{3}-\frac{p_{0}-1}{2}\epsilon_{4}$,
$\eta_{2}=-aD\epsilon_{1}-\epsilon_{4}$, $\eta_{3}=\epsilon_{1}$,
$\eta_{4}=\epsilon_{2}$ (note that $\psi_{0}$ we consider above is equal to
the pairing $(x,y)\mapsto\mathrm{tr}(xiy^{\dagger})$ in [Has95, (3)]). Denote
$\eta_{1}^{\vee},\eta_{2}^{\vee},\eta_{3}^{\vee},\eta_{4}^{\vee}$ the dual
basis of $W^{\vee}$, and let $\underline{\eta}^{\vee}$ be the column vector
with entries
$\eta_{1}^{\vee},\eta_{2}^{\vee},\eta_{3}^{\vee},\eta_{4}^{\vee}$. The
elements $\eta_{i}^{\vee}$ give rise to elements of
$\mathcal{H}^{1}_{\mathrm{dR}}(\mathcal{G})$, denoted with the same symbol,
which are horizontal with respect to $\nabla^{\mathrm{rig}}_{\mathcal{G}}$,
namely $\nabla_{\mathcal{G}}^{\mathrm{rig}}(\eta_{i}^{\vee})=0$. Write
$d\tau=\Pi(z)\cdot\underline{\eta}^{\vee}$. A simple calculation shows that

(31) $\Pi(z)=\left({\frac{\alpha^{-}}{2\sqrt{p_{0}}}(\alpha^{+}a\Delta
z+1)},{\frac{-1}{\sqrt{p_{0}}}(\alpha^{+}a\Delta
z+1)},z,{\frac{1}{2}\alpha^{+}z}\right).$

Since $\eta_{i}^{\vee}$ are horizontal sections of
$\nabla_{\mathcal{G}}^{\mathrm{rig}}$, using (31) to calcolate
$\frac{d\Pi(z)}{dz}$ shows that (25) becomes

(32)
$\nabla_{\mathcal{G}}^{\mathrm{rig}}(d\tau)=\left({\frac{\alpha^{-}\alpha^{+}a\Delta}{2\sqrt{p_{0}}}},{\frac{-\alpha^{+}a\Delta
z}{\sqrt{p_{0}}}},1,{\frac{1}{2}\alpha^{+}}\right)\cdot\underline{\eta}^{\vee}\otimes
dz.$

The recipe (19) to compute the Kodaira-Spencer map combined with (29) and (32)
gives then

(33) $\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}(d\tau\otimes
d\tau)=\frac{1}{t_{p}}\frac{d\Pi(z)}{dz}\begin{pmatrix}0&I_{2}\\\
-I_{2}&0\end{pmatrix}\Pi(z)^{T}dz=\frac{1}{t_{p}}dz.$

In particular, (33) shows that $\mathrm{KS}_{\mathcal{G}}^{\mathrm{rig}}$ is
an isomorphism, and therefore the same is true for
$\mathrm{KS}_{\mathcal{G}}^{\ast}$. This allows us to define the operator

$\textstyle{\tilde{\nabla}_{\mathcal{G},n}^{\ast}:\Lambda_{\mathcal{G},n}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\nabla_{\mathcal{G},n}^{\ast}}$$\textstyle{\Lambda_{\mathcal{G},n}^{\ast}\otimes_{\mathcal{A}^{\ast}}\Omega^{1}_{\mathcal{A}^{\ast}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\mathrm{KS}_{\mathcal{G}}^{\ast})^{-1}}$$\textstyle{\Lambda_{\mathcal{G},n}^{\ast}\otimes_{\mathcal{A}^{\ast}}w_{\mathcal{G},2}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{\mathcal{G},n}^{\ast}\otimes_{\mathcal{A}^{\ast}}\Lambda_{\mathcal{G},2}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{\mathcal{G},n+2}^{\ast}}$

where $\nabla_{\mathcal{G},n}^{\ast}$ is obtained from
$\nabla_{\mathcal{G}}^{\ast}$ using the Leibniz rule (as before, see for
example [HB15, §3.2]). The splitting in Theorem 4.7 induces a morphism of
$\mathcal{A}^{\ast}$-modules
$\Psi_{p,n}^{\ast}:\Lambda_{\mathcal{G},n}^{\ast}\rightarrow
w^{\ast}_{\mathcal{G},n}$. The composition

$\textstyle{\Theta_{p,n}^{\ast}:w_{\mathcal{G},n}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{\mathcal{G},n}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\nabla}_{\mathcal{G},n}^{\ast}}$$\textstyle{\Lambda_{\mathcal{G},n+2}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{p,n+2}^{\ast}}$$\textstyle{w^{\ast}_{\mathcal{G},n+2}}$

is the _$p$ -adic Shimura-Maass operator_. We also need to consider iterates
of this operator. Define

$\tilde{\nabla}_{n}^{j,\ast}=\tilde{\nabla}_{n+2j}^{\ast}\circ\dots\circ\tilde{\nabla}_{n+2}^{\ast}\circ\tilde{\nabla}_{n}^{\ast}.$

Define then

(34)
$\textstyle{\Theta_{p,n}^{j,\ast}:w_{\mathcal{G},n}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\Lambda_{\mathcal{G},n}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{\nabla}_{n}^{j,\ast}}$$\textstyle{\Lambda_{\mathcal{G},n+2j}^{\ast}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\Psi_{p,n+2j}^{\ast}}$$\textstyle{w^{\ast}_{\mathcal{G},n+2j}}$

where the morphism of $\mathcal{A}^{\ast}$-modules
$\Psi_{p,n+2j}^{\ast}:\Lambda_{\mathcal{G},n+2j}^{\ast}\rightarrow
w^{\ast}_{\mathcal{G},n+2j}$ is induced as above by the splitting in Theorem
4.7. We call $\Theta_{p,n}^{\ast,j}$ the $j$-th iterate of the $p$-adic
Shimura-Maass operator.

The work accomplished so far allows us to explicitly describe
$\Theta_{p,k}^{\ast,j}$. We first introduces some differential operators,
similar in shape to the Shimura-Maass operator in the real analytic setting.
For each integer $k\geq 0$, we may then define the
$\hat{\mathbb{Q}}_{p}^{\mathrm{unr}}$-linear function

(35)
$\delta_{p,k}=\frac{\partial}{\partial{z}}+\frac{k}{z-z^{\ast}}:\mathcal{A}^{\ast}\longrightarrow\mathcal{A}^{\ast}.$

For each integer $j\geq 0$ we get a map
$\delta_{p,k}^{j}:\mathcal{A}^{\ast}\rightarrow\mathcal{A}^{\ast}$ defined by

$\delta_{p,k}^{j}=\delta_{p,k+2(j-1)}\circ\dots\circ\delta_{p,k+2}\circ\delta_{p,k}.$

We call $\delta_{p,k}$ the _Shimura-Maass_ operator, and $\delta_{p,k}^{j}$
its $j$-th iteration. Applying (33) to compute the inverse of the Kodaira-
Spencer map to (27), we obtain

$\tilde{\nabla}_{k}^{\ast}\left(f(z)\otimes
d\tau^{\otimes{k}}\right)=\frac{1}{t_{p}}\cdot\left(\frac{\partial}{\partial
z}f(z)\otimes
d\tau^{\otimes{k+2}}+kf(z)\otimes\frac{d\tau-d\sigma(\tau)}{z-\sigma(z)}\otimes
d\tau^{\otimes{k+1}}\right).$

Applying the splitting ${\Psi}_{p,k+2}^{\ast}$ of the Hodge filtration which
annihilate $d\tau^{\ast}$, we finally obtain

(36) $\Theta_{p,k}^{\ast}\left(f(z)\otimes
d\tau^{\otimes{k}}\right)=\frac{1}{t_{p}}\cdot\left(\frac{\partial}{\partial
z}f(z)+\frac{kf(z)}{z-\sigma(z)}\right)\otimes
d\tau^{\otimes{k+2}}=\frac{1}{t_{p}}\cdot\delta_{p,k}(f)\otimes
d\tau^{\otimes{k+2}}.$

Iterating (36) we obtain

(37) $\Theta_{p,k}^{j,\ast}\left(f(z)\otimes
d\tau^{\otimes{k}}\right)=\left(\frac{1}{t_{p}}\right)^{j}\delta_{p,k}^{j}\left(f(z)\right)\otimes
d\tau^{\otimes{k+2j}}.$

##  5\. The $p$-adic Shimura-Maass operator on Shimura curves

The aim of this Section is to apply the results in Section 4 to define a
$p$-adic Shimura-Maass operator on Shimura curves.

###  5.1. $p$-adic uniformization of Shimura curves

In this subsection we review the Cerednik-Drinfel’d Theorem. Let
$B/\mathbb{Q}$ be the quaternion algebra obtained from $\mathcal{B}$ by
interchanging the invariants at $\infty$ and $p$; so $B$ is the definite
quaternion algebra over $\mathbb{Q}$ of discriminant $N^{-}$. For a subgroup
$U\subseteq\hat{B}^{\times}$, let $U^{(p)}$ the elements outside of the place
$p$. Fix isomorphisms $\mathcal{B}_{\ell}\simeq B_{\ell}$ for all primes
$\ell\neq p$, so that we can view $V_{1}(N^{+})^{(p)}$ as a subgroup of
$(\hat{B}^{\times})^{(p)}$. Define $\tilde{\Gamma}_{p}=B^{\times}\cap
V_{1}(N^{+})^{(p)}$. We still denote $\tilde{\Gamma}_{p}$ the image of
$\tilde{\Gamma}_{p}$ in $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ via a fixed
isomorphism
$i_{p}:B\otimes_{\mathbb{Q}}\mathbb{Q}_{p}\simeq\operatorname{M}_{2}(\mathbb{Q}_{p})$,
and we let ${\Gamma}_{p}$ denote the subgroup of $\tilde{\Gamma}_{p}$
consisting of elements whose determinant has even $p$-power order.

Base changing from $\mathbb{Z}_{p}$ to the valuation ring $\mathbb{Z}_{p^{2}}$
of $\mathbb{Q}_{p^{2}}$ gives a $\mathbb{Z}_{p^{2}}$-formal scheme
$\hat{\mathcal{H}}_{p^{2}}$, whose generic fiber $\mathcal{H}_{p^{2}}$ is the
base change of the $\mathbb{Q}_{p}$-rigid analytic space $\mathcal{H}_{p}$ to
$\mathbb{Q}_{p^{2}}$. The group $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ acts
on the $\mathbb{Z}_{p}$-formal scheme $\hat{\mathcal{H}}_{p}$ ([BC91, Chapitre
I, §6]) and acts on $\operatorname{Spf}({\mathbb{Z}}_{p^{2}})$ via the inverse
of the arithmetic Frobenius raised to the determinant map ([BC91, Chapitre II,
§9]). Therefore, the group $\operatorname{GL}_{2}(\mathbb{Q}_{p})$ also acts
on the $\mathbb{Z}_{p^{2}}$-formal scheme $\hat{\mathcal{H}}_{p^{2}}$ and the
$\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$-formal scheme
$\mathcal{\mathcal{H}}_{p}^{\mathrm{unr}}$, and the associated rigid analytic
spaces. We may then form the quotient
$\Gamma_{p}\backslash{\hat{\mathcal{H}}}_{p^{2}}$, in the category of
$\mathbb{Z}_{p^{2}}$-formal schemes, and the quotient
$\Gamma_{p}\backslash{\hat{\mathcal{H}}}_{p}^{\mathrm{unr}}$, in the category
of $\hat{\mathbb{Z}}_{p}^{\mathrm{unr}}$-formal schemes, and similarly for the
associated rigid analytic spaces. The formal completion