L-invariants for Hilbert modular forms

TL;DR

This paper proves that, under certain conditions, the Fontaine--Mazur and Teitelbaum type L-invariants for Hilbert eigenforms are equal, confirming a conjecture by Chida, Mok, and Park.

Contribution

It establishes the equality of two different L-invariants for Hilbert eigenforms under specific conditions, confirming a conjecture in the field.

Findings

Proved the equality of Fontaine--Mazur and Teitelbaum L-invariants for Hilbert eigenforms.

Confirmed a conjecture of Chida, Mok, and Park.

Provided conditions under which the invariants coincide.

Abstract

In this paper we show that under certain condition the Fontaine--Mazur $L$-invariant for a Hilbert eigenform coincides with its Teitelbaum type $L$-invariant, and thus prove a conjecture of Chida, Mok and Park.

Equations

$$(\text{CMP})\mathbf{f}\text{ is new at }\mathfrak{p}\text{ and }{U}_{\mathfrak{p}}\mathbf{f}=\mathcal{N}{\mathfrak{p}}^{w/2}\mathbf{f}.$$

$$\text{ there is no place other than }\mathfrak{p}\text{ above }p.$$

$$({\mathrm{B}}_{\mathrm{cris},F_{\mathfrak{p}}}{)}^{G_K}=({\mathrm{B}}_{\mathrm{st},F_{\mathfrak{p}}}{)}^{G_K}=({\mathrm{B}}_{\mathrm{dR}}{)}^{G_K}=K.$$

$$D_{\mathrm{st},F_{\mathfrak{p}}}(V):=(V{\otimes }_{{\mathbb{Q}}_p}{\mathrm{B}}_{\mathrm{st},F_{\mathfrak{p}}}{)}^{G_K}.$$

$$T_{F_{\mathfrak{p}}}\stackrel{\sim }{\to }\underset{\sigma }{⨁}{T}_{\sigma },t\otimes a\mapsto (\sigma (a)t{)}_{\sigma },$$

$${}^{\prime }{\rho }_{f_{\infty },\lambda ,\mathfrak{p}}^{\mathrm{F}\text{-}\mathrm{ss}}\simeq \check{\sigma }({\pi }_{f_{\infty },\mathfrak{p}}){\otimes }_L{L}_{\lambda }$$

$${\nabla }_X:E_X^{\mathrm{an}}\to E_X^{\mathrm{an}}\otimes {\Omega }_{X^{\mathrm{an}}}^1.$$

$$0\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}\text{ignorespaces}$$

$$\bigoplus_{e\in\mathrm{E}(\overline{{\mathcal{X}}})}{\mathrm{Res}}_{e}:\hskip 10.0pt\Big{(}\bigoplus_{e\in\mathrm{E}(\overline{{\mathcal{X}}})}H^{1}_{\mathrm{dR}}(]e[,\mathscr{E})\Big{)}^{+}\rightarrow\Big{(}\bigoplus_{e\in\mathrm{E}(\overline{{\mathcal{X}}})}H^{0}_{\mathrm{dR}}(]e[,\mathscr{E})\Big{)}^{-}.$$

$$\iota\circ(\bigoplus_{e\in\mathrm{E}(\overline{{\mathcal{X}}})}{\mathrm{Res}}_{e})\circ\Big{(}H^{1}_{\mathrm{dR}}(X^{\mathrm{an}},\mathscr{E})\rightarrow(\bigoplus_{e\in\mathrm{E}(\overline{{\mathcal{X}}})}H^{1}_{\mathrm{dR}}(]e[,\mathscr{E}))^{+}\Big{)}$$

$${\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathbb{Z}}_p}\mathrm{W}(\bar{k})=\prod _{j=0}^{v_p(q)-1}{\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathcal{O}}_{F_{\mathfrak{p}}^0},{\phi }^j\circ \iota }\mathrm{W}(\bar{k}).$$

$$u_{\varphi^{j}\circ\iota}=\left\{\begin{array}[]{ll}{\it\Pi}\otimes 1&\text{ if }j=0,\\ 1\otimes 1&\text{ if }j=1,\dots,v_{p}(q)-1.\end{array}\right.$$

$$\tilde{\mathrm{F}}x=(1\otimes \phi )x\cdot u,x\in {\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathbb{Z}}_p}\mathrm{W}(\bar{k}).$$

$$({\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathbb{Z}}_p}\mathrm{W}(\bar{k}),\tilde{\mathrm{V}},\tilde{\mathrm{F}})$$

$$<\cdot,\cdot>:\mathscr{O}_{B_{\mathfrak{p}}}\times\mathscr{O}_{B_{\mathfrak{p}}}\rightarrow{\mathbb{Z}}_{p},(x,y)\mapsto{\mathrm{tr}}_{F_{\mathfrak{p}}/{\mathbb{Q}}_{p}}\Big{(}\delta_{F_{\mathfrak{p}}/{\mathbb{Q}}_{p}}^{-1}{\mathrm{tr}}_{B_{\mathfrak{p}}/F_{\mathfrak{p}}}(xy^{t})\Big{)},$$

$${\phi }^j{\iota }_0,{\phi }^j{\iota }_1(0\le j\le v_p(q)-1)$$

$${\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathbb{Z}}_p}\mathrm{W}(\bar{k})=\prod _{j=0}^{v_p(q)-1}{\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathcal{O}}_{F_{\mathfrak{p}0}^{(2)}},{\phi }^j\circ {\iota }_0}\mathrm{W}(\bar{k})\times \prod _{j=0}^{v_p(q)-1}{\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathcal{O}}_{F_{\mathfrak{p}0}^{(2)}},{\phi }^j\circ {\iota }_1}\mathrm{W}(\bar{k}),$$

$$({\mathcal{O}}_{B_{\mathfrak{p}}}{\otimes }_{{\mathbb{Z}}_p}\mathrm{W}(\bar{k}),\tilde{\mathrm{F}})[1/p]=(B_{\mathfrak{p}}{\otimes }_{{\mathbb{Q}}_p}\mathrm{W}(\bar{k})[1/p],\tilde{\mathrm{F}})$$

$$(B_{\mathfrak{p}}{\otimes }_{{\mathbb{Q}}_p}\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}},{\tilde{\mathrm{F}}}^{v_p(q)})$$

$${\tilde{\mathrm{D}}}_0{\otimes }_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}{\mathcal{O}}_{\mathcal{H},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$$

$$1\otimes \mathrm{d}:{\tilde{\mathrm{D}}}_0{\otimes }_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}{\mathcal{O}}_{\mathcal{H},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}\to {\tilde{\mathrm{D}}}_0{\otimes }_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}{\Omega }_{\mathcal{H},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}.$$

$${\eta }_M:=N(M{)}^{{\phi }_M},T_M:=M/\mathrm{V}M;$$

$$N(M)\to M/\mathrm{V}M,((x_0,x_1))\mapsto x_0\mathrm{mod}\mathrm{V}M.$$

$$\{((g^0,0)),((g^1,0)),((\mathrm{V}{g}^0,0)),((\mathrm{V}{g}^1,0))\},$$

$$\rho :{\eta }_{M(\Phi )}^0{\otimes }_{{\mathcal{O}}_{F_{\mathfrak{p}}}}{F}_{\mathfrak{p}}\stackrel{\sim }{\to }{\eta }_{M(G)}^0{\otimes }_{{\mathcal{O}}_{F_{\mathfrak{p}}}}{F}_{\mathfrak{p}}.$$

$$z(G,\rho )=\frac{u_M^{\prime }\circ \rho ((\mathrm{V}{g}^1,0))}{u_M^{\prime }\circ \rho ((g^0,0))},$$

$${\mathrm{Fil}}^0{B}_{\mathfrak{p}}{\otimes }_{F_{\mathfrak{p}}}{\mathcal{O}}_{\mathcal{H},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$$

$${\mathrm{Fil}}^1{B}_{\mathfrak{p}}{\otimes }_{F_{\mathfrak{p}}}{\mathcal{O}}_{\mathcal{H},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$$

$${\mathrm{Fil}}^2{B}_{\mathfrak{p}}{\otimes }_{F_{\mathfrak{p}}}{\mathcal{O}}_{\mathcal{H},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$$

$$B_{\mathfrak{p}}{\otimes }_{F_{\mathfrak{p}}}\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}=B_{\mathfrak{p}}{\otimes }_{F_{\mathfrak{p}}^{(2)},{\tilde{\iota }}_0}\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}\oplus B_{\mathfrak{p}}{\otimes }_{F_{\mathfrak{p}}^{(2)},{\tilde{\iota }}_1}\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}},$$

$${\mathrm{Fil}}^0{B}_{\mathfrak{p}}{\otimes }_{F_{\mathfrak{p}}}{\mathcal{O}}_{\mathcal{H},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$$

Full text

Introduction

In the remarkable paper [17] Greenberg and Stevens proved a formula for the derivative at $s=1$ of the $p$-adic $L$-function of an elliptic curve $E$ over ${\mathbb{Q}}$ (i.e. a modular form of weight $2$) when $p$ is a prime of split multiplicative reduction, which is the exceptional zero conjecture proposed by Mazur, Tate, and Teitelbaum [21]. An important quantity in this formula is the $L$-invariant, namely ${\mathcal{L}}(E)=\log_{p}(q_{E})/v_{p}(q_{E})$ where $q_{E}\in p{\mathbb{Z}}_{p}$ is the Tate period for $E$.

For higher weight modular forms $f$, since [21], a number of different candidates for the $L$-invariant ${\mathcal{L}}(f)$ have been proposed. These include:

(1) Fontaine-Mazur’s $L$-invariant ${\mathcal{L}}_{FM}$ using $p$-adic Hodge theory,

(2) Teitelbaum’s $L$-invariant ${\mathcal{L}}_{T}$ built by the theory of $p$-adic uniformization of Shimura curves,

(3) an invariant ${\mathcal{L}}_{C}$ by Coleman’s theory of $p$-adic integration on modular curves,

(4) an invariant ${\mathcal{L}}_{DO}$ due to Darmon and Orton using “modular-form valued distributions”,

(5) Breuil’s $L$-invariant ${\mathcal{L}}_{B}$ by $p$-adic Langlands theory.

Now, all of these invariants are known to be equal [2, 4, 10, 19]. The readers are invited to consult Colmez’s paper [11] for some historical account.

The exceptional zero conjecture for higher weight modular forms has been proved by Steven using ${\mathcal{L}}_{C}$ [34], by Kato–Kurihara–Tsuji using ${\mathcal{L}}_{FM}$ (unpublished), by Orton using ${\mathcal{L}}_{DO}$ [25], by Emerton using ${\mathcal{L}}_{B}$ [14] and by Bertolini–Darmon–Iovita using ${\mathcal{L}}_{T}$ [2].

In [23] Mok addressed special cases of the exceptional zero conjecture in the setting of Hilbert modular forms. In [7] Chida, Mok and Park introduced the Teitelbaum type $L$-invariant for Hilbert modular forms, and conjectured that Teitelbaum type $L$-invariant coincides with the Fontaine-Mazur $L$-invariant. We state this conjecture precisely below.

Fix a prime number $p$. Let $F$ be a totally real field, $g=[F:{\mathbb{Q}}]$ and ${\mathfrak{p}}$ a prime ideal of $F$ above $p$. Let $f_{\infty}$ be a Hilbert eigen newform with even weight $(k_{1},\cdots,k_{g},w)$ and level divided exactly by ${\mathfrak{p}}$ (i.e. not by ${\mathfrak{p}}^{2}$). Here “even weight” means that $k_{1},\cdots,k_{g},w$ are all even.

On one hand, by Carayol’s result [6] we can attach to $f_{\infty}$ a $p$-adic representation of $G_{{\mathbb{Q}}}=\mathrm{Gal}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})$. This Galois representation (restricted to $G_{{\mathbb{Q}}_{p}}$) is semistable and thus we can attach to it a Fontaine-Mazur $L$-invariant ${\mathcal{L}}_{FM}(f_{\infty})$.

On the other hand, Chida, Mok and Park attached to an automorphic form $\mathbf{f}$ on a totally definite quaternion algebra over $F$ (of the same weight $(k_{1},\cdots,k_{g},w)$) a Teitelbaum type $L$-invariant ${\mathcal{L}}_{T}(\mathbf{f})$ under the following assumption

[TABLE]

Both ${\mathcal{L}}_{FM}(f_{\infty})$ and ${\mathcal{L}}_{T}(\mathbf{f})$ are vector valued. See Section 1.2 and Section 8.2 for their precise definitions.

Conjecture 0.1**.**

If $f_{\infty}$ and $\mathbf{f}$ are associated to each other by the Jacquet-Langlands correspondence, then ${\mathcal{L}}_{FM}(f_{\infty})={\mathcal{L}}_{T}(\mathbf{f})$.

Our main result is the following

Theorem 0.2**.**

$($=Theorem 9.4$)$ Assume that $F$ satisfies the following condition:

[TABLE]

Let $f_{\infty}$ and $\mathbf{f}$ be as above. Then ${\mathcal{L}}_{FM}(f_{\infty})={\mathcal{L}}_{T}(\mathbf{f})$.

We sketch the proof of Theorem 0.2. Our method is similar to that in [19]. The Galois representation attached to $f_{\infty}$ comes from the étale cohomology $H^{1}_{\mathrm{et}}$ of some local system on a Shimura curve. The technical part of our paper is the computation of the filtered $\varphi_{q}$-isocrystal attached to this local system. On one hand, Coleman and Iovita [10] provided a precise description of the monodromy operator, which is helpful for computing Fontaine-Mazur’s $L$-invariant. On the other hand, the Teitelbaum type $L$-invariant is closely related to the de Rham cohomology of the filtered $\varphi_{q}$-isocrystal by Coleman integration and Schneider integration. Our precise description of the filtered $\varphi_{q}$-isocrystal allows us to compute Fontaine-Mazur’s $L$-invariant and the Teitelbaum type $L$-invariant. Finally, analyzing the relation among the monodromy operator, Coleman integration and Schneider integration finishes the proof.

When $F$ has more than one place (say $r$ places) above $p$, our method of computing filtered $\varphi_{q}$-isocrystals is not valid. To make it work, one might have to consider the Shimura variety studied by Rapoport and Zink [27, Chapter 6] (which is of dimension $r$) instead of the Shimura curve. Coleman and Iovita’s result is valid only for curves, and so can not be applied directly. We plan to address this problem in a future work.

Our paper is organized as follows. Fontaine-Mazur’s $L$-invariant is introduced in Section 1. Coleman and Iovita’s result is recall in Section 2. Section 3 is devoted to compute the filtered $\varphi_{q}$-isocrystal attached to the universal special formal module. We introduce various Shimura curves, and study their $p$-adic uniformizations following Rapoport and Zink respectively in Section 4 and Section 5. In Section 6 we use the result in Section 3 to determine the filtered $\varphi_{q}$-isocrystals attached to various local systems on Shimura curves. In Section 7 we recall the theory of de Rham cohomology of certain local systems, and in Section 8 we recall Chida, Mok and Park’s construction of Teitelbaum type $L$-invariant. Finally in Section 9 we combine results in Section 2, Section 6 and Section 7 to prove our main theorem.

Acknowledgement

This paper is supported by the National Natural Science Foundation of China (grant 11671137).

Notations

For two ${\mathbb{Q}}$-algebras $A$ and $B$, write $A\otimes B$ for $A\otimes_{{\mathbb{Q}}}B$. For a ring $R$ let $R^{\times}$ denote the multiplicative group of invertible elements in $R$. For a linear algebraic group over ${\mathbb{Q}}$ we will identify it with its ${\mathbb{Q}}$-valued points.

Let $F$ be a totally real number field, $g=[F:{\mathbb{Q}}]$. Let $p$ be a fixed prime. Suppose that $p$ is inertia in $F$, i.e. there exists exactly one place of $F$ above $p$, denoted by ${\mathfrak{p}}$. If $q$ is a power of $p$, we use $v_{p}(q)$ to denote $\log_{p}q$.

Let ${\mathbb{A}}_{f}$ denote ${\mathbb{Q}}\otimes_{\mathbb{Z}}\widehat{{\mathbb{Z}}}$ and let ${\mathbb{A}}_{f}^{p}$ denote ${\mathbb{Q}}\otimes_{\mathbb{Z}}(\prod_{\ell\neq p}{\mathbb{Z}}_{\ell})$. Similarly for any number field $E$ let ${\mathbb{A}}_{E,f}$ denote $E\otimes_{\mathbb{Z}}\widehat{{\mathbb{Z}}}$, the group of finite adèles of $E$.

Fix an algebraic closure of $F_{\mathfrak{p}}$, denoted by $\overline{F_{\mathfrak{p}}}$, and let ${\mathbb{C}}_{p}$ be the completion of $\overline{F_{\mathfrak{p}}}$ with respect to the $p$-adic topology. By this way we have fixed an embedding $F_{\mathfrak{p}}\hookrightarrow{\mathbb{C}}_{p}$. The Galois group $G_{F_{\mathfrak{p}}}={\mathrm{Gal}}(\overline{F_{\mathfrak{p}}}/F_{\mathfrak{p}})$ can be naturally identified with the group of continuous $F_{\mathfrak{p}}$-automorphisms of ${\mathbb{C}}_{p}$.

1 Fontaine-Mazur invariant

1.1 Monodromy modules and Fontaine-Mazur $L$-invariant

Let $F_{{\mathfrak{p}}0}$ be the maximal absolutely unramified subfield of $F_{\mathfrak{p}}$. Let $q$ be the cardinal number of the residue field of $F_{\mathfrak{p}}$.

Let ${\mathrm{B}}_{\mathrm{cris}},{\mathrm{B}}_{\mathrm{st}}$ and ${\mathrm{B}}_{\mathrm{dR}}$ be Fontaine’s period rings [16]. As is well known, there are operators $\varphi$ and $N$ on ${\mathrm{B}}_{\mathrm{st}}$, and a descending ${\mathbb{Z}}$-filtration on ${\mathrm{B}}_{\mathrm{dR}}$; ${\mathrm{B}}_{\mathrm{cris}}$ is a $\varphi$-stable subring of ${\mathrm{B}}_{\mathrm{st}}$ and $N$ vanishes on ${\mathrm{B}}_{\mathrm{cris}}$. Put ${\mathrm{B}}_{{\mathrm{st}},F_{\mathfrak{p}}}:={\mathrm{B}}_{\mathrm{st}}\otimes_{F_{{\mathfrak{p}}0}}F_{\mathfrak{p}}$; ${\mathrm{B}}_{{\mathrm{st}},F_{\mathfrak{p}}}$ can be considered as a subring of ${\mathrm{B}}_{\mathrm{dR}}$. We extend the operators $\varphi_{q}=\varphi^{v_{p}(q)}$ and $N$ $F_{\mathfrak{p}}$-linearly to ${\mathrm{B}}_{{\mathrm{st}},F_{\mathfrak{p}}}$.

Let $K$ be either a finite unramified extension of $F_{\mathfrak{p}}$ or the completion of the maximal unramified extension of $F_{\mathfrak{p}}$ in ${\mathbb{C}}_{p}$. Write $G_{K}$ for the group of continuous automorphisms of ${\mathbb{C}}_{p}$ fixing elements of $K$. By our assumption on $K$ we have

[TABLE]

Let $L$ be a finite extension of ${\mathbb{Q}}_{p}$. For an $L$-linear representation $V$ of $G_{K}$, we put

[TABLE]

This is a finite rank $L\otimes_{{\mathbb{Q}}_{p}}K$-module. If $V$ is semistable, then $D_{{\mathrm{st}},F_{\mathfrak{p}}}(V)$ is a filtered $(\varphi_{q},N)$-module: the $(\varphi_{q},N)$-module structure is induced from the operators $\varphi_{q}=1_{V}\otimes\varphi_{q}$ and $N=1_{V}\otimes N$ on $V\otimes_{{\mathbb{Q}}_{p}}{\mathrm{B}}_{{\mathrm{st}},F_{\mathfrak{p}}}$; the filtration comes from that on $V\otimes_{{\mathbb{Q}}_{p}}{\mathrm{B}}_{\mathrm{dR}}$. Note that $\varphi_{q}$ and $N$ are $L\otimes_{{\mathbb{Q}}_{p}}K$-linear.

If $L$ splits $F_{\mathfrak{p}}$, then $L\otimes_{{\mathbb{Q}}_{p}}K$ is isomorphic to $\bigoplus_{\sigma}L\otimes_{\sigma,F_{\mathfrak{p}}}K$, where $\sigma$ runs through all embeddings of $F_{\mathfrak{p}}$ into $L$. Here the subscript $\sigma$ under $\otimes$ indicates that $F_{\mathfrak{p}}$ is considered as a subfield of $L$ via $\sigma$. Let $e_{\sigma}$ be the unity of the subring $L\otimes_{\sigma,F_{\mathfrak{p}}}K$.

We shall need the notion of monodromy modules. This notion is introduced in [20]. However we will use the slightly different definition given in [19].

Let $T$ be a finite-dimensional commutative semisimple ${\mathbb{Q}}_{p}$-algebra. A $T$-object $D$ in the category of filtered $(\varphi_{q},N)$-modules, is called a $2$-dimensional monodromy $T$-module, if the following hold:

$\bullet$ $D$ is a free $T_{F_{\mathfrak{p}}}$-module of rank $2$ ($T_{F_{\mathfrak{p}}}=T\otimes_{{\mathbb{Q}}_{p}}F_{\mathfrak{p}}$),

$\bullet$ the sequence $D\xrightarrow{N}D\xrightarrow{N}D$ is exact,

$\bullet$ there exists an integer $j_{0}$ such that ${\mathrm{Fil}}^{j_{0}}D$ is a free $T_{F_{\mathfrak{p}}}$-submodule of rank $1$ and ${\mathrm{Fil}}^{j_{0}}D\cap\ker(N)=0$.

Lemma 1.1**.**

$($[19, Lemma 2.3]$)$* If $D$ is a monodromy $T$-module, then there exists a decomposition $D=D^{(1)}\oplus D^{(2)}$ where $D^{(1)}$ and $D^{(2)}$ are $\varphi_{q}$-stable free rank one $T_{F_{\mathfrak{p}}}$-submodules such that $N:D\rightarrow D$ induces an isomorphism $N|_{D^{(2)}}:D^{(2)}\xrightarrow{\sim}D^{(1)}$.*

Let $D$ be a monodromy $T$-module and let $j_{0}$ be as above. The Fontaine-Mazur $L$-invariant of $D$ is defined to be the unique element in $T_{F_{\mathfrak{p}}}$, denoted as ${\mathcal{L}}_{FM}(D)$, such that $x-{\mathcal{L}}_{FM}(D)N(x)\in{\mathrm{Fil}}^{j_{0}}D$ for every $x\in D^{(2)}$.

What we are interested in is the case when $T$ is an $L$-algebra, where $L$ is a field splitting $F_{\mathfrak{p}}$. Note that we have an decomposition of $T_{F_{\mathfrak{p}}}$:

[TABLE]

where $\sigma$ runs through all embeddings of $F_{\mathfrak{p}}$ in $L$. The index $\sigma$ in $T_{\sigma}$ indicates that $T$ is considered as an $F_{\mathfrak{p}}$-algebra via $\sigma$. Then we have a decomposition of $D$ by $D\simeq\bigoplus_{\sigma}D_{\sigma}$, where $D_{\sigma}=e_{\sigma}\cdot D$. Each $D_{\sigma}$ is stable under $\varphi_{q}$ and $N$. Note that, for every $j$, ${\mathrm{Fil}}^{j}D$ is a $T_{F_{\mathfrak{p}}}$-submodule. Thus the filtration on $D$ restricts to a filtration on $D_{\sigma}$ for each $\sigma$, and satisfies ${\mathrm{Fil}}^{j}D=\bigoplus_{\sigma}{\mathrm{Fil}}^{j}D_{\sigma}$ for all $j\in{\mathbb{Z}}$.

Using the decomposition (1.1) we may write ${\mathcal{L}}_{FM}(D)$ in the form $({\mathcal{L}}_{FM,\sigma}(D))_{\sigma}$. It is easy to see that ${\mathcal{L}}_{FM,\sigma}(D)$ is the unique element in $T$ such that $x-{\mathcal{L}}_{FM,\sigma}(D)N(x)\in{\mathrm{Fil}}^{j_{0}}D_{\sigma}$ for every $x\in D^{(2)}_{\sigma}$. We also call $({\mathcal{L}}_{FM,\sigma}(D))_{\sigma}$, a vector with values in $T$, the Fontaine-Mazur $L$-invariant of $D$.

1.2 Fontaine-Mazur $L$-invariant for Hilbert modular forms

Let $\{\tau_{1},\cdots,\tau_{g}\}$ be the set of real embeddings $F\hookrightarrow{\mathbb{R}}$. Fix a multiweight $\mathrm{k}=(k_{1},\cdots,k_{g},w)\in{\mathbb{N}}^{g+1}$ satisfying $k_{i}\geq 2$ and $k_{i}\equiv w\hskip 6.0pt\mathrm{mod}\hskip 3.0pt2$.

Let $\pi=\otimes_{v}\pi_{v}$ be a cuspidal automorphic representation of ${\mathrm{GL}}(2,{\mathbb{A}}_{F})$ such that for each infinite place $\tau_{i}$, the $\tau_{i}$-component $\pi_{\tau_{i}}$ is a holomorphic discrete series representation $D_{k_{i}}$. Let ${\mathfrak{n}}$ be the level of $\pi$.

Carayol [6] attached to such an automorphic representation (under a further condition) an $\ell$-adic Galois representation, which will be recalled below.

Let $L$ be a sufficiently large number field of finite degree over ${\mathbb{Q}}$ such that the finite part $\pi^{\infty}=\otimes_{{\mathfrak{p}}\nmid\infty}\pi_{\mathfrak{p}}$ of $\pi$ admits an $L$-structure $\pi_{L}^{\infty}$. The fixed part $(\pi^{\infty}_{L})^{K_{1}({\mathfrak{n}})}$ is of dimension $1$ and generated by an eigenform $f_{\infty}$. In this case we write $\pi_{f_{\infty}}$ for $\pi$.

The local Langlands correspondence associates to every irreducible admissible representation $\pi$ of ${\mathrm{GL}}(2,F_{\mathfrak{p}})$ defined over $L$ a $2$-dimensional $L$-rational Frobenius-semisimple representation $\sigma(\pi)$ of the Weil-Deligne group $WD(\overline{F}_{\mathfrak{p}}/F_{\mathfrak{p}})$. Let $\check{\sigma}(\pi)$ denote the dual of $\sigma(\pi)$.

For an $\ell$-adic representation $\rho$ of ${\mathrm{Gal}}(\overline{F}/F)$, let $\rho_{\mathfrak{p}}$ denote its restriction to ${\mathrm{Gal}}(\overline{F}_{\mathfrak{p}}/F_{\mathfrak{p}})$, ${}^{\prime}\hskip-3.0pt\rho_{\mathfrak{p}}$ the Weil-Deligne representation attached to $\rho_{\mathfrak{p}}$ and ${}^{\prime}\hskip-3.0pt\rho_{\mathfrak{p}}^{\text{F-ss}}$ the Frobenius-semisimplification of ${}^{\prime}\hskip-3.0pt\rho_{\mathfrak{p}}$.

Theorem 1.2**.**

[6]** Let $f_{\infty}$ be an eigenform of multiweight $k$ satisfying the following condition:

If $g=[F:{\mathbb{Q}}]$ is even, then there exists a finite place ${\mathfrak{q}}$ such that the ${\mathfrak{q}}$-factor $\pi_{f_{\infty},{\mathfrak{q}}}$ lies in the discrete series.

Then for any prime number $\ell$ and a finite place $\lambda$ of $L$ above $\ell$, there exists a $\lambda$-adic representation $\rho=\rho_{f_{\infty},\lambda}:{\mathrm{Gal}}(\overline{F}/F)\rightarrow{\mathrm{GL}}_{L_{\lambda}}(V_{f_{\infty},\lambda})$ satisfying the following property:

For any finite place ${\mathfrak{p}}\nmid\ell$ there is an isomorphism

[TABLE]

of representations of the Weil-Deligne group $WD(\overline{F}_{\mathfrak{p}}/F_{\mathfrak{p}})$.

Saito [28] showed that when ${\mathfrak{p}}|\ell$, $\rho_{f_{\infty},\lambda,{\mathfrak{p}}}$ is potentially semistable.

Now we assume that $\ell=p$, ${\mathfrak{p}}$ is the prime ideal of $F$ above $p$, and $L$ contains $F$. Let $\mathfrak{P}$ be a prime ideal of $L$ above ${\mathfrak{p}}$.

Theorem 1.3**.**

$($=Theorem 9.2$)$ Let $f_{\infty}$ be as in Theorem 1.2, $\ell=p$ and $\lambda=\mathfrak{P}$. If $f_{\infty}$ is new at ${\mathfrak{p}}$ (when $[F:{\mathbb{Q}}]$ is odd, we demand that $f_{\infty}$ is new at another prime ideal), then $\rho_{f_{\infty},\mathfrak{P},{\mathfrak{p}}}$ is a semistable $($non-crystalline$)$ representation of ${\mathrm{Gal}}(\overline{F}_{\mathfrak{p}}/F_{\mathfrak{p}})$, and the filtered $(\varphi_{q},N)$-module $D_{{\mathrm{st}},F_{\mathfrak{p}}}(\rho_{f_{\infty},\mathfrak{P},{\mathfrak{p}}})$ is a monodromy $L_{\mathfrak{P}}$-module.

Remark 1.4*.*

The conditions in Theorem 1.2 and Theorem 1.3 are used to ensure that via the Jacquet-Langlands correspondence $f_{\infty}$ corresponds to a modular form on the Shimura curve $M$ associated to a quaternion algebra $B$ that splits at exactly one real place; in Theorem 1.3 the quaternion algebra $B$ is demanded to be ramified at ${\mathfrak{p}}$. See Section 4.1 for the construction of $M$.

Thus $D_{{\mathrm{st}},F_{\mathfrak{p}}}(\rho_{f_{\infty},\mathfrak{P},{\mathfrak{p}}})$ is associated with the Fontaine-Mazur $L$-invariant. We define the Fontaine-Mazur $L$-invariant of $f_{\infty}$, denoted by ${\mathcal{L}}_{FM}(f_{\infty})$, to be that of $D_{{\mathrm{st}},F_{\mathfrak{p}}}(\rho_{f_{\infty},\mathfrak{P},{\mathfrak{p}}})$.

2 Local systems and the associated filtered

$\varphi_{q}$-isocrystals

Let $X$ be a $p$-adic formal $\mathscr{O}_{F_{\mathfrak{p}}}$-scheme. Suppose that $X$ is analytically smooth over $\mathscr{O}_{F_{\mathfrak{p}}}$, i.e. the generic fiber $X^{\mathrm{an}}$ of $X$ is smooth.

The filtered convergent $\varphi$-isocrystals attached to local systems are studied in [15, 10]. It is more convenience for us to compute the filtered convergent $\varphi_{q}$-isocrystals attached to the local systems that we will be interested in. From now on, we will ignore “convergent” in the notion.

Filtered $\varphi_{q}$-isocrystal is a natural analogue of filtered $\varphi$-isocrystal. To define it one needs the notion of $F_{{\mathfrak{p}}}$-enlargement. An $F_{{\mathfrak{p}}}$-enlargement of $X$ is a pair $(T,x_{T})$ consisting of a flat formal $\mathscr{O}_{F_{\mathfrak{p}}}$-scheme $T$ and a morphism of formal $\mathscr{O}_{F_{\mathfrak{p}}}$-scheme $x_{T}:T_{0}\rightarrow X$, where $T_{0}$ is the reduced closed subscheme of $T$ defined by the ideal $\pi\mathscr{O}_{T}$.

An isocrystal $\mathscr{E}$ on $X$ consists of the following data:

$\bullet$ for every $F_{\mathfrak{p}}$-enlargement $(T,x_{T})$ a coherent $\mathscr{O}_{T}\otimes_{\mathscr{O}_{F_{\mathfrak{p}}}}F_{\mathfrak{p}}$-module $\mathscr{E}_{T}$,

$\bullet$ for every morphism of $F_{\mathfrak{p}}$-enlargements $g:(T^{\prime},x_{T^{\prime}})\rightarrow(T,x_{T})$ an isomorphism of

$\mathscr{O}_{T^{\prime}}\otimes_{\mathscr{O}_{F_{\mathfrak{p}}}}F_{\mathfrak{p}}$-modules $\theta_{g}:g^{*}(\mathscr{E}_{T})\rightarrow\mathscr{E}_{T^{\prime}}$.

The collection of isomorphisms $\{\theta_{g}\}$ is required to satisfy certain cocycle condition. If $T$ is an $F_{\mathfrak{p}}$-enlargement of $X$, then $\mathscr{E}_{T}$ may be interpreted as a coherent sheaf $E_{T}^{\mathrm{an}}$ on the rigid space $T^{\mathrm{an}}$.

As $X$ is analytically smooth over $\mathscr{O}_{F_{\mathfrak{p}}}$, there is a natural integrable connection

[TABLE]

Note that an isocrystal on $X$ depends only on $X_{0}$. Let $\varphi_{q}$ denote the absolute $q$-Frobenius of $X_{0}$. A $\varphi_{q}$-isocrystal on $X$ is an isocrystal $\mathscr{E}$ on $X$ together with an isomorphism of isocrystals $\varphi_{q}:\varphi_{q}^{*}\mathscr{E}\rightarrow\mathscr{E}$. A filtered $\varphi_{q}$-isocrystal on $X$ is a $\varphi_{q}$-isocrystal $\mathscr{E}$ with a descending ${\mathbb{Z}}$-filtration on $E_{X}^{\mathrm{an}}$.

The following well known result compares the de Rham cohomology of a filtered $\varphi_{q}$-isocrystal and the étale cohomology of the ${\mathbb{Q}}_{p}$-local system associated to it.

Proposition 2.1**.**

[15, Theorem 3.2]** Suppose that $X$ is a semistable proper curve over $\mathscr{O}_{F_{\mathfrak{p}}}$. Let $\mathscr{E}$ be a filtered $\varphi_{q}$-isocrystal over $X$ and ${\mathcal{E}}$ be a ${\mathbb{Q}}_{p}$-local system over $X_{\overline{F}_{\mathfrak{p}}}$ that are attached to each other. Then the Galois representation $H^{i}_{\mathrm{et}}(X_{\overline{F}_{\mathfrak{p}}},{\mathcal{E}})$ of $G_{F_{\mathfrak{p}}}$ is semistable, and the filtered $(\varphi_{q},N)$-module $D_{{\mathrm{st}},F_{\mathfrak{p}}}(H^{i}_{\mathrm{et}}(X_{\overline{F}_{\mathfrak{p}}},{\mathcal{E}}))$ is isomorphic to $H^{i}_{\mathrm{dR}}(X^{\mathrm{an}},\mathscr{E})$.

Coleman and Iovita [10] gave a precise description of the monodromy $N$ on $H^{1}_{\mathrm{dR}}(X^{\mathrm{an}},\mathscr{E})$.

Now let $X$ be a connected, smooth and proper curve over $F_{\mathfrak{p}}$ with a regular semistable model ${\mathcal{X}}$ over $\mathscr{O}_{F_{\mathfrak{p}}}$ such that all irreducible components of its special fiber $\overline{{\mathcal{X}}}$ are smooth. For a subset $U$ of $\overline{{\mathcal{X}}}$ let $]U[$ denote the tube of $U$ in $X^{\mathrm{an}}$. We associate to $\overline{{\mathcal{X}}}$ a graph $\mathrm{Gr}(\overline{{\mathcal{X}}})$. Let $\mathrm{n}:\overline{{\mathcal{X}}}^{\mathrm{n}}\rightarrow\overline{{\mathcal{X}}}$ be the normalization of $\overline{{\mathcal{X}}}$. The vertices $\mathrm{V}(\overline{{\mathcal{X}}})$ of $\mathrm{Gr}(\overline{{\mathcal{X}}})$ are irreducible components of $\overline{{\mathcal{X}}}$. For every vertex $v$ let $C_{v}$ be the irreducible component corresponding to $v$. The edges $\mathrm{E}(\overline{{\mathcal{X}}})$ of $\mathrm{Gr}(\overline{{\mathcal{X}}})$ are ordered pairs $\{x,y\}$ where $x$ and $y$ are two different liftings in $\overline{{\mathcal{X}}}^{\mathrm{n}}$ of a singular point. Let $\tau$ be the involution on $\mathrm{E}(\overline{{\mathcal{X}}})$ such that $\tau\{x,y\}=\{y,x\}$. Below, for a module $M$ on which $\tau$ acts, set $M^{\pm}=\{m\in M:\tau(m)=\pm m\}$.

Let $\mathscr{E}$ be a filtered $\varphi_{q}$-isocrystal over $X$. For any $e=\{x,y\}\in\mathrm{E}(\overline{{\mathcal{X}}})$ let $H^{i}_{\mathrm{dR}}(]e[,\mathscr{E})$ denote $H^{i}_{\mathrm{dR}}(]\mathrm{n}(x)[,\mathscr{E})$. Then $\tau$ exchanges $H^{i}_{\mathrm{dR}}(]e[,\mathscr{E})$ and $H^{i}_{\mathrm{dR}}(]\bar{e}[,\mathscr{E})$ where $\bar{e}=\{y,x\}$. Note that $\{C_{v}\}_{v\in\mathrm{V}(\overline{{\mathcal{X}}})}$ is an admissible covering of $X^{\mathrm{an}}$. From the Mayer-Vietorus exact sequence with respect to this admissible covering we obtain the following short exact sequence

[TABLE]

For any $e\in\mathrm{E}(\overline{{\mathcal{X}}})$ there is a natural residue map ${\mathrm{Res}}_{e}:H^{1}_{\mathrm{dR}}(]e[,\mathscr{E})\rightarrow H^{0}_{\mathrm{dR}}(]e[,\mathscr{E})$ [10, Section 4.1]. These residue maps induce a map

[TABLE]

Proposition 2.2**.**

[10, Theorem 2.6, Remark 2.7]** The monodromy operator $N$ on $H^{1}_{\mathrm{dR}}(X^{\mathrm{an}},\mathscr{E})$ coincides with the composition

[TABLE]

where H^{1}_{\mathrm{dR}}(X^{\mathrm{an}},\mathscr{E})\rightarrow\Big{(}\bigoplus_{e\in\mathrm{E}(\overline{{\mathcal{X}}})}H^{1}_{\mathrm{dR}}(]e[,\mathscr{E})\Big{)}^{+} is the restriction map.

3 The universal special formal module

3.1 Special formal modules and Drinfeld’s moduli theorem

Let $B_{\mathfrak{p}}$ be the quaternion algebra over $F_{\mathfrak{p}}$ with invariant $1/2$. So $B_{\mathfrak{p}}$ is isomorphic to $F_{\mathfrak{p}}^{(2)}[{\it\Pi}]$; ${\it\Pi}^{2}=\pi$ and ${\it\Pi}a=\bar{a}{\it\Pi}$ for all $a\in F^{(2)}_{\mathfrak{p}}$. Here, $\pi$ is a fixed uniformizer of $F_{\mathfrak{p}}$, $F_{\mathfrak{p}}^{(2)}$ is the unramified extension of $F_{\mathfrak{p}}$ of degree $2$, and $a\mapsto\bar{a}$ denotes the nontrivial $F_{\mathfrak{p}}$-automorphism of $F^{(2)}_{\mathfrak{p}}$.

Let $\mathscr{O}_{B_{\mathfrak{p}}}$ be the ring of integers in $B_{\mathfrak{p}}$. Let $F_{{\mathfrak{p}}0}$ be the maximal absolutely unramified subfield of $F_{\mathfrak{p}}$, $k$ the residue field of $F_{\mathfrak{p}}$, and $F_{{\mathfrak{p}}0}^{(2)}$ the unramified extension of $F_{{\mathfrak{p}}0}$ of degree $2$.

Let $\mathscr{O}^{\mathrm{ur}}$ denote the maximal unramified extension of $\mathscr{O}_{F_{\mathfrak{p}}}$, $\widehat{\mathscr{O}^{\mathrm{ur}}}$ its $\pi$-adic completion. Fix an algebraic closure $\bar{k}$ of $k$. We identify $\widehat{\mathscr{O}^{\mathrm{ur}}}/\pi\widehat{\mathscr{O}^{\mathrm{ur}}}$ with $\bar{k}$. Then $\mathrm{W}(\bar{k})\otimes_{\mathscr{O}_{F_{\mathfrak{p}}^{0}}}\mathscr{O}_{F_{\mathfrak{p}}}\cong\widehat{\mathscr{O}^{\mathrm{ur}}}$. Let $\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}$ be the fractional field of $\widehat{\mathscr{O}^{\mathrm{ur}}}$.

We use the notion of special formal $\mathscr{O}_{B_{\mathfrak{p}}}$-module in [13].

First we fix a special formal $\mathscr{O}_{B_{\mathfrak{p}}}$-module over $\bar{k}$, $\Phi$, as in [27, (3.54)]. Let $\iota$ denote the natural embedding of $F_{{\mathfrak{p}}0}$ into $\mathrm{W}(\bar{k})[1/p]$. Then all embeddings of $F_{{\mathfrak{p}}0}$ into $\mathrm{W}(\bar{k})[1/p]$ are $\varphi^{j}\circ\iota$ ($0\leq j\leq v_{p}(q)-1$). We have the decomposition

[TABLE]

Let $u\in\mathscr{O}_{B_{\mathfrak{p}}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k})$ be the element whose $\varphi^{j}\circ\iota$-component with respect to this decomposition is

[TABLE]

Let $\widetilde{\mathrm{F}}$ be the $1\otimes\varphi$-semilinear operator on $\mathscr{O}_{B_{\mathfrak{p}}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k})$ defined by

[TABLE]

Let $\widetilde{\mathrm{V}}$ be the $1\otimes\varphi^{-1}$-semilinear operator on $\mathscr{O}_{B_{\mathfrak{p}}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k})$ such that $\widetilde{\mathrm{F}}\widetilde{\mathrm{V}}=p$. Then

[TABLE]

is a Dieudonne module over $\mathrm{W}(\bar{k})$ with an action of $\mathscr{O}_{B_{\mathfrak{p}}}$ by the left multiplication. Let $\Phi$ be the special formal $\mathscr{O}_{B_{\mathfrak{p}}}$-module over $\bar{k}$ whose contravariant Dieudonne crystal is $(\mathscr{O}_{B_{\mathfrak{p}}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k}),\widetilde{\mathrm{V}},\widetilde{\mathrm{F}})$. 111The Dieudonne crystal in [27, (3.54)] is the covariant Dieudonne crystal of $\Phi$. The duality between our contravariant Dieudonne crystal and the covariant Dieudonne crystal is induced by the trace map

<\cdot,\cdot>:\mathscr{O}_{B_{\mathfrak{p}}}\times\mathscr{O}_{B_{\mathfrak{p}}}\rightarrow{\mathbb{Z}}_{p},(x,y)\mapsto{\mathrm{tr}}_{F_{\mathfrak{p}}/{\mathbb{Q}}_{p}}\Big{(}\delta_{F_{\mathfrak{p}}/{\mathbb{Q}}_{p}}^{-1}{\mathrm{tr}}_{B_{\mathfrak{p}}/F_{\mathfrak{p}}}(xy^{t})\Big{)},

where $\delta_{F_{\mathfrak{p}}/{\mathbb{Q}}_{p}}$ is the different of $F_{\mathfrak{p}}$ over ${\mathbb{Q}}_{p}$, ${\mathrm{tr}}_{B_{\mathfrak{p}}/F_{\mathfrak{p}}}$ is the reduced trace map, and $y\mapsto y^{t}$ is the involution of $B_{\mathfrak{p}}$ such that ${\it\Pi}^{t}={\it\Pi}$ and $a^{t}=\bar{a}$ if $a\in F_{\mathfrak{p}}^{(2)}$. Then we have $<b\cdot x,y>=<x,b^{t}\cdot y>$ for any $b\in\mathscr{O}_{B_{\mathfrak{p}}}$.

Let $\iota_{0}$ and $\iota_{1}$ be the extensions of $\iota$ to $F_{{\mathfrak{p}}0}^{(2)}$. Then

[TABLE]

are all embeddings of $F_{{\mathfrak{p}}0}^{(2)}$ into $\mathrm{W}(\bar{k})[1/p]$. We have

[TABLE]

where $\mathscr{O}_{B_{\mathfrak{p}}}$ is considered as an $\mathscr{O}_{F_{\mathfrak{p}}^{(2)}}$-module by the left multiplication. Let $X$ be the element of $\mathscr{O}_{B_{\mathfrak{p}}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k})$ whose $\varphi^{j}\circ\iota_{0}$-component ($0\leq j\leq v_{p}(q)-1$) is $1\otimes 1$ and whose $\varphi^{j}\circ\iota_{1}$-component ($0\leq j\leq v_{p}(q)-1$) is ${\it\Pi}\otimes 1$. Similarly, let $Y$ be the element whose $\varphi^{j}\circ\iota_{0}$-component ($0\leq j\leq v_{p}(q)-1$) is ${\it\Pi}\otimes 1$ and whose $\varphi^{j}\circ\iota_{1}$-component ($0\leq j\leq v_{p}(q)-1$) is $\pi\otimes 1$. Then $\{X,Y\}$ is a basis of $\mathscr{O}_{B_{\mathfrak{p}}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k})$ over $\mathscr{O}_{F_{\mathfrak{p}}^{(2)}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k})$.

Note that ${\mathrm{GL}}(2,F_{\mathfrak{p}})=({\mathrm{End}}^{0}_{\mathscr{O}_{B_{\mathfrak{p}}}}\Phi)^{\times}$ [27, Lemma 3.60]. We normalize the isomorphism such that the action on the $\varphi$-module

[TABLE]

is given by {\scriptsize{\big{[}\!\!\begin{array}[]{cc}a\!\!\!&\!\!\!b\\ c\!\!\!&\!\!\!d\end{array}\!\!\big{]}}}X=(a\otimes 1)X+(c\otimes 1)Y and {\scriptsize{\big{[}\!\!\begin{array}[]{cc}a\!\!\!&\!\!\!b\\ c\!\!\!&\!\!\!d\end{array}\!\!\big{]}}}Y=(b\otimes 1)X+(d\otimes 1)Y.

Let $\widetilde{\mathrm{D}}_{0}$ denote the $\varphi_{q}$-module

[TABLE]

coming from the $\varphi$-module $(\mathscr{O}_{B_{\mathfrak{p}}}\otimes_{{\mathbb{Z}}_{p}}\mathrm{W}(\bar{k}),\widetilde{\mathrm{F}})[1/p]$.

We describe Drinfeld’s moduli problem. Let $\mathrm{Nilp}$ be the category of $\widehat{\mathscr{O}^{{\mathrm{ur}}}}$-algebras on which $\pi$ is nilpotent. For any $A\in\mathrm{Nilp}$, let $\psi$ be the homomorphism $\bar{k}\rightarrow A/\pi A$; let $\mathrm{SFM}(A)$ be the set of pairs $(G,\rho)$ where $G$ is a special formal $\mathscr{O}_{B_{\mathfrak{p}}}$-module over $A$ and $\rho:\Phi_{A/\pi A}=\psi_{*}\Phi\rightarrow G$ is a quasi-isogeny of height zero.

We state a part of Drinfeld’s theorem [13] as follows.

Let ${\mathcal{H}}$ be the Drinfeld upper half plane over $F_{\mathfrak{p}}$, i.e. the rigid analytic $F_{\mathfrak{p}}$-variety whose ${\mathbb{C}}_{p}$-points are ${\mathbb{C}}_{p}-F_{\mathfrak{p}}$.

Theorem 3.1**.**

The functor $\mathrm{SFM}$ is represented by a formal scheme $\widehat{{\mathcal{H}}}\hat{\otimes}\widehat{\mathscr{O}^{{\mathrm{ur}}}}$ over $\widehat{\mathscr{O}^{{\mathrm{ur}}}}$ whose generic fiber is ${\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}={\mathcal{H}}\hat{\otimes}\widehat{F_{\mathfrak{p}}^{{\mathrm{ur}}}}$.

Let ${\mathcal{G}}$ be the universal special formal $\mathscr{O}_{B_{\mathfrak{p}}}$-module over $\widehat{{\mathcal{H}}}\hat{\otimes}\widehat{\mathscr{O}^{{\mathrm{ur}}}}$. There is an action of ${\mathrm{GL}}(2,F_{\mathfrak{p}})$ on ${\mathcal{G}}$ (see [3, Chapter II (9.2)]): The group ${\mathrm{GL}}(2,F_{\mathfrak{p}})$ acts on the functor $\mathrm{SFM}$ by $g\cdot(\psi;G,\rho)=(\psi\circ\mathrm{Frob}^{-n};G,\rho\circ\psi_{*}(g^{-1}\circ\mathrm{Frob}^{n}))$ if $v_{\mathfrak{p}}(\det g)=n$. Here, $v_{\mathfrak{p}}$ is the valuation of ${\mathbb{C}}_{p}$ normalized such that $v_{\mathfrak{p}}(\pi)=1$.

3.2 The filtered $\varphi_{q}$-isocrystal attached to the universal special formal

module

It is rather difficult to describe ${\mathcal{G}}$ precisely. 222See [35] for some information about ${\mathcal{G}}$ and [38] for a higher rank analogue. However, we can determine the associated (contravariant) filtered $\varphi_{q}$-isocrystal ${\mathcal{M}}$.

In the following, we write $\mathscr{O}_{{\mathcal{H}},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ for $\mathscr{O}_{{\mathcal{H}}\hat{\otimes}\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ and $\Omega_{{\mathcal{H}},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ for the differential sheaf $\Omega_{{\mathcal{H}}\hat{\otimes}\widehat{F_{\mathfrak{p}}^{{\mathrm{ur}}}}}$.

As is observed in [15] and [27], except for the filtration, the $\varphi_{q}$-isocrystal ${\mathcal{M}}$ is constant. So it is naturally isomorphic to the $\varphi_{q}$-isocrystal

[TABLE]

with the $q$-Frobenius being $\mathrm{F}^{v_{p}(q)}\otimes\varphi_{q,{\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}}$ and the connection being

[TABLE]

Next we determine the filtration on $\widetilde{\mathrm{D}}_{0}\otimes_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}\mathscr{O}_{{\mathcal{H}},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$. For any $F_{\mathfrak{p}}$-algebras $K$ and $L$, $L\otimes_{{\mathbb{Q}}_{p}}K$ is isomorphic to $L\otimes_{F_{\mathfrak{p}}}K\oplus(L\otimes_{{\mathbb{Q}}_{p}}K)_{\mathrm{non}}$, where $(L\otimes_{{\mathbb{Q}}_{p}}K)_{\mathrm{non}}$ is the kernel of the homomorphism $L\otimes_{{\mathbb{Q}}_{p}}K\rightarrow L\otimes_{F_{\mathfrak{p}}}K$, $\ell\otimes a\mapsto\ell\otimes a$. If $L$ is a field extension of $F_{\mathfrak{p}}$ containing all embeddings of $F_{\mathfrak{p}}$, then $L\otimes_{{\mathbb{Q}}_{p}}K=\bigoplus_{\tau:F_{\mathfrak{p}}\hookrightarrow L}L\otimes_{\tau,F_{\mathfrak{p}}}K$ and $(L\otimes_{{\mathbb{Q}}_{p}}K)_{\mathrm{non}}$ corresponds to the non-canonical embeddings. We apply this to $L=F_{\mathfrak{p}}$ and $K=\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$; consider $\widetilde{\mathrm{D}}_{0}=B_{\mathfrak{p}}\otimes_{{\mathbb{Q}}_{p}}\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$ as an $F_{\mathfrak{p}}\otimes_{{\mathbb{Q}}_{p}}\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$-module. Then $\widetilde{\mathrm{D}}_{0}$ splits into two parts: one is the canonical part which corresponds to the natural embedding $\mathrm{id}:F_{\mathfrak{p}}\hookrightarrow F_{\mathfrak{p}}$, and the other is the non-canonical part. Correspondingly, $\widetilde{\mathrm{D}}_{0}\otimes_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}\mathscr{O}_{{\mathcal{H}},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ splits into two parts, the canonical part $B_{\mathfrak{p}}\otimes_{F_{\mathfrak{p}}}\mathscr{O}_{{\mathcal{H}},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ and the non-canonical part. Since $F_{\mathfrak{p}}$ acts on the Lie algebra of any special formal $\mathscr{O}_{B_{\mathfrak{p}}}$-module through the natural embedding, the filtration on the non-canonical part is trivial.

The filtration on the canonical part is closely related to the period morphism [15, 27]. Let us recall the definition of the period morphism [38, Section 2.2]. We will use the notations in [38].

Let $M(\Phi)$ be the Cartier module of $\Phi$, a $\mathbb{Z}/2\mathbb{Z}$-graded module. The $\mathbb{Z}/2\mathbb{Z}$-grading depends on a choice of $F_{\mathfrak{p}}$-embedding of $F_{\mathfrak{p}}^{(2)}$ into $\widehat{F_{\mathfrak{p}}^{{\mathrm{ur}}}}$. We choose the one, $\tilde{\iota}_{0}$, that restricts to $\iota_{0}$, and denote the other $F_{\mathfrak{p}}$-embedding by $\tilde{\iota}_{1}$. We fix a graded $\mathrm{V}$-basis $\{g^{0},g^{1}\}$ of $M(\Phi)$ such that $\mathrm{V}g^{0}={\it\Pi}g^{0}$ and $\mathrm{V}g^{1}={\it\Pi}g^{1}$. Then $\{g^{0},g^{1},\mathrm{V}g^{0},\mathrm{V}g^{1}\}$ is a basis of $M(\Phi)[1/p]$ over $\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$; $F_{\mathfrak{p}}^{(2)}\subset B_{\mathfrak{p}}$ acts on $\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}g^{0}\oplus\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}\mathrm{V}g^{1}$ by $\tilde{\iota}_{0}$, and acts on $\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}\mathrm{V}g^{0}\oplus\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}g^{1}$ by $\tilde{\iota}_{1}$. See [13] for the definition of Cartier module and the meaning of graded $\mathrm{V}$-basis. See [27, (3.55)] for the relation between $M(\Phi)$ and the covariant Dieudonne module attached to $\Phi$. In loc. cit. Cartier module is called $\tau$-$W_{F}(L)$-crystal.

Let $R$ be any $\pi$-adically complete $\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$-algebra. Drinfeld constructed for each $(\psi;G,\rho)\in\mathrm{SFM}(R)$ a quadruple $(\eta,T,u,\rho)$. Let $M=M(G)$ be the Cartier module of $G$, $N(M)$ the auxiliary module that is the quotient of $M\oplus M$ by the submodule generated by elements of the form $(\mathrm{V}x,-{\it\Pi}x)$ and $\beta_{M}$ the quotient map $M\oplus M\rightarrow N(M)$. For $(x_{0},x_{1})\in M\oplus M$, we write $((x_{0},x_{1}))$ for $\beta_{M}(x_{0},x_{1})$. Then we have a map $\varphi_{M}:N(M)\rightarrow N(M)$. See [38, Definition 4] for its definition. Put

[TABLE]

both $\eta_{M}$ and $T_{M}$ are $\mathbb{Z}/2\mathbb{Z}$-graded. Let $u_{M}:\eta_{M}\rightarrow T_{M}$ be the $\mathscr{O}_{F_{\mathfrak{p}}}[{\it\Pi}]$-linear map of degree [math] that is the composition of the inclusion $\eta_{M}\hookrightarrow N(M)$ and the map

[TABLE]

Then $\eta_{M(\Phi)}$ is a free $\mathscr{O}_{F_{\mathfrak{p}}}$-module of rank $4$ with a basis

[TABLE]

where $((g^{0},0))$, $((\mathrm{V}g^{1},0))$ are in degree [math], and $((g^{1},0))$, $((\mathrm{V}g^{0},0))$ are in degree $1$. The quasi-isogeny $\rho:\psi_{*}\Phi\rightarrow G_{R/\pi R}$ induces an isomorphism

[TABLE]

Then the period of $(G,\rho)$ is defined by

[TABLE]

where $u^{\prime}_{M}$ is the the map $\eta^{0}_{M(G)}\otimes_{\mathscr{O}_{F_{\mathfrak{p}}}}F_{\mathfrak{p}}\rightarrow T_{M}^{0}\otimes_{R}R[1/p]$ induced by $u_{M}$.

Note that considered as a $\varphi_{q}$-module, $M(\Phi)[1/p]$ is the dual of $B_{\mathfrak{p}}\otimes_{F_{{\mathfrak{p}}}}\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$, the canonical part of $\widetilde{\mathrm{D}}_{0}$. Let $\{v_{0},v_{1},v_{2},v_{3}\}$ be the basis of $B_{\mathfrak{p}}\otimes_{F_{{\mathfrak{p}}}}\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$ over $\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$ dual to $\{\pi g^{1},g^{0},\mathrm{V}g^{0},\mathrm{V}g^{1}\}$. Then

[TABLE]

Here $z$ is the canonical coordinate on ${\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$.

We decompose $B_{\mathfrak{p}}\otimes_{F_{\mathfrak{p}}}\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}$ into two direct summands:

[TABLE]

where $B_{\mathfrak{p}}$ is considered as an $F_{\mathfrak{p}}^{(2)}$-module by left multiplication. Let $e_{0}$ and $e_{1}$ denote the projection to the first summand and that to the second, respectively. We may choose $g^{0},g^{1}$ such that $v_{0}=e_{0}X,\ v_{1}=e_{1}Y,\ v_{2}=e_{0}Y,\ v_{3}=e_{1}X$. Thus

[TABLE]

Finally we note that the induced action of $\mathrm{GL}(2,F_{\mathfrak{p}})$ on ${\mathcal{H}}$ is given by {\scriptsize{\big{[}\!\!\begin{array}[]{cc}a\!\!\!&\!\!\!b\\ c\!\!\!&\!\!\!d\end{array}\!\!\big{]}}}z=\frac{az+b}{cz+d}.

4 Shimura curves

Fix a real place $\tau_{1}$ of $F$. Let $B$ be a quaternion algebra over $F$ that splits at $\tau_{1}$ and is ramified at other real places $\{\tau_{2},\cdots,\tau_{g}\}$ and ${\mathfrak{p}}$.

4.1 Shimura curves $M$, $M^{\prime}$ and $M^{\prime\prime}$

We will use three Shimura curves studied by Carayol [5] and recall their constructions below (see also [28]).

Let $G$ be the reductive algebraic group over ${\mathbb{Q}}$ such that $G(R)=(B\otimes R)^{\times}$ for any ${\mathbb{Q}}$-algebra $R$. Let $Z$ be the center of $G$; it is isomorphic to $T={\mathrm{Res}}_{F/{\mathbb{Q}}}{\mathbb{G}}_{m}$. Let $\nu:G\rightarrow T$ be the morphism obtained from the reduced norm $\mathrm{Nrd}_{B/F}$ of $B$. The kernel of $\nu$ is $G^{\mathrm{der}}$, the derived group of $G$, and thus we have a short exact sequence of algebraic groups

[TABLE]

Let $X$ be the $G({\mathbb{R}})$-conjugacy class of the homomorphism

[TABLE]

where ${\mathbb{H}}$ is the Hamilton quaternion algebra. The conjugacy class $X$ is naturally identified with the union of upper and lower half planes. Let $M=M(G,X)=(M_{U}(G,X))_{U}$ be the canonical model of the Shimura variety attached to the Shimura pair $(G,X)$; the canonical model is defined over $F$, the reflex field of $(G,X)$. There is a natural right action of $G({\mathbb{A}}_{f})$ on $M(G,X)$. Here and in what follows, by abuse of terminology we call a projective system of varieties simply a variety.

Take an imaginary quadratic field $E_{0}={\mathbb{Q}}(\sqrt{-a})$ ($a$ a square-free positive integer) such that $p$ splits in $E_{0}$. Put $E=FE_{0}$ and $D=B\otimes_{F}E=B\otimes_{{\mathbb{Q}}}E_{0}$. We fix a square root $\rho$ of $-a$ in ${\mathbb{C}}$. Then the prolonging of $\tau_{i}$ to $E$ by $x+y\sqrt{-a}\mapsto\tau_{i}(x)+\tau_{i}(y)\rho$ (resp. $x+y\sqrt{-a}\mapsto\tau_{i}(x)-\tau_{i}(y)\rho$) is denoted by $\tau_{i}$ (resp. $\bar{\tau}_{i}$).

Let $T_{E}$ be the torus ${\mathrm{Res}}_{E/{\mathbb{Q}}}{\mathbb{G}}_{m}$, $T_{E}^{1}$ the subtorus of $T_{E}$ such that $T_{E}^{1}({\mathbb{Q}})=\{z\in E:z\bar{z}=1\}$. We consider the amalgamate product $G^{\prime\prime}=G\times_{Z}T_{E}$, and the morphism $G^{\prime\prime}=G\times_{Z}T_{E}\xrightarrow{\nu^{\prime\prime}}T^{\prime\prime}=T\times T_{E}^{1}$ defined by $(g,z)\mapsto(\nu(g)z\bar{z},z/\bar{z})$. Consider the subtorus $T^{\prime}={\mathbb{G}}_{m}\times T_{E}^{1}$ of $T^{\prime\prime}$, and let $G^{\prime}$ be the inverse image of $T^{\prime}$ by the map $\nu^{\prime\prime}$. The restriction of $\nu^{\prime\prime}$ to $G^{\prime}$ is denoted by $\nu^{\prime}$. Both the derived group of $G^{\prime}$ and that of $G^{\prime\prime}$ are identified with $G^{\mathrm{der}}$, and we have two short exact sequences of algebraic groups

[TABLE]

and

[TABLE]

The complex embeddings $\tau_{1},\cdots,\tau_{g}$ of $E$ identify $G^{\prime\prime}({\mathbb{R}})$ with ${\mathrm{GL}}_{2}({\mathbb{R}})\cdot{\mathbb{C}}^{\times}\times{\mathbb{H}}^{\times}\cdot{\mathbb{C}}^{\times}\times\cdots\times{\mathbb{H}}^{\times}\cdot{\mathbb{C}}^{\times}$. We consider the $G^{\prime}({\mathbb{R}})$-conjugacy class $X^{\prime}$ (resp. $G^{\prime\prime}({\mathbb{R}})$-conjugacy class $X^{\prime\prime}$) of the homomorphism

[TABLE]

Let $M^{\prime}=M(G^{\prime},X^{\prime})$ and $M^{\prime\prime}=M(G^{\prime\prime},X^{\prime\prime})$ be the canonical models of the Shimura varieties defined over their reflex field $E$. There are natural right actions of $G^{\prime}({\mathbb{A}}_{f})$ and $G^{\prime\prime}({\mathbb{A}}_{f})$ on $M^{\prime}$ and $M^{\prime\prime}$, respectively.

Put $T_{E_{0}}={\mathrm{Res}}^{E_{0}}_{{\mathbb{Q}}}{\mathbb{G}}_{m}$. Using the complex embeddings $\tau_{1},\cdots,\tau_{g}$ of $E$, we identify $T_{E}({\mathbb{R}})$ with ${\mathbb{C}}^{\times}\times\cdots\times{\mathbb{C}}^{\times}$; similarly via the embedding $x+y\sqrt{-a}\rightarrow x+y\rho$ we identify $T_{E_{0}}({\mathbb{R}})$ with ${\mathbb{C}}^{\times}$. Consider the homomorphisms

[TABLE]

Let $N_{E}=M(T_{E},h_{E})$ and $N_{E_{0}}=M(T_{E_{0}},h_{E_{0}})$ be the canonical models attached to the pairs $(T_{E},h_{E})$ and $(T_{E_{0}},h_{E_{0}})$ respectively. Then $N_{E}$ is defined over $E$, and $N_{E_{0}}$ is defined over $E_{0}$.

Consider the homomorphism $\alpha:G\times T_{E}\rightarrow G^{\prime\prime}$ of algebraic groups inducing

[TABLE]

on ${\mathbb{Q}}$-valued points, and the homomorphism $\beta:G\times T_{E}\rightarrow T_{E_{0}}$ inducing

[TABLE]

on ${\mathbb{Q}}$-valued points. Here, $\mathrm{N}_{E/E_{0}}$ denotes the norm map $E^{\times}\rightarrow E_{0}^{\times}$. Since $h^{\prime}=\alpha\circ(h\times h_{E})$ and $h_{E_{0}}=\mathrm{N}_{E/E_{0}}\circ h_{E}$, $\alpha$ and $\beta$ induce morphisms of Shimura varieties $M\times N\rightarrow M^{\prime\prime}$ and $M\times N_{E}\rightarrow N_{E_{0}}$ again denoted by $\alpha$ and $\beta$ respectively. We have the following diagram

[TABLE]

4.2 Connected components of $M$, $M\times N_{E}$, $M^{\prime}$ and $M^{\prime\prime}$

We write $\widetilde{G}$ for $G\times T_{E}$ and write $\widetilde{M}$ for $M\times N_{E}$. For $\natural=\widetilde{},\emptyset,\ ^{\prime},^{\prime\prime}$, as $B$ is ramified at ${\mathfrak{p}}$, there exists a unique maximal compact open subgroup $U^{\natural}_{p,0}$ of $G^{\natural}({\mathbb{Q}}_{p})$. Then $U^{\prime}_{p,0}=U^{\prime\prime}_{p,0}\cap G^{\prime}({\mathbb{Q}}_{p})$ and $U^{\prime\prime}_{p,0}=\alpha(\widetilde{U}_{p,0})$.

If $U^{\natural}$ is a subgroup of $G^{\natural}({\mathbb{A}}_{f})$ of the form $U^{\natural}_{p,0}U^{\natural,p}$ where $U^{\natural,p}$ is a compact open subgroup of $G^{\natural}({\mathbb{A}}_{f}^{p})$, we will write $M^{\natural}_{0,U^{\natural,p}}$ for $M^{\natural}_{U^{\natural}}$. Let $M^{\natural}_{0}$ denote the projective system $(M^{\natural}_{0,U^{\natural,p}})_{U^{\natural,p}}$; this projective system has a natural right action of $G^{\natural}({\mathbb{A}}_{f}^{p})$.

Lemma 4.1**.**

1. (a)

For any sufficiently small $U^{\natural,p}$, each geometrically connected component of $M^{\natural}_{0,U^{\natural,p}}$ is defined over a field that is unramified at all places above $p$. 2. (b)

Let $\widetilde{U}^{p}$ be a sufficiently small compact open subgroup of $\widetilde{G}({\mathbb{A}}_{f}^{p})$. Then the morphism

[TABLE]

induced by $\alpha$ is an isomorphism onto its image when restricted to every geometrically connected component.

Proof.

When $U^{\natural,p}$ is sufficiently small, $M^{\natural}_{0,U^{\natural,p}}$ is smooth. Let $\pi_{0}(M^{\natural}_{0,U^{\natural,p}})$ denote the group of geometrically irreducible components of $M^{\natural}_{0,U^{\natural,p}}$ over $\overline{{\mathbb{Q}}}$ (which must be connected since $M^{\natural}_{0,U^{\natural,p}}$ is smooth). Then ${\mathrm{Gal}}(\overline{{\mathbb{Q}}}/E)$ acts on $\pi_{0}(M^{\natural}_{0,U^{\natural,p}})$. This action is explicitly described by Deligne [12, Theorem 2.6.3], from which we deduce (a).

As $\alpha$ induces an isomorphism from the derived group of $\widetilde{G}$ to that of $G^{\prime\prime}$, by [12, Remark 2.1.16] or [22, Proposition II.2.7] we obtain (b). ∎

4.3 Modular interpolation of $M^{\prime}$

Let $\ell\mapsto\bar{\ell}$ be the involution on $D=B\otimes_{{\mathbb{Q}}}E_{0}$ that is the product of the canonical involution on $B$ and the complex conjugate on $E_{0}$. Choose an invertible symmetric element $\delta\in D$ ($\delta=\bar{\delta}$). Then we have another involution $\ell\mapsto\ell^{*}:=\delta^{-1}\bar{\ell}\delta$ on $D$.

Let $V$ denote $D$ considered as a left $D$-module. Let $\psi$ be the non-degenerate alternating form on $V$ defined by $\psi(x,y)=\mathrm{Tr}_{E/{\mathbb{Q}}}(\sqrt{-a}\ \mathrm{Trd}_{D/E}(x\delta y^{*}))$, where $\mathrm{Tr}_{E/{\mathbb{Q}}}$ is the trace map and $\mathrm{Trd}_{D/E}$ is the reduced trace map. For $\ell\in D$ put

[TABLE]

where ${\mathrm{Fil}}^{\bullet}$ is the Hodge structure defined by $h^{\prime}$. We have

[TABLE]

for $\ell\in D$. The subfield of ${\mathbb{C}}$ generated by $t(\ell)$, $\ell\in D$, is exactly $E$.

Choose an order $\mathscr{O}_{D}$ of $D$, $T$ the corresponding lattice in $V$. With a suitable choice of $\delta$, we may assume that $\mathscr{O}_{D}$ is stable by the involution $\ell\mapsto\ell^{*}$ and that $\psi$ takes integer values on $T$. Put $\hat{\mathscr{O}}_{D}:=\mathscr{O}_{D}\otimes\hat{{\mathbb{Z}}}$ and $\hat{T}:=T\otimes\hat{{\mathbb{Z}}}$.

In Section 5 when we consider the $p$-adic uniformization of the Shimura curves, we need to make the following assumption.

Assumption 4.2*.*

We assume that $\delta$ is chosen such that $\hat{T}$ is stable by $U^{\prime}_{p,0}$.

If $U^{\prime}$ is a sufficiently small compact open subgroup of $G^{\prime}({\mathbb{A}}_{f})$ keeping $\hat{T}$, then $M^{\prime}_{U^{\prime}}$ represents the following functor ${\mathcal{M}}_{U^{\prime}}$ [28, Section 5]:

For any $E$-algebra $R$, ${\mathcal{M}}_{U^{\prime}}(R)$ is the set of isomorphism classes of quadruples $(A,\iota,\theta,\kappa)$ where

$\bullet$ $A$ is an isomorphism class of abelian schemes over $R$ with an endomorphism $\iota:\mathscr{O}_{D}\rightarrow{\mathrm{End}}(A)$ such that ${\mathrm{tr}}(\iota(\ell),{\mathrm{Lie}}A)=t(\ell)$ for all $\ell\in\mathscr{O}_{D}$.

$\bullet$ $\theta$ is a polarization $A\rightarrow\check{A}$ whose associated Rosati involution sends $\iota(\ell)$ to $\iota(\ell^{*})$.

$\bullet$ $\kappa$ is a $U^{\prime}$-orbit of $\mathscr{O}_{D}\otimes\hat{{\mathbb{Z}}}$-linear isomorphisms $\hat{T}(A):=\prod\limits_{\ell}T_{\ell}(A)\rightarrow\hat{T}$ such that there exists a $\hat{{\mathbb{Z}}}$-linear isomorphism $\kappa^{\prime}:\hat{T}(1)\rightarrow\hat{{\mathbb{Z}}}$ making the diagram

[TABLE]

commutative.

Let ${\mathcal{A}}_{U^{\prime}}$ be the universal $\mathscr{O}_{D}$-abelian scheme over $M^{\prime}_{U^{\prime}}$.

5 $p$-adic Uniformizations of Shimura curves

5.1 Preliminaries

We provide two simple facts, which will be useful later.

(i) Let $X$ be a scheme with a discrete action of a group $C$ on the right hand side, and let $Z$ be a group that contains $C$ as a normal subgroup of finite index. Fix a set of representatives $\{g_{i}\}_{i\in C\backslash Z}$ of $C\backslash Z$ in $Z$. We define a scheme $X*_{C}Z$ with a right action of $Z$ below. As a scheme $X*_{C}Z$ is $\bigsqcup_{C\backslash Z}X^{(g_{i})}$, where $X^{(g_{i})}$ is a copy of $X$. For any $g\in Z$ and $x^{(g_{i})}\in X^{(g_{i})}$, if $g_{i}g=hg_{k}$ with $h\in C$, then $x^{(g_{i})}\cdot g=(x\cdot h)^{(g_{k})}$. It is easy to show that up to isomorphism $X*_{C}Z$ and the right action of $Z$ are independent of the choice of $\{g_{i}\}_{i\in C\backslash Z}$.

(ii) Let $X_{1}$ and $X_{2}$ be two schemes whose connected components are all geometrically connected. Suppose that each of $X_{1}$ and $X_{2}$ has an action of an abelian group $Z$; $Z$ acts freely on the set of components of $X_{1}$ (resp. $X_{2}$). Let $C$ be a closed subgroup of $Z$. Then the $Z$-actions on $X_{1}$ and $X_{2}$ induce $Z/C$-actions on $X_{1}/C$ and $X_{2}/C$.

Lemma 5.1**.**

If there exists a $Z/C$-equivariant isomorphism $\gamma:X_{1}/C\rightarrow X_{2}/C$, then there exists a $Z$-equivariant isomorphism $\tilde{\gamma}:X_{1}\rightarrow X_{2}$ such that the following diagram

[TABLE]

is commutative, where $\pi_{1}$ and $\pi_{2}$ are the natural projections.

Proof.

We identify $X_{1}/C$ with $X_{2}/C$ by $\gamma$, and write $Y$ for it. The condition on $Z$-actions implies that the action of $Z/C$ on the set of connected components of $Y$ is free and that the morphism $\pi_{1}$ (resp. $\pi_{2}$) maps each connected component of $X_{1}$ (resp. $X_{2}$) isomorphically to its image.

We choose a set of representatives $\{Y_{i}\}_{i\in I}$ of the $Z/C$-orbits of components of $Y$. Then $\{\bar{g}Y_{i}:\bar{g}\in Z/C,i\in I\}$ are all different connected components of $Y$. For each $i\in I$ we choose a connected component $\tilde{Y}_{i}^{(1)}$ (resp. $\tilde{Y}_{i}^{(2)}$) of $X_{1}$ (resp. $X_{2}$) that is a lifting of $Y_{i}$. Then $\{g\tilde{Y}_{i}^{(1)}:g\in Z,i\in I\}$ (resp. $\{g\tilde{Y}_{i}^{(2)}:g\in Z,i\in I\}$) are all different connected components of $X_{1}$ (resp. $X_{2}$).

As $\pi_{1}|_{\tilde{Y_{i}}^{(1)}}:\tilde{Y_{i}}^{(1)}\rightarrow Y_{i}$ and $\pi_{2}|_{\tilde{Y_{i}}^{(2)}}:\tilde{Y_{i}}^{(2)}\rightarrow Y_{i}$ are isomorphisms, there exists an isomorphism $\tilde{\gamma}_{i}:\tilde{Y_{i}}^{(1)}\rightarrow\tilde{Y_{i}}^{(2)}$ such that $\pi_{1}|_{\tilde{Y_{i}}^{(1)}}=\pi_{2}|_{\tilde{Y_{i}}^{(2)}}\circ\tilde{\gamma}_{i}$. We define the morphism $\tilde{\gamma}:X^{(1)}\rightarrow X^{(2)}$ as follows: $\tilde{\gamma}$ maps $g\tilde{Y}_{i}^{(1)}$ to $g\tilde{Y}_{i}^{(2)}$, and $\tilde{\gamma}|_{g\tilde{Y}_{i}^{(1)}}=g\circ\tilde{\gamma}_{i}\circ g^{-1}$. Then $\tilde{\gamma}$ is a $Z$-equivariant isomorphism and $\pi_{1}=\pi_{2}\circ\tilde{\gamma}$. ∎

5.2 Some Notations

Fix an isomorphism ${\mathbb{C}}\cong{\mathbb{C}}_{p}$. Combining the isomorphism ${\mathbb{C}}\cong{\mathbb{C}}_{p}$ and the inclusion $E_{0}\hookrightarrow{\mathbb{C}}$, $x+y\sqrt{-a}\mapsto x+y\rho$, we obtain an inclusion $E_{0}\hookrightarrow{\mathbb{Q}}_{p}$ and $E\hookrightarrow F_{\mathfrak{p}}$. Thus $D\otimes{\mathbb{Q}}_{p}$ is isomorphic to $B_{\mathfrak{p}}\oplus B_{\mathfrak{p}}$.

Note that $G({\mathbb{Q}}_{p})$ is isomorphic to $B_{\mathfrak{p}}^{\times}$, $G^{\prime}({\mathbb{Q}}_{p})$ is isomorphic to the subgroup

[TABLE]

of $B_{\mathfrak{p}}^{\times}\times B_{\mathfrak{p}}^{\times}$, and $G^{\prime\prime}({\mathbb{Q}}_{p})$ is isomorphic to

[TABLE]

where $a\mapsto\bar{a}$ is the canonical involution on $B$. Note that $T_{E}({\mathbb{Q}}_{p})$ is isomorphic to $F_{\mathfrak{p}}^{\times}\times F_{\mathfrak{p}}^{\times}$, and $T_{E_{0}}({\mathbb{Q}}_{p})$ is isomorphic to ${\mathbb{Q}}_{p}^{\times}\times{\mathbb{Q}}_{p}^{\times}$. We normalize these isomorphisms such that $G^{\prime}({\mathbb{Q}}_{p})\hookrightarrow G^{\prime\prime}({\mathbb{Q}}_{p})$ becomes the natural inclusion

[TABLE]

$\alpha:G({\mathbb{Q}}_{p})\times T_{E}({\mathbb{Q}}_{p})\rightarrow G^{\prime\prime}({\mathbb{Q}}_{p})$ becomes

[TABLE]

and $\beta:G({\mathbb{Q}}_{p})\times T_{E}({\mathbb{Q}}_{p})\rightarrow T_{E_{0}}({\mathbb{Q}}_{p})$ becomes

[TABLE]

Let $\bar{B}$ be the quaternion algebra over $F$ such that

[TABLE]

With $\bar{B}$ instead of $B$ we can define analogues of $G$, $G^{\prime}$ and $G^{\prime\prime}$, denoted by $\bar{G}$, $\bar{G}^{\prime}$ and $\bar{G}^{\prime\prime}$ respectively. For $\natural=\emptyset,^{\prime},^{\prime\prime}$ we have $\bar{G}^{\natural}({\mathbb{A}}_{f}^{p})=G^{\natural}({\mathbb{A}}_{f}^{p})$; $\bar{G}({\mathbb{Q}}_{p})$ is isomorphic to ${\mathrm{GL}}(2,F_{\mathfrak{p}})$; $\bar{G}^{\prime}({\mathbb{Q}}_{p})$ is isomorphic to the subgroup

[TABLE]

of ${\mathrm{GL}}(2,F_{\mathfrak{p}})\times{\mathrm{GL}}(2,F_{\mathfrak{p}})$, and $\bar{G}^{\prime\prime}({\mathbb{Q}}_{p})$ is isomorphic to

[TABLE]

If $\natural=\emptyset$, let $\bar{G}({\mathbb{Q}}_{p})$ act on ${\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}$ as in Section 3. If $\natural=\widetilde{}$, let $\widetilde{\bar{G}}=\bar{G}\times T_{E}$ act on ${\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}$ by the projection to the first factor. If $\natural=^{\prime}$ or ′′, let $\bar{G}^{\natural}({\mathbb{Q}}_{p})$ act on ${\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}$ by the first factor. Let $\bar{G}^{\natural}({\mathbb{Q}})$ act on ${\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}$ via its embedding into $\bar{G}^{\natural}({\mathbb{Q}}_{p})$.

The center of $\bar{G}^{\natural}$, $Z(\bar{G}^{\natural})$, is naturally isomorphic to the center of $G^{\natural}$, $Z(G^{\natural})$; we denote both of them by $Z^{\natural}$.

5.3 The $p$-adic uniformizations

Let $\natural$ be either $\widetilde{},\ ^{\prime}$ or ′′. For any compact open subgroup $U^{\natural,p}$ of $G^{\natural}({\mathbb{A}}_{f}^{p})$, let $X^{\natural}_{U^{\natural,p}}$ denote $M^{\natural}_{0,U^{\natural,p}}\times_{{\mathrm{Spec}}(F_{\mathfrak{p}})}{\mathrm{Spec}}(\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}})$.

Proposition 5.2**.**

Suppose that Assumption 4.2 holds.

1. (a)

Assume that $\natural=\widetilde{}\ ,^{\prime}$ or ′′. For any sufficiently small compact open subgroup $U^{\natural,p}$ of $G^{\natural}({\mathbb{A}}_{f}^{p})$, writing $U^{\natural}=U^{\natural}_{p,0}U^{\natural p}$, we have a $Z^{\natural}({\mathbb{Q}})\backslash Z^{\natural}({\mathbb{A}}_{f})/(Z^{\natural}({\mathbb{A}}_{f})\cap U^{\natural})$-equivariant isomorphism

[TABLE]

Here, $\bar{G}^{\natural}({\mathbb{Q}})$ acts on ${\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}$ as mentioned above and acts on $G^{\natural}({\mathbb{A}}_{f}^{p})/U^{\natural,p}$ by the embedding $\bar{G}^{\natural}({\mathbb{Q}})\hookrightarrow\bar{G}^{\natural}({\mathbb{A}}_{f}^{p})\xrightarrow{\simeq}G^{\natural}({\mathbb{A}}_{f}^{p})$; in the case of $\natural=^{\prime}$ or ′′, if $g\in\bar{G}^{\natural}({\mathbb{Q}})$ satisfies $g_{p}=(a,b)$ with $a,b\in{\mathrm{GL}}(2,F_{\mathfrak{p}})$, then $g$ acts on $G^{\natural}({\mathbb{Q}}_{p})/U^{\natural}_{p,0}$ via the left multiplication by $({\it\Pi}^{v_{{\mathfrak{p}}}(\det a)},{\it\Pi}^{v_{\mathfrak{p}}(\det b)})$; while, in the case of $\natural=\widetilde{}$, $\widetilde{g}=(g,t)\in\widetilde{\bar{G}}({\mathbb{Q}})$ $(g\in G({\mathbb{Q}}),t\in T_{E}({\mathbb{Q}}))$ acts on $\widetilde{G}({\mathbb{Q}}_{p})/\widetilde{U}_{p,0}$ via the left multiplication by $({\it\Pi}^{v_{\mathfrak{p}}(\det g_{p})},t_{p})$; the group $Z^{\natural}({\mathbb{Q}})\backslash Z^{\natural}({\mathbb{A}}_{f})/(Z^{\natural}({\mathbb{A}}_{f})\cap U^{\natural})$ acts on the right hand side of $($5.1$)$ by right multiplications on $\bar{G}^{\natural}({\mathbb{A}}_{f})$. 2. (b)

The isomorphisms in $($a$)$ can be chosen such that, for either $\sharp=\widetilde{}$ and $\natural=^{\prime\prime}$, or $\sharp=^{\prime}$ and $\natural=^{\prime\prime}$, we have a commutative diagram

[TABLE]

*compatible with the $Z^{\sharp}({\mathbb{Q}})\backslash Z^{\sharp}({\mathbb{A}}_{f})/(Z^{\sharp}({\mathbb{A}}_{f})\cap U^{\sharp})$-actions on the upper and the *

$Z^{\natural}({\mathbb{Q}})\backslash Z^{\natural}({\mathbb{A}}_{f})/(Z^{\natural}({\mathbb{A}}_{f})\cap U^{\natural})$-actions on the lower, where the left vertical arrow is induced from the morphism $M^{\sharp}\rightarrow M^{\natural}$, and the right vertical arrow is induced by the identity morphism ${\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}\rightarrow{\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ and the homomorphism $\alpha:\widetilde{G}=G\times T_{E}\rightarrow G^{\prime\prime}$ or the inclusion $G^{\prime}\hookrightarrow G^{\prime\prime}$. Here, in the case of $\sharp=\widetilde{}$ and $\natural=^{\prime\prime}$, $U^{\natural}=\alpha(U^{\sharp})$; in the case of $\sharp=^{\prime}$ and $\natural=^{\prime\prime}$, $U^{\sharp}=U^{\natural}\cap G^{\prime}({\mathbb{A}}_{f})$.*

The conclusions of Proposition 5.2 especially (a) are well known [27, 37]. However, the author has no reference for (b), so we provide some detail of the proof.

Proof.

Assertion (a) in the case of $\natural=^{\prime}$ comes from [27, Theorem 6.50].

For the case of $\sharp=^{\prime}$ and $\natural=^{\prime\prime}$ we put

[TABLE]

and

[TABLE]

Then $X^{\prime\prime}_{U^{\prime\prime p}}$ is $Z$-equivariantly isomorphic to $X^{\prime}_{U^{\prime p}}*_{C}Z$, and $\bar{G}^{\prime\prime}({\mathbb{Q}})\backslash({\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}\times G^{\prime\prime}({\mathbb{A}}_{f})/U^{\prime\prime})$ is $Z$-equivariantly isomorphic to \Big{(}\bar{G}^{\prime}({\mathbb{Q}})\backslash({\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}\times G^{\prime}({\mathbb{A}}_{f})/U^{\prime})\Big{)}*_{C}Z. So (a) in the case of $\natural=^{\prime\prime}$ and (b) in the case of $\sharp=^{\prime}$, $\natural=^{\prime\prime}$ follow.

Now we consider the rest cases. Let $H$ be the kernel of the homomorphism $\alpha:\widetilde{G}=G\times T_{E}\rightarrow G^{\prime\prime}$. Put

[TABLE]

Put $X_{1}=\widetilde{X}_{\widetilde{U}^{p}}$ and $X_{2}=\widetilde{\bar{G}}({\mathbb{Q}})\backslash({\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}\times\widetilde{G}({\mathbb{A}}_{f})/\widetilde{U}_{p,0}\widetilde{U}^{p})$. By Lemma 4.1 (a), all connected components of $X_{1}$ are geometrically connected; it is obvious that all connected components of $X_{2}$ are geometrically connected. Thus $Z$ acts freely on the set of components of $X_{1}$ (resp. $X_{2}$). Furthermore $X_{1}/C$ is isomorphic to $X^{\prime\prime}_{\alpha(\widetilde{U}^{p})}$, and $X_{2}/C$ is isomorphic to $\bar{G}^{\prime\prime}({\mathbb{Q}})\backslash({\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}\times G^{\prime\prime}({\mathbb{A}}_{f})/U^{\prime\prime}_{p,0}\alpha(\widetilde{U}^{p}))$. We have already proved that $X_{1}/C$ is $Z/C$-equivariantly isomorphic to $X_{2}/C$. Applying Lemma 5.1 we obtain (a) in the case of \natural=\widetilde{}\ and (b) in the case of $\sharp=\widetilde{}\ ,\natural=^{\prime\prime}$. ∎

Remark 5.3*.*

By [37] the similar conclusion of Proposition 5.2 (a) holds for the case of $\natural=\emptyset$. We use $X_{U^{p}}$ to denote $\bar{G}({\mathbb{Q}})\backslash({\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}\times G({\mathbb{A}}_{f})/U_{p,0}U^{p})$, where the action of $\bar{G}({\mathbb{Q}})$ on ${\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}\times G({\mathbb{A}}_{f})/U_{p,0}U^{p}$ is defined similarly.

6 Local systems and the associated filtered

$\varphi_{q}$-isocrystals on Shimura Curves

6.1 Local systems on Shimura curves

We choose a number field $L$ splitting $F$ and $B$. We identify $\{\tau_{i}:F\rightarrow L\}$ with $I=\{\tau_{i}:F\rightarrow{\mathbb{C}}\}$ by the inclusion $L\rightarrow{\mathbb{C}}$. Fix an isomorphism $L\otimes_{{\mathbb{Q}}}B=M(2,L)^{I}$. Then we have a natural inclusion $G({\mathbb{Q}})\hookrightarrow{\mathrm{GL}}(2,L)^{I}$. Let $\mathfrak{P}$ be a place of $L$ above ${\mathfrak{p}}$.

For a multiweight $\mathrm{k}=(k_{1},\cdots,k_{g},w)$ with $k_{1}\equiv\cdots k_{g}\equiv w\hskip 6.0pt\mathrm{mod}\hskip 3.0pt2$ and $k_{1}\geq 2,\cdots,k_{g}\geq 2$, we define the morphism $\rho^{(\mathrm{k})}:G\rightarrow{\mathrm{GL}}(n,L)$ $(n=\prod_{i=1}^{g}(k_{i}-1))$ to be the product $\otimes_{i\in I}[({\mathrm{Sym}}^{k_{i}-2}\otimes\det^{(w-k_{i})/2})\circ\check{pr}_{i}]$. Here $\check{pr}_{i}$ denotes the contragradient representation of the $i$th projection $pr_{i}:{\mathrm{GL}}(2,L)^{I}\rightarrow{\mathrm{GL}}(2,L)$. The algebraic group denoted by $G^{c}$ in [22, Chapter III] is the quotient of $G$ by $\ker(\mathrm{N}_{F/{\mathbb{Q}}}:F^{\times}\rightarrow{\mathbb{Q}}^{\times})$. As the restriction of $\rho^{(\mathrm{k})}$ to the center $F^{\times}$ is the scalar multiplication by $\mathrm{N}_{F/{\mathbb{Q}}}^{-(w-2)}(\cdot)$, $\rho^{(\mathrm{k})}$ factors through $G^{c}$, so we can attach to the representation $\rho^{(\mathrm{k})}$ a $G({\mathbb{A}}_{f})$-equivariant smooth $L_{\mathfrak{P}}$-sheaf ${\mathcal{F}}{(\mathrm{k})}$ on $M$.

Let $p_{2}:G^{\prime\prime}_{E_{0}}\rightarrow G_{E_{0}}$ be the map induced by the second projection on $(D\otimes_{\mathbb{Q}}E_{0})^{\times}=D^{\times}\times D^{\times}$ corresponding to the conjugate $E_{0}\rightarrow E_{0}$. As the algebraic representation $\rho^{\prime\prime(\mathrm{k})}=\rho^{(\mathrm{k})}\circ p_{2}$ factors through $G^{\prime\prime c}$, we can attach to it a $G^{\prime\prime}({\mathbb{A}}_{f})$-equivariant smooth $L_{\mathfrak{P}}$-sheaf ${\mathcal{F}}^{\prime\prime}(\mathrm{k})$ on $M^{\prime\prime}$. Let ${\mathcal{F}}^{\prime}(\mathrm{k})$ be the restriction of ${\mathcal{F}}^{\prime\prime}{(\mathrm{k})}$ to $M^{\prime}$.

We define a charcater $\bar{\chi}:T_{0}\rightarrow{\mathbb{G}}_{m}$ such that on ${\mathbb{C}}$-valued point $\bar{\chi}$ is the inverse of the second projection $T_{0{\mathbb{C}}}={\mathbb{C}}^{\times}\times{\mathbb{C}}^{\times}\rightarrow{\mathbb{C}}^{\times}$. Let ${\mathcal{F}}(\bar{\chi})$ be the $L_{\mathfrak{P}}$-sheaf attached to the representation $\bar{\chi}$. By [28] one has the following $G({\mathbb{A}}_{f})\times T({\mathbb{A}}_{f})$-equivariant isomorphism of $L_{\mathfrak{P}}$-sheaves

[TABLE]

on $M\times N$, where $\mathrm{pr}_{1}$ is the projection $M\times N\rightarrow M$.

Note that $L\otimes_{{\mathbb{Q}}}D\simeq(\mathrm{M}_{2}(L)\times\mathrm{M}_{2}(L))^{I}$. For each $i\in I$, the first component $\mathrm{M}_{2}(L)$ corresponds to the embedding $E_{0}\subset L\subset{\mathbb{C}}$ and the second $\mathrm{M}_{2}(L)$ to its conjugate. Let ${\mathcal{F}}^{\prime}$ be the local system $R^{1}g_{*}L_{\mathfrak{P}}$ where $g:{\mathcal{A}}\rightarrow M^{\prime}$ is the universal $\mathscr{O}_{D}$-abelian scheme; it is a sheaf of $L\otimes_{{\mathbb{Q}}}D$-modules. For each $i\in I$, let $e_{i}\in L\otimes_{{\mathbb{Q}}}D$ be the idempotent whose $(2,i)$-th component is a rank one idempotent e.g. {\scriptsize{\big{[}\!\!\begin{array}[]{cc}1\!\!\!&\!\!\!0\\ 0\!\!\!&\!\!\!0\end{array}\!\!\big{]}}} and the other components are zero. Let ${\mathcal{F}}^{\prime}_{i}$ denote the $e_{i}$-part $e_{i}\cdot R^{1}g_{*}L_{\mathfrak{P}}$. Note that ${\mathcal{F}}^{\prime}_{i}$ does not depend on the choice of the rank one idempotent. By [28] we have an isomorphism of local systems

[TABLE]

We can define more local systems on $M^{\prime}$. For $(\mathrm{k},\mathrm{v})=(k_{1},\cdots,k_{g};v_{1},\cdots,v_{g})$, let ${\mathcal{F}}^{\prime}{(\mathrm{k},\mathrm{v})}$ be the local system \bigotimes_{i\in I}\Big{(}{\mathrm{Sym}}^{k_{i}-2}{\mathcal{F}}^{\prime}_{i}\bigotimes(\det{\mathcal{F}}^{\prime}_{i})^{v_{i}}\Big{)}.

6.2 Filtered $\varphi_{q}$-isocrystals associated to

the local systems

We use $\tilde{\mathrm{k}}$ uniformly to denote $(\mathrm{k},\mathrm{v})=(k_{1},\cdots,k_{g};v_{1},\cdots,v_{g})$ (resp. $\mathrm{k}=(k_{1},\cdots,k_{g},w)$) in the case of $\natural=^{\prime}$ (resp. $\natural=\emptyset,^{\prime},^{\prime\prime}$).

We shall need the filtered $\varphi_{q}$-isocrystal attached to ${\mathcal{F}}{(\tilde{\mathrm{k}})}$. However we do not know how to compute it. Instead, we compute that attached to $\mathrm{pr}_{1}^{*}{\mathcal{F}}{(\tilde{\mathrm{k}})}$. As a middle step we determine the filtered $\varphi_{q}$-isocrystals associated to ${\mathcal{F}}^{\prime}{(\tilde{\mathrm{k}})}$ and ${\mathcal{F}}^{\prime\prime}{(\tilde{\mathrm{k}})}$.

For any integers $k$ and $v$ with $k\geq 2$, and any inclusion $\sigma:F_{\mathfrak{p}}\rightarrow L_{\mathfrak{P}}$, let $V_{\sigma}(k,v)$ be the space of homogeneous polynomials in two variables $X_{\sigma}$ and $Y_{\sigma}$ of degree $k-2$ with coefficients in $L_{\mathfrak{P}}$; let ${\mathrm{GL}}(2,F_{\mathfrak{p}})$ act on $V_{\sigma}(k,v)$ by

[TABLE]

For $(\mathrm{k},\mathrm{v})=(k_{1},\cdots,k_{g};v_{1},\cdots,v_{g})$ we put

[TABLE]

where the tensor product is taken over $L_{\mathfrak{P}}$.

Let $\bar{G}^{\natural}$ (\natural=\emptyset,^{\prime},^{\prime\prime},\widetilde{}\) be the groups defined in Section 5.2. For $\natural=\widetilde{}$, via the projection $\bar{G}^{\natural}({\mathbb{Q}}_{p})\rightarrow{\mathrm{GL}}(2,F_{\mathfrak{p}})$, $V(\tilde{\mathrm{k}})$ becomes a $\bar{G}^{\natural}({\mathbb{Q}}_{p})$-module. For $\natural=^{\prime},^{\prime\prime}$, via the projection of $\bar{G}^{\natural}({\mathbb{Q}}_{p})\subset\mathrm{GL}(2,F_{\mathfrak{p}})\times\mathrm{GL}(2,F_{\mathfrak{p}})$ to the second factor, $V(\tilde{\mathrm{k}})$ becomes a $\bar{G}^{\natural}({\mathbb{Q}}_{p})$-module. In each case via the inclusion $\bar{G}^{\natural}({\mathbb{Q}})\hookrightarrow\bar{G}^{\natural}({\mathbb{Q}}_{p})$, $V(\tilde{\mathrm{k}})$ becomes a $\bar{G}^{\natural}({\mathbb{Q}})$-module. Using the $p$-adic uniformization of $X^{\natural}=X^{\natural}_{U^{\natural,p}}$ we attach to this $\bar{G}^{\natural}({\mathbb{Q}})$-module a local system ${\mathcal{V}}^{\natural}(\tilde{\mathrm{k}})$ on $X^{\natural}$.

Let $\varphi_{q,\mathrm{k},\mathrm{v}}$ be the operator on $V(\mathrm{k},\mathrm{v})$

[TABLE]

For $\mathrm{k}=(k_{1},\cdots,k_{g},w)$ we put

[TABLE]

and

[TABLE]

Let $\mathscr{F}^{\natural}(\tilde{\mathrm{k}})$ be the filtered $\varphi_{q}$-isocrystal ${\mathcal{V}}^{\natural}(\tilde{\mathrm{k}})\otimes_{{\mathbb{Q}}_{p}}\mathscr{O}_{X^{\natural}}$ on $X^{\natural}$ with the $q$-Frobenius $\varphi_{q,\tilde{\mathrm{k}}}\otimes\varphi_{q,\mathscr{O}_{X^{\natural}}}$ and the connection $1\otimes\mathrm{d}:{\mathcal{V}}^{\natural}(\tilde{\mathrm{k}})\otimes_{{\mathbb{Q}}_{p}}\mathscr{O}_{X^{\natural}}\rightarrow{\mathcal{V}}^{\natural}(\tilde{\mathrm{k}})\otimes_{{\mathbb{Q}}_{p}}\Omega^{1}_{X^{\natural}}$; the filtration on

[TABLE]

is given by

[TABLE]

with the convention that $v_{\tau}=\frac{w-k_{\tau}}{2}$ in the case of $\tilde{\mathrm{k}}=(k_{1},\cdots,k_{g},w)$, where $z$ is the canonical coordinate on ${\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$.

Lemma 6.1**.**

When $k_{1}=\cdots=k_{i-1}=k_{i+1}=\cdots=k_{g}=2$, $k_{i}=3$, and $v_{1}=\cdots=v_{g}=0$, the filtered $\varphi_{q}$-isocrystal attached to ${\mathcal{F}}^{\prime}{(\mathrm{k},\mathrm{v})}$ is isomorphic to $\mathscr{F}^{\prime}(\mathrm{k},\mathrm{v})$.

Proof.

Let $\tilde{e}_{i}\in L\otimes_{{\mathbb{Q}}}D$ be the idempotent whose $(2,i)$-th component is {\scriptsize{\big{[}\!\!\begin{array}[]{cc}1\!\!\!&\!\!\!0\\ 0\!\!\!&\!\!\!1\end{array}\!\!\big{]}}} and the other component are zero. Let ${\mathcal{A}}$ be the universal $\mathscr{O}_{D}$-abelian scheme over $M^{\prime}$, $\widehat{{\mathcal{A}}}$ the formal module on $X^{\prime}$ attached to ${\mathcal{A}}$. Note that $\tilde{e}_{i}(\mathfrak{o}_{L_{\mathfrak{P}}}\otimes_{{\mathbb{Z}}_{p}}\widehat{{\mathcal{A}}})$ is just the pullback of $\mathfrak{o}_{L_{\mathfrak{P}}}\otimes_{\tau_{i},\mathfrak{o}_{F_{\mathfrak{p}}}}{\mathcal{G}}$ by the projection $X^{\prime}_{U^{\prime p}}\rightarrow(\bar{G}^{\prime}({\mathbb{Q}})\cap U^{\prime p}U^{\prime}_{p,0})\backslash{\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}$ [27, 6.43], where ${\mathcal{G}}$ is the universal special formal $\mathscr{O}_{B_{\mathfrak{p}}}$-module (forgetting the information of $\rho$ in Drinfeld’s moduli problem).

As $L_{\mathfrak{P}}$ splits $B_{\mathfrak{p}}$, $L_{\mathfrak{P}}$ contains all embeddings of $F^{(2)}_{\mathfrak{p}}$. The embedding $\tau_{i}:F_{\mathfrak{p}}\hookrightarrow L_{\mathfrak{P}}$ extends in two ways to $F^{(2)}_{\mathfrak{p}}$ denoted respectively by $\tau_{i,0}$ and $\tau_{i,1}$. Then

[TABLE]

We decompose $\mathfrak{o}_{L_{\mathfrak{P}}}\otimes_{\tau_{i},\mathfrak{o}_{F_{\mathfrak{p}}}}{\mathcal{G}}$ into the sum of two direct summands according to the action of ${\mathfrak{o}}_{F_{\mathfrak{p}}^{(2)}}\subset{\mathfrak{o}}_{B_{\mathfrak{p}}}$: ${\mathfrak{o}}_{F_{\mathfrak{p}}^{(2)}}$ acts by $\tau_{i,0}$ on the first direct summand, and acts by $\tau_{i,1}$ on the second. Without loss of generality we may assume that $e_{i}$ in the definition of ${\mathcal{F}}^{\prime}_{i}$ (see Section 6.1) is chosen such that $e_{i}$ is the projection onto the first direct summand. So $e_{i}(\mathfrak{o}_{L_{\mathfrak{P}}}\otimes_{{\mathbb{Z}}_{p}}\widehat{{\mathcal{A}}})$ is just the pullback of $\mathfrak{o}_{L_{\mathfrak{P}}}\otimes_{\tau_{i,0},\mathfrak{o}_{F_{\mathfrak{p}}^{(2)}}}{\mathcal{G}}$ by the projection $X^{\prime}_{U^{\prime p}}\rightarrow(\bar{G}^{\prime}({\mathbb{Q}})\cap U^{\prime p}U^{\prime}_{p,0})\backslash{\mathcal{H}}_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}}$. Now the statement of our lemma follows from the discussion in Section 3.2. ∎

Proposition 6.2**.**

The filtered $\varphi_{q}$-isocrystal attached to ${\mathcal{F}}^{\prime}{(\mathrm{k},\mathrm{v})}$ is isomorphic to $\mathscr{F}^{\prime}(\mathrm{k},\mathrm{v})$.

Proof.

Let $\mathscr{F}^{\prime}_{i}$ denote the filtered $\varphi_{q}$-isocrystal attached to $\mathcal{F}^{\prime}_{i}$. By (6.5) the filtered $\varphi_{q}$-isocrystal attached to ${\mathcal{F}}^{\prime}(\mathrm{k},\mathrm{v})$ is isomorphic to

[TABLE]

By Lemma 6.1 a simple computation implies our conclusion. ∎

Corollary 6.3**.**

The filtered $\varphi_{q}$-isocrystal attached to ${\mathcal{F}}^{\prime\prime}{(\mathrm{k})}$ is isomorphic to $\mathscr{F}^{\prime\prime}(\mathrm{k})$.

Proof.

This follows from Proposition 6.2 and [28, Lemma 6.1]. ∎

Lemma 6.4**.**

The filtered $\varphi_{q}$-isocrystal associated to the local system ${\mathcal{F}}(\bar{\chi})$ over $(N_{E_{0},0})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ is ${\mathcal{F}}(\bar{\chi})\otimes\mathscr{O}_{(N_{E_{0},0})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}}$ with the $q$-Frobenius being $1\otimes\varphi_{q,(N_{E_{0},0})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}}$ and the filtration being trivial.

Proof.

We only need to show that any geometric point of $(N_{E_{0},0})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ is defined over $\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}$.

Let $h_{E_{0}}$ be as in Section 4.1, $\mu$ the cocharacter of $T_{0}$ defined over $E_{0}$ attached to $h_{E_{0}}$. Let $r$ be the composition

[TABLE]

Let

[TABLE]

be the reciprocal of the reciprocity map from class field theory. For any compact open subgroup $U$ of $T_{0}({\mathbb{A}}_{f})$, ${\mathrm{Gal}}(\overline{{\mathbb{Q}}}/E_{0})$ acts on $(N_{E_{0}})_{U}(\overline{{\mathbb{Q}}})=T_{0}({\mathbb{Q}})\backslash T_{0}({\mathbb{A}}_{f})/U$ by $\sigma(T_{0}({\mathbb{Q}})aU)=T_{0}({\mathbb{Q}})r_{f}(s_{\sigma})aU$, where $s_{\sigma}$ is any idèle such that $\mathrm{art}_{E_{0}}(s_{\sigma})=\sigma|E^{\mathrm{ab}}$, and $r_{f}$ is the composition

[TABLE]

of $r$ and the projection map $T_{0}({\mathbb{A}})\rightarrow T_{0}({\mathbb{A}}_{f})$. Let $\mathcal{I}$ be the subgroup of ${\mathrm{Gal}}(\overline{{\mathbb{Q}}}/E_{0})$ consisting of $\sigma$ such that $s_{\sigma}\in r_{f}^{-1}(U)$. Put $\mathcal{K}=\overline{{\mathbb{Q}}}^{\mathcal{I}}$. Then any geometric point of $(N_{E_{0}})_{U}$ is defined over $\mathcal{K}$. Observe that, when $U$ is of the form $U_{p,0}U^{p}$ with $U^{p}$ a compact open subgroup of $T_{0}({\mathbb{A}}_{f}^{p})$ and $U_{p,0}$ the maximal compact open subgroup of $T_{0}({\mathbb{Q}}_{p})$, $\mathcal{K}$ is unramified over $p$. Therefore, any geometric point of $(N_{E_{0},0})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ is already defined over $\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}$. ∎

Corollary 6.5**.**

The filtered $\varphi_{q}$-isocrystal attached to $\mathrm{pr}_{1}^{*}{\mathcal{F}}(\mathrm{k})$ is $\mathrm{pr}_{1}^{*}\mathscr{F}(\mathrm{k})$.

Proof.

By (6.1) the filtered $\varphi_{q}$-isocrystal attached to $\mathrm{pr}_{1}^{*}{\mathcal{F}}{(\mathrm{k})}$ is the tensor product of the filtered $\varphi_{q}$-isocrystal attached to $\alpha^{*}{\mathcal{F}}^{\prime\prime}{(\mathrm{k})}$ and that attached to $\beta^{*}{\mathcal{F}}(\bar{\chi}^{-1})^{(g-1)(w-2)}$. Our conclusion follows from Proposition 5.2 (b) (in the case of $\sharp=\widetilde{}$ and $\natural=^{\prime\prime}$), Corollary 6.3 and Lemma 6.4. ∎

It is rather possible that the filtered $\varphi_{q}$-isocrystal attached to ${\mathcal{F}}(\mathrm{k})$ is $\mathscr{F}(\mathrm{k})$. But the author does not know how to descent the conclusion of Corollary 6.5 to $X_{U^{p}}$.

7 The de Rham cohomology

7.1 Covering filtration and Hodge filtration for de Rham

cohomology

We fix an arithmetic Schottky group $\Gamma$ that is cocompact in ${\mathrm{PGL}}(2,F_{\mathfrak{p}})$. Then $\Gamma$ acts freely on ${\mathcal{H}}$, and the quotient $X_{\Gamma}=\Gamma\backslash{\mathcal{H}}$ is the rigid analytic space associated with a proper smooth curve over $F_{\mathfrak{p}}$. Here we write ${\mathcal{H}}$ for ${\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$.

We recall the theory of de Rham cohomology of local systems over $X_{\Gamma}$ [30, 31, 32, 33].

We denote by $\widehat{{\mathcal{H}}}$ the canonical formal model of ${\mathcal{H}}$. The curve $X_{\Gamma}$ has a canonical semistable module ${\mathcal{X}}_{\Gamma}=\Gamma\backslash\widehat{{\mathcal{H}}}$; the special fiber ${\mathcal{X}}_{\Gamma,s}$ of ${\mathcal{X}}_{\Gamma}$ is isomorphic to $\Gamma\backslash\widehat{{\mathcal{H}}}_{s}$.

The graph $\mathrm{Gr}({\mathcal{X}}_{\Gamma,s})$ is closely related to the Bruhat-Tits tree ${\mathcal{T}}$ for ${\mathrm{PGL}}(2,F_{\mathfrak{p}})$. The group $\Gamma$ acts freely on the tree ${\mathcal{T}}$. Let ${\mathcal{T}}_{\Gamma}$ denote the quotient tree. The set of connected components of the special fiber ${\mathcal{X}}_{\Gamma,s}$ is in one-to-one correspondence to the set $\mathrm{V}({\mathcal{T}}_{\Gamma})$ of vertices of ${\mathcal{T}}_{\Gamma}$, each component being isomorphic to the projective line over $k(=$the resider field of $F_{\mathfrak{p}}$). Write $\{{\mathrm{P}}^{1}_{v}\}_{v\in\mathrm{V}({\mathcal{T}}_{\Gamma})}$ for the set of components of ${\mathcal{X}}_{\Gamma,s}$. The singular points of ${\mathcal{X}}_{\Gamma,s}$ are ordinary $k$-rational double singular points; they correspond to (unoriented) edges of ${\mathcal{T}}_{\Gamma}$. Two components ${\mathrm{P}}^{1}_{u}$ and ${\mathrm{P}}^{1}_{v}$ intersect if and only if $u$ and $v$ are adjacent; in this case, they intersect at a singular point. For simplicity we will use the edge $e$ joining $u$ and $v$ to denote this singular point. There is a reduction map from $X_{\Gamma}^{\mathrm{an}}$ to ${\mathcal{X}}_{\Gamma,s}$. For a closed subset $U$ of ${\mathcal{X}}_{\Gamma,s}$ let $]U[$ denote the tube of $U$ in $X_{\Gamma}^{\mathrm{an}}$. Then $\{]{\mathrm{P}}^{1}_{v}[\}_{v\in\mathrm{V}({\mathcal{T}}_{\Gamma})}$ is an admissible covering of $X_{\Gamma}^{\mathrm{an}}$. Clearly $]{\mathrm{P}}^{1}_{o(e)}[\cap]{\mathrm{P}}^{1}_{t(e)}[=]e[$.

Let $L$ be a field that splits $F_{\mathfrak{p}}$. Fix an embedding $\tau:F_{\mathfrak{p}}\hookrightarrow L$. Let $V$ be an $L[\Gamma]$-module that comes from an algebraic representation of ${\mathrm{PGL}}(2,F_{\mathfrak{p}})$ of the form $V(\mathrm{k})$ with $\mathrm{k}=(k_{1},\cdots,k_{g};2)$. We will regard $V$ as an $F_{\mathfrak{p}}$-vector space by $\tau$. Let $\mathscr{V}=\mathscr{V}(\mathrm{k})$ be the local system on $X_{\Gamma}$ associated with $V$. Let $H^{*}_{{\mathrm{dR}},\tau}(X_{\Gamma},\mathscr{V})$ be the hypercohomology of the complex $\mathscr{V}\otimes_{\tau,F_{\mathfrak{p}}}\Omega_{X_{\Gamma}}^{\bullet}$.

We consider the Mayer-Vietorus exact sequence attached to $H^{*}_{{\mathrm{dR}},\tau}(X_{\Gamma},\mathscr{V})$ with respect to the admissible covering $\{]{\mathrm{P}}^{1}_{v}[\}_{v\in\mathrm{V}({\mathcal{T}}_{\Gamma})}$. As a result we obtain an injective map

[TABLE]

As $]{\mathrm{P}}^{1}_{v}[$ and $]e[$ are quasi-Stein, a simple computation shows that $H^{0}_{{\mathrm{dR}},\tau}(]{\mathrm{P}}^{1}_{v}[,\mathscr{V})$ and $H^{0}_{{\mathrm{dR}},\tau}(]e[,\mathscr{V})$ are isomorphic to $V$. Let $C^{0}(V)$ be the space of $V$-valued functions on $\mathrm{V}({\mathcal{T}})$, $C^{1}(V)$ the space of $V$-valued functions on $\mathrm{E}({\mathcal{T}})$ such that $f(e)=-f(\bar{e})$. Let $\Gamma$ act on $C^{i}(V)$ as $f\mapsto\gamma\circ f\circ\gamma^{-1}$. Then we have a $\Gamma$-equivariant short exact sequence

[TABLE]

where $\partial(f)(e)=f(o(e))-f(t(e))$. Observe that

[TABLE]

and the map

[TABLE]

coincides with $\partial$. Thus

[TABLE]

is isomorphic to $C^{1}(V)^{\Gamma}/\partial C^{0}(V)^{\Gamma}$. From (7.1) we get the injective map

[TABLE]

Let $C^{1}_{\mathrm{har}}(V)$ be the space of harmonic forms

[TABLE]

Fixing some $v\in\mathrm{V}({\mathcal{T}})$, let $\epsilon$ be the map $C^{1}_{\mathrm{har}}(V)^{\Gamma}\rightarrow H^{1}(\Gamma,V)$ [7, (2.26)] defined by

[TABLE]

where the sum runs over the edges joining $v$ and $\gamma v$; $\epsilon$ does not depend on the choice of $v$. By [7, Appendix A] $\epsilon$ is minus the composition

[TABLE]

and is an isomorphism. Combining this with the injectivity of $\delta$ we obtain that both the natural map $C^{1}_{\mathrm{har}}(V)^{\Gamma}\rightarrow C^{1}(V)^{\Gamma}/\partial C^{0}(V)^{\Gamma}$ and $\delta$ are isomorphisms. Below, we will identify $C^{1}_{\mathrm{har}}(V)^{\Gamma}$ with $C^{1}(V)^{\Gamma}/\partial C^{0}(V)^{\Gamma}$.

By [33] we have

[TABLE]

In loc. cit, de Shalit only considered a special case, but his argument is valued for our general case. If $\omega$ is a $\Gamma$-invariant $V$-valued differential of the second kind on ${\mathcal{H}}$, let $F_{\omega}$ be a primitive of it [32], which is defined by Coleman’s integral [8]. 333Precisely we choose a branch of Coleman’s integral. Let $P$ be the map

[TABLE]

Note that $P\circ\iota$ coincides with $\delta$. Thus $P$ splits the inclusion $\iota\circ\delta^{-1}:H^{1}(\Gamma,V)\rightarrow H^{1}_{{\mathrm{dR}},\tau}(X_{\Gamma},\mathscr{V})$.

Let $I$ be the map

[TABLE]

Now, we suppose that $\Gamma$ is of the form in [7, Appendix A]. We do not describe it precisely, but only point out that $\Gamma_{i,0}$ in Section 7.2 is of this form.

Proposition 7.1**.**

We have an exact sequence called the covering filtration exact sequence

[TABLE]

Proof.

What we need to prove is that the map

[TABLE]

is an isomorphism. When $V$ is the trivial module, this is already proved in [33]. So we assume that $V$ is not the trivial module. First we prove the injectivity of the above map. For this we only need to repeat the argument in [33, Theorem 1.6]. Let $\omega$ be a $\Gamma$-invariant differential form of second kind on ${\mathcal{H}}$ such that $P([\omega])=I([\omega])=0$, where $[\omega]$ denotes the class of $\omega$ in $H^{1}_{{\mathrm{dR}},\tau}(X_{\Gamma},\mathscr{V})$. Let $F_{\omega}$ be a primitive of $\omega$. As $I(\omega)=0$, the residues of $\omega$ vanish, and thus $F_{\omega}$ is meromorphic. As $P(\omega)=0$, we may adjust $F_{\omega}$ by a constant so that it is $\Gamma$-invariant. By (7.3) we have $[\omega]=0$. To show the surjectivity we only need to compare the dimensions. By [7, Appendix A] we have

[TABLE]

and

[TABLE]

where $V^{*}={\mathrm{Hom}}_{F_{\mathfrak{p}}}(V,F_{\mathfrak{p}})$ is the dual $F_{\mathfrak{p}}[\Gamma]$-module. By [30, Theorem 1] we have

[TABLE]

Hence

[TABLE]

as desired. ∎

We have also a Hodge filtration exact sequence

[TABLE]

This exact sequence and the covering filtration exact sequence fit into the commutative diagram

[TABLE]

The diagonal arrows are isomorphisms. Indeed, by [32, Section 3] the south-west arrow is an isomorphism; one easily deduces from this that the north-east arrow is also an isomorphism. In particular, we have a Hodge-like decomposition

[TABLE]

7.2 De Rham cohomology of $\mathscr{F}(\mathrm{k})$

Let $\bar{B}$ be as in Section 5.2. Write $\widehat{\bar{B}}^{\times}:=(\bar{B}\bigotimes{\mathbb{A}}_{f})^{\times}$ and $\widehat{\bar{B}}^{p,\times}:=(\bar{B}\bigotimes{\mathbb{A}}_{f}^{p})^{\times}$. Let $U$ be a compact open subgroup of $G({\mathbb{A}}_{f})$ that is of the form $U_{p,0}U^{p}$. We identify $U^{p}=\prod\limits_{{\mathfrak{l}}\neq{\mathfrak{p}}}U_{\mathfrak{l}}$ with a subgroup of $\widehat{\bar{B}}^{p,\times}$.

Write $\widehat{\bar{B}}^{p,\times}=\sqcup_{i=1}^{h}\bar{B}^{\times}x_{i}U^{p}$. For $i=1,\cdots,h$, put

[TABLE]

Then $X_{U^{p}}$ is isomorphic to

[TABLE]

Here we identify ${\mathbb{Z}}$ with $G({\mathbb{Q}}_{p})/U_{p,0}$. Note that $\widetilde{\Gamma}_{i}$ acts transitively on $G({\mathbb{Q}}_{p})/U_{p,0}$.

Note that, for every point in ${\mathbb{Z}}=G({\mathbb{Q}}_{p})/U_{p,0}$ it is fixed by $\gamma\in\bar{B}^{\times}$ if and only if $\gamma_{\mathfrak{p}}$ is in $\mathrm{GL}(2,\mathscr{O}_{F_{\mathfrak{p}}})$. Put

[TABLE]

Let $\Gamma_{i,0}$ be the image of $\widetilde{\Gamma}_{i,0}$ in $\mathrm{PGL}(2,F_{\mathfrak{p}})$. Then we have an isomorphism

[TABLE]

Applying the constructions in Section 7.1 to each part $\Gamma_{i,0}\backslash{\mathcal{H}}$ of $X_{U^{p}}$, we obtain operators $\iota$, $P$ and $I$.

8 Automorphic Forms on totally definite quaternion algebras and Teitelbaum type

$L$-invariants

In this section we recall Chida, Mok and Park’s definition of Teitelbaum type $L$-invariant [7].

8.1 Automorphic Forms on totally definite quaternion algebras

We recall the theory of automorphic forms on totally definite quaternion algebras.

Let $\bar{B}$ be as in Section 5, which is a totally definite quaternion algebra over $F$. Let $\Sigma=\prod_{\mathfrak{l}}\Sigma_{\mathfrak{l}}$ be a compact open subgroup of $\widehat{\bar{B}}^{\times}$.

Let $\chi_{F,\mathrm{cyc}}:{\mathbb{A}}_{F}^{\times}/F^{\times}\rightarrow{\mathbb{Z}}_{p}^{\times}$ be the Hecke character obtained by composing the cyclotomic character $\chi_{{\mathbb{Q}},\mathrm{cyc}}:{\mathbb{A}}_{\mathbb{Q}}^{\times}/{\mathbb{Q}}^{\times}\rightarrow{\mathbb{Z}}_{p}^{\times}$ and the norm map from ${\mathbb{A}}_{F}^{\times}$ to ${\mathbb{A}}_{\mathbb{Q}}^{\times}$.

Definition 8.1**.**

An automorphic form on $\bar{B}^{\times}$, of weight $\mathrm{k}=(k_{1},\cdots,k_{g},w)$ and level $\Sigma$, is a function $\mathbf{f}:\widehat{\bar{B}}^{\times}\rightarrow V(\mathrm{k})$ that satisfies

[TABLE]

for all $\gamma\in\bar{B}^{\times}$, $u\in\Sigma$, $b\in\widehat{\bar{B}}^{\times}$ and $z\in\widehat{F}^{\times}$. Denote by $S^{\bar{B}}_{\mathrm{k}}(\Sigma)$ the space of such forms. Remark that our $S^{\bar{B}}_{\mathrm{k}}(\Sigma)$ coincides with $S^{\bar{B}}_{\mathrm{k}^{\prime},\mathrm{v}}(\Sigma)$ for $\mathrm{k}^{\prime}=(k_{1}-2,\cdots,k_{g}-2)$ and $\mathrm{v}=(\frac{w-k_{1}}{2},\frac{w-k_{2}}{2},\cdots,\frac{w-k_{g}}{2})$ in [7].

Observe that a form $\mathbf{f}$ of level $\Sigma$ is determined by its values on the finite set $\bar{B}^{\times}\backslash\widehat{\bar{B}}^{\times}/\Sigma$. As in Section 7.2 write $\widehat{\bar{B}}^{\times}=\sqcup_{i=1}^{h}\bar{B}^{\times}x_{i}{\mathrm{GL}}(2,F_{\mathfrak{p}})\Sigma$; for $i=1,\cdots,h$, put

[TABLE]

Then we have a bijection

[TABLE]

The class of $g$ in $\widetilde{\Gamma}_{i}\backslash{\mathrm{GL}}(2,F_{\mathfrak{p}})/\Sigma_{\mathfrak{p}}$ corresponds to the class of $x_{i,{\mathfrak{p}}}g_{\mathfrak{p}}$ in $\bar{B}^{\times}\backslash\widehat{\bar{B}}^{\times}/\Sigma$, where $g_{\mathfrak{p}}$ is the element of $\widehat{\bar{B}}^{\times}$ that is equal to $g$ at the place ${\mathfrak{p}}$, and equal to the identity at each other place. Using this we can attach to an automorphic form $\mathbf{f}$ of weight $\mathrm{k}$ and level $\Sigma$ an $h$-tuple of functions $(f_{1},\cdots,f_{h})$ on ${\mathrm{GL}}(2,F_{\mathfrak{p}})$ with values in $V(\mathrm{k})$ defined by $f_{i}(g)=\mathbf{f}(x_{i,{\mathfrak{p}}}g_{\mathfrak{p}})$. The function $f_{i}$ satisfies

[TABLE]

for all $\gamma_{\mathfrak{p}}\in\widetilde{\Gamma}_{i}$, $g\in{\mathrm{GL}}(2,F_{\mathfrak{p}})$, $u\in\Sigma_{\mathfrak{p}}$ and $z\in F_{\mathfrak{p}}^{\times}$.

For each prime ${\mathfrak{l}}$ of $F$ such that $\bar{B}$ splits at ${\mathfrak{l}}$, ${\mathfrak{l}}\neq{\mathfrak{p}}$, and $\Sigma_{\mathfrak{l}}$ is maximal, one define a Hecke operator $T_{\mathfrak{l}}$ on $S^{\bar{B}}_{\mathrm{k}}(\Sigma)$ as follows. Fix an isomorphism $\iota_{\mathfrak{l}}:B_{\mathfrak{l}}\rightarrow M_{2}(F_{\mathfrak{l}})$ such that $\Sigma_{\mathfrak{l}}$ becomes identified with $\mathrm{GL}_{2}(\mathfrak{o}_{F_{\mathfrak{l}}})$. Let $\pi_{\mathfrak{l}}$ be a uniformizer of $\mathfrak{o}_{F_{\mathfrak{l}}}$. Given a double coset decomposition

[TABLE]

we define the Hecke operator $T_{\mathfrak{l}}$ on $S^{\bar{B}}_{\mathrm{k}}(\Sigma)$ by

[TABLE]

We define $U_{\mathfrak{p}}$ similarly. Let ${\mathbb{T}}_{\Sigma}$ be the Hecke algebra generated by $U_{\mathfrak{p}}$ and these $T_{\mathfrak{l}}$.

Denote by $\mathfrak{o}_{F}^{({\mathfrak{p}})}$ the ring of ${\mathfrak{p}}$-integers of $F$ and $(\mathfrak{o}_{F}^{({\mathfrak{p}})})^{\times}$ the group of ${\mathfrak{p}}$-units of $F$. We have $\widetilde{\Gamma}_{i}\cap F^{\times}=(\mathfrak{o}_{F}^{({\mathfrak{p}})})^{\times}$. For $i=1,\cdots,h$, put $\Gamma_{i}=\widetilde{\Gamma}_{i}/(\mathfrak{o}_{F}^{({\mathfrak{p}})})^{\times}$. Consider the following twisted action of $\widetilde{\Gamma_{i}}$ on $V(\mathrm{k})$:

[TABLE]

Then $(\mathfrak{o}_{F}^{({\mathfrak{p}})})^{\times}$ is trivial on $V(\mathrm{k})$, so we may consider $V(\mathrm{k})$ as a $\Gamma_{i}$-module via the above twisted action.

8.2 Teitelbaum type $L$-invariants

Chida, Mok and Park [7] defined Teitelbaum type $L$-invariant for automorphic forms $\mathbf{f}\in S^{\bar{B}}_{\mathrm{k}}(\Sigma)$ satisfying the condition (CMP) given in the introduction:

[TABLE]

We recall their construction below.

We attach to each $f_{i}$ a $\Gamma_{i}$-invariant $V(\mathrm{k})$-valued cocycle $c_{f_{i}}$, where $\Gamma_{i}$ acts on $V(\mathrm{k})$ via $\star$. For $e=(s,t)\in\mathrm{E}({\mathcal{T}})$, represent $s$ and $t$ by lattices $L_{s}$ and $L_{t}$ such that $L_{s}$ contains $L_{t}$ with index ${\mathcal{N}}\mathfrak{p}$. Let $g_{e}\in{\mathrm{GL}}(2,F_{\mathfrak{p}})$ be such that $g_{e}(\mathfrak{o}_{F_{\mathfrak{p}}}^{2})=L_{s}$. Then we define $c_{f_{i}}(e):=|\det(g)|_{\mathfrak{p}}^{\frac{w-2}{2}}g_{e}\star f_{i}(g_{e})$. If $\mathbf{f}$ satisfies (CMP), then $c_{f_{i}}$ is in $C^{1}_{\mathrm{har}}(V(\mathrm{k}))^{\Gamma_{i}}$ [7, Proposition 2.7]. Thus we obtain a vector of harmonic cocycles $c_{\mathbf{f}}=(c_{f_{1}},\cdots,c_{f_{h}})$.

For each $c\in C^{1}_{\mathrm{har}}(V(\mathrm{k}))^{\Gamma_{i}}$ we define $\kappa^{\mathrm{sch}}_{c}$ to be the following function on $\Gamma_{i}$ with values in $V(\mathrm{k})$: fixing some $v\in\mathrm{V}({\mathcal{T}})$, for each $\gamma\in\Gamma_{i}$, we put

[TABLE]

where $e$ runs over the edges in the geodesic joining $v$ and $\gamma v$. As $c$ is $\Gamma_{i}$-invariant, $\kappa_{c}^{\mathrm{sch}}$ is a $1$-cocycle. Furthermore the class of $\kappa^{\mathrm{sch}}_{c}$ in $H^{1}(\Gamma_{i},V(\mathrm{k}))$ is independent of the choice of $v$. Hence we obtain a map

[TABLE]

By [7, Proposition 2.9] $\kappa^{\mathrm{sch}}$ is an isomorphism.

For each $\sigma:F_{\mathfrak{p}}\rightarrow L_{\mathfrak{P}}$, let $L_{\mathfrak{P},\sigma}(k,v)$ be the dual of $V_{\sigma}(k,v)$ with the right action of ${\mathrm{GL}}(2,F_{\mathfrak{p}})$: if $g\in{\mathrm{GL}}(2,F_{\mathfrak{p}})$, $P^{\prime}\in L_{\mathfrak{P},\sigma}(k,v)$ and $P\in V_{\sigma}(k,v)$, then $\langle P^{\prime},g\cdot P\rangle=\langle P^{\prime}|_{g},P\rangle$. We realize $L_{\mathfrak{P},\sigma}(k,v)$ by the same space as $V_{\sigma}(k,v)$, with the pairing

[TABLE]

and the right ${\mathrm{GL}}(2,F_{\mathfrak{p}})$-action

[TABLE]

Put

[TABLE]

with the right action of ${\mathrm{GL}}(2,F_{\mathfrak{p}})$, where the tensor product is taken over $L_{\mathfrak{P}}$.

For each harmonic cocycle $c\in C^{1}_{\mathrm{har}}(V(\mathrm{k}))^{\Gamma_{i}}$ the method of Amice-Velu and Vishik allows one to define the $V(\mathrm{k})^{\tau}$-valued rigid analytic distribution $\mu_{c}^{\tau}$ on ${\mathrm{P}}^{1}(F_{\mathfrak{p}})$ such that the value of $\int_{U_{e}}t^{j}\mu_{c}^{\tau}(t)\in V(\mathrm{k})^{\tau}$ ($0\leq j\leq k_{\tau}-2$) satisfies

[TABLE]

for each $Q\in L_{\mathfrak{P}}(\mathrm{k})^{\tau}$. 444There are minors in the definitions of $\mu_{c}^{\tau}$ and $\lambda^{\tau}_{c}$ in [7]. See [36] Definition 6 and the paragraph before Proposition 9.

Using $\mu_{c}^{\tau}$ we obtain a $V(\mathrm{k})^{\tau}$-valued rigid analytic function $g^{\tau}_{c}$, precisely a global section of $V(\mathrm{k})^{\tau}\otimes_{\tau,F_{\mathfrak{p}}}{\mathcal{O}}_{{\mathcal{H}},\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ by

[TABLE]

for $z\in{\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$. The function $g_{c}^{\tau}$ satisfies the transformation property: for $\gamma\in\tilde{\Gamma}_{i}$, let {\scriptsize{\big{[}\!\!\begin{array}[]{cc}a\!\!\!&\!\!\!b\\ c\!\!\!&\!\!\!d\end{array}\!\!\big{]}}} be the image of $\gamma$ in $\bar{B}_{\mathfrak{p}}\cong{\mathrm{GL}}(2,F_{\mathfrak{p}})$; then

[TABLE]

Consider $V(\mathrm{k})$ as an $F_{\mathfrak{p}}$-module via $\tau:F_{\mathfrak{p}}\hookrightarrow L_{\mathfrak{P}}$. We define the $V(\mathrm{k})$-valued cocycle $\lambda^{\tau}_{c}$ as follows. Fix a point $z_{0}\in{\mathcal{H}}$. For each $\gamma\in\Gamma_{i}$ the value $\lambda_{c}^{\tau}(\gamma)$ is given by the formula: for $Q\in L_{\mathfrak{P}}(\mathrm{k})^{\tau}$,

[TABLE]

$(0\leq j\leq k_{\tau}-2)$, where the integral is the branch of Coleman’s integral chosen in Section 7. Then $\lambda_{c}^{\tau}$ is a $1$-cocycle on $\Gamma_{i}$, and the class of $\lambda_{c}^{\tau}$ in $H^{1}(\Gamma_{i},V(\mathrm{k}))$, denoted by $[\lambda_{c}^{\tau}]$, is independent of the choice of $z_{0}$. This defines a map

[TABLE]

As $\kappa^{\mathrm{sch}}$ is an isomorphism, for each $\tau$ there exists a unique $\ell_{\tau}\in L_{\mathfrak{P}}$ such that

[TABLE]

The Teitelbaum type $L$-invariant of $\mathbf{f}$, denoted by ${\mathcal{L}}_{T}(\mathbf{f})$, is defined to be the vector $(\ell_{\tau})_{\tau}$ [7, Section 3.2]. We also write ${\mathcal{L}}_{T,\tau}(\mathbf{f})$ for $\ell_{\tau}$.

9 Comparing $L$-invariants

Let $B$, $\bar{B}$, $G$ and $\bar{G}$ be as before. Let ${\mathfrak{n}}^{-}$ be the conductor of $\bar{B}$. By our assumption on $\bar{B}$, ${\mathfrak{p}}\nmid{\mathfrak{n}}^{-}$ and the conductor of $B$ is ${\mathfrak{p}}{\mathfrak{n}}^{-}$. Let ${\mathfrak{n}}^{+}$ be an ideal of $\mathfrak{o}_{F}$ that is prime to ${\mathfrak{p}}{\mathfrak{n}}^{-}$, and put ${\mathfrak{n}}:={\mathfrak{p}}{\mathfrak{n}}^{+}{\mathfrak{n}}^{-}$.

For any prime ideal ${\mathfrak{l}}$ of $\mathfrak{o}_{F}$, put

[TABLE]

Let $\bar{\Sigma}=\Sigma({\mathfrak{p}}{\mathfrak{n}}^{+},{\mathfrak{n}}^{-})$ be the level $\prod_{\mathfrak{l}}\bar{R}_{\mathfrak{l}}$. We write $S^{\bar{B}}_{\mathrm{k}}({\mathfrak{p}}{\mathfrak{n}}^{+},{\mathfrak{n}}^{-})$ for $S^{\bar{B}}_{\mathrm{k}}(\Sigma({\mathfrak{p}}{\mathfrak{n}}^{+},{\mathfrak{n}}^{-}))$. Similarly we define $\Sigma=\Sigma({\mathfrak{n}}^{+},{\mathfrak{p}}{\mathfrak{n}}^{-})$, a compact open subgroup of $G({\mathbb{A}}_{f})$. Let $S_{\mathrm{k}}^{B}({\mathfrak{n}}^{+},{\mathfrak{p}}{\mathfrak{n}}^{-})$ be the space of modular forms on the Shimura curve $M$ of weight $\mathrm{k}$ and level $\Sigma$.

Let $\mathrm{k}=(k_{1},\cdots,k_{g},w)$ be a multiweight such that $k_{1}\equiv\cdots k_{g}\equiv w\hskip 6.0pt\mathrm{mod}\hskip 3.0pt2$ and $k_{1},\cdots,k_{g}$ are all even and larger than $2$. Let $f_{\infty}$ be a (Hilbert) eigen cusp newform of weight $\mathrm{k}$ and level ${\mathfrak{n}}$ that is new at $\mathfrak{p}\mathfrak{n}^{-}$. Let $\mathbf{f}\in S_{\mathrm{k}}^{\bar{B}}({\mathfrak{p}}{\mathfrak{n}}^{+},{\mathfrak{n}}^{-})$ (resp. $f_{B}\in S_{\mathrm{k}}^{B}({\mathfrak{n}}^{+},{\mathfrak{p}}{\mathfrak{n}}^{-})$) be an eigen newform corresponding to $f_{\infty}$ by the Jacquet-Langlands correspondence; $\mathbf{f}$ (resp. $f_{B}$) is unique up to scalars.

We further assume that $\mathbf{f}$ satisfies (CMP), so that we can attach to $\mathbf{f}$ the Teitelbaum type $L$-invariant ${\mathcal{L}}_{T}(\mathbf{f})$. We define ${\mathcal{L}}_{T}(f_{\infty})$ to be ${\mathcal{L}}_{T}(\mathbf{f})$. The goal of this section is to compare ${\mathcal{L}}_{FM}(f_{\infty})$ and ${\mathcal{L}}_{T}(f_{\infty})$.

Let $L$ be a (sufficiently large) finite extension of $F$ that splits $B$ and contains all Hecke eigenvalues acting on $f_{\infty}$. Let $\lambda$ be an arbitrary place of $L$.

Lemma 9.1**.**

[28, Lemma 3.1]** There is an isomorphism

[TABLE]

of representations of $G({\mathbb{A}}_{f})\times{\mathrm{Gal}}(\overline{F}/F)$ over $L_{\lambda}$. Here $f^{\prime}$ runs through the conjugacy classes over $L$, up to scalars, of eigen newforms of multiweight $\mathrm{k}$ that are new at primes dividing ${\mathfrak{p}}{\mathfrak{n}}^{-}$. The extension of $L$ generated by the Hecke eigenvalues acting on $f^{\prime}$ is denoted by $L(f^{\prime})$, and $\lambda^{\prime}$ runs through places of $L(f^{\prime})$ above $\lambda$.

By the strong multiplicity one theorem (cf. [26]) there exists a primitive idempotent $e_{\mathbf{f}}\in{\mathbb{T}}_{\bar{\Sigma}}$ such that $e_{\mathbf{f}}{\mathbb{T}}_{\bar{\Sigma}}=Le_{\mathbf{f}}$ and $e_{\mathbf{f}}\cdot S_{\mathrm{k}}^{\bar{B}}(\Sigma({\mathfrak{p}}{\mathfrak{n}}^{+},{\mathfrak{n}}^{-}))=L\cdot\mathbf{f}$. Lemma 9.1 tells us that $e_{\mathbf{f}}\cdot H^{1}_{\mathrm{et}}(M_{\overline{F}},{\mathcal{F}}(\mathrm{k})_{\lambda})^{\Sigma}$ is exactly $\rho_{f_{B},\lambda}$, the $\lambda$-adic representation of ${\mathrm{Gal}}(\overline{F}/F)$ attached to $f_{B}$. By Carayol’s construction of $\rho_{f_{\infty},\lambda}$ [6] $\rho_{f_{\infty},\lambda}$ coincides with $\rho_{f_{B},\lambda}$.

Now we take $\lambda$ to be a place above ${\mathfrak{p}}$, denoted by $\mathfrak{P}$.

Recall that in Section 7.2 and Section 8 we associate to $\bar{\Sigma}$ the groups $\widetilde{\Gamma}_{i,0},\widetilde{\Gamma}_{i},\Gamma_{i},\Gamma_{i,0}$ ($i=1,\cdots,h$). By (7.5) $X_{\Sigma}$ is isomorphic to $\coprod_{i}X_{\Gamma_{i,0}}$, where $X_{\Gamma_{i,0}}=\Gamma_{i,0}\backslash{\mathcal{H}}_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$.

Theorem 9.2**.**

Let $f_{B}$ be as above. Then $\rho_{f_{B},\mathfrak{P},{\mathfrak{p}}}$ is a semistable $($non-crystalline$)$ representation of ${\mathrm{Gal}}(\overline{F}_{\mathfrak{p}}/F_{\mathfrak{p}})$, and the filtered $(\varphi_{q},N)$-module $D_{{\mathrm{st}},F_{\mathfrak{p}}}(\rho_{f_{B},\mathfrak{P},{\mathfrak{p}}})$ is a monodromy $L_{\mathfrak{P}}$-module.

Proof.

To show that $\rho_{f_{B},\mathfrak{P},{\mathfrak{p}}}$ is semistable, we only need to prove that $H^{1}_{\mathrm{et}}((X_{\Sigma})_{{\bar{F}}_{\mathfrak{p}}},{\mathcal{F}}(\mathrm{k}))$ is semistable, since $\rho_{f_{B},\mathfrak{P},{\mathfrak{p}}}$ is a subrepresentation of $H^{1}_{\mathrm{et}}((X_{\Sigma})_{{\bar{F}}_{\mathfrak{p}}},{\mathcal{F}}(\mathrm{k}))$. But this follows from Proposition 2.1 and the fact that $X_{\Sigma}$ is semistable.

Next we prove that $D_{{\mathrm{st}},F_{\mathfrak{p}}}(\rho_{f_{B},\mathfrak{P},{\mathfrak{p}}})$ is a monodromy $L_{\mathfrak{P}}$-module. We only need to consider $D_{{\mathrm{st}},{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}}(\rho_{f_{B},\mathfrak{P},{\mathfrak{p}}})$ instead.

Twisting $f_{B}$ by a central character we may assume that $w=2$.

Being a Shimura variety, $N_{E}$ is a family of varieties. But in the following we will use $N_{E}$ to denote any one in this family that corresponds to a level subgroup whose ${\mathfrak{p}}$-factor is $\mathscr{O}_{E_{\mathfrak{p}}}^{\times}$. By the proof of Lemma 6.4, any geometric point of $(N_{E})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ is defined over $\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}$. In other words, $(N_{E})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}$ is the product of several copies of ${\mathrm{Spec}}(\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}})$.

Let $\mathrm{pr}_{1}$ be the projection $X_{\Sigma}\times(N_{E})_{\widehat{F_{\mathfrak{p}}^{\mathrm{ur}}}}\rightarrow X_{\Sigma}$. By Corollary 6.5 the filtered $\varphi_{q}$-isocrystal attached to $\mathrm{pr}_{1}^{*}\mathcal{F}(\mathrm{k})$ is $\mathrm{pr}_{1}^{*}\mathscr{F}(\mathrm{k})$. Note that

[TABLE]

The ${\mathrm{Gal}}(\overline{F}_{\mathfrak{p}}/F_{\mathfrak{p}}^{\mathrm{ur}})$-representation $H^{0}_{\mathrm{et}}((N_{E})_{\widehat{\overline{F}}_{\mathfrak{p}}},{\mathbb{Q}}_{p})$ is crystalline and the associated filtered $\varphi_{q}$-module is $H^{0}_{\mathrm{dR}}((N_{E})_{\widehat{F^{\mathrm{ur}}_{\mathfrak{p}}}},{\mathbb{Q}}_{p})$ with trivial filtration. Let $H^{0}$ denote this filtered $\varphi_{q}$-module for simplicity. As a consequence, we have an isomorphism of filtered $(\varphi_{q},N)$-modules

[TABLE]

Using the decomposition (6.3), for each embedding $\tau:F_{\mathfrak{p}}\hookrightarrow L_{\mathfrak{P}}$ we put

[TABLE]

In Section 8.2 we attached to $\mathbf{f}=(f_{1},\cdots,f_{h})$ an $h$-tuple $g^{\tau}=(g_{1}^{\tau},\cdots,g_{h}^{\tau})$. Let $M_{\tau}(\mathbf{f})$ denote the $L_{\mathfrak{P}}$-subspace of $\bigoplus_{i}H^{1}_{{\mathrm{dR}},\tau}(X_{\Gamma_{i,0}},\mathscr{F}(\mathrm{k}))$ generated by the element

[TABLE]

(Note that, when $w=2$, the twisted action $\star$ in Section 8 coincides with the original action.) Therefore, we have

[TABLE]

Consider the pairing $<\cdot,\cdot>$ on $V(\mathrm{k})$ defined by

[TABLE]

It is perfect and induces a perfect pairing on $H^{1}_{{\mathrm{dR}},\tau}(X_{\Sigma},\mathscr{F}(\mathrm{k}))$. With respect to this pairing ${\mathrm{Fil}}^{\frac{w-k_{\tau}}{2}+1}H^{1}_{{\mathrm{dR}},\tau}(X_{\Sigma},\mathscr{F}(\mathrm{k}))$ is orthogonal to ${\mathrm{Fil}}^{\frac{w+k_{\tau}}{2}-2}H^{1}_{{\mathrm{dR}},\tau}(X_{\Sigma},\mathscr{F}(\mathrm{k}))$. As $e_{\mathbf{f}}\cdot H^{1}_{{\mathrm{dR}},\tau}(X_{\Sigma},\mathscr{F}(\mathrm{k}))$ is of rank $2$ over $L_{\mathfrak{P}}$, we obtain

[TABLE]

Thus

[TABLE]

Let $\iota^{\tau}$ and $I^{\tau}$ be the operators attached to the sheaf $\mathscr{F}(\mathrm{k})$ over $X_{\Sigma}$ (see Section 7.2). Here, the superscript $\tau$ is used to emphasize the embedding $\tau:F_{\mathfrak{p}}\hookrightarrow L_{\mathfrak{P}}$. Proposition 2.2 tells us that the monodromy $N$ on $H^{1}_{{\mathrm{dR}},\tau}(X_{\Sigma},\mathscr{F}(\mathrm{k}))$ coincides with $\iota^{\tau}\circ I^{\tau}$. By Proposition 7.1 the kernel of $N$ is

[TABLE]

So, by (7.4) the restriction of $N$ to $M_{\tau}(\mathbf{f})$ is injective. Hence,

[TABLE]

as desired. ∎

Let $P^{\tau}$ be the operator attached to $\mathscr{F}(\mathrm{k})$ (see Section 7.2).

Lemma 9.3**.**

Let $\omega_{\mathbf{f}}^{\tau}$ be as in the proof of Theorem 9.2. Then

[TABLE]

Proof.

The first formula comes from the definitions.

The proof of the second formula is similar to that of [36, Theorem 3]. Let $\mu^{\tau}_{i}$ ($i=1,\cdots,h$) be the rigid analytic distributions on $\mathrm{P}^{1}(F_{\mathfrak{p}})$ coming from $c_{\mathbf{f}}$ (see Section 8). Recall that

[TABLE]

For each edge $e$ of ${\mathcal{T}}$ let $B(e)$ be the affinoid open disc in ${\mathrm{P}}^{1}({\mathbb{C}}_{p})$ that corresponds to $e$. Assume that $B(e)$ meets the limits set ${\mathrm{P}}^{1}(F_{\mathfrak{p}})$ in a compact open subset $U(e)$. Put

[TABLE]

Let $a(e)$ be a point in $U(e)$. Expanding $\frac{1}{z-t}$ at $a(e)$ we obtain that

[TABLE]

and thus $g^{\tau}_{i,e}(z)$ converges on the complement of $B(e)$. Note that $g^{\tau}_{i}-g^{\tau}_{i,e}$ is analytic on $B(e)$. So, we have

[TABLE]

where the fourth equality follows from the fact that ${\mathrm{Res}}_{e}$ commutes with $\int_{U(e)}\ \cdot\ \mu_{i}^{\tau}(t)$. ∎

Theorem 9.4**.**

Let $f_{\infty}$ be as above. Then ${\mathcal{L}}_{FM}(f_{\infty})={\mathcal{L}}_{T}(f_{\infty})$.

Proof.

Twisting $f_{\infty}$ by a central character we may assume that $w=2$.

Put $D_{\tau}=H^{0}\otimes_{{\mathbb{Q}}_{p}}e_{\mathbf{f}}H^{1}_{{\mathrm{dR}},\tau}(X_{\Sigma},\mathscr{F}(\mathrm{k}))$. Note that $N=\iota^{\tau}\circ I^{\tau}$. As the kernel of $N$ coincides with the image of $\iota^{\tau}\circ\delta^{-1}$ and $P^{\tau}$ splits $\iota^{\tau}\circ\delta^{-1}$, we have $D_{\tau}=\ker(N)\oplus\ker(P^{\tau})$. Write $\omega_{\mathbf{f}}^{\tau}=x+y$ according to this decomposition. Then

[TABLE]

By the proof of Theorem 9.2, $y$ is non-zero and so $N(y)\neq 0$.

By Lemma 9.3 and the definition of Teitelbaum type $L$-invariant, ${\mathcal{L}}_{T,\tau}(f_{\infty})$ is characterized by the property

[TABLE]

where $\epsilon$ is the map defined by (7.2) which coincides with $\kappa^{\mathrm{sch}}$. As $\delta^{-1}\circ\epsilon=-\mathrm{id}$ and $\iota^{\tau}\circ I^{\tau}=N$, we have

[TABLE]

By (9.2), (9.3) and (9.4) we get

[TABLE]

By the definition of Fontaine-Mazur $L$-invariant, ${\mathcal{L}}_{FM,\tau}(f_{\infty})$ is characterized by the property

[TABLE]

Combining (9.5) and (9.6) we obtain

[TABLE]

But $N(y)$ is in $\ker(N)$ and is non-zero, and by Theorem 9.2

[TABLE]

Therefore

[TABLE]

as wanted. ∎