Galois representations modulo $p$ and cohomology of Hilbert modular varieties
Mladen Dimitrov
The aim of this paper is to extend some arithmetic results on elliptic modular forms to the case of Hilbert modular forms. Among these results let's mention : $\relbar$ the control of the image of the Galois representation modulo $p$ \cite{serre2}\cite{ribet3}, $\relbar$ Hida's congruence criterion outside an explicit set of primes $p$ \cite{hida63}, $\relbar$ the freeness of the integral cohomology of the Hilbert modular variety over certain local components of the Hecke algebra and the Gorenstein property of these local algebras \cite{Ma}\cite{FaJo}. \medskip We study the arithmetic of the Hilbert modular forms by studying their modulo $p$ Galois representations and our main tool is the action of the inertia groups at the primes above $p$. In order to determine this action, we compute the Hodge-Tate (resp. the Fontaine-Laffaille) weights of the $p$-adic (resp. the modulo $p$) {\'e}tale cohomology of the Hilbert modular variety. The cohomological part of our paper is inspired by the work of Mokrane, Polo and Tilouine \cite{MoTi,PoTi} on the cohomology of the Siegel modular varieties and builds upon the geometric constructions of \cite{dimdg,dimtildg}.
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\begin{document}

\title[{\tiny   Mod $p$ Galois representations  and
cohomology of Hilbert  modular  varieties}]{Galois 
representations modulo $p$ and
cohomology of Hilbert modular varieties}

\author{Mladen Dimitrov}
\date{\today}


\begin{abstract} 
The aim of this paper is to extend some  arithmetic results 
on elliptic modular forms to the case of Hilbert modular  forms.
Among these results let's mention :

$\relbar$ the control of the image of the Galois representation modulo $p$  
\cite{serre2}\cite{ribet3},

$\relbar$ Hida's congruence criterion outside an explicit set of primes $p$  
\cite{hida63}, 

$\relbar$ the freeness of the integral cohomology of the Hilbert modular 
variety  over certain local components of the Hecke algebra and 
the Gorenstein property of these local algebras \cite{Ma}\cite{FaJo}.

\medskip 
We study the arithmetic of the  Hilbert modular  forms  by studying 
their modulo $p$ Galois representations and our main tool is 
the action of the inertia groups at the primes above $p$.
In order to determine  this action,  we compute the 
Hodge-Tate  (resp. the Fontaine-Laffaille) weights of the 
$p$-adic (resp. the modulo $p$) {\'e}tale  cohomology of the 
 Hilbert modular variety. The cohomological part of our paper is
inspired  by the work of Mokrane,  Polo and  Tilouine \cite{MoTi,PoTi} 
on the cohomology of the Siegel modular varieties and  builds upon 
the geometric constructions  of \cite{dimdg,dimtildg}.

\end{abstract}

\maketitle
   

\setcounter{tocdepth}{1}
\tableofcontents
\setcounter{tocdepth}{3}

\newpage
\section*{Introduction}

Let $F$ be a totally real number field of degree $d$,  ring of integers 
$\go$ and different $\gd$. Denote by   $\widetilde{F}$   
the Galois  closure of  $F$ in $\overline{\Q}$ and by  $J_F$ 
 the set of   all embeddings of $F$ into  $\overline{\Q}\subset \C$.
 
We fix an ideal $\gn\!\subset\! \go$ and we put  $\Delta= \Nm(\gn\gd)$. 

For a weight  $k=\sum_{\tau\in J_F}k_\tau\tau\in\Z[J_F]$ as in Def.\ref{weight} 
we put  $k_0=\max\{k_\tau |\tau \in J_F\}$. If $\psi$ is 
a Hecke character  of $F$  of conductor dividing $\gn$ and  type 
$2-k_0$ at infinity, we denote  by  $ S_k(\gn,\psi)$ the corresponding 
space of  Hilbert modular cuspforms (see  Def.\ref{space-hmf}).


Let  $f\!\in \!S_k(\gn,\psi)$ be  a  newform (that is a primitive
 normalized eigenform). For all  ideals $\ga\!\subset\! \go$,
 we denote by  $c(f,\ga)$ the eigenvalue of  the standard Hecke
operator $T_{\ga}$ on $f$. 

Let $p$ be a prime number and let 
$\iota_p:\overline{\Q}\hookrightarrow \overline{\Q}_p$ be an embedding.

Denote by   $E$  a sufficiently large  
$p$-adic field,   of ring of integers $\cO$, maximal ideal $\cP$ and
residue field $\kappa$. 


\subsection{Galois image results.} 
The  absolute Galois  group of a field $L$ is denoted  by $\G_L$.
  By results of  Taylor \cite{Ta} and   Blasius-Rogawski  \cite{BR} 
there exists a continuous representation 
$\rho=\rho_{f,p}:\G_F \rightarrow \GL_2(E)$
 which is absolutely irreducible, totally odd, 
unramified outside $\gn p$ and 
such that for each prime ideal $v$ of $\go$,
not dividing $p\gn$, we have:
\begin{equation*}
\tr(\rho(\Frob_v))=\iota_p(c(f,v)), \enspace 
\det(\rho(\Frob_v))=\iota_p(\psi(v))\Nm(v),
\end{equation*}

\noindent where  $\Frob_v$ denotes a geometric Frobenius at $v$.

 By taking a Galois stable $\cO$-lattice, we define 
$\overline{\rho}=\rho\mod{\cP}:\G_F
\rightarrow \GL_2(\kappa)$,  whose  semi-simplification is   
independent of the particular choice of a lattice.


The following proposition is a generalization  to the Hilbert modular case,
of results of  Serre \cite{serre2} and  Ribet \cite{ribet3}
on elliptic modular forms (see Prop.\ref{Irr}, Prop.\ref{LI} and 
Prop.\ref{LI-Ind}).

\begin{prop} 
{\rm(i)} For all but finitely many primes $p$,

\medskip
$\mathrm{\bf (Irr_{\overline{\rho}})}$ $\overline{\rho}=\overline{\rho}_{f,p}$ is absolutely
 irreducible.

\smallskip
{\rm(ii)} If $f$ is not a theta series, then for 
all but finitely many primes $p$,

\medskip
$\mathrm{\bf (LI_{\overline{\rho}})}$ there exists a power $q$ of $p$ 
such that $\SL_2(\F_q)\!\subset\! 
\im(\overline{\rho})
\!\subset\! \kappa^\times \GL_2(\F_q)$.

\smallskip 
{\rm(iii)} Assume that $f$ is not a twist by a character of any of 
 its $d$ internal conjugates   and  is not a theta series. Then  
for  all but finitely many primes $p$, 
there exist a power $q$ of $p$,  a partition $J_F=\coprod_{i\in I} J_F^i $ 
and  $\sigma_{i,\tau}\!\in \!\Gal(\F_q/\F_p)$, $\tau\!\in \!J_F^i $,
such that $(\tau\neq \tau'\Rightarrow \sigma_{i,\tau}\neq \sigma_{i,\tau'})$
 and

$\mathrm{{\bf (LI}_{Ind \overline{\rho}}{\bf )}}$ 
$\Ind_F^{\Q}\overline{\rho}:\G_{\widehat{F}''}\rightarrow 
\SL_2(\F_q)^{J_F}$ factors as  a surjection 
$\G_{\widehat{F}''}\twoheadrightarrow 
\SL_2(\F_q)^{I}$ followed by the  map $(M_i)_{i\in I}\mapsto 
(M_i^{\sigma_{i,\tau}})_{i\in I,\tau\in J_F^i }$,
where  $\widehat{F}''$ denotes the compositum of   $\widetilde{F}$ and 
the fixed field of $(\Ind_F^{\Q}\overline{\rho})^{-1}(\SL_2(\F_q)^{J_F})$.
\end{prop}

\subsection{Cohomological results.} \label{coh-results}
Let  $Y_{/\Z[\frac{1}{\Delta}]}$ be  the Hilbert modular  variety
of level $K_1(\gn)$ (see \S\ref{hmv}). Consider the $p$-adic  {\'e}tale  cohomology
$\rH^\bullet(Y_{\overline{\Q}},\bV_n(\overline{\Q}_p))$, where 
 $\bV_n(\overline{\Q}_p)$  denotes the local system of weight 
$n=\sum_{\tau\in J_F}(k_\tau-2)\tau\in\n[J_F]$ (see \S\ref{MWC}). 
By a result of Brylinski and Labesse  \cite{BL} the subspace 
$W_f:=\underset{\ga\subset \go}{\bigcap} \ker(T_{\ga}-c(f,\ga))$ of 
 $\rH^d(Y_{\overline{\Q}},\bV_n(\overline{\Q}_p))$  is 
isomorphic,  as $\G_{\widetilde{F}}$-module and after
semi-simplification, to the tensor induced representation 
$\otimes\Ind_F^{\Q}\rho$.  

\newpage 

Assume that 

{\bf (I) }$p$ does not divide $\Delta$. 

\medskip 
Then $Y$ has  smooth toroidal compactifications  
over $\Z_p$ (see \cite{dimdg}). For each   $J\!\subset\! J_F$,  we put 
$|p(J)|=\sum_{\tau\in J}(k_0\!-\!m_\tau\!-\!1)+
\sum_{\tau\in J_F\bs J}m_\tau$, where $m_\tau=(k_0-k_\tau)/2\in\n$.
 By applying a  method 
of Chai and Faltings \cite{FaCh} Chap.VI,  one can  prove  
(see \cite{dimtildg} Thm.7.8, Cor.7.9)

\begin{theo} Assume that $p$ does not divide $\Delta$. Then 

{\rm(i)} the Galois representation 
 $\rH^j(Y_{\overline{\Q}},\bV_n(\overline{\Q}_p))$ is 
crystalline at $p$ and its Hodge-Tate weights 
belong to the set   $\{|p(J)|, J\!\subset\! J_F\enspace |J|\leq j \}$, and

\smallskip
{\rm(ii)} the   Hodge-Tate weights of  $W_f$ are given by the 
multiset  $\{|p(J)|, J\!\subset\! J_F\}$.
\end{theo}

For our main  arithmetic applications we need to establish a modulo $p$ version
 of the above theorem. This is achieved under the  following
 additional assumption  : 
 
\medskip 
{\bf (II) } $p\!-\!1>\underset{\tau\in J_F}{\sum}(k_\tau\!-\!1)$.

\smallskip
\noindent The integer  $\sum_{\tau\in J_F}(k_\tau\!-\!1)$ is equal to the difference  
$|p(J_F)|-|p(\varnothing)|$ 
between the largest and the smallest  Hodge-Tate weights of the 
cohomology of the Hilbert modular  variety.
We use  {\bf (I) } and {\bf (II) } in order to apply 
Fontaine-Laffaille's Theory \cite{FoLa},
as well as  Faltings' Comparison Theorem modulo $p$ \cite{Fa-jami}.
By  adapting to the case of Hilbert  modular varieties  some techniques  developed  by 
Mokrane, Polo and Tilouine \cite{MoTi, PoTi} for  Siegel modular
varieties, such as the construction of an  integral Bernstein-Gelfand-Gelfand 
complex for distribution algebras, we compute the  Fontaine-Laffaille 
weights of $\rH^\bullet(Y_{\overline{\Q}},\bV_n(\kappa))$
(see Thm.\ref{bgg-modp}).




\subsection{Arithmetic results.}
Consider the $\cO$-module of interior   cohomology $\rH_{!}^d(Y,\bV_n(\cO))'$, 
defined as the image of $\rH_{c}^d(Y,\bV_n(\cO))$ in $\rH^d(Y,\bV_n(E))$. 
Let $\T=\cO[T_{\ga},\ga\!\subset\! \go]$ 
be the full Hecke  algebra acting on it, and let $\T'\subset \T$ be the subalgebra generated by the Hecke operators
outside a finite set of places containing those dividing $\gn p$. 
Denote by $\gm$ the maximal ideal of $\T$  corresponding to  $f$ and $\iota_p$
and  put $\gm'=\gm\cap \T'$.

\begin{theo}\label{cohom}
Assume that the conditions  $\mathrm{{\bf (I)}}$ and 
$\mathrm{{\bf (II)}}$ from \S\ref{coh-results}  hold.

 {\rm (i)} If $\mathrm{\bf (Irr_{\overline{\rho}})}$ holds, 
 $d(p\!-\!1)>5\sum_{\tau\in J_F}(k_\tau\!-\!1)$ and 

\medskip
 $\mathrm{\bf (MW)}$  the middle weight  
$\frac{|p(J_F)|+|p(\varnothing)|}{2}=\frac{d(k_0-1)}{2}$ 
does not belong to   $\{|p(J)|,J\!\subset\! J_F\}$,

\vspace{2mm}
then  the local component  $\rH^\bullet_{\partial}(Y,\bV_n(\cO))_{\gm'}$ 
of the boundary  cohomology   vanishes, and  the  Poincar{\'e}  pairing  
$\rH_{!}^d(Y,\bV_n(\cO))'_{\gm'} \times
\rH_{!}^d(Y,\bV_n(\cO))'_{\gm'}\rightarrow \cO$ is a perfect duality.


{\rm (ii)} If $\mathrm{\bf (LI_{\Ind\overline{\rho}})}$ holds, then 
$\rH^\bullet(Y,\bV_n(\cO))_{\gm'}=\rH^d(Y,\bV_n(\cO))_{\gm'}$ is a free  
$\cO$-module  of  finite rank  and its Pontryagin dual is isomorphic to 
 $\rH^d(Y,\bV_n(E/\!\cO))_{\gm'}$.
\end{theo}

 The proof involves a ``local-global'' Galois argument. For the (i), 
we use lemma \ref{key2}(ii) and a  theorem of Pink \cite{Pi} on the 
{\'e}tale cohomology   of a local system  restricted to the boundary of
$Y$ (see Thm.\ref{boundary}). For the (ii),  we use lemma \ref{key}
and the  computation of the Fontaine-Laffaille weights of the
cohomology of Thm.\ref{bgg-modp} (see Thm.\ref{coh-loc}). 


\medskip
Let $\Lambda^*(\Ad^0(f),s)$  be the imprimitive adjoint $L$-function  of $f$, 
completed  by its Euler factors at infinity and let 
$W(f)$ be the  complex constant from the functional equation 
of the  standard $L$-function  of $f$ (see {\S}\ref{L-ad}).
We denote  by  $\Omega_f^{\pm}\in \C^\times\!\!/\cO^\times$ any 
two complementary  periods
defined  by the  Eichler-Shimura-Harder isomorphism  (see {\S}\ref{periodes}). 


\medskip
{\bf Theorem A} (Thm.\ref{theo-A})
{\it  Let  $f$ and  $p$ be such that {\bf (I)}, $\mathrm{\bf (Irr_{\overline{\rho}})}$ and {\bf (MW)} hold,
and $p-1>\max(1,\frac{5}{d})\sum_{\tau\in J_F}(k_\tau-1)$.  Assume
that  $\iota_p(\frac{W(f)\Lambda^*(\Ad^0(f),1)}{\Omega_f^+\Omega_f^-})\in\cP$.
Then there  exists another normalized eigenform  $g\in  S_k(\gn,\psi)$
such that $f\equiv g\!\pmod{\cP}$, in the sense that 
$c(f,\ga)\equiv c(g,\ga)\!\pmod{\cP}$ for  each  ideal $\ga\!\subset\! \go$.}

\medskip
The proof follows closely the original one given  by  Hida  \cite{hida63}
in the elliptic modular case, and uses Thm.\ref{cohom}(i)
as well as a formula of Shimura  relating 
$\Lambda^*(\Ad^0(f),1)$ to the Petersson inner product of $f$ (see 
(\ref{adjoint})). Let us note that Ghate  \cite{ghate1} has obtained a
very similar  result when the weight $k$ is parallel. A converse for 
Thm A is provided by the (ii) of the following 


\medskip

{\bf Theorem B} (Thm.\ref{theo-B})
{\it Let  $f$ and  $p$ be such that   $\mathrm{{\bf (I)}}$, 
$\mathrm{{\bf (II)}}$ and $\mathrm{\bf (LI_{\Ind\overline{\rho}})}$  hold. Then

{\rm(i)} $\rH^\bullet(Y,\bV_n(\kappa))[\gm]=
\rH^d(Y,\bV_n(\kappa))[\gm]$ is a  $\kappa$-vector 
space  of dimension $2^d$. 

{\rm(ii)} $\rH^\bullet(Y,\bV_n(\cO))_{\gm}=
\rH^d(Y,\bV_n(\cO))_{\gm}$ 
is free of rank  $2^d$ over  $\T_{\gm}$.  

{\rm(iii)} $\T_{\gm}$ is Gorenstein.}


\medskip

By \cite{Ma} it is enough to prove (i), which is a consequence of 
  Thm.\ref{cohom}(ii) and the  $q$-expansion principle \S\ref{qdev}.

\medskip
The last theorem is due,  under milder assumptions, 
to Mazur \cite{Ma} for $F=\Q$ and $k=2$, and to 
Faltings and Jordan \cite{FaJo} for $F=\Q$. The Gorenstein
property is proved  by Diamond \cite{diamond2} when  $F$ is  quadratic and 
$k=(2,2)$, under the assumptions {\bf (I)}, {\bf (II)} and $\mathrm{\bf (Irr_{\overline{\rho}})}$.
We expect that Diamond's approach via intersection cohomology 
could be generalized in order to  prove the Gorenstein property 
of $\T_{\gm}$ under the assumptions 
{\bf (I)}, {\bf (II)} and $\mathrm{\bf (LI_{\overline{\rho}})}$ 
(see lemma \ref{key2}(i) and  remark \ref{key-diamond}). 


\medskip
When $f$ is ordinary at $p$ (see Def.\ref{ord-autom}) we can replace 
 the assumptions {\bf (I)} and  {\bf (II)} of theorems A and B by the weaker 
assumptions that $p$ does not divide $\Nm(\gd)$ and that $k\pmod{p-1}$
satisfies {\bf (II)} (see Cor.\ref{thm-ord}). The proof uses  Hida's 
 families of $p$-adic ordinary   Hilbert modular forms. We prove an exact 
control theorem for the ordinary part of the cohomology of the 
 Hilbert modular variety, and give a new proof of Hida's  exact 
control theorem for the ordinary Hecke algebra (see Prop.\ref{control}). 

\medskip
 Theorems A and B relate  $\cO/\iota_p
(\frac{W(f)\Lambda^*(\Ad^0(f),1)}{\Omega_f^+\Omega_f^-})$ to the 
congruence module associated to the $\cO$-algebra homomorphism 
$\T\rightarrow \cO$, $T_{\ga}\mapsto \iota_p(c(f,\ga))$. In a subsequent paper
 \cite{dim-ihara} we relate these two
with the cardinality of the Selmer group of $\Ad^0(\rho)\otimes E/\cO$. 
An interesting question is whether $\Omega_f^{\pm}$ are the periods 
involved in the Bloch-Kato conjecture for $\Ad^0(f)$ (see
the work of Diamond, Flach and Guo \cite{dfg} for the elliptic modular case). 


\subsection{Explicit results.} 
By a classical theorem of Dickson, the image in  $\PGL_2(\kappa)$
 of an irreducible subgroup of $\GL_2(\kappa)$  not 
satisfying $\mathrm{\bf (LI_{\overline{\rho}})}$, is either a  
 dihedral, a  tetrahedral, an octahedral or  an icosahedral group.  
In the next proposition we consider the later exceptional cases
for the image of $\overline{\rho}$ in $\PGL_2(\kappa)$.

\medskip
Denote by $\go^{\times}_+$ (resp. by $\go^{\times}_{\gn,1}$) be the group of 
 totally positive (resp. congruent to $1$ modulo $\gn$) units of $\go$. 

\newpage 

\begin{prop}
Assume   $p$ does not divide $\Delta$ and   $p>k_0$. 

{\rm(i)} Assume that $k$  is non-parallel.  If  for all  $J\!\subset\! J_F$, 
there exists an unit  $\epsilon\in\go^{\times}_+\cap\go^{\times}_{\gn,1}$,  such that  $p$ 
  does not divide  $\N_{\widetilde{F}\!/\!\Q}
\left(\underset{\tau\in J}{\prod}\tau(\epsilon)^{k_0-m_\tau-1}-
\underset{\tau\in J_F\!\bs\!J}{\prod}\tau(\epsilon)^{m_\tau}\right)
\neq 0$,  then  $\mathrm{\bf (Irr_{\overline{\rho}})}$ holds.

\smallskip
{\rm(ii)} If $d(p\!-\!1) >  5\underset{\tau\in J_F}{\sum}(k_\tau\!-\!1)$,
then the image of $\overline{\rho}$ in $\PGL_2(\kappa)$ is not 
a tetrahedral, an  octahedral nor  an  icosahedral group.

\smallskip
{\rm(iii)}  Assume  that for all $\tau\in J_F$, $p\neq 2k_\tau-1$ and the   following condition :

\medskip
\noindent {\bf (non-CM)}  for  each  quadratic extension 
 $K$  of  $F$ of discriminant dividing  $\gn$ and splitting  
all the primes  $\gp$ of $F$ above  $p$, one of the following
holds :  
\begin{enumerate}
\item 
$  K \text{ is CM and   there does not exist a 
Hecke character }   \varphi \text{ of }   K \text{ of   conductor dividing}\\
\gn\Delta_{K/F}^{-1} \text{ and   infinity type }
(m_\tau,k_0\!-\!1\!-\!m_\tau)_{\tau\in J_F}\text{, such that }
\rho\equiv \Ind_{K}^{F} \varphi \pmod{\cP}, $

\item 
$  K\text{  is not  CM and for all extensions }
 \widetilde{\tau}  \text{   of } \tau\in J_F \text{ to } K\text{,
 there exists a unit }
 \epsilon\in\gO^{\times},\\  \epsilon-1\in\gn  \text{ such that } 
 p \text{  does not divide }
 \displaystyle \N_{K/\Q}\left(\prod_{\tau\in J_F}
\widetilde{\tau}(\epsilon)^{m_{\tau}}\widetilde{\tau}(c(\epsilon))^{k_0-m_{\tau}-1}\!-\!1\right). $
\end{enumerate}

Then   the image   of  $\overline{\rho}$ in $\PGL_2(\kappa)$ 
 is not a  dihedral group. 

\smallskip
{\rm(iv)} Assume $\mathrm{\bf (LI_{\overline{\rho}})}$ and 
that $k$ is not induced from a weight for a strict subfield 
$F'$ of $F$. Assume moreover that for all $\tau,\tau'\in J_F$,
 $p\neq k_\tau+k_{\tau'}-1$. Then  $\mathrm{\bf (LI_{\Ind\overline{\rho}})}$ holds. 
\end{prop}

By the last proposition, we obtain the following  corollary to theorems A and B.


\begin{cor} Let   $\epsilon$ be any element of 
$\go^{\times}_+\cap\go^{\times}_{\gn,1}$.

{\rm(i)} Assume $d\!=\!2$ and $k\!=\!(k_0,k_0\!-\!2m_1)$,  with  $m_1\neq 0$. 
If $p\nmid \Delta 
\Nm((\epsilon^{m_1}\!-\!1)(\epsilon^{k_0\!-\!m_1\!-\!1}\!-\!1))$ and 
$p\!-\!1>4(k_0\!-\!m_1\!-\!1)$, then theorem A holds.  If additionally  
we have the  {\bf (non-CM)}  condition then   theorem B  also holds.

{\rm(ii)} Assume $d\!=\!3$, $\id\!\neq\!\tau\in J_F$ and 
$k\!=\!(k_0,k_0-2m_1,k_0-2m_2)$, 
with $0\leq m_1\leq m_2\neq 0$. 
If $p\nmid \Delta \N_{\widetilde{F}\!/\!\Q}(
(\tau(\epsilon)^{m_1}-\epsilon^{-m_2})
(\tau(\epsilon)^{m_1}-\epsilon^{m_2+1-k_0})
(\tau(\epsilon)^{m_1+1-k_0}-\epsilon^{m_2})
(\tau(\epsilon)^{k_0-m_1-1}-\epsilon^{m_2+1-k_0})
)$ and  $p-1>\frac{5}{3}(3k_0-2m_1-2m_2-3)$, 
then theorem A holds.  If additionally  we
have $p\neq 2k_0-1$ and the  {\bf (non-CM)}  condition then   theorem B  also holds.
\end{cor}

\subsection{Acknowledgements.}
 I would like to thank J. Bella{\"\i}che, D. Blasius, G. Chenevier,
 L. Dieulefait, E. Ghate, M. Kisin, 
V. Lafforgue, A. Mokrane, E. Urban, J. Wildeshaus and J.-P. Wintenberger for helpful 
conversations.
This article was completed during my visit at UCLA on an invitation by  H. Hida. 
I would like to thank him heartily for his  hospitality and for many inspiring 
discussions. 
I am grateful to G. Kings and R. Taylor for their interesting comments 
on an earlier version of this paper that have much improved it.
Finally, I would like to thank J. Tilouine who suggested me to study this problem 
and supported me during the  preparation of this article.

\newpage

\section{Hilbert modular forms and varieties.}\label{hmfv}

We define the algebraic groups  $D_{\!/\!\Q}=\Res^{F}_{\Q} \Gm$,
 $G_{\!/\!\Q}=\Res^{F}_{\Q} \GL_2$ and   $G^*_{/\Q}=G\times_{D}\Gm$,  
where  the fiber product is relative to  the reduced norm map $\nu: G \rightarrow D$. The standard 
Borel subgroup  of $G$, its unipotent  radical  and  its standard maximal 
torus    are denoted  by $B$, $U$ and  $T$, respectively. 
We identify $D \times D$ with $T$, by $(u,\epsilon)\mapsto 
\begin{pmatrix} u\epsilon & 0 \\ 0 & u^{-1} \end{pmatrix}$.

\subsection{Analytic Hilbert modular varieties.}

Let $D(\R)_+$ (resp. $G(\R)_+$) be the identity component  of  
$D(\R)=(F\otimes \R)^\times$  (resp. of $G(\R)$). The group $G(\R)_+$ 
acts  by linear fractional transformations  on     the space  
$\mathfrak{H}_F=\{z\in F\otimes\C \enspace |\enspace \im(z)\in 
D(\R)_+\}$. We have $\mathfrak{H}_F\cong \mathfrak{H}^{J_F}$, where 
$\mathfrak{H}=\{z\in \C\enspace |\enspace \im(z)>0\} $ is  the 
 Poincar{\'e}'s upper half-plane (the isomorphism being  given by 
$\xi\otimes z \mapsto (\tau(\xi)z)_{\tau\in J_F}$, for $\xi\in F$, $z\in \C$).
We consider the unique group action  of $G(\R)$ on  the space  
$\mathfrak{H}_F$ extending  the action of $G(\R)_+$  and such that, 
on each copy of $\mathfrak{H}$ 
the element  $\begin{pmatrix} -1 & 0\\ 0 & 1\end{pmatrix}$ acts 
 by $z\mapsto -\overline{z}$.
 We put  $\underline{i}=(\sqrt{-1},...,\sqrt{-1})\in \mathfrak{H}_F$,  
$K_{\infty}^+=\Stab_{G(\R)_+}(\underline{i})=\SO_2(F\otimes\R)D(\R)$ and 
$K_{\infty}=\Stab_{G(\R)}(\underline{i})=\mathrm{O}_2(F\otimes \R)D(\R)$.


We denote by $\widehat{\Z}=\prod_l \Z_l$ the profinite completion  of $\Z$ 
and we put  $\widehat{\go}=\widehat{\Z}\otimes \go=
\prod_v \go_v$, where $v$ runs over all the finite places  of $F$.
Let  $\A$ (resp. $\A_f$) be  the ring of the  ad{\`e}les 
(resp. of the finite ad{\`e}les) of  $\Q$. We 
consider the following open compact subgroup of $G(\A_f)$ :
$$K_1(\gn)=\left\{\begin{pmatrix}a& b \\ c & d\end{pmatrix}\in
G(\widehat{\Z}) | d-1\in \gn, c\in \gn \right\}.$$

 The ad{\'e}lic Hilbert modular variety of level 
$K_1(\gn)$ is defined as 
$$Y^{\an}=Y_1(\gn)^{\an}=
G(\Q)\bs G(\A)/K_1(\gn)K_{\infty}^+.$$

By the Strong Approximation Theorem, the 
 connected components of $Y^{\an}$ are indexed by the narrow ideal class group 
$\Cl_F^{+}\!=\!D(\A)/D(\Q)D(\widehat{\Z})D(\R)_+$
of $F$. For each fractional ideal $\gc$ of $F$, we put $\gc^*=\gc^{-1}\gd^{-1}$.
We define the following congruence subgroup of $G(\Q)$ :
$$\Gamma_1(\gc,\gn)=\Big{\{}
\begin{pmatrix}a &b \\c &d\end{pmatrix} \in G(\Q)\cap
\begin{pmatrix} \go & \gc^* \\ 
\mathfrak{cdn} & \go\end{pmatrix}
\enspace \Big{|} \enspace ad-bc\in \go_+^\times, \enspace  d\equiv 1 \pmod{\gn} \Big{\}}.$$

Put $M^{\an}=M_1(\gc,\gn)^{\an}=
\Gamma_1(\gc,\gn)\bs  \mathfrak{H}_F$.
Then we have $Y_1(\gn)^{\an}\simeq\coprod_{i=1}^{h^+} 
M_1(\gc_i,\gn)^{\an}$,  where the ideals $\gc_i$, $1\leq i\leq h^+$, 
form  a set of representatives of  $\Cl_F^{+}$.

\medskip
Put $\mathfrak{H}_F^*= \mathfrak{H}_F\coprod \p^1(F)$. The minimal 
compactification $M^{*\an}$  of  $M^{\an}$  is defined as $M^{*\an}=\Gamma\bs \mathfrak{H}_F^*$. It is an analytic  normal projective  space whose
boundary $M^{*\an}\bs M^{\an}$ is a finite union of closed points,
called the {\it cusps} of $M^{\an}$.  

\medskip
The same way, by replacing $G$ by $G^*$, we define   
$\Gamma_1^1(\gc,\gn)$, $M^{1,\an}=M_1^1(\gc,\gn)^{\an}$ and $M^{1*,\an}$.

%$K_1^1(\gn)$, $Y^{1,\an}=Y_1^1(\gn)^{\an}$,  
% Note that $\A^\times=\Q^\times
%\widehat{\Z}{}^\times\R_+^\times$, so $Y^{1,\an}$ is connected. 

 

\subsection{Analytic Hilbert modular forms.}
\label{fmh-hecke}

For the definition of the $\C$-vector space of  Hilbert modular forms we 
follow   \cite{hida128}.


\begin{defin}\label{weight} An element 
$k=\sum_{\tau\in J_F}k_\tau\tau\in\Z[J_F]$ is called a  weight. We
always assume that the  $k_\tau$ have the same parity  and 
are all $\geq 2$.  We put 
$k_0=\max\{k_\tau |\tau \in J_F\}$, $n_0=k_0-2$, $t=\sum_{\tau\in J_F}\tau$,
$n=\sum_{\tau\in J_F}n_\tau\tau=k-2t$ and 
$m=\sum_{\tau\in J_F}m_\tau\tau=(k_0t-k)/2$.
\end{defin}

 For $z\in \mathfrak{H}_F$, $\gamma=\begin{pmatrix}a&b\\c&d\end{pmatrix}$
we put $j_J(\gamma,z)=c\cdot z^J+d \in D(\C)$, 
where $z^J_\tau=\begin{cases} z_\tau\textrm{, } \tau\in J,\\  
\overline{z}_\tau\textrm{, } \tau\in J_F\bs J.\end{cases}$ 

% For  every subset  $J$ of $J_F$ and   $\gamma\in G(\R)$, we put 
%$$ J^\gamma=\{\tau \in J \,\,|\,\, 
% \nu(\gamma_\tau)>0\}\cup\{\tau\in J_F \bs J \,\,|\,\, \nu(\gamma_\tau)<0)\}.$$


\begin{defin}  \label{hmaf} The space $G_{k,J}(K_1(\gn))$ of  ad{\'e}lic Hilbert modular 
 forms of weight $k$, level $K_1(\gn)$ and type $J\!\subset\! J_F$ 
at infinity, is  the $\C$-vector space of the   functions 
$g:G(\A) \rightarrow \C$  satisfying the    followings three
conditions : 

(i) $g(axy)=g(x)$ for all $a\in G(\Q)$, $y\in K_1(\gn)$ and 
$x\in G(\A)$.

(ii) $g(x\gamma)= \nu(\gamma)^{k+m-t}j_J(\gamma,\underline{i})^{-k}
g(x)$, for all  $\gamma\in K_{\infty}^+$ and $x\in G(\A)$.

\medskip
 For all $x\in G(\A_f)$ define  $g_x:\mathfrak{H}_F\rightarrow \C$,   by 
$z\mapsto \nu(\gamma)^{t -k-m} j_J(\gamma,\underline{i})^{k}g(x\gamma),$
\noindent where $\gamma\in G(\R)_+$ is such that  $z=\gamma\cdot \underline{i}$. 
By (ii) $g_x$ does not depend on the  particular choice  of  $\gamma$.

\medskip
(iii)  $g_x$ is holomorphic at  $z_\tau$, for $\tau\in J$, and anti-holomorphic 
at  $z_\tau$, for $\tau\in J_F\bs J$ (when $F=\Q$ an extra condition of 
holomorphy at cusps is needed).

The space $S_{k,J}(K_1(\gn))$ of  ad{\'e}lic Hilbert modular  cuspforms
is the subspace of $G_{k,J}(K_1(\gn))$ consisting of functions satisfying 
the following additional condition : 

(iv) $\int_{U(\Q)\bs U(\A)}g(ux)du=0$, for all $x\in G(\A)$ and
 all additive Haar measures    $du$.
\end{defin}

The conditions  (i) and (ii) of the above  definition imply that for all 
$g\in G_{k,J}(K_1(\gn))$ there exists  a  Hecke character
  $\psi$  of  $F$ of conductor  dividing $\gn$ and  of type 
$-n_0t $ at infinity, such that   for all $x\in G(\A)$ and for all 
$z\in D(\Q)D(\widehat{\Z})D(\R)$,  we have $g(zx)=\psi(z)^{-1}g(x)$. 

\begin{defin}  \label{space-hmf}
Let $\psi$ be a Hecke character  of  $F$ of conductor 
dividing $\gn$ and  of type  $-n_0t $ at infinity.  The space  
$S_{k,J}(\gn,\psi)$ (resp. $G_{k,J}(\gn,\psi)$) is defined as the subspace 
of  $S_{k,J}(K_1(\gn))$ (resp. $G_{k,J}(K_1(\gn))$) of elements  $g$ satisfying
$g(zx)=\psi(z)^{-1}g(x)$, for all $x\in G(\A)$ and for all $z\in D(\A)$.
When $J=J_F$ this  space is denoted by $S_k(\gn,\psi)$ (resp. by $G_k(\gn,\psi)$).
\end{defin}


As the characters of the ideal class  group
$\Cl_F= D(\A)/D(\Q)D(\widehat{\Z})D(\R)$  of $F$ form a basis
of the   complex valued functions on this set, we have :
\begin{equation} \label{decomp-fmh}
G_{k,J}(K_1(\gn))=\bigoplus_{\psi} G_{k,J}(\gn,\psi),\enspace\enspace
S_{k,J}(K_1(\gn))=\bigoplus_{\psi} S_{k,J}(\gn,\psi)
\end{equation}
where $\psi$ runs over  the   Hecke characters of  $F$, 
of conductor dividing $\gn$ and infinity type $-n_0t$.
Let $\Gamma$ be a congruence subgroup of $G(\Q)$. We recall the
classical definition :


\begin{defin} \label{def-hmf} The space $G_{k,J}(\Gamma;\C)$ of Hilbert modular 
  forms of weight $k$, level $\Gamma$ and type $J\!\subset\! J_F$ 
at infinity, is  the $\C$-vector space of the   functions 
$g:\mathfrak{H}_F \rightarrow \C$  which are holomorphic at  
$z_\tau$, for $\tau\in J$, and anti-holomorphic at  $z_\tau$, for $\tau\in
 J_F\bs J$, and such that for every $\gamma\in \Gamma$
we have $g(\gamma(z))=\nu(\gamma)^{t-k-m}j_J(\gamma,z)^{-k}g(z)$. 

The space $S_{k,J}(\Gamma;\C)$ of  Hilbert modular 
 cuspforms is the subspace of $G_{k,J}(\Gamma;\C)$, consisting of 
functions vanishing at all cusps.
\end{defin}

Put $x_i=\begin{pmatrix}\eta_i & 0 \\0 &1\end{pmatrix}$, where  $\eta_i$ is 
the id{\`e}le associated to the ideal $\gc_i$, $1\leq i\leq h^+$.  The map 
$g\mapsto (g_{x_i})_{1\leq i\leq h^+}$ (see Def.\ref{hmaf}) induces isomorphisms :
\begin{equation}\label{ad-cl}
G_{k,J}(K_1(\gn))\simeq\bigoplus_{1\leq i\leq h^+} G_{k,J}(\Gamma_1(\gc_i,\gn);\C), 
\enspace\enspace 
S_{k,J}(K_1(\gn))\simeq\bigoplus_{1\leq i\leq h^+} S_{k,J}(\Gamma_1(\gc_i,\gn);\C).
\end{equation}

 Let  $d\mu(z)=\underset{\tau\in J_F}{\prod}y_\tau^{-2}dx_\tau dy_\tau$
be the standard  Haar  measure on  $\mathfrak{H}_F$.

\begin{defin}\label{peterson}
The Petersson inner product of two cuspforms $g,h\in S_{k,J}(K_1(\gn))$
is given by the formula 
$$(g,h)_{\gn}=\sum_{i=1}^{h^+} \int_{\Gamma_1(\gc_i,\gn)\bs\mathfrak{H}_F}
\overline{g_i(z)} h_i(z) y^k d\mu(z),$$
where $(g_i)_{1\leq i\leq h^+}$ (resp. $(h_i)_{1\leq i\leq h^+}$)
is the image of $g$ (resp. $h$) under the isomorphism (\ref{ad-cl}).
\end{defin}



\subsection{Hilbert-Blumenthal abelian varieties.}\label{hbav}
A sheaf over a scheme $S$ which is locally free of rank one over 
 $\go\otimes \cO_S$, is called an {\it invertible $\go$-bundle} on $S$.

\begin{defin} A Hilbert-Blumenthal abelian variety (HBAV) over a 
$\Z[\frac{1}{\Nm(\gd)}]$-scheme $S$
is an abelian scheme $\pi:A\rightarrow S$ of relative dimension $d$ 
together with an injection  $\iota:\go\hookrightarrow \End(A/S)$, 
such that  $\underline{\omega}_{A/S}:=\pi_*
\Omega^1_{A/S}$ is  an invertible $\go$-bundle on $S$.
\end{defin}

Let $\gc$ be a fractional ideal of $F$ and  $\gc_+$ be the cone of 
totally positive elements in  $\gc$.   Given a HBAV $A/S$, the 
functor assigning  to a $S$-scheme $X$ the set  $A(X)\otimes_{\go}\gc$
is representable by  another HBAV, denoted by $A\otimes_{\go}\gc$.
Then $\iota$ yields $\gc\hookrightarrow  \Hom_{\go}(A,A\otimes_{\go}\gc)$. 
The dual of a HBAV $A$ is denoted by $A^t$.


\begin{defin}
{\rm (i)} A $\gc$-polarization on a HBAV  $A/S$ is an $\go$-linear 
isomorphism $\lambda:A\otimes_{\go}\gc \overset{\sim}{\longrightarrow} A^t$, 
such that under the  induced isomorphism $\Hom_{\go}(A,A\otimes_{\go}\gc)
\cong \Hom_{\go}(A,A^t)$ elements of $\gc$ (resp. $\gc_+$)
correspond exactly to symmetric elements (resp. polarizations).


{\rm (ii)} A $\gc$-polarization class $\overline{\lambda}$
is an orbit of $\gc$-polarizations under  $\go_+^\times$. 
\end{defin}

 Let  $(\Gm\otimes \mathfrak{d}^{-1})[\gn]$
be the reduced  subscheme of $\Gm\otimes \mathfrak{d}^{-1}$,  defined
as the intersection 
of the kernels of multiplications  by elements of $\gn$. Its  Cartier dual 
is isomorphic to the   finite    group scheme $\go\!/\!\gn$.

\begin{defin}
 A $\mu_{\gn}$-level structure on a HBAV $A/S$
is an $\go$-linear closed immersion $\alpha:(\Gm\otimes \mathfrak{d}^{-1})[\gn]
\hookrightarrow A$ of  group schemes over $S$.
\end{defin}


\subsection{Hilbert modular varieties.}\label{hmv}
We consider the contravariant functor $\underline{\cM}^1$
(resp. $\underline{\cM}$) from the category of 
$\Z[\frac{1}{\Delta}]$-schemes to the category of sets, 
assigning  to a scheme $S$  the set of isomorphism classes
of triples $(A,\lambda,\alpha)$ (resp. 
 $(A,\overline{\lambda},\alpha)$) where 
$A$ is a HBAV over $S$, endowed with a $\gc$-polarization  $\lambda$
(resp. a  $\gc$-polarization  class $\overline{\lambda}$)  and 
a  $\mu_{\gn}$-level structure $\alpha$. 
Assume the following condition :

\medskip
{\bf (NT)  } $\gn$  does not divide $2$, nor $3$, nor $\Nm(\mathfrak{d})$.

\smallskip
Then $\Gamma_1(\gc,\gn)$ is torsion free, and 
 the functor $\underline{\cM}^1$ is representable 
by a quasi-projective, smooth,  geometrically connected 
$\Z[\frac{1}{\Delta}]$-scheme $M^1=M^1_1(\gc,\gn)$, 
endowed with  an universal HBAV $\pi:\cA\rightarrow M^1$. 
By definition,  the sheaf $\underline{\omega}_{\cA/M^1}=\pi_*
\Omega^1_{\cA/M^1}$ is an invertible $\go$-bundle on $M^1$. 
Consider the first de Rham cohomology sheaf 
$\cH^1_{\dR}(\cA/M^1)=R^1\pi_*\Omega^\bullet_{\cA/M^1}$ on $M^1$.  
The Hodge filtration yields an exact sequence :
$$0 \rightarrow \underline{\omega}_{\cA/M^1} \rightarrow 
\cH^1_{\dR}(\cA/M^1)  \rightarrow 
\underline{\omega}_{\cA/M^1}^{\vee}\otimes\mathfrak{cd}^{-1}  \rightarrow 0.$$
Therefore $\cH^1_{\dR}(\cA/M^1)$ 
is locally free of rank two over $\go\otimes \cO_{M^1}$. 


The  functor $\underline{\cM}$ admits a coarse moduli space 
$M=M_1(\gc,\gn)$, which is a  quasi-projective, smooth, geometrically 
connected  $\Z[\frac{1}{\Delta}]$-scheme. The finite group 
$\go_+^\times/\go^{\times 2}_{\gn,1}$ acts properly and discontinuously on
  $M^1$ by  $[\epsilon]:(A,\iota,\lambda,\alpha)/S\mapsto 
(A,\iota,\epsilon\lambda,\alpha)/S$ and the quotient is given by $M$. 
This group acts also on $\underline{\omega}_{\cA/M^1}$ 
 and  on $\cH^1_{\dR}(\cA/M^1)$  by acting on the de Rham 
complex  $\Omega^\bullet_{\cA/M^1}$ ($[\epsilon]$ acts on  
$\underline{\omega}_{\cA/M^1}$ by  $\epsilon^{-1/2}[\epsilon]^*$).

These actions are defined over the integer ring of the number field
$F(\epsilon^{1/2}, \epsilon\in \go_+^\times)$. 

Let   $\go'$ be the integer ring of 
$\widetilde{F}(\epsilon^{1/2}, \epsilon\in \go_+^\times)$. For 
every $\Z[\frac{1}{\Delta}]$-scheme $X$ we put 
$$\textstyle X'=X\times \Spec(\go'[\frac{1}{\Delta}]).$$  

The sheaf of $\go_+^\times/\go^{\times 2}_{\gn,1}$- invariants  of 
 $\underline{\omega}_{\cA/M^1}$ (resp. of $\cH^1_{\dR}(\cA/M^1)$) 
 is locally free of rank  one (resp. two)  over $\go\otimes \cO_{M'}$, and 
is denoted  by $\underline{\omega}$ (resp. $\cH^1_{\dR}$). 

We put $Y=Y_1(\gn)=\coprod_{i=1}^{h^+} M_1(\gc_i,\gn)$ and
$Y^1=Y_1^1(\gn)=\coprod_{i=1}^{h^+} M_1^1(\gc_i,\gn)$,
  where the ideals $\gc_i$, $1\leq i\leq h^+$, form  
 a set of representatives of  $\Cl_F^{+}$.
\subsection{Geometric Hilbert modular forms.}

Under the action of $\go$, the invertible $\go$-bundle  $\underline{\omega}$
on $M'$ decomposes as a direct sum of line bundles 
$\underline{\omega}_{\tau}$, $\tau\in J_F$.  For every 
$k=\sum_\tau k_\tau\tau \in \Z[J_F]$ we define the line bundle 
$\underline{\omega}^k=\underset{\tau}{\otimes}
\underline{\omega}_{\tau}^{\otimes k_\tau}$ on $M'$. 

One should be careful to observe, that the global section of 
$\underline{\omega}^k$ on $M^{\an}$ are given by the cocycle
$\gamma\mapsto \nu(\gamma)^{-k/2}j(\gamma,z)^k$, meanwhile we are interested
in finding a geometric interpretation of the cocycle 
$\gamma\mapsto \nu(\gamma)^{t-k-m}j(\gamma,z)^k$, used in Def.\ref{def-hmf}.

\medskip
The universal polarization class $\overline{\lambda}$ 
endows $\cH^1_{\dR}$ with  a perfect symplectic $\go$-linear pairing. 
Consider the invertible $\go$-bundle  $\underline{\nu}:=
\wedge^2_{\go\otimes \cO_{M'}}\cH^1_{\dR}$ on $M'$. 
Note that $(k+m-t)-\frac{k}{2}=\frac{n_0}{2}t$. 

\begin{defin} Let $R$ be an $\go'[\frac{1}{\Delta}]$-algebra.
A Hilbert modular forms of weight $k$, level $\Gamma$ and
coefficients in $R$, is a global section of $\underline{\omega}^k\otimes 
\underline{\nu}^{-n_0t/2}$ over  $M\times_{\Spec(\Z[\frac{1}{\Delta}])}\Spec(R)$.
 We denote by $G_k(\Gamma;R)= \rH^0(M\times_{\Spec(\Z[\frac{1}{\Delta}])} 
\Spec(R),\underline{\omega}^k\otimes \underline{\nu}^{-n_0t/2})$ the 
$R$-module of these Hilbert modular forms.
\end{defin}

\subsection{Toroidal  compactifications.}\label{toroidal}

The toroidal compactifications of the moduli space of $\gc$-polarized 
HBAV with {\it principal} level structure have been constructed
by Rapoport \cite{rapoport}. Several modifications  need to 
be made in  order  to treat the case of $\mu_{\gn}$-level structure. These 
are described  in \cite{dimdg}Thm.7.2.

Let  $\Sigma$ be a smooth $\Gamma_1^1(\gc,\gn)$-admissible collection of fans 
(see \cite{dimdg}Def.7.1).
 Then,  there exists an open immersion of $M^1$ into a proper and  smooth
$\Z[\frac{1}{\Delta}]$-scheme $\overline{M^1}=M^1_{\Sigma}$, called the 
 toroidal  compactification of $M^1$ with respect to   $\Sigma$. 
 The universal HBAV   $\pi:\cA\rightarrow M^1$ extends uniquely to
a semi-abelian scheme $\overline{\pi}:\mathfrak{G} \rightarrow \overline{M^1}$.
The group scheme $\mathfrak{G} $ is endowed with an action of 
$\go$ and its restriction to $\overline{M^1}\bs M^1$ is a torus.
Moreover, the sheaf $\underline{\omega}_{\mathfrak{G}/\overline{M^1}}$ of
$\mathfrak{G}$-invariants sections of 
$\overline{\pi}_*\Omega^1_{\mathfrak{G}/\overline{M^1}}$ is an 
invertible $\go$-bundle on $\overline{M^1}$,  extending
 $\underline{\omega}_{\cA/M^1}$. 


The scheme  $\overline{M^1}\bs M^1$ is a divisor with normal crossings, 
and the formal completion of $\overline{M^1}$ along this divisor
can be  completely determined in terms of $\Sigma$ (see \cite{dimdg}Thm.7.2).
For the sake of simplicity, we will only describe the 
 completion of $\overline{M^1}$ along the connected component 
of $\overline{M^1}\bs M^1$ corresponding to the 
standard cusp at $\infty$. Let $\Sigma^\infty\in \Sigma$ be the 
 fan corresponding to the   cusp at $\infty$. It is a  complete, smooth 
fan of $\gc_+^*\cup \{0\}$, stable by the action of $\go^{\times 2}_{\gn,1}$, 
and containing a finite number of cones modulo this action. 
Put $R_\infty=\Z[q^\xi,\xi\in\gc]$ and 
$S_\infty=\Spec(R_\infty)=\Gm\otimes \gc^*$. Associated to the 
fan $\Sigma^\infty$, there is a  toroidal embedding  
$S_\infty\hookrightarrow S_{\Sigma^\infty}$ (it is obtained 
by gluing the affine toric  embeddings $S_\infty\hookrightarrow 
S_{\infty,\sigma}=\Spec(\Z[q^\xi,\xi\in\gc\cap\check{\sigma}])$ for 
$\sigma\in \Sigma^\infty$). 
Let $S_{\Sigma^\infty}^{\wedge}$ be the formal completion of
 $S_{\Sigma^\infty}$ along $S_{\Sigma^\infty}\bs S_\infty$. 
By construction, the formal completion of $\overline{M^1}$ 
along the connected component  of $\overline{M^1}\bs M^1$ corresponding to the 
standard cusp at $\infty$, is isomorphic to $S_{\Sigma^\infty}^{\wedge}/
\go^{\times 2}_{\gn,1}$. 

Assume that  $\Sigma$ is  $\Gamma_1(\gc,\gn)$-admissible (for the cusp at $\infty$, 
it means that $\Sigma^\infty$ is stable under the action of 
$\go_+^{\times}$). Then  the finite group $\go_+^\times/\go^{\times 2}_{\gn,1}$ 
acts properly and discontinuously  on $\overline{M^1}$, and  the quotient 
$\overline{M}=M_{\Sigma}$ is a proper and smooth $\Z[\frac{1}{\Delta}]$-scheme,
containing $M$ is an open subscheme. Again by construction, the formal
 completion of $\overline{M}$  along the connected component  of 
$\overline{M}\bs M$ corresponding to the  standard cusp at $\infty$, 
is isomorphic to $S_{\Sigma^\infty}^{\wedge}/ \go_{+}^{\times}$.


The invertible $\go$-bundle  $\underline{\omega}_{\mathfrak{G}/\overline{M^1}}$
on $\overline{M^1}$ descends to an invertible $\go$-bundle on $\overline{M}{}'$, 
extending  $\underline{\omega}$. We still denote this extension by 
$\underline{\omega}$. For each $k\in \Z[J_F]$ this gives us an extension of
 $\underline{\omega}^k$ to a line bundle on $\overline{M}{}'$,  still 
denoted by  $\underline{\omega}^k$.

\subsection{$q$-expansion and Koecher Principles.}\label{qdev}

The Koecher Principle states (see \cite{dimdg}Thm.8.3)
\begin{equation}\label{koecher}
\rH^0(M\times\Spec(R), \underline{\omega}^k\otimes \underline{\nu}^{-n_0t/2})=
\rH^0(\overline{M}\times\Spec(R),
\underline{\omega}^k\otimes \underline{\nu}^{-n_0t/2})
\end{equation}

For  simplicity, we will only describe the $q$-expansion at the 
standard (unramified) cusp at $\infty$. For every $\sigma\in \Sigma^\infty$, 
and every $\go'[\frac{1}{\Delta}]$-algebra $R$, the pull-back of 
$\underline{\omega}$ to $S_{\Sigma^\infty}^{\wedge}\times\Spec(R)$ is 
canonically isomorphic to 
$\go\otimes \cO_{S_{\Sigma^\infty}^{\wedge}}\otimes R$.  Thus 
$$\hspace{-5mm}\rH^0(S_{\Sigma^\infty}^{\wedge}\times\Spec(R)/\go_+^\times,
\underline{\omega}^k\otimes \underline{\nu}^{-n_0t/2})\!=\!
\left\{\! \sum_{\xi\in\gc_+\cup\{0\}}\!\!\!\! a_\xi q^\xi\Big{|}
 a_\xi\!\in\! R, a_{u^2\epsilon\xi}=u^k\epsilon^{k+m-t} a_\xi, 
\forall (u,\epsilon)\! \in \! \go^{\times}_{\gn,1} \!\times\!\go_+^\times \right\}$$


By the above construction, to each $g\in G_k(\Gamma;R)$, 
we can associate an element $g_\infty= \sum_{\xi\in\gc_+\cup\{0\}}a_\xi(g)q^\xi$, 
 called the $q$-expansion of  $g$ at the cusp at $\infty$. 
The element $a_0(g)\in R$ is the value of  $g$ at the cusp at $\infty$.  

\begin{prop} Let $R$ be a $\go'[\frac{1}{\Delta}]$-algebra. 

{\rm (i)($q$-expansion Principle)}  $ G_k(\Gamma;R)\rightarrow 
 R[[q^\xi,\xi\in\gc_+\cup\{0\}]],  \enspace g\mapsto g_\infty$ is injective.

(ii) If there exists $g\in G_k(\Gamma;R)$, such that 
$a_0(g)\neq 0$, then  $\epsilon^{k+m-t}-1$ is a zero-divisor in $R$, 
for  all $\epsilon\in \go_+^\times $. 
\end{prop}



\subsection{The minimal compactification.}
There exist a projective, normal $\Z[\frac{1}{\Delta}]$-scheme
$M^{1*}$, containing $M^1$ as an open dense subscheme and 
such that the scheme $M^{1*}\bs M^1$ is  finite and {\'e}tale 
over  $\Z[\frac{1}{\Delta}]$.   Moreover, for each toroidal 
compactification $\overline{M^1}$  of $M^1$ there is a 
natural surjection $\overline{M^1}\rightarrow M^{1*}$, inducing 
the identity map on  $M^1$. 
The scheme $M^{1*}$ is called the  minimal compactification of $M^1$. 
The action of  $\go_+^\times/\go^{\times 2}_{\gn,1}$
on $M^1$ extends to an action on $M^{1*}$, and the 
minimal compactification $M^*$ of $M$ is defined as
the quotient for this action. In general 
 $M^{1*}\rightarrow M^*$ is not {\'e}tale.

We summarize the above discussion in the following commutative diagram :
$$\xymatrix@C=20pt@R=8pt{
\mathfrak{G}\ar[rr]^{\overline{\pi}} && \overline{M^1}\ar@{->>}[rr]
\ar@{->>}[rd] && \overline{M}\ar@{->>}[rd] & \\
              & &                &M^{1*} \ar@{-->>}[rr] &      & M^*\\
\cA \ar[rr]^{\pi}\ar@{^{(}->}[uu]   &&      M^1 \ar@{->>}[rr]\ar@{^{(}->}[uu]\ar@{^{(}->}[ru]    &&   M \ar@{^{(}->}[uu] \ar@{^{(}->}[ru]     & }$$

\subsection{Toroidal compactifications of the Kuga-Sato varieties.}\label{kugasato}
Let $s$ be a positive integer. Let  $\pi_s:\cA^s\rightarrow M^1$ be 
the $s$-fold fiber product of $\pi:\cA\rightarrow M^1$, and 
$(\overline{\pi})_s: \mathfrak{G}^s\rightarrow  \overline{M^1}$
be the $s$-fold fiber product of $\overline{\pi} :\mathfrak{G}
\rightarrow  \overline{M^1}$. 

Let $\widetilde{\Sigma}$ be a  $(\go\oplus\gc)\rtimes 
\Gamma_1^1(\gc,\gn)$-admissible, polarized, equidimensional,  smooth collection of fans, 
above the $\Gamma_1^1(\gc,\gn)$-admissible collection of fans $\Sigma$ of \S\ref{toroidal}. 
Using Faltings-Chai's method
\cite{FaCh}, the main result of \cite{dimtildg} Sect.6 is the following :
there exists an open immersion of a $\cA^s$ into a 
projective smooth $\Z[\frac{1}{\Delta}]$-scheme 
$\overline{\cA^s}=\cA^s_{\widetilde{\Sigma}}$, 
and a  proper, semi-stable homomorphism  
$\overline{\pi_s}:\overline{\cA^s}\rightarrow  \overline{M^1}$
extending $\pi_s:\cA^s\rightarrow M^1$, and such that 
$\overline{\cA^s}\bs\cA^s$ is a relative normal crossing divisor 
above $ \overline{M^1}\bs M^1$. Moreover, $\overline{\cA^s}$
contains $\mathfrak{G}^s$ as an open dense subscheme and 
$\mathfrak{G}^s$ acts on $\overline{\cA^s}$
extending the translation action of $\cA^s$ on itself. 


The  sheaf  $\cH^1_{\ldR}(\overline{\cA}/\overline{M^1})=
R^1\overline{\pi_1}_*\Omega^\bullet_{\overline{\cA}/\overline{M^1}}
(\dlog\infty)$ is independent of the particular choice of $\widetilde{\Sigma}$
 above   $\Sigma$ and is endowed with a filtration :
$$0 \rightarrow \underline{\omega}_{\mathfrak{G}/\overline{M^1}} \rightarrow 
\cH^1_{\ldR}(\overline{\cA}/\overline{M^1})  \rightarrow 
\underline{\omega}_{\mathfrak{G}/\overline{M^1}}^{\vee}
\otimes\mathfrak{cd}^{-1}  \rightarrow 0.$$
It descends to a sheaf $\cH^1_{\ldR}$ on $\overline{M}$  which fits
in the following exact sequence : 
$$ 0 \rightarrow \underline{\omega}\rightarrow \cH^1_{\ldR}
 \rightarrow \underline{\omega}^{\vee}
\otimes\mathfrak{cd}^{-1}  \rightarrow 0.$$

\subsection{Hecke operators on modular forms.}\label{hecke} Let 
 $\Z[K_1(\gn)\bs G(\A_f)/K_1(\gn)]$ be the free abelian group
with basis the double cosets of $K_1(\gn)$ in $G(\A_f)$. It is endowed with 
algebra structure, where the product of two basis elements is given by : 
\begin{equation}\label{convolution}
[K_1(\gn)xK_1(\gn)]\cdot [K_1(\gn)yK_1(\gn)]=\sum_i [K_1(\gn)x_iyK_1(\gn)],
\end{equation}
where   $[K_1(\gn)xK_1(\gn)]=\coprod_i K_1(\gn) x_i$. 
For $g\in S_{k,J}(K_1(\gn))$ we put :
$$g|_{[K_1(\gn)xK_1(\gn)]}(\cdot)=\sum_i g(\cdot x_i^{-1}).$$

This defines an action of the algebra $\Z[K_1(\gn)\bs G(\A_f)/K_1(\gn)]$ on 
$S_{k,J}(K_1(\gn))$ (resp. on $G_{k,J}(K_1(\gn))$). Unfortunately, this algebra 
is not commutative  when $\gn\neq\go$.  We will now define a commutative 
subalgebra.  Consider the semi-group :
$$\Delta(\gn)=\left\{\begin{pmatrix} a & b \\ c &
    d\end{pmatrix}\in G(\A_f)\cap 
    \mathrm{M}_2(\widehat{\go})\enspace  |\enspace  d_v\in \go_v^\times,  \enspace 
c_v\in \gn_v\text{, for all } v\text{ dividing } \gn\right\}.$$

The abstract Hecke algebra of level $K_1(\gn)$, is defined as  
$\Z[K_1(\gn)\bs \Delta(\gn)/K_1(\gn)]$ endowed with the convolution product 
$(\ref{convolution})$. This algebra has the following explicit description.


For each ideal $\ga\!\subset\! \go$ we define the  Hecke 
operator $T_{\ga}$ as the finite sum of double  cosets
 $[K_1(\gn)xK_1(\gn)]$ contained  in  the set  $\{x\in
\Delta(\gn)|\nu(x)\go=\ga\}$.
In the  same way, for a prime to $\gn$ ideal $\ga\!\subset\! \go$, 
we define the Hecke operator
$S_{\ga}$ by the double  coset for $K_1(\gn)$
containing the scalar matrix of the id{\`e}le attached  to the ideal 
$\ga$.

 For each finite   place  $v$ of $F$, we have 
$T_v=K_1(\gn)\begin{pmatrix} \varpi_v & 0 \\ 0 & 1 \end{pmatrix} 
K_1(\gn)$, and for each $v$ not  dividing $\gn$ 
we have $S_v=K_1(\gn)\begin{pmatrix} \varpi_v & 0 \\ 0 & \varpi_v 
\end{pmatrix} K_1(\gn)$, where $\varpi_v$ is an uniformizer of $F_v$.

Then, the abstract Hecke algebra of level $K_1(\gn)$ is isomorphic to the 
polynomial algebra in  the  variables   $T_v$, where $v$ 
runs over the prime ideals of $F$, and the variables $S_v$,
where $v$  runs over the prime ideals of $F$ not dividing  $\gn$.
The  action of Hecke algebra obviously  preserves  the  decomposition 
 (\ref{decomp-fmh}) and moreover, $S_v$ acts on $S_{k,J}(\gn,\psi)$
as the scalar $\psi(v)$. 

\medskip
Let $\T(\C)=\T_k(\gn,\psi;\C)$ be the subalgebra of $\End_{\C}(S_{k,J}(\gn,\psi))$
 generate by   the operators $S_v$ for $v \nmid \gn$  and $T_v$ for all $v$ 
(we will see in   \S\ref{conjugates} that  $\T(\C)$ does not  depend on $J$). 

The algebra $\T(\C)$  is commutative, but  not semi-simple in  
general. Nevertheless, for   $v \nmid \gn$  the operators $S_v$ and $T_v$
 are normal with respect to the Petersson inner product (see Def.\ref{peterson}). 
Denote by  $\T'(\C)$ be the subalgebra of $\T(\C)$   generated by the Hecke operators
outside a finite set of places containing those dividing $\gn$. 
The algebra $\T'(\C)$  is  semi-simple, that is to say  $S_{k,J}(\gn,\psi)$ has a basis
 made of eigenvectors for   $\T'(\C)$.

\medskip

We will now describe the relation between Fourier coefficients and eigenvalues 
for the Hecke operators. By (\ref{ad-cl}) we can associate to $g\in S_k(K_1(\gn))$ 
a family of classical cusp forms  $g_i\in S_k(\Gamma_1(\gc_i,\gn);\C)$, where 
$\gc_i$ are  representatives  of the narrow ideal class group   $\Cl_F^+$.


Each form $g_i$ is determined by its $q$-expansion 
at the cusp $\infty$ of $M_1(\gc_i,\gn)^{\an}$.
For each fractional ideal  $\ga=\gc_i \xi$, 
with $\xi\in F_+^\times$,  we put  $c(g,\ga)=\xi^m a_\xi(g_i)$.
By \S\ref{qdev} for each $\epsilon \in \go_+^\times$,  we have 
$a_{\epsilon\xi}=\epsilon^{k+m-t}a_{\xi}$
and  therefore the definition of $c(g,\ga)$ does not depend 
on the  choice of $\xi$ (nor on  the particular choice  of the 
ideals   $\gc_i$; see \cite{hida-padic} IV.4.2.9.). 

\begin{defin}
 We say that $g\in S_k(\gn,\psi)$ is an {\it eigenform}, if it is an  
eigenvector for  $\T(\C)$. We say that an eigenform  $g$ is  {\it normalized}
if  $c(g,\go)=1$.
\end{defin} 

\begin{lemma}{\rm (\cite{hida128} Prop.4.1, \cite{hida-padic} (4.64))}
If $g\in S_k(\gn,\psi)$ is a normalized eigenform, then the  eigenvalue of 
$T_{\ga}$ on $g$ is  equal  to the   Fourier coefficient $c(g,\ga)$.
\end{lemma}

A consequence of this lemma and the $q$-expansion Principle 
(see \S\ref{qdev}), is the Weak Multiplicity One Theorem stating
 that two normalized  eigenforms having  the same   eigenvalues are equal.

\medskip
\noindent{\bf Ordinary modular forms.}
When the weight $k$ is non-parallel, the definition of the 
Hecke operators should be slightly modified. We 
 put $T_{0,v}=\varpi_v^{-m}T_v$ and 
$S_{0,v}=\varpi_v^{-2m}S_v$ (see \cite{hida128} Sect.3 ;
in the applications our base ring will be the  $p$-adic ring
$\cO$ which satisfies  the assumptions of this reference).

The advantage of the Hecke operators $T_{0,v}$ and $S_{0,v}$ is that 
they preserve in an optimal way the  $\cO$-integral 
structures on the space of Hilbert modular forms and on the 
cohomology of the  Hilbert modular variety.

\begin{defin} \label{ord-autom}
A Hilbert modular eigenform is {\it ordinary } at $p$
if, for all primes $\gp$ of $F$ dividing $p$, the image by $\iota_p$ of
its  $T_{0,\gp}$-eigenvalue   is a  $p$-adic unit.
\end{defin}

\medskip
\noindent{\bf Primitive modular forms.}
For each $\gn_1$ dividing $\gn$ and divisible 
by the conductor of $\psi$,  and for all $\gn_2$
dividing $\mathfrak{nn}_1^{-1}$ we consider  the linear map 
$$S_k(\gn_1,\psi)\rightarrow S_k(\gn,\psi),
\enspace g\mapsto g|\gn_2,$$

\noindent where $g|\gn_2$ is determined by the relation 
$c(\ga,g|\gn_2)=c(\ga\gn_2^{-1},g)$. 


We define  the subspace $S_k^{\mathrm{old}}(\gn,\psi)$
of  $S_k(\gn,\psi)$ as the subspace  
generated by the images of all these  linear maps. This 
space is preserved  by the Hecke operators outside $\gn$.
We define  the space  $S_k^{\mathrm{new}}(\gn,\psi)$ 
of the primitive modular forms as the orthogonal of 
$S_k^{\mathrm{old}}(\gn,\psi)$
in $S_k(\gn,\psi)$ with respect to the Petersson
inner product (see Def.\ref{peterson}). Because the 
Hecke operators  outside  $\gn$ are normal 
for the Petersson inner  product, the direct sum decomposition 
$S_k(\gn,\psi)=S_k^{\mathrm{new}}(\gn,\psi)
\oplus S_k^{\mathrm{old}}(\gn,\psi)$ is preserved
 by  $\T'(\C)$.
The  Strong Multiplicity One  Theorem, due to
 Miyake in the Hilbert modular case, asserts  that if 
$f\in S_k^{\mathrm{new}}(\gn,\psi)$ 
 is an eigenform  $\T'(\C)$, then it is an eigenform  for $\T(\C)$. 

A normalized primitive eigenform is called a {\it newform}.

\medskip

The pairing $\T(\C)\times S_k(\gn,\psi)\rightarrow \C$, 
$(T,g)\mapsto c(g|_{T},\go)$ is   a perfect  duality (see \cite{hida128} Thm.5.2).




\subsection{External  and  Weyl group  conjugates.}\label{conjugates}

For an element $\sigma\in \Aut(\C)$ we define the
 {\it external conjugate} of $g\in S_k(K_1(\gn))$, 
as the unique element $g^\sigma \in S_k(K_1(\gn))$
satisfying $c(g^\sigma,\ga)\!=\!c(g,\ga)^\sigma$,
for each ideal $\ga$ of $\go$. 



\medskip
We  identify  $\{\pm 1\}^{J_F}$ with the Weyl group  $K_{\infty}/K_{\infty}^+$ 
of $G$,  by sending $\epsilon_J\!=\!(-1_J, 1_{J_F\!\bs\!J})$
to $c_JK_{\infty}^+$, where for all $\tau\in J_F$,  $\det(c_{J,\tau})< 0$ 
if and only if $\tau\in J$. The length of $\epsilon_J$ is $|J|$.


We have an action of the Weyl group on the space of Hilbert modular forms.
More precisely, $\epsilon_J$ acts as the double class 
 $[K_1(\gn)c_JK_1(\gn)]$, and  maps   bijectively 
$S_k(K_1(\gn))$ onto $S_{k,J_F\!\bs\!J}(K_1(\gn))$.
The action of $\epsilon_J$ commutes with the action of the  Hecke operators.
For an element $g\in S_k(K_1(\gn))$  we put $g_J=\epsilon_{J_F\!\bs\!J}\cdot g$. 



\subsection{Eichler-Shimura-Harder isomorphism.}\label{ESHI}
Let $R$ be  an $\cO$-algebra  and  $V_n(R)$ be the polynomial ring 
over  $R$ in the  variables $(X_\tau,Y_\tau)_{\tau\in J_F}$ which are  
homogeneous  of degree ${n}_\tau$ in $(X_\tau,Y_\tau)$. 
We have a pairing (perfect if $n_0!$ is invertible in $R$)
\begin{equation} \label{theta-n}
\langle\quad,\quad\rangle:V_n(R)\times V_n(R)\rightarrow R
\text{, given by}
\end{equation}
\begin{displaymath}\left\langle\sum_{0\leq j\leq  {n}}
a_jX^{{n}-j}Y^j,\sum_{0\leq j\leq {n}}
b_jX^{{n}-j}Y^j\right\rangle=
\sum_{0\leq j\leq {n}}(-1)^j a_j b_{{n}-j}\binom{{n}}{j}
\text{,where } \binom{{n}}{j}=\prod_{\tau\in
  J_F}\binom{{n}_\tau}{j_\tau},
\end{displaymath} 

The   $R$-module $V_n(R)$  realizes   the  algebraic 
representation  $V_n=\bigotimes_{\tau}(\Sym^{n_\tau}\otimes \det^{m_\tau}) $ of 
$G(R)$. We endow  $V_n(R)$  with an   action of $(M_2(\cO)\cap \GL_2(E))^{J_F}$
given by
$$\gamma.P((X_\tau,Y_\tau)_{\tau\in
  J_F})=\nu(\gamma)^m P((\det(\gamma)\gamma^{-1})^t
(X_\tau,Y_\tau)_{\tau\in J_F}).$$

%\begin{rque}\label{Vn-an} Let $\Gamma$ be a congruence subgroup of $G(\Q)$. 
%Consider the universal covering  
%$\mathfrak{H}_F\rightarrow \Gamma\bs\mathfrak{H}_F=M^{\an}$. We have an 
%equivalence of categories between the category
%of representations  of $\Gamma$ on finite dimensional 
%$K$-vector spaces, with trivial action of the center,
%and the category of local systems in $K$-vector spaces
% on $M^{\an}$, which associates to  such a representation $V$, 
%the local system  $\bV^{\an}$ of continuous sections of 
%$\Gamma\bs(\mathfrak{H}_F\times V) \rightarrow \Gamma\bs\mathfrak{H}_F $
%(the space $V$ being endowed with the discrete topology).
%\end{rque}


%As the weight $n+2m=n_0t $ is parallel, the center  $\Phi_n$ of $\Gamma$ 
%acts trivially on $V_n(R)$, and therefore we can consider the local system
%$\bV^{\an}_n$ on $M^{\an}$ associated to  $V_n$.

%We denote by $\psi_0$ the restriction of $\psi$ to 
%$\widehat{\go}{}^\times$ and we consider 
%the $R$-module  $V_n^{\psi_0}(R)$, obtained
%from $V_n(R)$ by twisting the   action of $\Delta(\gn)$  by $\psi_0$
%(see \cite{hida74} Sect.3). 

Let $\bV_n(R)$ be the sheaf of continuous (thus locally constant) sections of  
$$G(\Q)\bs G(\A)\times V_n(R)/K_1(\gn)K_{\infty}^+\rightarrow
G(\Q)\bs G(\A)/K_1(\gn)K_{\infty}^+=Y^{\an},$$
 where $y\in K_1(\gn)K_{\infty}^+$ acts on $V_n(R)$ via its $p$-part $y_p$.

For each $y\in \Delta(\gn)$ the map $[y]:G(\A)\times V_n(R)\rightarrow 
G(\A)\times V_n(R)$,  $(x,v)\mapsto (xy,y_p.v)$ is a homomorphism of sheaves. 
This induces   an action of the   Hecke operator $[K_1(\gn)yK_1(\gn)]$ on   
$\rH^d(Y^{\an},\bV_n(R))$  preserving the cuspidal cohomology 
$\rH_{\mathrm{cusp}}^d(Y^{\an},\bV_n(R))$. 


The action  of $\epsilon_J$ on $(M^{\an },\bV^{\an})$ given by 
$\epsilon_J \cdot ((z_J,z_{J_F\!\bs\!J}),v)=((-\overline{z}_J,z_{J_F\!\bs\!J}),v)$,
induces an action of the Weyl group  on $\rH^d (Y^{\an} , \bV^{\an})$
commuting  with the Hecke action.

By  Harder \cite{harder} we know that, if $n\neq 0$, then
$\rH^d_{!}(Y^{\an},\bV_n(\C))=\rH_{\mathrm{cusp}}^d(Y^{\an},\bV_n(\C))$.
%We denote  $\T_k(\gn,\psi;R)$ the $R$-algebra generated 
%by all  the Hecke operators acting on $\rH^d_{!}(Y^{\an},\bV_n(R))$.

By  (\ref{theta-n}) we have  a  Poincar{\'e} pairing
$\langle\quad,\quad \rangle: \rH^d(Y^{\an},\bV_n(R))
 \times \rH^d_c(Y^{\an},\bV_n(R))  \rightarrow R$.

Let $\eta$ be the id{\`e}le corresponding to the ideal $\gn$ and  let  
$\iota=\begin{pmatrix}  0 & 1 \\ -\eta & 0 \end{pmatrix}$ be the Atkin-Lehner 
involution. By putting $[x,y]=\langle x,\iota y \rangle $ we obtain a new pairing
\begin{equation}\label{accoupl-twisted}
[\quad,\quad]: \rH_{!}^d(Y^{\an},\bV_n(R))\times\rH_{!}^d(Y^{\an},\bV_n(R))
\rightarrow R,
\end{equation}
which is Hecke-equivariant. We call it the {\it twisted Poincar{\'e} pairing}.

\medskip

Now we state the  Eichler-Shimura-Harder  isomorphism :

\begin{theo} 
{\rm (Hida, \cite{hida74})}
If  $n\neq 0$, then there exists an isomorphism :
\begin{equation}\label{ESH}
\delta:\bigoplus_{\psi}\bigoplus_{J\subset J_F}S_{k,J}(\gn,\psi)\cong
\rH_{!}^d(Y^{\an},\bV_n(\C)),
\end{equation}
\noindent where $\psi$ runs over the Hecke  characters of conductor dividing 
$\gn$ and  type $-n_0t$ at infinity. This  isomorphism is equivariant for 
the actions of the  Hecke algebra and  the Weyl  group. 
\end{theo}

\medskip
For  each $J\!\subset\! J_F$ let $\widehat{\epsilon}_J:
\{\pm 1\}^{J_F}\rightarrow \{\pm 1\}$  be the  unique character of the Weyl  
group  sending  $\epsilon_\tau=(-1_\tau,1^\tau)$ 
to  $1$, if $\tau\in J$, and to  $-1$  if   $\tau\in J_F\!\bs\!J$. 
The restriction of the Eichler-Shimura-Harder isomorphism (\ref{ESH})
to $S_{k,J}(\gn,\psi)$, followed by the projection on the 
$(\psi,\widehat{\epsilon}_J)$-part yields a Hecke equivariant isomorphism
\begin{equation}\label{ESH-epsilon}
\delta_J:S_{k,J}(\gn,\psi)\cong \rH_{!}^d
(Y,\bV_n(\C))[\psi,\widehat{\epsilon}_J].
\end{equation}



\medskip
%Put $\rH^{J}(M^{\an }, \bV_n(\C))=
%\{x \enspace  | \enspace
%(\epsilon_\tau\otimes c) (x)=-(-1)^{\psi_{J}(\tau)}x \}$, 
%where $\psi_J$ denotes the characteristic function of  $J$ of $J_F$.
Moreover, after twisting by the complex conjugation $c$ on the coefficients, 
we still have   a direct sum decomposition :
\begin{equation}\label{hodge-decomp}
\rH^d(M^{\an }, \bV_n(\C))=\bigoplus_{J\subset J_F} 
\rH^d(M^{\an }, \bV_n(\C))[\widehat{{\epsilon}_J\!\otimes\! c}].
\end{equation}

This  decomposition is  finer than the usual Hodge decomposition,
whose graded are given by ($0\leq a\leq d$) :
$$ \gr^a\rH^d(M^{\an }, \bV_n(\C))=
\bigoplus_{J\!\subset\! J_F, |J|=a} \rH^d(M^{\an },
 \bV_n(\C))[\widehat{{\epsilon}_J\!\otimes\! c}].$$
 

The transcendental decomposition (\ref{hodge-decomp}) has an algebraic 
interpretation, via the so-called BGG complex, that we will
describe in the next section.


\section{Hodge-Tate weights of the Hilbert modular varieties.}

 The aim of this  section  is to determine   the   Hodge-Tate weights 
of the $p$-adic {\'e}tale  cohomology of the Hilbert modular variety 
$\rH^\bullet(M_{\overline{\Q}_p},\bV_n(\overline{\Q}_p))$, as well as
those of the $p$-adic Galois representation associated to a
Hilbert modular form. In all this section we assume 

\medskip
{\bf (I)} $p$ does not divide $\Delta= \Nm(\gn\gd)$.

\medskip
The proof relies on Faltings'  Comparison Theorem \cite{Fa-jami}
relating  the {\'e}tale cohomology of $M$ with coefficients in the 
local system $\bV_n(\Q_p)$ to the de Rham logarithmic cohomology
of the corresponding vector bundle $\overline{\cV}_n$ over 
a smooth toroidal compactification $\overline{M}$ of $M$.
  The Hodge-Tate weights are given  by the jumps of 
the Hodge filtration of the associated de Rham complex. 
These are computed, following 
\cite{FaCh},  using the so-called  Bernstein-Gelfand-Gelfand complex (BGG complex).

Instead of using Faltings'  Comparison Theorem, one can 
use Tsuji's result for  the  {\'e}tale cohomology with 
constant coefficients of the Kuga-Sato variety  $\cA^s$ 
($s$-fold fiber product of the universal abelian variety $\cA$ 
above the fine moduli space $M^1$ associated to $M$; see
\cite{dimtildg} Sect.6 for the construction of  toroidal compactifications
of  $\cA^s$).

\bigskip
For each  subset $J$ of $J_F$ we put 
$p(J)=\sum_{\tau\in J}(k_0\!-\!m_\tau\!-\!1)\tau+
\sum_{\tau\in J_F\bs J}m_\tau\tau\in\Z[J_F]$ and for each
$a=\sum_{\tau\in J_F}a_\tau\tau\in\Z[J_F]$ we put
$|a|=\sum_{\tau\in J_F}a_\tau\in \Z$.  
 

\subsection{Motivic weight of the  cohomology.} \label{MWC}
Consider the smooth sheaf $R^1\pi_* \overline{\Q}_p$
on $M^1$, where $\pi:\cA\rightarrow M^1$ is the universal HBAV. It corresponds
to a representation of the fundamental group of $M^1$ in $G(\overline{\Q}_p)$.
By composing this representation with the  algebraic representation $V_n$  of $G$
of highest weight $n$ (see \S\ref{ESHI}),
we obtain a smooth sheaf on $M^1$ (thus on $Y^1$). It descends to  
a smooth sheaf on $Y$ denoted  $\bV_n(\overline{\Q}_p)$. 

Let  $W_f=\underset{\ga\subset \go}{\bigcap}
\ker(T_{\ga}-c(f,\ga))$ be the subspace 
of  $\rH^d(Y_{\overline{\Q}},\bV_n(\overline{\Q}_p))$ 
corresponding  to the   Hilbert modular newform  $f\in S_k(\gn,\psi)$.
Put $s=\sum_\tau(n_\tau+2m_\tau)=d n_0$.

\begin{prop}  $W_f$ is pure of weight $d+s$, that is to say  for all prime 
$l\nmid p\Delta$ the eigenvalues  of the geometric Frobenius 
 $\Frob_l$ at $l$ are Weil numbers of  absolute  value  $l^{\frac{d+s}{2}}$. 
\end{prop}

{\bf Proof : } As $f$ is cuspidal $W_f\!\subset\! 
\rH^d_!(Y_{\overline{\Q}},\bV_n(\overline{\Q}_p))$. We recall that 
 $Y_{\overline{\Q}}$ is a disjoint union  of its connected components 
 $M_{\overline{\Q}}=M_1(\gc_i,\gn)_{\overline{\Q}}$,
where the $\gc_i$'s form  a set of representatives of $\Cl_F^+$. 
Let $\gc$ be one of the $\gc_i$'s and  $M^1=M_1^1(\gc,\gn)$. 
For $*=\varnothing,c$ we have 
$$\rH^0(\go_+^\times/\go^{\times 2}_{\gn,1}, 
\rH^d_*(M_{\overline{\Q}}^1,\bV_n(\overline{\Q}_p)))=
\rH^d_*(M_{\overline{\Q}},\bV_n(\overline{\Q}_p)),$$
 and therefore, it is enough to 
prove that $\rH^d_!(M_{\overline{\Q}}^1,\bV_n(\overline{\Q}_p))$
is pure of weight $d+s$. We use  Deligne's method
\cite{De2}. Let $\pi: \cA\rightarrow  M^1$ be the universal
abelian variety (see \S\ref{hmv}). The  sheaf 
$\bV_n(\overline{\Q}_p)$ corresponds to the representation 
$\bigotimes_{\tau\in J_F}\Sym^{n_\tau}\otimes \det^{m_\tau}$
of the  group $G^*$ and can therefore be cut out by algebraic 
correspondences in   $(R^1\pi_*\overline{\Q}_p)^{\otimes s}$. Let  
 $\pi_s: \cA^s\rightarrow  M^1$ be the Kuga-Sato variety. By the 
Kunneth's formula we have
$$\rH^d_!(M_{\overline{\Q}}^1,(R^1\pi_*\overline{\Q}_p)^{\otimes s})\!\subset\!
\rH^d_!(M_{\overline{\Q}}^1,R^s\pi_{s*}\overline{\Q}_p)\!\subset\!
\rH^{d+s}_!(\cA_{\overline{\Q}}^s,\overline{\Q}_p)\!\subset\!
\rH^{d+s}(\overline{\cA_{\overline{\Q}}^s},\overline{\Q}_p), $$

\noindent where the middle inclusion comes from the degeneration 
of the of the Leray spectral sequence  $\rE_{2*}^{i,j}=
\rH^i_*(M_{\overline{\Q}}^1,R^j\pi_{s*}\overline{\Q}_p)\Rightarrow 
\rH^{i+j}_*(\cA_{\overline{\Q}}^s,\overline{\Q}_p)$ 
for $*=\varnothing,c$ (see \cite{De2}).
The proposition is then a consequence of the Weil conjectures
for the eigenvalues of the Frobenius, proved by  Deligne  \cite{De1}.
\hfill$\square$


\subsection{The Bernstein-Gelfand-Gelfand complex over $\overline{\Q}$.}
In  this and the next sections we  give, 
following Faltings \cite{Fa},   an algebraic construction 
of the  transcendental  decomposition of the 
Betti cohomology described in (\ref{hodge-decomp}).

In this section all the objects are defined over a
characteristic zero field  splitting  $G$.


\medskip
Let $\mathfrak{g}$, $\gb$, $\gt$ and $\gu$ 
denote the Lie algebras of  $G$, $B$, $T$ and $U$,  respectively.
Consider the canonical splitting 
$\mathfrak{g}=\gb\oplus \gu^-$.
Let  $U(\mathfrak{g})$, $U(\gb)$ be  the enveloping  algebras of 
$\mathfrak{g}$ and  $\gb$, respectively.


\medskip
 The aim of this section is to write down a  resolution of $V_n$  of the type :
$$ 0 \leftarrow  V_n\leftarrow U(\mathfrak{g})
\otimes_{ U(\gb)}K_n^{\bullet},$$
where the  $K_n^j$  are finite dimensional semi-simple 
$\gb$-modules, with explicit simple components.

We start  by the case $n=0$.
If we put $K_0^j=\wedge^j(\mathfrak{g}/\gb)$ we 
obtain the so-called   {\it bar-resolution} of $V_0$.
Note that $\wedge^i(\mathfrak{g}/\gb)$ is a $\gb$-module
with trivial action of  $\gu$, therefore 
$K_0^j=\oplus W_{\mu}$, where $\mu$  runs over the weights  of $B$
that are  sum of $j$ distinct negative roots.

 By tensoring  this resolution with $V_n$ we  obtain the Koszul's complex :
\begin{equation}\label{bgg0}
 0 \leftarrow  V_n\leftarrow U(\mathfrak{g})
\otimes_{ U(\gb)} \left(\wedge^j(\mathfrak{g}/
\gb)\otimes V_n|_{\gb}\right),
\end{equation}
which is a resolution of  $V_n$ by   $\gb$-modules 
$\wedge^i(\mathfrak{g}/ \gb)\otimes V_n|_{\gb}$,  
 not   semi-simple in  general. 

\medskip

The BGG complex that we are going to define is  a direct factor 
of the Koszul's complex, cut by the action of the center 
$U(\mathfrak{g})^G$ of $U(\mathfrak{g})$.

Denote by   $\chi_n$ the character of  $U(\mathfrak{g})^G$
 corresponding to the  weight $n$. It is a classical  result  that 

\begin{lemma} 
$\chi_n=\chi_{\mu}$, if and only if there 
exists $J\!\subset\! J_F$ such that $\mu=\epsilon_J(n+t)-t $.
\end{lemma}

By taking the $\chi_n$-part of the bar resolution (\ref{bgg0}) 
of  $V_n$ we obtain the complex  :
\begin{equation}\label{bgg}
 0 \leftarrow V_n \leftarrow U(\mathfrak{g})
\otimes_{ U(\gb)}K_n^{\bullet}, \enspace
\mathrm{with}  \enspace K_n^i=
\bigoplus_{J\!\subset\! J_F, |J|=i} W_{\epsilon_J(n+t )-t },
\end{equation}
which  is  still  a resolution of $V_n$, as it is a direct factor 
of a   resolution. We call this resolution  the BGG complex.


\subsection{Hodge-Tate decomposition of  $\rH^\bullet(M\otimes 
\overline{\Q}_p,\bV_n(\overline{\Q}_p))$.} \label{hodge-tate} ${}$
In this paragraph we summarize the results  of \cite{dimtildg} Sect.7. 
The algebraic groups $G$, $B$, $T$ and $D$ of \S\ref{hmfv}  
have models over $\Z$, denoted by the same letters. 
For every scheme $X$, we put $X'=X\times \Spec(\go'[\frac{1}{\Delta}])$.

\medskip
By \S\ref{kugasato},  we can extend the  vector bundles $\underline{\omega}$ and 
$\cH^1_{\dR}$ to $\overline{M}{}'$. Only the 
construction depends on a choice of a toroidal compactification
 $\overline{\pi}:\overline{\cA}\rightarrow \overline{M^1}$  of 
$\pi:\cA\rightarrow M^1$. 

The sheaf  $\underline{\gM_D}=\underline{\Isom}_{\go\otimes 
\cO_{\overline{M}{}'}} (\underline{\omega},\go\otimes \cO_{\overline{M}{}'})$ 
is a $D'$-torsor over $\overline{M}{}'$ (for the Zariski topology).
We have a functor $\cF_D$ from the category of algebraic representations
of $D'$  to the  category of  vector bundles on $\overline{M}{}'$
which are direct sum of invertible bundles. To an  algebraic representation
$W$ of $D'$, $\cF_D$ associates the fiber product 
$\overline{\cW}:=\gM_D\overset{D'}{\times} W$. 

\medskip
The sheaf  $\underline{\gM_B}=\underline{\Isom}_{\go\otimes 
\cO_{\overline{M}{}'}}^{\fil}(\cH^1_{\ldR},
(\go\otimes \cO_{\overline{M}{}'})^2)$ is a $B'$-torsor over $\overline{M}{}'$.
We have a functor $\cF_B$ from the category of algebraic representations
of $B'$  to the  category of filtered vector bundles on $\overline{M}{}'$
whose graded are sums of invertible bundles. To an  algebraic representation
$V$ of $B'$, $\cF_B$ associates the fiber product 
$\overline{\cV}:=\gM_B\overset{B'}{\times} V$. 

A representation of $G$ (resp. $T$) can be considered as a representation 
of $B$ by restriction (resp. by making $U$ act trivially). Thus, we may define 
the  filtered vector  bundle $\overline{\cV}_n$ on $\overline{M}{}'$
associated to the algebraic representation $V_n$ of $G$, and the 
invertible bundle  $\overline{\cW}_{n,n_0}$ on  $\overline{M}{}'$ associated 
to the algebraic representation of $T=D\times D$, given by $(u,\epsilon)\mapsto 
u^n\epsilon^m$.


\medskip
The sheaf  $\underline{\gM_G}=\underline{\Isom}_{\go\otimes 
\cO_{\overline{M}{}'}}(\cH^1_{\ldR},
(\go\otimes \cO_{\overline{M}{}'})^2)$ is a $G'$-torsor over $\overline{M}{}'$.
We have a functor $\cF_G$ from the category of algebraic representations
of $G'$  to the  category of flat vector bundles on $\overline{M}{}'$
(that is  vector bundles  endowed with an 
integrable quasi-nilpotent logarithmic connection). 
To any algebraic representation $V$ of $B'$, $\cF_G$ associates the 
fiber product  $\overline{\cV}{}^{\nabla}:=\gM_G\overset{G'}{\times} V$.
For $j\!\in\!\n$, we  put
$\cH_{\ldR}^{j}(\overline{M}',\overline{\cV})=
R^j\phi_*(\overline{\cV}{}^{\nabla}\otimes \Omega^\bullet_{\overline{M}'}(\dlog\infty))$,
where $\phi: \overline{M}'\rightarrow \Spec(\go'[\frac{1}{\Delta}])$
denotes the structural homomorphism.



\medskip
By the Faltings' Comparison Theorem \cite{Fa-jami}, 
the $\G_{\Q_p}$-representation $\rH^\bullet 
(M^1_{\overline{\Q}_p},\bV_n(\overline{\Q}_p))$ is crystalline, hence de Rham, 
and we have a canonical isomorphism
$$\rH^\bullet(M^1_{\overline{\Q}_p},\bV_n(\overline{\Q}_p))\otimes B_{\dR}\cong
\cH_{\ldR}^\bullet(\overline{M^1}_{/\overline{\Q}_p},\overline{\cV}_n)\otimes B_{\dR}.$$


By \cite{dimtildg} Sect.7, the  Hodge to de Rham spectral sequence
$$\rE_1^{i,j}=\rH^{i+j}(\overline{M^1}_{/\overline{\Q}_p},\gr^i 
(\overline{\cV}_n\otimes 
\Omega^\bullet_{\overline{M^1}}(\dlog\,\infty)))
\Rightarrow \cH_{\ldR}^{i+j}(\overline{M^1}_{/\overline{\Q}_p},\overline{\cV}_n), $$
degenerates at $\rE_1$ (the filtration is 
the tensor product of the two Hodge filtrations). 
In order to compute the  jumps of the resulting filtration we introduce 
the  BGG complex :
$$\overline{\mathcal{K}}_n^i=\bigoplus_{J \!\subset\! J_F, |J|=i} 
\overline{\cW}_{\epsilon_J(n+t)-t,n_0}.$$
The fact that $\overline{\mathcal{K}}_n^\bullet$ is a complex
follows from  (\ref{bgg}) and from the following isomorphism
(see  \cite{FaCh} Prop.VI.5.1)
\begin{equation}\label{op-diff}
\Hom_{U(\mathfrak{g})}(U(\mathfrak{g})
\otimes_{U(\gb)}W_{1}),U(\mathfrak{g}) \otimes_{U(\gb)}W_{2})\rightarrow 
\mathrm{Diff.Op.}(\overline{\cW}_{2},\overline{\cW}_{1}).
\end{equation}
Define a  filtration on  $\overline{\mathcal{K}}_n^\bullet$  by 
$\Fil^i\overline{\mathcal{K}}_n^\bullet=\bigoplus_{J \!\subset\! J_F,
 |p(J)|\geq i}  \overline{\cW}_{\epsilon_J(n+t)-t,n_0}.$

As  the image  of the  Koszul's complex (\ref{bgg0}) by the 
contravariant functor  $W\mapsto \overline{\cW}$ 
is  equal to the de Rham  complex, and as the BGG complex
is  a direct (filtered) factor   of the Koszul's complex, we
have :


\begin{theo}\label{hodge-pad} {\rm (\cite{dimtildg} Thm.7.8)}
{\rm (i)} There is a  quasi-isomorphism of filtered complexes 
$$\overline{\mathcal{K}}_n^\bullet\hookrightarrow \overline{\cV}_n\otimes 
\Omega^\bullet_{\overline{M^1}}(\dlog\,\infty).$$
 
{\rm (ii)} The spectral sequence  given by the  Hodge filtration 

$$\rE_1^{i,j}=\bigoplus_{J \!\subset\! J_F, |p(J)|=i} 
\rH^{i+j-|J|} (\overline{M^1}_{/\overline{\Q}_p}, 
\overline{\cW}_{\epsilon_J(n+t)-t,n_0})
\Rightarrow
\cH_{\ldR}^{i+j}(\overline{M^1}_{/\overline{\Q}_p},\overline{\cV}_n)$$
degenerates at $\rE_1$.

{\rm (iii)} For all  $j\leq d$, the Hodge-Tate  weights 
 of the  $p$-adic  representation
$\rH^j(M^1_{\overline{\Q}_p},\bV_n(\overline{\Q}_p))$ 
belong to the set  $\{|p(J)|\enspace , \enspace |J|\leq j\}$.
\end{theo}


\subsection{Hecke operators on the cohomology.}\label{hecke2}
We describe  the standard Hecke  operator $T_{\ga}$ as a correspondence 
on $Y^1$. We are indebted to M. Kisin for pointing us out that the usual 
definition of Hecke operators  on $Y$ extends to $Y^1$ (see \cite{KiLa}\S1.9-1.11). 
Note that the  corresponding Hecke action on  analytic modular forms for $G^*$ 
(see \S\ref{hecke}) is not easy to write down, because the double 
class for the Hecke operator $T_v$ does not belong to $G^*(\A_f)$, 
unless $v$ is inert in $F$. 


Recall that $Y_1^1(\gn)=\coprod_{i=1}^{h^+} M_1^1(\gc_i,\gn)$,
where $\gc_1$,..,$\gc_{h^+}$ are a set of representatives of $\Cl_F^+$.

Assume that $\gc_i\ga$ and $\gc_j$ have the same class in $\Cl_F^+$. 
Then, consider the contravariant functor $\underline{\cM}^1_{\ga}$
 from the category of  $\Z[\frac{1}{\Delta}]$-schemes to the category of sets, 
assigning to a scheme $S$  the set of isomorphism classes of quintuples 
$(A,\lambda,\alpha,C,\beta)$ where  $(A,\lambda,\alpha)/S$ is a $\gc_i$-polarized 
HBAV  with $\mu_{\gn}$-level structure, $C$ is a closed subscheme of $A[\ga]$ 
which is $\go$-stable, disjoint from  $\alpha(\Gm\otimes \mathfrak{d}^{-1})$ and 
locally isomorphic to the constant group scheme $\go/\ga$ over $S$, and $\beta$ is an 
$\go^{\times 2}_{\gn,1}$-orbit of isomorphisms $(\gc_i\ga,(\gc_i\ga)_+) 
\overset{\sim}{\longrightarrow} (\gc_j,\gc_{j+})$.

We have a projection  $\underline{\cM}^1_{\ga}\rightarrow
\underline{\cM}^1$, $(A,\lambda,\alpha,C,\beta)\mapsto (A,\lambda,\alpha)$
which is relatively representable by $\pi_1:M^1_{\ga}(\gc_i,\gn)\rightarrow
M^1_1(\gc_i,\gn)$. We have also a projection $\pi_2:M^1_{\ga}(\gc_i,\gn)
\rightarrow M^1_1(\gc_j,\gn)$, coming from 
$(A,\lambda,\alpha,C,\beta)\mapsto(A/C,\lambda',\alpha')$, where 
$\alpha'$  is the composed map of $\alpha$ and $A\rightarrow A/C$, and 
$\lambda'$ is a $\gc_j$-polarization  of $A/C$ (defined via $\lambda$ and $\beta$).

Put $Y^1_{\ga}=\coprod_{i=1}^{h^+} M^1_{\ga}(\gc_i,\gn)$. 
As $\gc_i\mapsto \gc_j\simeq \gc_i\ga$ is a permutation of $\Cl_F^+$, we get 
two finite projections $\pi_1,\pi_2:Y^1_{\ga}\rightarrow Y^1$. 
$$\xymatrix@C=30pt@R=20pt{
\cA\ar[d]_{\pi}  &\cA_{\ga}\ar[d]^{\pi_{\ga}} \ar[r]\ar[l] & \cA \ar[d]^{\pi}\\
Y^1 & Y^1_{\ga}\ar[r]^{\pi_2}\ar[l]_{\pi_1} & Y^1 }$$

From this diagram we obtain 
$\pi_2^*\cH^1_{\dR}\rightarrow \pi_1^*\cH^1_{\dR}$.
Therefore, for every  algebraic representation $V$ of $G$, we have 
$\pi_2^*\cV^\nabla\rightarrow \pi_1^*\cV^\nabla$. 
By composing this morphisms by  $\pi_{1*}$ and taking  the trace, we obtain 
$\cV^\nabla \rightarrow  \pi_{1*}\pi_2^*\cV^\nabla\rightarrow
 \pi_{1*}\pi_1^*\cV^\nabla \rightarrow \cV^\nabla$, and thus 
we obtain an action of $T_{\ga}$ on $\rH^\bullet(Y^1,\cV^\nabla)$.

The same way from the above diagram we obtain 
$\pi_2^*\underline{\omega}\rightarrow \pi_1^*\underline{\omega}$ and 
 $\pi_2^*\underline{\nu}\rightarrow \pi_1^*\underline{\nu}$.
Therefore, for each  algebraic  representation  $W$ of $T$, we get 
$\pi_2^*\cW\rightarrow \pi_1^*\cW$.
In order to define the good action of  $T_{\ga}$ on on Hilbert modular forms, we 
should modify slightly the last arrow :
 we decompose  $\cW$ as $(\cW\underline{\omega}^{-2t})\underline{\omega}^{2t}$ 
and  we define  $\pi_{2*}(\cW\underline{\omega}^{-2t})
\rightarrow\pi_{1*}(\cW\underline{\omega}^{-2t})$ as above and 
$\pi_{2*}\underline{\omega}^{2t}\rightarrow\pi_{1*}\underline{\omega}^{2t}$
as in \cite{KiLa}\S1.11 (via the Kodaira-Spencer isomorphism 
$\Omega^1_{Y^1}\simeq \underline{\omega}^2\otimes_{\go}\gd\gc^{-1}$). 
Thus we obtain 
$\cW \rightarrow  \pi_{1*}\pi_2^*\cW\rightarrow \pi_{1*}\pi_1^*\cW\rightarrow\cW,$
and an  action of  $T_{\ga}$  on  $\rH^\bullet(Y^1,\cW)$. 

In particular, we obtain an action of  $T_{\ga}$  on the space 
of geometric Hilbert modular forms for $G^*$, 
 $\rH^0(Y^1,\underline{\omega}^k\otimes \underline{\nu}^{-n_0t/2})$.  As it 
has been observed in \cite{KiLa}1.11.8, this action is given by the projection 
$$\textstyle\frac{1}{[\go_+^\times:\go_{\gn,1}^\times]}\sum_{[\epsilon]\in
\go_+^\times/\go_{\gn,1}^\times} [\epsilon]\cdot:\rH^0(Y^1,\underline{\omega}^k\otimes \underline{\nu}^{-n_0t/2}) \rightarrow \rH^0(Y,\underline{\omega}^k\otimes \underline{\nu}^{-n_0t/2}), $$
followed by the usual Hecke operator on the space of  Hilbert modular forms 
(see \S\ref{hecke}).


\subsection{Hodge-Tate weights of $\otimes\Ind_F^{\Q}\rho$
 in the crystalline case.} \label{HT-induced}


We first recall the notion of induced representation.
Let $V_0$ be vector space over a field $L$, and let
 $\rho_0:\G_F\rightarrow \GL(V_0)$ be a linear representation.
The  induced representation $\Ind_F^{\Q}\rho_0$ of $\rho_0$ from $F$ to  $\Q$ 
is by definition  the $L$-vector space  
 $$\Hom_{\G_F}(\G_{\Q},V_0):= \{\phi_0:\G_{\Q}\rightarrow V_0\enspace 
|\enspace  \forall h\in\G_F,\enspace g\in \G_{\Q},\enspace 
\phi_0(gh)=\rho_0(h^{-1})(\phi_0(g))\},$$ 
where  $g\in \G_{\Q}$ acts on  $\phi_0\in \Hom_{\G_F}(\G_{\Q},V_0)$ by 
 $g\cdot \phi_0 (\cdot)=\phi_0(g^{-1}\cdot)$. 

\smallskip
For any fixed decomposition  $\G_{\Q}= \underset{\tau\in J_F}{\coprod}\widetilde{\tau}\G_F$,
the map  $\phi\mapsto (\phi(\widetilde{\tau}))_{\tau}$
gives an isomorphism between  $\Hom_{\G_F}(\G_{\Q},V_0)$ and 
 the direct sum  $\underset{\tau}{\oplus} V_\tau$ 
(where each $V_\tau$ is isomorphic to  $V_0$).
Via this identification,  the action of $\G_{\Q}$ on
 $\underset{\tau}{\oplus} V_\tau$ is given by :
$$ (\Ind_F^{\Q}\rho_0)(g)((v_\tau)_{\tau})=
(\rho_0(\widetilde{\tau}^{-1}g \widetilde{\tau_g})(v_{\tau_g}))_{\tau},$$
where $g^{-1}\widetilde{\tau}\in \widetilde{\tau_g}\G_F$. In fact
$(\phi_0(\widetilde{\tau}))_{\tau}\overset{g}{\mapsto}
(\phi_0(g^{-1}\widetilde{\tau}))_{\tau}=
(\rho_0(\widetilde{\tau}^{-1}g\widetilde{\tau_g})
(\phi(\widetilde{\tau_g})))_{\tau}$.

Keeping  the same  notations we define, following  Yoshida \cite{yoshida}, 
the tensor induced  representation $\otimes\Ind_F^{\Q}\rho_0:
\G_{\Q}\rightarrow \GL(\underset{\tau}{\otimes} V_\tau)$ as :
$$(\otimes\Ind_F^{\Q}\rho_0)(g)(\underset{\tau}{\otimes}  v_\tau)=
\underset{\tau}{\otimes} \rho_0(\widetilde{\tau}^{-1}g\widetilde{\tau_g})(v_{\tau_g}).$$


\begin{rque} For each $g\in \G_{\Q}$ the map  $\tau\mapsto \tau_g$
is a  permutation of $J_F$, and it  is trivial if and only if 
$g\in \G_{\widetilde{F}}$. Therefore, for each  $g\in \G_{\widetilde{F}}$,
we have  $(\otimes\Ind_F^{\Q}\rho_0)(g)=\underset{\tau}{\otimes}
\rho_0(\widetilde{\tau}^{-1}g\widetilde{\tau})$.  
Moreover for all $g,g'\in \G_{\Q}$  we have $(\tau_g)_{g'}=\tau_{gg'}$.
\end{rque}


\begin{defin} The {\it internal conjugate} $g_\tau$ 
of $g$ by $\tau\in J_F$, is defined as the unique element $g_\tau \in 
S_{k^\tau,J^\tau}(\tau(\gn),\psi_\tau)/\tau(F)$
satisfying $c(g_\tau,\ga)\!=\!c(g,\tau(\ga))$,
for each ideal $\ga$ of $\go$, where $k^\tau=\sum_{\tau'} k_{\tau\tau'}\tau'$ 
and  $\psi_\tau(\ga)\!=\!\psi(\tau(\ga))$. 
\end{defin} 

If  $\rho=\rho_{f,p}$ by the  previous remark we have 
$(\otimes\Ind_F^{\Q}\rho)(g)=
\underset{\tau}{\otimes} \rho_{f_{\tau}}(g)$, for each  $g\in \G_{\widetilde{F}}$.

 Brylinski and Labesse \cite{BL} have shown (see \cite{Ta} for 
this formulation)

\begin{theo} {\rm (Brylinski-Labesse)}\label{BL-thm}
The restrictions to $\G_{\widetilde{F}}$ of the two $\G_{\Q}$-representations  
 $W_f$ and $\otimes\Ind_F^{\Q}\rho$ have the same characteristic polynomial.
\end{theo}



\begin{cor}\label{hodge-complex}  {\rm(\cite{dimtildg} Cor.7.9)}
{\rm (i)} The spectral sequence  given by the  Hodge filtration 

$$\rE_1^{i,j}=\bigoplus_{J \!\subset\! J_F, |p(J)|=i} 
\rH^{i+j-|J|} (\overline{M}_{/\overline{\Q}_p}, \overline{\cW}_{\epsilon_J(n+t)-t,n_0}) \Rightarrow
\cH_{\ldR}^{i+j}(\overline{M}_{/\overline{\Q}_p},\overline{\cV}_n)$$
degenerates at $\rE_1$ and is Hecke equivariant.

{\rm (ii)}
 The   Hodge-Tate weights of $W_f$ are the integers
$\vert p(J)\vert$, $J\!\subset\! J_F$, counted with multiplicity.
\end{cor}

{\bf Proof :}  
(i) By taking the invariants of the Hodge filtration of  $\overline{\cV}_n \otimes 
\Omega^\bullet_{\overline{M^1}}(\dlog\,\infty)$ by the Galois group
 of  the {\'e}tale covering  $\overline{M^1}\rightarrow \overline{M}$,
we  obtain a filtration of the complex $\overline{\cV}_n \otimes 
\Omega^\bullet_{\overline{M}}(\dlog\,\infty)$ on $\overline{M}{}'$, 
 still called  the  Hodge filtration. The  same way, we define 
the BGG complex over $\overline{M}{}'$ by taking the  invariants of the BGG complex
over  $ \overline{M^1}$. The   associated spectral sequence is given by 
the invariants of the spectral sequence of  Thm.\ref{hodge-pad}(ii). 
We have now to  see that it is Hecke equivariant.


The Hecke operator $T_{\ga}$ extends to $\overline{Y^1}$. One way to define it 
is to take the schematic closure of $T_{\ga}\subset Y^1\times Y^1$ in 
$\overline{Y^1}\times \overline{Y^1}$. Another way is to take a 
toroidal compactification $\overline{Y^1_{\ga}}$ of $Y^1_{\ga}$ over  the
toroidal compactification  $\overline{Y^1}$ of $Y^1$. In both cases we obtain 
an action of  $T_{\ga}$ on   $\rH^\bullet(\overline{Y^1},\overline{\cW})$ and 
on $\rH^\bullet(\overline{Y^1},\overline{\cV}{}^\nabla)$.  Although it is not clear
in general that these extended Hecke operators commute. Nevertheless, they
commute on the right  hand side  of  Thm.\ref{hodge-pad}(ii), because
by Faltings' Comparison Theorem this side  is independent
of the  toroidal compactification.
Since the spectral sequence  of Thm.\ref{hodge-pad}(ii) degenerates at $\rE_1$, 
 they should also commute  on the left hand side. 

(ii) We have   $\overline{\cW}_{\epsilon_J(n+t)-t,n_0}=
\underline{\omega}^{-\epsilon_J(n+t)+t}\otimes \underline{\nu}^{p(J)}$. 
 It follows from Thm.\ref{hodge-pad} (as in \cite{FaCh} Thm.5.5
 and \cite{MoTi} Sect.2.3) that the jumps of the Hodge filtration are 
among $|p(J)|$, $J\!\subset\! J_F$. 

Moreover $ \gr^{|p(J)|}\rH^{d}(\overline{M}_{/\overline{\Q}_p},\overline{\cV}_n\otimes 
\Omega^\bullet_{\overline{M}}(\dlog\,\infty))= \rH^{d-|J|}(\overline{M}_{/\overline{\Q}_p},
\underline{\omega}^{-\epsilon_J(n+t)+t}\otimes \underline{\nu}^{p(J)})$.

It is enough to see that the  $\overline{\Q}_p$-vector space   
$\rH^{d-|J|}(\overline{Y}_{\overline{\Q}_p},
\underline{\omega}^{-\epsilon_J(n+t)+t}\otimes \underline{\nu}^{p(J)})
[f]$ is of dimension $1$, for all $J\!\subset\! J_F$.

Because of the existence of a  BGG complex over $\overline{\Q}$ 
 giving by base change the BGG complexes  over $\overline{\Q}_p$ and 
$\C$, we have an Hecke-equivariant isomorphism
$$\rH^{d-|J|}(\overline{Y}_{\overline{\Q}_p},
\underline{\omega}^{-\epsilon_J(n+t)+t}\otimes 
\underline{\nu}^{p(J)})  \otimes_{\overline{\Q}_p} {\C}=
\rH^d(Y^{\an }, \bV_n(\C))[\widehat{{\epsilon}_J\!\otimes\! c}].$$

Finally,   the 
$f$-part  $\rH^d(Y^{\an }, \bV_n(\C))[\widehat{{\epsilon}_J\!\otimes\! c},f]$ is 
equal to $\rH^d_{!}(Y^{\an }, \bV_n(\C))[\widehat{{\epsilon}_J\!\otimes\! c},f]$
and is therefore one dimensional by (\ref{ESH-epsilon}), for all $J\!\subset\! J_F$. 
\hfill $\square$


\begin{rque}
1) We proved that  $W_f$ is pure of weight $d(k_0-1)$. 
The set  of its   Hodge-Tate weights is stable by the symmetry
$h \mapsto d(k_0-1)-h$, corresponding  $|p(J)|\mapsto |p(J_F\!\bs\!J)|$. 
This symmetry is induced by the  Poincar{\'e} duality 
 $W_f\times W_f\rightarrow \overline{\Q}_p(-d(k_0-1))$.

2) If $F$ is a real quadratic field and  $\tau$ denotes the non-trivial embedding of $F$, then
  the Hodge-Tate weights of   $W_f$ are given by $m_\tau, k_0-m_\tau-1, k_0+m_\tau-1, 2k_0-m_\tau-2$.
\end{rque}





\subsection{Hodge-Tate weights of $\rho$ in the crystalline case.}
\label{HT-crys}
The embedding $\iota_p:\overline{\Q}\hookrightarrow \overline{\Q}_p$
allows us to identify $J_F$ with
$\Hom_{\Q\mathrm{-alg.}}(F,\overline{\Q}_p)$.
For each prime $\gp$ of $F$ dividing $p$, we put 
$J_{F,\gp}=\Hom_{\Q_p\!\mathrm{-alg.}}
(F_{\gp},\overline{\Q}_p)$. Thus we get a partition 
$J_F=\underset{\gp}{\coprod}J_{F,\gp}$.
Let $D_{\gp}$ (resp. $I_{\gp}$) be a decomposition (resp.  inertia) subgroup of $\G_F$  at $\gp$.

The following result is due to Wiles  if $k$ is parallel, and 
to Hida  in the  general case. 

\begin{theo} {\rm (Wiles \cite{wiles}, Hida \cite{hida-jami})}\label{ord}
Assume that  $f$ is  ordinary  at $p$ (see  Def.\ref{ord-autom}).

Then  $\rho_{|D_{\gp}}$ is reducible and  :
$$\hspace{-7cm}\mathrm{\bf (ORD)}\hspace{3cm} \rho_{f|I_{\gp}}\sim
\begin{pmatrix} \varepsilon_{\gp}  & * \\
0  & \delta_{\gp} \end{pmatrix},$$
where $\delta_{\gp}$ (resp. $\varepsilon_{\gp}$) is obtained  by  composing the  
class field theory map  $I_{\gp}\rightarrow \go_{\gp}^\times$
with the map  $\go_{\gp}^\times \rightarrow  \overline{\Q}_p^\times$, \enspace
 $x\mapsto \underset{\tau\in J_{F,\gp} }{\prod}\tau(x)^{-m_\tau}$
(resp. $x\mapsto \underset{\tau\in J_{F,\gp} }{\prod}\tau(x)^{-(k_0-m_\tau-1)}$). 
\end{theo}

\medskip
 Breuil \cite{breuil} has shown that  if  $p>k_0$ and  
$p$ does not divide $\Delta$, then $\rho$ 
is crystalline at each prime $\gp$ of $F$ dividing $p$, 
with   Hodge-Tate weights  between $0$ and $k_0-1$. 

\begin{cor} Assume  $p>k_0$ and  that  $p$ does not divide $\Delta$. Then 
for each prime $\gp$ of $F$ dividing $p$, $\rho_{|D_{\gp}}$ is crystalline with 
 Hodge-Tate weights the  $2[F_{\gp}:\Q_p]$ integers 
$(m_{\tau},k_0-m_{\tau}-1)_{\tau\in J_{F,\gp}}$.
\end{cor}

{\bf Proof : }
Assume first  that $n\neq 0$. Let $K$ be a CM  quadratic extension   of $F$,
 in which each   prime $\gp$ of $F$ 
(dividing $p$) splits as $\gp=\mathfrak{P}\mathfrak{P}^c$.
Blasius and Rogawski \cite{BR} have then  constructed  a pure motive   
over  $K$ with coefficients in $E$, of Hodge weights 
$(m_{\tau},k_0-m_{\tau}-1)_{\tau\in J_F}$
and  whose $p$-adic  realization  is isomorphic  to the 
restriction of $\rho$ to $\G_K$. This shows that $\rho_{|D_{\gp}}$ 
is de Rham, and even   crystalline for   $p$ big enough. 

By Faltings' Comparison  Theorem the Hodge  weights  of this   
motif correspond naturally  
(via   $\iota_p:\overline{\Q} \hookrightarrow \overline{\Q}_p$) 
to the  Hodge-Tate weights of its  $\mathfrak{P}$-adic realization,
 which are the same as the Hodge-Tate weights of $\rho$. This proves
 the corollary for  $n\neq 0$.

If $n=0$ (or more generally if  $k$ is parallel) we
can complete the proof using the   following
\begin{lemma} 
Let $a$ and $b$ be two positive integers  and let 
$(a_{\tau})_{\tau\in J_F}$ (resp.  
$(b_{\tau})_{\tau\in J_F}$)  be  integers  
satisfying  $0\leq 2a_{\tau}<a$ (resp. $0\leq 2b_{\tau}<b$).
Assume that the following two sets are equal (with multiplicities)

$\left\{\sum_{\tau\in J} a_{\tau}+\sum_{\tau\in J_F\bs J} 
(a-a_{\tau}) \text{ , } J\!\subset\! J_F\right\}=
\left\{ \sum_{\tau\in J} b_{\tau}+\sum_{\tau\in J_F\bs J} 
(b-b_{\tau}) \text{ , } J\!\subset\! J_F\right\}.$

 Then $a=b$ and  we have equality (with multiplicities)
 $\{a_{\tau} \text{ , } \tau\in J_F\}=
\{b_{\tau} \text{ , } \tau\in J_F\}$.
\end{lemma}

Using this lemma together with  Thm.\ref{BL-thm} and  Cor.\ref{hodge-complex}(ii) 
we obtain the Hodge-Tate weights of $\rho$  at the primes   $\gp$
dividing $p$,  up to permutation.
In particular, we know exactly the Hodge-Tate weights of $\rho$ when
$k$ is parallel. \hfill $\square$ 



\subsection{Fontaine-Laffaille weights of $\overline{\rho}$ 
in the crystalline case.} 

Our aim is to  find the weights of   $\overline{\rho}_{|I_{\gp}}$  for 
$\gp$ dividing  $p$. If $f$ is ordinary at $p$ we know  by 
Thm.\ref{ord}, that   $\rho_{|D_{\gp}}$ is reducible  and  by a simple 
 reduction modulo $\cP$ we obtain the weights of  $\overline{\rho}_{|I_{\gp}}$.

\begin{prop}  Assume  $p>k_0$ and  that  $p$ does not divide $\Delta$.
 Then  $\overline{\rho}$ is  crystalline  at each $\gp$
dividing $p$ with  Fontaine-Laffaille weights  $(m_{\tau},k_0-m_{\tau}-1)_{\tau\in J_{F,\gp}}$.
\end{prop}

{\bf Proof  : } It is a consequence of the theory of Fontaine and Laffaille
\cite{FoLa},  and of the computation of  Hodge-Tate weights of 
 $\rho_{|D_{\gp}}$ in   {\S}\ref{HT-crys}.


Consider a Galois stable lattice $\cO^2$ in the crystalline representation 
$\rho$, as well as the  sub-lattice   $\cP^2$. The representation 
 $\overline{\rho}$  is equal to the  quotient  of these  two lattices. 
It is crystalline, as a sub-quotient of a  crystalline representation.
Its     weights are determined by the associated 
filtered  Fontaine-Laffaille module. Since  the Fontaine-Laffaille's 
functor is exact, it is a  quotient of the  Fontaine-Laffaille's filtered modules 
 associated to the two lattices. By compatibility 
of the filtrations on these two lattices, and by the condition $p>k_0$, 
the graded of the quotient have  the right   dimension. See the 
Appendix for more details. \hfill$\square$

\begin{cor} \label{poids-modere}
Let $\gp$ be a  prime of $F$ above  $p$.
 Then
$$\overline{\rho}_{|I_{\gp}}\sim
\begin{pmatrix} \varepsilon_{\gp}  & * \\
0  & \delta_{\gp} \end{pmatrix},$$
\noindent
where $\varepsilon_{\gp},\delta_{\gp}:
I_{\gp}\rightarrow 
\overline{\F}_p^\times $ are two tame characters of 
level $|J_{F,\gp}|$, whose 
product equals the $(1-k_0)$th power  of the modulo $p$ cyclotomic character 
and whose sum has   Fontaine-Laffaille weights 
 $(m_{\tau},k_0-m_{\tau}-1)_{\tau\in J_{F,\gp}}$.
\end{cor}




\section{Study  of the images  of $\overline{\rho}$
and   $\Ind_F^{\Q}\overline{\rho}$.}

In all this section we assume that $p \!>\! k_0$ and 
that $p$ does not divide $6\Delta$.
 
Let  $\omega:\G_{\Q}\rightarrow \F_p^\times$ be the   modulo $p$
cyclotomic character and let $\pr:\GL_2(\kappa)\rightarrow \PGL_2(\kappa)$
be  the canonical  projection.  

\subsection{Lifting of characters and irreducibility criterion 
for $\overline{\rho}$.}

\begin{prop}\label{Irr}
{\rm(i)} For all but finitely many primes $p$ $\mathrm{\bf (Irr_{\overline{\rho}})}$ holds, that is 
 $\overline{\rho}=\overline{\rho}_{f,p}$  is absolutely irreducible.

{\rm(ii)}  Assume that $k$  is non-parallel.  If  for all  $J\!\subset\! J_F$
there exists  $\epsilon\in\go_+^{\times}$,
$\epsilon-1\in\gn$ such that  $p$   does not divide  the non zero 
integer  $\Nm(\epsilon^{p(J)}\!-\!1)$, then  $\mathrm{\bf (Irr_{\overline{\rho}})}$ holds.
\end{prop}

\begin{rque} Assume that   $k=k_0t$ is parallel and that for all 
$\varnothing\!\subsetneq\!J\!\subsetneq\! J_F$, there exists  $\epsilon\in\go_+^{\times}$,
$\epsilon-1\in\gn$ such that  $p$   does not divide  the non zero 
integer  $\Nm(\epsilon^{p(J)}\!-\!1)$. Then 
we expect  $\overline{\rho}=\overline{\rho}_{f,p}$ to be   absolutely
irreducible, unless $p$
 divides 
the  constant term  of an Eisenstein   series  of weight $k$ and level
dividing $\gn$, that is the numerator of the value  at $1-k_0$ of the  $L$-function of
a  finite order Hecke character  of  $F$ of conductor dividing
$\gn$ (see \cite{FaJo}\S3.2 for the case $F=\Q$). 
 \end{rque}

{\bf Proof : } As  $\overline{\rho}$ is  totally odd, 
if it is  irreducible, then it is  absolutely irreducible.
 Assume   that   $\overline{\rho}$  is reducible :  
$\overline{\rho}^{\mathrm{s.s.}}=\varphi_{\gal}\oplus
 \varphi_{\gal}'$. The characters  $\varphi_{\gal},\varphi_{\gal}':
\G_F\rightarrow \kappa^\times$ are unramified outside $\gn p$ and  
$\varphi_{\gal}\varphi_{\gal}'=\det(\overline{\rho})=\overline{\psi}_{\gal}
\omega^{-1}$  (recall that  $\psi$ is a Hecke character of infinity type $-n_0t$). 
Denote by $\widehat{\go}^{\times}_{\gn,1}$
the subgroup of $\widehat{\go}^{\times}$ of elements 
$\equiv 1\pmod{\gn}$. Then $\widehat{\go}^{\times}_{\gn,1}$ is a product of
its $p$-part $\prod_{\gp\mid p}\go_{\gp}^\times$
and its part  outside $p$, denoted by  $\widehat{\go}^{\times(p)}_{\gn,1}$.

By the global class field theory, the Galois group of  the  maximal 
$\gn$-ramified (resp. $\gn p^\infty$-ramified)
abelian extension  of $F$ is isomorphic to 
$\Cl^+_{F,\gn}=\A_F^\times/ F^\times 
\widehat{\go}^{\times}_{\gn,1}  D(\R)_+$
(resp.  $\Cl^+_{F,\gn p^\infty}:=\limproj 
\Cl^+_{F,\gn p^r}=\A_F^\times/\overline{ F^\times 
\widehat{\go}^{\times(p)}_{\gn,1}D(\R)_+}$). 
We choose the convention in which an uniformizer corresponds to a 
geometric Frobenius. We have the following  exact sequence  
\begin{equation}\label{Irr2}
1 \rightarrow 
(\prod_{\gp\mid p}\go_{\gp}^\times)/
\overline{\{\epsilon\in\go_+^\times | \epsilon-1\in \gn\}}
\rightarrow  \Cl^+_{F,\gn p^\infty}
\rightarrow  \Cl^+_{F,\gn}\rightarrow 1.
\end{equation}
By  Cor.\ref{poids-modere}, for each $\gp\mid
 p$, $\varphi_{\gal}\oplus  \varphi_{\gal}'$ is  crystalline at  $\gp$ 
of  weights $(m_{\tau},k_0\!-\!m_{\tau}\!-\!1)_{\tau\in J_{F,\gp}}$.

By (\ref{Irr2}) for every  $\epsilon\in\go_+^\times $, 
$\epsilon-1\in\gn$ we have the following equality in $\kappa$ :
\begin {equation*}
1=\varphi_{\gal}(\epsilon)=\prod_{\gp\mid p}
\varphi_{\gal,\gp}(\epsilon)=
\underset{\gp\mid p}{\prod}\enspace
\underset{\tau\in J_{F,\gp}}{\prod}\tau(\epsilon)^{m_\tau
\text{ or } (k_0-m_\tau-1)}=\epsilon^{p(J)},
\end{equation*}
for some subset $J\subset J_F$. Because of the assumption $p>k_0$,
if $k$ is  non-parallel, then  $\epsilon^{p(J)}\!\neq\!1$ for all
 $J\subset J_F$. Thus   we obtain (ii)  and (i)  when 
$k$ is  non-parallel.

\bigskip
Assume now that  $k=k_0t$ is parallel and that for all 
$\varnothing\!\subsetneq\!J\!\subsetneq\! J_F$, there exists  $\epsilon\in\go_+^{\times}$,
$\epsilon-1\in\gn$ such that  $p$   does not divide  the non zero 
integer  $\Nm(\epsilon^{p(J)}\!-\!1)$. The same arguments
 as above show that the  restriction  to 
$\prod_{\gp\mid p}\go_{\gp}^\times$ 
 of the character  $\varphi_{\gal}$ (resp. $\varphi_{\gal}'$) 
$ \Cl^+_{F,\gn p^\infty} \rightarrow \kappa^\times$ 
is trivial (resp.  given by the  $(1-k_0)$-th  power of the norm).
By the following lemma (applied to $P=\Cl^+_{F,\gn p^\infty}$ and 
$Q=(\prod_{\gp\mid p}\go_{\gp}^\times)/
\overline{\{\epsilon\in\go_+^\times | \epsilon-1\in \gn\}} $)  there exists an  unique character
 $\widetilde{\varphi}_{\gal}$ (resp. $\widetilde{\varphi}_{\gal}'$)
$:\Cl^+_{F,\gn p^\infty} \rightarrow \mathcal{O}^\times$
lifting  $\varphi_{\gal}$ (resp. $\varphi_{\gal}'$) and  whose  restriction to 
$\prod_{\gp\mid p}\go_{\gp}^\times$ is also
trivial (resp.   given by the $(1-k_0)$-th  power of the norm).

\begin{lemma} \label{lifting}
Let $P$ be an abelian group  and  $Q$ be a subgroup, such that 
the  factor  group $P/Q$ is finite. Let $\varphi_{P}:P \rightarrow 
\kappa^\times$  and $\widetilde{\varphi_{Q}}:Q \rightarrow 
\mathcal{O}^\times$  be two  characters such  that 
$\varphi_{P}|_{Q}= \widetilde{\varphi_{Q}}\mod{p}$. Then, there  exists an unique 
character  $\widetilde{\varphi_{P}}:P \rightarrow \mathcal{O}^\times$, 
whose restriction to $Q$ is $\widetilde{\varphi_{Q}}$ and such that 
$\widetilde{\varphi_{P}}\mod{p}=\varphi_{P}$.
\end{lemma}

For $x\in \A_F^\times$, we put   $\varphi(x):= \widetilde{\varphi}_{\gal}(x)$ and 
 $\varphi'(x):= \widetilde{\varphi}_{\gal}'(x)x_p^{-k}
x_{\infty}^{k}$.  Then $\varphi$ (resp. $\varphi'$)  is a  
 Hecke character of $F$, of conductor dividing $\gn$ 
and infinity type $0$ (resp. $(1-k_0)t$).  It is crucial to observe that
 there are only finitely many such  $\varphi$ and  $\varphi'$.


Assume now that for infinitely many primes $p$,  
$\overline{\rho}$   is reducible. Then there exist 
Hecke characters $\varphi$ and  $\varphi'$ as above, such that 
for infinitely many  primes $p$ we have  $\overline{\rho}^{\mathrm{s.s.}}\equiv
\varphi_{\gal}\oplus\varphi_{\gal}'\pmod{\cP}$.
Hence for each prime $v$ of $F$ not dividing $\gn$ we have  
$c(f,v)\equiv\varphi(\varpi_{v})+\varphi'(\varpi_{v})\pmod{\cP}$ 
for infinitely many  $\cP$'s and  hence
$c(f,v)=\varphi(\varpi_{v})+\varphi'(\varpi_{v})$. By the Cebotarev
Density Theorem  we obtain
$\rho^{\mathrm{s.s.}}=\varphi\oplus\varphi'$.
This  contradicts the absolute irreducibility of $\rho$ (see \cite{Ta2}). \hfill$\square$



\subsection{The  exceptional  case.}\label{exceptional}

The aim of this paragraph is to find a bound for the primes $p$ such
that  $\pr(\overline{\rho}_{f,p}(\G_F))$ is  isomorphic to one  of the groups 
$A_4$, $S_4$ or $A_5$. We will only use the fact that 
 the elements of these groups   are of order  at most  $5$.

Assume that $\pr(\overline{\rho}_{f,p}(\G_F))\cong A_4, S_4$ or $A_5$.
By Cor.\ref{poids-modere} there exist
$\epsilon_\tau\!\in\!\{\pm 1\}$, $\tau\in J_F$, such that   for all 
$\gp\mid p$ and  for any generator $x$ of 
$\F_{p^h}^\times$, where $h=|J_{F,\gp}|$,  the element
$$\prod_{\tau\in \Gal(\F_{p^h}\!/\!\F_{p})}\!\!\!
\tau(x)^{\epsilon_\tau(k_\tau-1)}\in \F_{p^h}^\times$$
\noindent belongs  to  $\pr(\overline{\rho}(I_{\gp}))$ 
and    is therefore of order   at most  $5$  (if {\bf (ORD)} holds
 we may   assume that   $\epsilon_\tau=1$ for all $\tau$). Denote by 
$\tau_1,...,\tau_{h}$ the elements of $J_{F,\gp}$. Then 
$$\epsilon_{\tau_1}(k_{\tau_1}-1)+\epsilon_{\tau_2} p(k_{\tau_2}-1)+...+
\epsilon_{\tau_{h}}  p^{h-1}(k_{\tau_{h}}-1)\in \Z/(p^h-1)$$
 is of order  $\leq 5$, hence 
$5((k_{\tau_1}-1)+p(k_{\tau_2}-1)+...+p^{h-1}
(k_{\tau_{h}}-1))\geq p^h-1$.

If  we replace the generator $x$  by $x^p,x^{p^2},...,x^{p^{h-1}}$
and then sum these  inequalities we find 
$5\underset{\tau\in J_{F,\gp}}{\sum}(k_\tau-1)
\geq |J_{F,\gp}|(p-1)$. Hence 
 $5\underset{\tau\in J_F}{\sum}(k_\tau-1) \geq d(p-1)$. 

We conclude that $\pr(\overline{\rho}(\G_F))$ cannot be 
isomorphic to $A_4$, $S_4$ or $A_5$ if 

$$d(p-1)>5\sum_{\tau\in J_F}(k_\tau-1).$$


Note  that this assumption follows from  {\bf (II)} if  $d\geq 5$.

\subsection{The dihedral case.}\label{dihedral}


In this paragraph we study the case when  $\pr(\overline{\rho}_{f,p}(\G_F))$
 is isomorphic to the dihedral group
 $D_{2n}$, where $n\geq 3$ is an  integer  prime to  $p$. Let
$C_n$ be  the cyclic subgroup  of order  $n$ of $D_{2n}$.
Since  $\pr^{-1}(C_n)$ is a commutative group containing only semi-simple
 elements ($p$ does not divide $n$), it is diagonalizable.
 Since  $\pr^{-1}(D_{2n}\bs C_n)$ is 
contained in the normalizer of $\pr^{-1}(C_n)$, it
is contained in the set of anti-diagonal   matrices. 

Let $\varepsilon:D_{2n}\rightarrow \{\pm 1\}$ be the signature map and 
let $K$ be the fixed field  of $\ker(\varepsilon\circ \pr 
\circ \overline{\rho})$. The extension $K/F$ is  quadratic  and 
 unramified outside $\gn p$. 


\medskip
Let $c$ be the  non-trivial  element of the Galois group  
$\Gal(K/F)$. As $\overline{\rho}$ is absolutely irreducible, 
but  $\overline{\rho}_{|\G_K}$ is not, there exists a 
character $\varphi_{\gal}:\G_K \rightarrow \kappa^\times$
distinct from  its Galois conjugate  $\varphi_{\gal}^c$ and such that 
$\overline{\rho}_{|\G_K}=\varphi_{\gal}\oplus \varphi_{\gal}^c$.

\begin{lemma}  Let  $\gp$ be a  prime of  $F$ dividing $p$.
Assume $p\neq 2k_\tau-1$,  for  $\tau\in J_{F,\gp}$. Then

\rm{(i)}  the field $K$ is  unramified at  $\gp$,

\rm{(ii)} the prime  $\gp$ splits in $K$ as $\mathfrak{P}\mathfrak{P}^c$, 
and  $\varphi_{\gal}$  is crystalline at $\mathfrak{P}$ (resp. $\mathfrak{P}^c$) 
of weights $(p_\tau)_{\tau\in J_{F,\gp}}$ (resp. 
$(q_\tau)_{\tau\in J_{F,\gp}}$), where $\{p_\tau,q_\tau\}=
\{m_{\tau},k_0-m_{\tau}-1\}$ for each  $\tau\in J_{F,\gp}$.
\end{lemma}

{\bf Proof : } (i)
Otherwise $\overline{\rho}(I_{\gp})$ would contain at least 
one anti-diagonal matrix and the  basis vectors would not be 
eigen for  $\overline{\rho}(I_{\gp})$. But the group 
$\overline{\rho}(I_{\gp})$ has  a common eigenvector. 
Hence, the elements  of  $\pr( \overline{\rho}(I_{\gp}))$ 
would be  of order $\leq 2$. Using the computations of 
\S\ref{exceptional} and $p>k_0$, we find that for all 
$\tau\in J_{F,\gp}$ we have  $2(k_\tau-1)= p-1$.
Contradiction.

(ii) By  Cor.\ref{poids-modere}, $\varphi_{\gal}\oplus\varphi_{\gal}^c$
is crystalline of  weights
$(m_{\tau},k_0-m_{\tau}-1)_{\tau\in J_{F,\gp}}$.
For each prime $\mathfrak{P}$ of $K$ dividing $\gp$, we have
 $(\varphi_{\gal}\varphi_{\gal}^c)_{|I_{\mathfrak{P}}}=
\omega^{1-k_0}_{|I_{\mathfrak{P}}}$. Since $m_{\tau}<k_0-m_{\tau}-1<p-1$
we deduce that ${\varphi_{\gal}}_{|I_{\mathfrak{P}}}\neq
{\varphi_{\gal}^c}_{|I_{\mathfrak{P}}}\cong
{\varphi_{\gal}}_{|I_{\mathfrak{P}^c}}$ and so $\gp$ splits in $K$. 
\hfill $\square$


\medskip

Let $\gO$ be the integer ring of $K$, and 
 $\widehat{\gO}$ its profinite completion.
Denote by $\widehat{\gO}^{\times}_{\gn,1}$
the subgroup of $\widehat{\gO}^{\times}$ of elements 
$\equiv 1\pmod{\gn}$. Then 
$\widehat{\gO}^{\times}_{\gn,1}$ is a product of
its $p$-part $\prod_{\mathfrak{P}\mid p}\gO_{\mathfrak{P}}^\times$
and its part  outside $p$, denoted by 
$\widehat{\gO}^{\times(p)}_{\gn,1}$.

By the global class field theory, the Galois group of  the  maximal 
$\gn$-ramified (resp. $\gn p^\infty$-ramified)
abelian extension  of $K$ is isomorphic to 
$\Cl_{K,\gn}:=\A_K^\times/ K^\times 
\widehat{\gO}^{\times}_{\gn,1}  K_{\infty}^\times$
(resp.  to $\Cl_{K,\gn p^\infty}:=\A_K^\times/\overline{ K^\times 
\widehat{\gO}^{\times(p)}_{\gn,1}K_{\infty}^\times}$). 
 We have the following  exact sequence  :
\begin{equation}\label{CM}
1 \rightarrow 
(\prod_{\mathfrak{P}\mid p}\gO_{\mathfrak{P}}^\times)/
\overline{\{\epsilon\in\gO^\times | \epsilon-1\in \gn\}}
\rightarrow  \Cl_{K,\gn p^\infty}
\rightarrow  \Cl_{K,\gn}\rightarrow 1
\end{equation}

\begin{prop} \label{not-CM}
{\rm (i)} Assume that  for all $\tau\in J_F$, $p\neq 2k_\tau-1$ and 
that $\pr(\overline{\rho}(\G_F))$ is dihedral. Let $K/F$ be the 
quadratic extension defined above. Then 

 $\relbar$ either $K$ is  CM and  there exists 
a  Hecke character   $\varphi$ of  $K$ of conductor of norm
dividing $\gn\Delta_{K/F}^{-1}$ and  infinity type 
$(m_\tau,k_0\!-\!1\!-\!m_\tau)_{\tau\in J_F}$ such that $\rho\equiv \Ind_{K}^{F}
\varphi \pmod{\cP}$,

 $\relbar$ either $K$ is not  CM and we can choose  
places $\widetilde{\tau}$ of $K$ above each $\tau\in J_F$
such that for all $\epsilon\in\gO^{\times}$, $\epsilon-1\in\gn$ the prime  $p$  divides 
 $\displaystyle \N_{K/\Q}\left(\prod_{\tau\in J_F}
\widetilde{\tau}(\epsilon)^{m_{\tau}}\widetilde{\tau}(c(\epsilon))^{k_0-m_{\tau}-1}\!-\!1\right)$. 

 {\rm (ii)} Assume that   $f$ is not a theta series. Then for 
all but finitely many primes $p$ the group
 $\pr(\overline{\rho}_{f,p}(\G_F))$ is not dihedral.
\end{prop}


\begin{rque}
(i) The  primes $p$ for which the   congruence $\rho\equiv \Ind_{K}^{F} 
\varphi\pmod{\cP}$ may occur should  be  controlled by the special value  
 of the $L$-function  associated to the  CM character   $\varphi/\varphi^c$ 
(in the  elliptic case it is proved by  Hida \cite{hida64} and 
Ribet \cite{ribet4}; see also  Thms A and B). 

(ii) We would like to thank E. Ghate for having pointed us out the
possible existence of dihedral primes for non CM fields $K$. It
would be interesting to explore the converse statement, that is to say 
to try to construct,  for a given prime $p$ dividing the above norms,  a newform $f$ 
such that $\pr(\overline{\rho}_{f,p}(\G_F))$ is dihedral. 
\end{rque}

{\bf Proof : } (i) By (\ref{CM}) and the above lemma, we have 
$\varphi_{\gal}: \Cl_{K,\gn p^\infty} \rightarrow \kappa^\times$
whose restriction to 
$\prod_{\mathfrak{P}\mid p}\gO_{\mathfrak{P}}^\times $
is given by the reduction modulo $p$ of   an algebraic character
$x\mapsto x^{\widetilde{k}}$, where  $\widetilde{k}=\sum_{\tau\in J_F}m_{\tau}
\widetilde{\tau}+(k_0-m_{\tau}-1)\widetilde{\tau}\circ c$, for some
choice of places $\widetilde{\tau}$ of $K$ above $\tau\in J_F$. 

We observe that the  character   $x\mapsto x^{\widetilde{k}}$  is
trivial on $\go_+^{\times}$,  whereas 
it is only trivial modulo $p$ on  $\{\epsilon\in\gO^\times | \epsilon-1\in \gn\}$. 
The case  when $K$ is not  CM follows imediately.

Assume now that $K$ is CM. In this case 
$\{\epsilon\in\go_+^\times | \epsilon-1\in \gn\}$ is 
a finite index subgroup of $\{\epsilon\in\gO^\times | \epsilon-1\in
\gn\}$. Since $\ker(\cO^\times\rightarrow \kappa^\times)$ does not
contain elements of finite order, the above  character is 
trivial on $\{\epsilon\in\gO^\times | \epsilon-1\in \gn\}$.


By the  lemma \ref{lifting} (applied to $P=\Cl_{K,\gn p^\infty}$ and 
$Q=(\prod_{\mathfrak{P}\mid p}\gO_{\mathfrak{P}}^\times)/
\overline{\{\epsilon\in\gO^\times | \epsilon-1\in \gn\}}$) there exists a lift  
$\widetilde{\varphi}_{\gal}:\Cl_{K,\gn p^\infty} \rightarrow
\cO^\times $ whose restriction to 
$\prod_{\mathfrak{P}\mid p}\gO_{\mathfrak{P}}^\times$ is given by 
$x\mapsto x^{\widetilde{k}}$. 

We  put    $\varphi(x):= \widetilde{\varphi}_{\gal}(x)x_p^{-\widetilde{k}}
x_{\infty}^{\widetilde{k}}$.  Then $\varphi$   is a  Hecke character of $K$ as desired.

(ii) There are finitely many fields $K$ as above. For those $K$ that are
not CM it is enough to choose $\epsilon\in\gO^\times$, $\epsilon-1\in
\gn$
of infinite order in $\gO^\times\!/\!\go^\times$. 

 For each of the CM fields  $K$ that are only finitely many
 characters $\varphi$ as above.
Therefore, if  $\pr(\overline{\rho}(\G_F))$ is dihedral
for infinitely many  primes $p$, then there would exist $K$ and 
$\varphi$ as above, such that the congruence 
$\rho\equiv \Ind_{K}^{F} \varphi \pmod{\cP}$ happens for  infinitely many  
$\cP$'s. Hence  $f$ would be  
equal to the theta series associated to $\varphi$. \hfill $\square$


\subsection{The image  of $\overline{\rho}$ is ``large''.}


\begin{theo}\label{dickson} $\mathrm{(Dickson)}$
{\rm(i)} An  irreducible subgroup  of  $\PSL_2(\kappa)$  of order divisible 
by  $p$ is  conjugated   inside  $\PGL_2(\kappa)$  to $\PSL_2(\F_{q})$
 or to $\PGL_2(\F_{q})$,  for some  power $q$  of $p$.

{\rm(ii)} An  irreducible subgroup  of  $\PSL_2(\kappa)$  of order prime to 
$p$  is either dihedral, either isomorphic to one of the 
groups $A_4$, $S_4$ or $A_5$.
\end{theo}


As an application of this theorem, Prop.\ref{Irr}, 
Prop.\ref{not-CM} and {\S}\ref{exceptional} we obtain  the 
following 

\begin{prop}\label{LI}
 Assume that $f\in S_k(\gn,\psi)$ is a newform, which is 
not a theta series. Then for  all but finitely many primes $p$,
the image of the $p$-adic representation 
$\overline{\rho}$ associated to $f$ is  large, in the following sense :

\smallskip
$\mathrm{\bf (LI_{\overline{\rho}})}$ there exists a power $q$ of $p$ 
such that $\SL_2(\F_q)\!\subset\! 
\im(\overline{\rho}) \!\subset\! \kappa^\times \GL_2(\F_q)$.
\end{prop}

\medskip
Let $\widehat{F}$ be the compositum of   $\widetilde{F}$
and  of the subfield of $\overline{\Q}$ fixed  by the Galois group 
$\ker(\overline{\psi}\omega^{n_0})$.
The extension $\widehat{F}/F$ is Galois and unramified at $p$,
because $\widetilde{F}$  is unramified at $p$ and $\psi$ is of conductor 
prime to $p$. Therefore  $\G_{\widehat{F}}$ is a normal 
subgroup  of $\G_F$  containing  the inertia subgroups  
$I_{\gp}$, for all $\gp$ dividing $p$.

We put $\mathcal{D}=\det(\overline{\rho}(\G_{\widehat{F}}))=
(\F_p^\times)^{1-k_0}$.


\begin{prop}\label{oneform} Assume $\mathrm{\bf (LI_{\overline{\rho}})}$.
Then  there exists a power $q$ of $p$ such that,

either   $\overline{\rho}(\G_{\widehat{F}})=\GL_2(\F_q)^\mathcal{D}:=
\{x\in \GL_2(\F_q)\enspace |\enspace  \det(x)\in \mathcal{D}\}$,

either  $\overline{\rho}(\G_{\widehat{F}})=
(\F_{q^2}^\times \GL_2(\F_q))^\mathcal{D}:=
\{x\in\F_{q^2}^\times \GL_2(\F_q) 
\enspace|\enspace \det(x)\in \mathcal{D}\}$ .
\end{prop}


{\bf Proof : }  We first show that  $\pr(\overline{\rho}(\G_{\widehat{F}}))$
is still  irreducible of order  divisible by  $p$.  By 
$\mathrm{\bf (LI_{\overline{\rho}})}$  the group $\pr(\overline{\rho}(\G_F))$ 
 is isomorphic  to   $\PSL_2(\F_q)$ or $\PGL_2(\F_q)$. The group 
$\pr(\overline{\rho}(\G_{\widehat{F}}))$
is a   non-trivial normal subgroup of $\pr({\im}(\overline{\rho}))$ 
(because it  contains  $\pr(\overline{\rho}(I_p))$ and  $p\!>\!k_0$; 
see  Cor.\ref{poids-modere}). As $\PSL_2(\F_q)$ is a simple group of
 index   $2$ in  the group $\PGL_2(\F_q)$, we deduce 
$$\PSL_2(\F_q)\subset \pr(\overline{\rho}(\G_{\widehat{F}}))\subset
 \pr(\overline{\rho}(\G_F)) \subset\PGL_2(\F_q).$$

\begin{lemma}\label{sl2}
Let $H$ be a group of center $Z$ and let $\pr:H\rightarrow H/Z$ the 
canonical projection. Let $P$ and $Q$ be two subgroups of $H$ such that 
 $\pr(P)\supset \pr(Q)$. Assume moreover that  $Q$ does not have 
non-trivial abelian quotients. Then $P\supset Q$.
\end{lemma}

It follows from this lemma that $\overline{\rho}(\G_{\widehat{F}})\supset
\SL_2(\F_q)$, hence 
 $$(\kappa^\times \GL_2(\F_q))^\mathcal{D}\supset 
 \overline{\rho}(\G_{\widehat{F}}) \supset \GL_2(\F_q)^\mathcal{D}.$$

Since  $[(\kappa^\times \GL_2(\F_q))^\mathcal{D}:
\GL_2(\F_q)^\mathcal{D}]\leq 2$ we're done . \hfill$\square$

\bigskip
Let $y\in \F_{q^2}\backslash \F_q$ be  such that   $y^2\in \F_q$. 
Then  $(\F_{q^2}^\times \GL_2(\F_q))^\mathcal{D}=
\GL_2(\F_q)^\mathcal{D}\amalg(y\GL_2(\F_q))^\mathcal{D}$ and  hence
$\tr((\F_{q^2}^\times \GL_2(\F_q))^\mathcal{D})= \F_q\cup y\F_q$.
Therefore,  the $\F_p$-algebra generated by the traces of the 
elements of $(\F_{q^2}^\times \GL_2(\F_q))^\mathcal{D}$ is 
 $\F_{q^2}$, while 
 $\pr((\F_{q^2}^\times \GL_2(\F_q))^\mathcal{D})\subset \PGL_2(\F_q)$.
This reflects the existence of a congruence with a form having  inner twists.



\subsection{ The image  of $\Ind_F^{\Q}\overline{\rho}$ is 
 ``large''.}\label{ind-image}

We assume in this paragraph that $\mathrm{\bf (LI_{\overline{\rho}})}$ holds.

\smallskip
By Prop.\ref{oneform} there exists a power $q$ of $p$ such that 
$\pr(\overline{\rho}(\G_{\widehat{F}}))=\PSL_2(\F_q)$ or $\PGL_2(\F_q)$. 
Consider the representation  $\pr(\Ind_F^{\Q}\overline{\rho}): 
\G_{\widehat{F}}\rightarrow\PGL_2(\F_q)^{J_F}$.

Any automorphism of the simple  group $\PSL_2(\F_q)$ is the
composition of  a conjugation  by an element of $\PGL_2(\F_q)$ and 
of a Galois automorphism of $\F_q$. By a lemma of Serre 
(see \cite{ribet3}), there exist a partition $J_F=\coprod_{i\in I}J_F^i$
 and for each  $i\in I$, and for each  $\tau\in J_F^i$   an element
  $\sigma_{i,\tau}\in \Gal(\F_q\!/\!\F_p)$ such that 
$$\pr(\phi(\SL_2(\F_q)^I))
\subset \pr(\Ind_F^{\Q}\overline{\rho}(\G_{\widehat{F}}))\subset
  \pr(\phi(\GL_2(\F_q)^I)),$$
$$ {\rm where }\enspace \phi=(\phi^i)_{i\in I} :\GL_2(\F_q)^I
\hookrightarrow  \GL_2(\F_q)^{J_F} \enspace \text{\rm is given by   }\enspace
 \phi^i(M_i)= (M_i^{\sigma_{i,\tau}})_{\tau\in J_F^i }.$$

Keeping these notations, we introduce the following assumption on the image of 
$\Ind_F^{\Q}\overline{\rho}$

\medskip 
\medskip 
$\mathrm{{\bf (LI}_{Ind \overline{\rho}}{\bf )}}$ the condition 
$\mathrm{\bf (LI_{\overline{\rho}})}$ holds and $\forall$
$i\in I$, $\forall$ $\tau,\tau'\in J_F^i$ $(\tau\neq \tau' \Rightarrow 
\sigma_{i,\tau}\neq \sigma_{i,\tau'})$.

\medskip 
We  now  introduce a genericity  assumption on the weight $k$.
\begin{defin}
We say that the weight $k\in \Z[J_F]$ is non-induced, if there does not exist 
a strict subfield $F'$ of $F$ and a weight $k'\in \Z[J_{F'}]$ such that 
for each $\tau\in J_F$, $k_\tau=k'_{\tau|_{F'}}$.
\end{defin}


\begin{rque}\label{non-IND2}
Define $\widetilde{k}=\sum_{\widetilde{\tau}\in J_{\widetilde{F}}}
k_{\widetilde{\tau}}\widetilde{\tau}\in \Z[J_{\widetilde{F}}]$
 by  putting $k_{\widetilde{\tau}}= k_{\widetilde{\tau}|_F}$, for all 
$\widetilde{\tau}\in J_{\widetilde{F}}$. 
The group   $\G_{\Q}$ acts on $\Z[J_{\widetilde{F}}]$ by 
$\widetilde{k}=\sum_{\widetilde{\tau}\in J_{\widetilde{F}}}
k_{\widetilde{\tau}}\widetilde{\tau}\mapsto 
\widetilde{k}^{\widetilde{\tau}'}=\sum_{\widetilde{\tau}\in J_{\widetilde{F}}}
k_{\widetilde{\tau}\widetilde{\tau}'}\widetilde{\tau}$. It is easy to see
that $k\in \Z[J_F]$ is non-induced, if and only if 
 $\{\widetilde{\tau}'\in \G_{\Q} \enspace |\enspace 
 \widetilde{k}=\widetilde{k}^{\widetilde{\tau}'}\}$ equals $\G_F$.
\end{rque}



\begin{prop}\label{k-not-IND}
Assume that $\mathrm{\bf (LI_{\overline{\rho}})}$ hold and $k$ is non-induced.
Assume moreover that for all $\tau\neq\tau'\in J_F$,
 $p\neq k_\tau+k_{\tau'}-1$. 
 Then $\mathrm{{\bf (LI}_{Ind \overline{\rho}}{\bf )}}$ hold.
\end{prop}


{\bf Proof : } Let $\widetilde{\tau}_1, \widetilde{\tau}_2\in \G_{\Q}$
be such that  for all $g\in \G_{\widehat{F}}$ we have 
$\pr(\overline{\rho}(\widetilde{\tau}_1^{-1}g\widetilde{\tau}_1))=
\pr(\overline{\rho}(\widetilde{\tau}_2^{-1}g\widetilde{\tau}_2))$. 
We have to prove that  $\widetilde{\tau}_1^{-1}\widetilde{\tau}_2\in \G_F$.
We put $\overline{\rho}_i(g)=
\overline{\rho}(\widetilde{\tau}_i^{-1}g\widetilde{\tau}_i)$  ($i=1,2$).


Let $\mathfrak{P}$ be  a prime ideal  of $\widehat{F}$ above 
a  prime ideal  $\gp$ of $F$ dividing  $p$. Denote by   $h_i'$ (resp.
 $h_i$) the residual degree of $\mathfrak{P}^{\widetilde{\tau}_i}$ (resp. of 
$\gp^{\tau_i}$), $i=1,2$. By   Cor.\ref{poids-modere} 
we have 
$\overline{\rho}_i|_{I_{\mathfrak{P}}}^{\enspace\mathrm{s.s.}}=
\varepsilon_{i}\oplus \delta_i,$ 
where $\varepsilon_{i}$ (resp. $\delta_i$)  $I_{\mathfrak{P}}
\rightarrow  I_{\mathfrak{P}^{\widetilde{\tau}_i}} \rightarrow 
\F_{p^{h_i'}}^\times \rightarrow \F_{p^{h_i}}^\times \rightarrow 
\kappa^\times$ is obtained by composing 
the conjugation by  $\widetilde{\tau_i}$, the projection on the tame 
inertia, the norm map, and the character 
$x\mapsto\underset{\tau\in J_{F,\gp^{\tau_i}}}\prod 
\tau(x)^{p_\tau}$ (resp. $x\mapsto\underset{\tau\in 
J_{F,\gp^{\tau_i}}} \prod  \tau(x)^{q_\tau}$),  
where $\{p_\tau,q_\tau\}=\{m_\tau,k_0-1-m_\tau\}$.
In the case {\rm(ORD)} we can even assume that for all
$\tau\in J_F$, we have $p_\tau=m_\tau$ and $q_\tau=k_0-1-m_\tau$. 

Note that $\varepsilon_1\delta_1= \varepsilon_2\delta_2=\omega^{1-k_0}$. 
Since  $I_{\mathfrak{P}}\subset \G_{\widehat{F}}$ and 
$\pr\circ\overline{\rho}_1=\pr\circ\overline{\rho}_2$ on $\G_{\widehat{F}}$,
we may assume that  $\varepsilon_1/\delta_1=\varepsilon_2/\delta_2$. 
By varying   $\mathfrak{P}$ 
we deduce that for all $\widetilde{\tau}\in J_{\widetilde{F}}$, 
$\widetilde{k}_{\widetilde{\tau}}=
\widetilde{k}_{\widetilde{\tau}\widetilde{\tau}_1^{-1}\widetilde{\tau}_2}$   
(here we use   $p>k_0$ and the assumption  $p\neq k_\tau+k_{\tau'}-1$). 
As $k$ is non-induced, it follows from the remark \ref{non-IND2} that 
 $\widetilde{\tau}_1^{-1}\widetilde{\tau}_2\in \G_F$. \hfill $\square$




The following corollary  generalizes a  result of Ribet \cite{ribet3} 
on the image of a Galois representation associated to a 
family of classical modular forms, to the case of the 
family of internal conjugates of a   Hilbert modular form. 


\begin{cor} Assume that $\mathrm{\bf (LI_{\overline{\rho}})}$ hold 
and $k$ is non-induced. Assume moreover that  $p>2k_0$ is totally 
split in $F$. Then, 
$$(\GL_2(\F_q)^{J_F})^{\mathcal{D}}
\!\subset\! \Ind_F^{\Q}\overline{\rho}(\G_{\widehat{F}})\!\subset\!
(\overline{\rho}(\G_{\widehat{F}})^{J_F})^{\mathcal{D}} \text{ , where } 
\mathcal{D}=\F_p^{\times(1-k_0)}.$$ 
\end{cor}



$${\rm Put } \enspace
 H(\F_q)=\left(\prod_{i\in I}\GL_2(\F_q)\right)^\mathcal{D}:=
\left\{ (M_i)_{i\in I}\in  \prod_{i\in I}\GL_2(\F_q)\Big{|}\enspace\exists
 \delta\in\mathcal{D},\enspace \forall i, \enspace  \det(M_i)=\delta\right\}.$$


\begin{lemma}\label{borel} Assume $p\!>\!2k_0$. Then,
 
{\rm(i)} for all $\gp$ dividing $p$, $\overline{\rho}( I_{\gp})$
is contained (possibly after conjugation by  an element of $\GL_2(\F_q)$), 
either in the Borel's   subgroup of   $\GL_2(\F_q)$, either in the 
 non-split torus  of $\GL_2(\F_q)$.
The second case cannot occur if  $f$ is ordinary at $p$. 
 
{\rm(ii)} $\Ind_F^{\Q}\overline{\rho}(I_p)\subset  \phi(H(\F_q))$.
\end{lemma}

{\bf Proof : }  (i) Put $h:=|J_{F,\gp}|$. By   
Cor.\ref{poids-modere}  we have 
$\overline{\rho}|_{I_{\gp}}^{\enspace\mathrm{s.s.}}=
\varepsilon_{\gp}\oplus \delta_{\gp}$, 
 where $\varepsilon_{\gp}$ (resp. $\delta_{\gp}$)
$I_{\gp}\rightarrow  \kappa^\times $ are obtained by
composing the tame inertia map $I_{\gp}\rightarrow
\F_{p^h}^\times$ and the character $\varepsilon: x\mapsto
\underset{\tau\in J_{F,\gp}}{\prod}  \tau(x)^{p_\tau}$ (resp. $\delta
:x\mapsto \underset{\tau\in J_{F,\gp}}{\prod}\tau(x)^{q_\tau}$),
where $\{p_\tau,q_\tau\}=\{m_\tau,k_0-1-m_\tau\}$.

Let $x_h$ be a generator of $\F_{p^h}^\times$. As the traces of the  elements 
of $\overline{\rho}(\G_{\widehat{F}})$ are  in $F_q\coprod y\F_q$,
we have  $(\varepsilon(x_h)+\delta(x_h))^2\in\F_q$ and therefore
$\varepsilon(x_h)^2+\delta(x_h)^2\in\F_q$.

If $\varepsilon(x_h)^2,\delta(x_h)^2\in\F_q^\times$, then 
it is easy to see that $\varepsilon(x_h),\delta(x_h)\in\F_q^\times$
(we use $p>k_0$ and $p\neq 2k_\tau-1$). Therefore $I_{\gp}$
fixes a  $\F_q$-rational line, and $\overline{\rho}( I_{\gp})$
 is contained  in a Borel subgroup of  $\GL_2(\F_q)$.

Otherwise $\varepsilon(x_h)^2$ and $\delta(x_h)^2$ are conjugated by the 
non-trivial element  of $\Gal(\F_{q^2}/\F_q)$, hence 
$\varepsilon(x_h)^2=\delta(x_h)^{2q}$. Since  $p\!>\!2k_0$, we have
$\varepsilon(x_h)=\delta(x_h)^q$ and so 
$\varepsilon(x_h)+\delta(x_h)^q\in \F_q^\times$. Hence 
$\tr(\overline{\rho}( I_{\gp}))\subset \F_q$, and therefore
$\overline{\rho}( I_{\gp})\!\subset\! \GL_2(\F_q)$.
In this case  $\overline{\rho}( I_{\gp})$ is contained in a 
non-split torus of $\GL_2(\F_q)$.
If  $f$ is ordinary at $p$, then for all $\tau$,  
$p_\tau=m_\tau<k_0-m_\tau-1=q_\tau$ and therefore
$\varepsilon(x_h)^2\neq \delta(x_h)^{2q}$, and 
so the second case cannot occur.

(ii) The determinant condition ${}^\mathcal{D}$ being satisfied,  all we 
have to check is the following : for all $i\in I$ and $\tau,\tau'\in J_F^i$
the character 
$$ I_p \rightarrow \{\pm 1\} \text{ , } g\mapsto (\overline{\rho}(\widetilde{\sigma}_{i,\tau}^{-1}g
\widetilde{\sigma}_{i,\tau}))^{-1} (\overline{\rho}(\widetilde{\sigma}_{i,\tau'}^{-1}g
\widetilde{\sigma}_{i,\tau'}))$$
is trivial. This follows, as in the proof of the 
Prop.\ref{k-not-IND}, from the assumption $p\!>\!2k_0$. \hfill $\square$


%\begin{lemma}\label{cyclic}
%Take $h\!\geq\! 1$ and let 
%$\delta:\F_{p^h}^\times \rightarrow\F_{p^h}^\times$ 
%be a group homomorphism given by $x\mapsto 
%\underset{\tau\in \Gal(\F_{p^h}\!/\!\F_p)}{\prod} \tau(x)^{a_\tau}$,
%where  for all  $\tau$, $0\leq a_\tau\!< \! \frac{p-1}{2}$. 
%If  $\im(\delta^2)\!\subset\! 
%\F_q^\times$, then   $\im(\delta)\!\subset\! \F_q^\times$.
%\end{lemma}


\begin{lemma}\label{big-image}
$\phi(H(\F_q)) \subset \Ind_F^{\Q}\overline{\rho}(\G_{\widehat{F}}) $,
\end{lemma}


{\bf Proof : } We have seen in the beginning of this paragraph that
$\pr(\phi(\SL_2(\F_q)^I))\subset 
\pr(\Ind_F^{\Q}\overline{\rho}(\G_{\widehat{F}}))$. 
By lemma \ref{sl2}, we deduce that 
$\phi(\PSL_2(\F_q)^I)\subset \Ind_F^{\Q}\overline{\rho}(\G_{\widehat{F}})$.

As $\phi(H(\F_q))=\phi(\SL_2(\F_q)^I) \Ind_F^{\Q}\overline{\rho}(I_p)$, we're done.
\hfill $\square$

\begin{prop}\label{LI-Ind}
Assume that $f$ is not a theta series and that 
$\mathrm{{\bf (LI}_{Ind \overline{\rho}}{\bf )}}$  does not hold
for infinitely many primes $p$. Then, there exists 
$\tau\in J_F$, $ \tau\neq \mathrm{id}$ and a finite order Hecke character
$\varepsilon$ of $\widetilde{F}$ of conductor dividing 
$\Nm(\gn)$, such that for all prime 
$v\nmid \Nm(\gn)$ which splits completely 
in $\widetilde{F}$, we have $c(f_{\tau},v)=\varepsilon(v)c(f,v)$.
\end{prop}


{\bf Proof : } Since $f$ is not a theta series, we know that 
$\mathrm{{\bf (LI}_{\overline{\rho}}{\bf )}}$ hold for all
but finitely many primes $\p$. Take such a $p$, and 
assume that $\mathrm{{\bf (LI}_{Ind \overline{\rho}}{\bf )}}$ does not hold.
 Then there exist $\widetilde{\tau}_1,\widetilde{\tau}_2 \in \G_{\Q}$
such that $\tau:=\widetilde{\tau}_2^{-1}\widetilde{\tau}_1|_F\neq\mathrm{id} $
and for all $g\in\G_{\widehat{F}}$, 
$\pr\overline{\rho}(\widetilde{\tau}_2^{-1}g\widetilde{\tau}_1)=
\pr\overline{\rho}(\widetilde{\tau}_2^{-1}g\widetilde{\tau}_2)$.
As $\mathrm{{\bf (LI}_{\overline{\rho}}{\bf )}}$ hold and 
$\G_{\widehat{F}}$ is a normal subgroup of $\G_{\widetilde{F}}$, 
the above relation hold for every $g\in \G_{\widetilde{F}}$. 
Therefore, there exist a character $\varepsilon_{\gal}:\G_{\widetilde{F}}
\rightarrow \kappa^\times$, such that for all $g\in \G_{\widetilde{F}}$, 
$\overline{\rho}_{f_\tau}(g)=\varepsilon_{\gal}(g)\overline{\rho}_{f}(g)$.
Assume that $p>2k_0$. Then, 
the same argument as in the proof of Prop.\ref{k-not-IND} shows that 
$\varepsilon_{\gal}$ is unramified at primes dividing $p$.
By lemma \ref{lifting} $\varepsilon_{\gal}$ can then be 
lifted to a  finite order  Hecke character $\varepsilon$ of  $\widetilde{F}$ 
of conductor dividing $\Nm(\gn)$.  Because of the 
determinant relation $\overline{\psi_{\tau}}=\varepsilon_{\gal}^2
\overline{\psi}$, there finitely many such $\varepsilon$'s.

For every prime $v\nmid \gn p$ which splits completely  in 
$\widetilde{F}$, we have $c(f_{\tau},v)\equiv\varepsilon(v)c(f,v)\pmod{\cP}$.
If $\mathrm{{\bf (LI}_{Ind \overline{\rho}}{\bf )}}$ fails for
infinitely many $\cP$'s, then the congruence above become an 
equality. \hfill $\square$


\begin{cor} Assume that $F$ is a Galois field of odd degree and the central 
character of $\psi$ of $f$ is trivial ($F=\widehat{F}$). Assume moreover that  
$f$ is not a theta series and that
$\mathrm{{\bf (LI}_{Ind \overline{\rho}}{\bf )}}$   does not hold
for infinitely many primes $p$. Then, there exist 
a subfield $F' \subsetneq F$ and a Hilbert modular form 
$f'$ on $F'$, such that the base change of $f'$ to $F$ is  a twist of $f$ 
by a quadratic character of conductor dividing $\Nm(\gn)$.
\end{cor}

{\bf Proof : } As in the proof of Prop.\ref{LI-Ind} there exist a  quadratic 
character $\varepsilon$ of $F$ of conductor dividing $\Nm(\gn)$ and 
$\mathrm{id}\neq \tau\in \Gal(F/\Q)$ such that  we 
have $\rho_{f_\tau}= \varepsilon_{\gal}\otimes\rho$. Let $F'\subset F$ (resp.
$F_i\supset F$)  be the  fixed field of $\tau$ (resp. of 
$\ker(\varepsilon_{\tau^i})$). By assumption $F/F'$ is a cyclic extension 
of {\it odd} degree $h$. Let $F''=\prod_{i=1}^{h} F_i$. Then we have
$$\Gal(F''/F')=\{(u_1,..,u_h)\in\{\pm 1\}^h \enspace | \enspace\prod_{i=1}^{h} 
u_i=1\}\rtimes \{\tau^i\enspace | \enspace 0\leq i\leq h-1\} ,$$
where $\tau$ acts on $(u_1,..,u_h)$ by cyclic permutation. When $h=3$ the group 
$\Gal(F''/F')$ is isomorphic to $A_4$. 

The representation $\rho|_{\G_{F''}}$ is invariant by $\Gal(F''/F')$, 
but Langlangs' Cyclic Descend does not apply directly, because 
the order of $\Gal(F''/F')$ is even. Consider the  quadratic character 
$\delta=\varepsilon\cdot \varepsilon_{\tau^2}\cdot ..\cdot 
\varepsilon_{\tau^{h-1}}$. Then  the $\G_F$-representation  $\delta_{\gal}\rho$ 
is invariant by $\Gal(F/F')$,  so extends  to a representation of $\G_{F'}$.
By applying Langlangs' Cyclic Descend to $\delta\otimes f$
 we obtain $f'$ as desired. \hfill $\square$


\section{Boundary cohomology and congruence criterion.}

We recall that  $f\in S_k(\gn,\psi)$ is a  Hilbert modular newform.

\begin{defin}
We say that a  normalized eigenform $g\in  S_k(\gn,\psi)$ is 
 {\it  congruent } to  $f$ modulo   $\cP$, if  
their respective eigenvalues  for the Hecke operators   
(that is  their Fourier coefficients)  are congruent
 modulo $\cP$.

We say that a   prime $\cP$ is a 
{\it congruence prime} for $f$, if there exists 
a  normalized eigenform $g\in  S_k(\gn,\psi)$ 
 distinct from $f$ and congruent to  $f$ modulo  $\cP$.
\end{defin}

One expects that, as in the elliptic modular case 
(carried out by Hida  \cite{hida63,hida64}
and Ribet \cite{ribet4}),  the congruence  primes
for $f$ are controlled by the special value at $1$ of the 
adjoint $L$-function of $f$. Such  results  have been obtained 
 by Ghate \cite{ghate1} when $k$ is parallel. 

 Following \cite{hida63}, \cite{ghate1} and  using a
vanishing result of  the boundary cohomology,
we obtain a new result in this direction (see Thm.\ref{theo-A} and Thm.\ref{theo-B}(ii)).

\subsection{Vanishing of certain local  components  of 
the boundary  cohomology.}

We  introduce the following condition :

\medskip
\noindent {\bf (MW)}  the middle weight 
 $\frac{|p(J_F)|+|p(\varnothing)|}{2}=\frac{d(k_0-1)}{2}$ 
does not belong to   $\{|p(J)|,J\!\subset\! J_F\}$.

\medskip
This condition is automatically satisfied when  the  motivic  weight
 $d(k_0-1)$ is odd, or when $d=2$ and $k$ is non-parallel.

\begin{lemma} \label{key2}
Let $\rho_0$ be  a  representation of $\G_{\widetilde{F}}$ on a 
finite dimensional $\kappa$-vector space $W$.  Assume that 
for every $g\in \G_{\widetilde{F}}$,  the characteristic polynomial  of 
$(\otimes\Ind_F^{\Q}\overline{\rho})(g)$ annihilates $\rho_0(g)$.

{\rm (i)} If {\bf (I)}, {\bf (II)} and $\mathrm{\bf (LI_{\overline{\rho}})}$
 hold,  then for all $h\in\Z$, the weights  $h$ and $d(k_0-1)-h$ occur
with the  same multiplicity in 
 each  $\G_{\widetilde{F}}$-irreducible  subquotient  of $\rho_0$.

{\rm (ii)} If   {\bf (I)}, $\mathrm{\bf (Irr_{\overline{\rho}})}$ and {\bf (MW)} hold, and  
$p-1>\max(1,\frac{5}{d})\sum_{\tau\in J_F}(k_\tau-1)$, then
 each $\G_{\widetilde{F}}$-irreducible  subquotient  of $\rho_0$ 
 contains at least  two different  weights  for the action of 
the tame inertia at $p$. 
\end{lemma}


{\bf Proof : } We may assume  that $\rho_0$ is  irreducible. 

(i) By the lemmas \ref{borel}(ii) and \ref{big-image} we have 
$\Ind_F^{\Q}\overline{\rho}(I_p)\subset  \phi(H(\F_q))\subset 
\Ind_F^{\Q}\overline{\rho}(\G_{\widehat{F}})$. Let 
$T'$ be the torus of $H(\F_q)$ containing the image of the 
tame inertia, and $N'$ be the normalizer of $T'$ in $H(\F_q)$.
The image by $\phi$ of  $N'/T'\cong \{\pm 1\}^I$ is
the subgroup of  the Weyl group  $N/T=\{\pm 1\}^{J_F}$ of $G$
containing the  elements  which are constant on the partition 
$J_F=\coprod_{i\in I}J_F^i$.  In particular, the longest Weyl 
element $\epsilon_{J_F}$ belongs to the  image of $ N'/T'$.

Let  $x\in W$ be an eigenvector for the action of $T'$. 
By the annihilation condition, there exists a subset $J_x\subset J_F$, 
such that $I_p$ acts on $x$ by the weight $|p(J_x)|$.

Let $g_{J_F}\in \G_{\widetilde{F}}$
be such that $\Ind_F^{\Q}\overline{\rho}(g_{J_F})=\epsilon_{J_F}\mod{T'}$. 
Then $\rho_0(g_{J_F})(x)$ is of weight $|p(J_x\Delta J_F)|=d(k_0-1)-|p(J_x)|$.
Therefore, for each $h\in\Z$, $\rho_0(g_{J_F})$ gives  a  bijection 
between the eigenspaces for the tame inertia of weight $h$ and $d(k_0-1)-h$. 


(ii) If $\mathrm {\bf (LI_{\overline{\rho}})}$ hold, then  the statement 
follows from  (i) and {\bf (MW)}. Otherwise, by Prop.\ref{LI} the image 
 $\pr(\overline{\rho}(\G_F))$ is dihedral.
Since $\widetilde{F}$ is totally real, $\pr(\overline{\rho}(\G_{\widetilde{F}}))$
is also dihedral  (see {\S}\ref{dihedral}). 

Denote by $N$ the normalizer of the standard torus $T$ in $G$. Put 
$N'=\Ind_F^{\Q}\overline{\rho}(\G_{\widetilde{F}}) \subset N(\kappa)$
and $T'=N'\cap T(\kappa)$. Then $N'/T'$ is a subgroup of the 
Weyl group $\{\pm 1\}^{J_F}=N/T$ of $G$. 

As we have seen in {\S}\ref{dihedral}, the representation 
 $\Ind_F^{\Q}\overline{\rho}$ is tamely ramified at $p$ and the image 
of the inertia group $I_p$ is contained in $T'$. 

Let  $x\in W$ be an eigenvector for the action of $T'$. 
By the annihilation condition, there exists a subset $J_x\subset J_F$, 
such that $I_p$ acts on $x$ by the weight $|p(J_x)|$.
For every element $\epsilon_J\in N'/T'$, ($J\subset J_F$), 
let $g_J\in \G_{\widetilde{F}}$
be such that $\Ind_F^{\Q}\overline{\rho}(g_J)=\epsilon_J\mod{T'}$. 
Then $\rho_0(g_J)(x)$ is of weight $|p(J_x\Delta J)|$. It remains to 
show that the $|p(J_x\Delta J)|$ are not all equal when 
$\epsilon_J$ runs over the elements of $N'/T'$. Note that,  for all 
$\tau\in J_F$, the $\tau$-projection  $N'/T'\rightarrow \{\pm 1\}$ 
is a surjective homomorphism (because the group 
$\pr(\overline{\rho}_{f_{\tau}}(\G_{\widetilde{F}}))$ is also dihedral). 
Therefore,  we have : 
$$\sum_{\epsilon_J\in N'/T'} |p(J_x\Delta J)|= |N'/T'| \frac{d(k_0-1)}{2}.$$
The statement now follows from the {\bf (MW)} assumption.
\hfill $\square$

\begin{rque}\label{key-diamond} 
The (i) in the previous lemma is a generalization, from the quadratic to 
the arbitrary degree case,   of the key lemma in  \cite{diamond2}.
This  lemma is  false in general under the only assumptions 
{\bf (I)}, {\bf (II)} and $\mathrm{\bf (Irr_{\overline{\rho}})}$, when the degree is $\geq 3$. In fact,
consider the following construction in the cubic case : let $L$ be a  
Galois extension  of $\Q$ of group $A_4$, such that the cubic subfield 
$F$ fixed by the Klein group is totally real; let $K$ be a quadratic extension 
of $F$ in $L$, and consider a theta series $f$ of weight $(2,2,2)$
attached to a Hecke character of $K$; then the tensor
induced representation $\otimes\Ind_F^{\Q}\rho$ has two
irreducible four dimensional subquotients of Hodge-Tate weights 
$(0,2,2,2)$ and $(1,1,1,3)$. 
\end{rque}


Let $\T'\subset \T$ be the subalgebra generated by the Hecke operators
outside a finite set of places containing those dividing $\gn p$. 


\begin{theo} \label{boundary}
Assume that {\bf (I)}, $\mathrm{\bf (Irr_{\overline{\rho}})}$ and {\bf (MW)} hold,
and $p-1\!>\!\max(1,\frac{5}{d})\underset{\tau\in J_F}{\sum}(k_\tau\!-\!1)$. 
Denote by $\gm$ the  maximal ideal   of $\T$ corresponding to 
$f$ and  $p$ and put $\gm'=\gm\cap \T'$. Then  

{\rm (i)} the $\gm'$-torsion of the  boundary  cohomology
$\rH^\bullet_{\partial}(Y,\bV_n)(\kappa)[\gm']$ vanishes,


{\rm (ii)}  the Poincar{\'e}   pairing     
$\rH_{!}^d(Y,\bV_n(\cO))'_{\gm'} \times
\rH_{!}^d(Y,\bV_n(\cO))'_{\gm'}\rightarrow \cO$
is  a perfect duality of free $\cO$-modules of finite rank,


$\mathrm{(iii)}$ 
$\rH^\bullet(Y,\bV_n(\cO))_{\gm'}=
\rH_c^\bullet(Y,\bV_n(\cO))_{\gm'}=
\rH_{!}^\bullet(Y,\bV_n(\cO))_{\gm'}$.
\end{theo}

{\bf Proof : } (i) Consider the minimal compactification 
$Y_{\overline{\Q}}\overset{j}{\hookrightarrow} 
Y_{\overline{\Q}}^*\overset{i}{\hookleftarrow}  \partial Y_{\overline{\Q}}^*$.
The Hecke correspondences extend to  $Y_{\overline{\Q}}^*$. 
By  the  Betti-{\'e}tale comparison isomorphism, we  identify  (in a  
Hecke-equivariant way) the following two  long exact cohomology sequences :

\xymatrix{ ...\ar[r] 
& \rH_c^r(Y,\bV_n(\kappa)) \ar[r]\ar@{=}[d]
& \rH^r(Y,\bV_n(\kappa)) \ar[r]\ar@{=}[d]
& \rH^r_{\partial}(Y,\bV_n(\kappa)) \ar[r]\ar@{=}[d] & ...\\ ...\ar[r] 
& \rH^r(Y_{\overline{\Q}}^*,j_{!}\bV_n(\kappa)) \ar[r]
& \rH^r(Y_{\overline{\Q}}^*,j_{*}\bV_n(\kappa)) \ar[r]
& \rH^r(\partial Y_{\overline{\Q}}^*, i^*Rj_{*}\bV_n(\kappa)) \ar[r]& ...}

Consider the $\G_{\Q}$-module $W^r_{\partial}=
\rH^r(\partial Y_{\overline{\Q}}^*, i^*Rj_{*}\bV_n(\kappa))$. 
We have to show that $W^r_{\partial}[\gm']=0$. 

By the Cebotarev Density Theorem and the congruence relations at 
totally split primes of $F$, we can apply lemma \ref{key2} to
$W^r_{\partial}[\gm']$.  Therefore  each $\G_{\widehat{F}}$-irreducible
subquotient  of $W^r_{\partial}[\gm']$  has  at least  two different
weights  for the action of the tame inertia at $p$. So it is enough 
to show that  each $\G_{\Q}$-irreducible subquotient  of  $W^r_{\partial}$ 
is pure (=contains a single weight for the action of the tame inertia at $p$).
Because $\partial M_{\overline{\Q}}^*$ is   zero dimensional, 
the spectral sequence  $\rH^\bullet(\partial Y_{\overline{\Q}}^*,
i^*R^\bullet j_{*}\bV_n(\kappa))\Rightarrow 
\rH^\bullet(\partial Y_{\overline{\Q}}^*,
i^*Rj_{*}\bV_n(\kappa))$ shows that   $W^r_{\partial}=
\rH^0(\partial Y_{\overline{\Q}}^*,i^*R^r j_{*}\bV_n(\kappa))$.

As  $\rH^0(\partial Y_{\overline{\Q}}^*,i^*R^r j_{*}\bV_n(\kappa))$
is a subquotient of 
$\rH^0(\partial Y_{\overline{\Q}}^{1,*},i^*R^r j_{*}\bV_n(\kappa))$
it is enough  to show that  each $\G_{\Q}$-irreducible subquotient  of 
this last is pure.

This will be done using a result of Pink\cite{Pi}. 
We had  to replace $Y$ by $Y^1$, because the group $G$ does not
satisfy the conditions of this reference, while $G^*$ satisfies them.

Consider the decomposition   $T=D_l\times D_h$, 
according to   $\begin{pmatrix} u\epsilon & 0\\ 0 & u^{-1}\end{pmatrix}=
\begin{pmatrix} u& 0\\ 0 & u^{-1}\end{pmatrix}
\begin{pmatrix} \epsilon  & 0\\ 0 & 1\end{pmatrix}$. 
Put $\Gamma^1=\Gamma_1^1(\gc,\gn)$. 
By \cite{Pi}Thm.5.3.1, the restriction  of the {\'e}tale sheaf 
$i^*R^r j_{*}\bV_n(\F_p)$ to a cusp  $\mathcal{C}=\gamma\infty$ of 
$Y_{\overline{\Q}}^{1,*}$, is the  image by the functor of  Pink of the 
$\gamma^{-1}\Gamma^1\gamma\cap B /\gamma^{-1}\Gamma^1\gamma\cap D_lU$-module
$$\underset{a+b=r}\oplus \rH^a(\gamma^{-1}\Gamma^1\gamma\cap D_l ,
\rH^b(\gamma^{-1}\Gamma^1\gamma\cap U,\bV_n(\F_p))).$$


Under the assumption {\bf (II)}, a modulo $p$ version of a theorem
of  Kostant   (see \cite{PoTi}) gives an  isomorphism of $T$-module 
$\rH^b(\gamma^{-1}\Gamma^1\gamma\cap U,\bV_n(\F_p))=\underset{|J|=b}{\oplus}
W_{\epsilon_J(n+t)-t}$. By decomposing $W_{\epsilon_J(n+t)-t}=
W_{\epsilon_J(n+t)-t,l}\otimes W_{\epsilon_J(n+t)-t,h}$ 
according to $T=D_l\times D_h$, we get 
$$ \rH^a(\gamma^{-1}\Gamma^1\gamma\cap D_l,
\rH^b(\gamma^{-1}\Gamma^1\gamma\cap U ,\bV_n(\F_p)))=
\underset{|J|=b}{\oplus} \rH^a(\gamma^{-1}\Gamma^1\gamma\cap D_l,
W_{\epsilon_J(n+t)-t,l}) \otimes W_{\epsilon_J(n+t)-t,h},$$ 
where  Galois acts only on the  second factors of the right hand side. 


Therefore $\rH^0(\partial M_{\overline{\Q}}^*,i^*R^r 
j_{*}\bV_n(\F_p))$ is a direct sum of subspaces 
$\rH^0(\mathcal{C},W_{\epsilon_J(n+t)-t,h}(\F_p))$, $|J|\leq r$,
each containing a single Fontaine-Laffaille weight, namely 
the weight $|p(J)|$.

(ii)  As the Poincar{\'e} duality   is perfect over $E$, it is enough 
to show that the $\gm'$-localization of natural map 
$\rH^d(\cO)/\rH_{!}^d(\cO)\rightarrow 
\rH^d(E)/\rH_{!}^d(E)$ is  injective. For this, it is sufficient to
show that   $\rH_{\partial}^d(\cO)_{\gm'}:=
 \rH^d({\partial}M,\bV_n(\cO))_{\gm'}$ is  torsion free,
which is a consequence of the vanishing of 
 $\rH_{\partial}^{d-1}(E/\!\cO)_{\gm'}$. 
By (i) and Nakayama'a lemma  $\rH_{\partial}^{d-1}(\kappa)_{\gm'}=0$ and 
moreover  we have a surjection $\rH_{\partial}^{d-1}(\kappa)_{\gm'}\twoheadrightarrow 
\rH_{\partial}^{d-1}(E/\!\cO)_{\gm'}[\varpi]$, 
where $\varpi$ is an uniformizer of $\cO$.  

(iii) The vanishing of $\rH_{\partial}^\bullet(\kappa)_{\gm'}$
gives the vanishing of $\rH_{\partial}^\bullet(\cO)_{\gm'}=0$.
\hfill $\square$





\subsection{Definition of periods.}\label{periodes}
 By taking the subspace  $\underset{\ga\subset 
\go}{\bigcap} \ker(T_{\ga}-c(f,\ga))$ 
of (\ref{ESH-epsilon})  we obtain  

$$\delta_J : \C f_J \overset{\sim}\longrightarrow \rH_{!}^d
(Y^{\an},\bV_n(\C))[\widehat{\epsilon}_J,f].$$

Fix an isomorphism $\C\cong\overline{\Q}_p$
compatible with $\iota_p$. We recall that   $\rH_{!}^d(Y^{\an},\bV_n(\cO))'$  
denotes the image  of the natural map  $\rH_c^d(Y^{\an},\bV_n(\cO))
\rightarrow \rH^d (Y^{\an},\bV_n(\C))$. 
As $\cO$ is principal, the  $\cO$-module   
$L_{f,J}:=\rH_{!}^d(Y^{\an},\bV_n(\cO))'[\widehat{\epsilon}_J,f]$ 
is free (of rank $1$). We fix a basis $\eta(f,J)$ of $L_{f,J}$. 

\begin{defin}
For each $J\!\subset\!  J_F$ we define the period
$\Omega(f,J)=\frac{\delta_J(f_J)}{\eta(f,J)}\in \C^\times\!/\!\cO^\times.$

\noindent We fix  $J_0\!\subset\!  J_F$ and
put $\Omega_f^+=\Omega(f,J_0)$ and  $\Omega_f^-= \Omega(f,J_F\!\bs\!J_0)$.
\end{defin}

\begin{rque}
The  periods $\Omega_f^{\pm}$  differ from  
the ones originally  introduces by Hida in \cite{hida63}.
The  Hida's  periods put together all the external conjugates of $f$.  
Our  slightly  different definition is  motivated  by the  
congruence criterion  that we want to show  (Thm.\ref{theo-A}). 
As we can prove the perfectness of the twisted Poincar{\'e} pairing 
only for certain local components of the middle degree cohomology
$\rH_{!}^d(Y^{\an},\bV_n(\cO))'$, and that 
in general $f$ and his external conjugates do not belong 
to the same local component,  we have to separate them 
in  the period's definition.
\end{rque}


\subsection{Computation of a  discriminant.}
The aim of this  paragraph is the computation of the discriminant 
$\disc(L_f)$ of the  $\cO$-lattice $L_f:=\rH_{!}^d
(Y^{\an},\bV_n(\cO))'[f] =
\oplus_{J\subset J_F} L_{f,J}$, with respect twisted Poincar{\'e}  pairing
 $[\enspace, \enspace]$. We follow \cite{ghate1} Sect.6. 

We have $\disc(L_f)=\det(([\eta(f,J),\eta(f,J')])_{J,J'\subset J_F})$. 

By \cite{ghate1} (41),  for every $\tau\in J_F$ and $x,y \in \rH_{!}^d
(Y^{\an},\bV_n(\C))$ we have  $[\epsilon_\tau\cdot x,y]=-[x,\epsilon_\tau\cdot y]$.


The  embedding  $\cO\hookrightarrow \C$ that we have  fixed  
gives   an embedding $\tau_0 :F \hookrightarrow \C$. We have 
$$\hspace{-4mm}\disc(L_f)=\!\prod_{\tau_0\in J\!\subset\! J_F}
\begin{vmatrix}  0  & \left[\eta(f,J),\eta(f,J_F\!\bs\!J)\right] \\ 
\left[\eta(f,J_F\!\bs\!J),\eta(f,J)\right] & 0 \end{vmatrix}=\!
\prod_{\tau_0\in J\!\subset\! J_F}-\left(
\frac{[\delta_J(f),\delta_{J_F\!\bs\!J}(f)]}{\Omega(f,J)\Omega(f,J_F\!\bs\!J)}
\right)^2 $$

We have  $[\delta_J(f),\delta_{J_F\!\bs\!J}(f)]\!=\!
2^d \langle \epsilon_{J_F}\delta(f),\iota\cdot\delta(f)\rangle\!=\!
2^d W(f)\langle \epsilon_{J_F}\delta(f),\delta(f^c)\rangle 
\!=\!2^d W(f)(f,f)_{\gn},$

\noindent  where $f^c$ denotes is the complex conjugate 
 of $f$, $\iota$ is the Atkin-Lehner involution  and $W(f)$ is  the 
complex constant  of the functional  equation  of the standard $L$-function 
 of $f$.  Thus we get the following equality in $E^\times\!/\cO^\times$  :
\begin{equation}\label{discriminant}
\disc(L_{f,J_0}\oplus L_{f,J_F\!\bs\!J_0})=
\left(\frac{W(f)(f,f)_{\gn}}{\Omega_f^+\Omega_f^-}\right)^2.
\end{equation}


\subsection{Shimura's  formula for $L(\Ad^0(f),1)$.} \label{L-ad}
For each  prime ideal $v$ of $F$  we define  
$\alpha_v$ and $\beta_v$ by the equations 
$$\alpha_v+\beta_v=c(f,v),\enspace \enspace\alpha_v\beta_v=\begin{cases}
\psi(v)\Nm(v) & \text{, if }v\nmid \gn, \\
0  & \text{, if }v\mid \gn. \end{cases}$$


The naive adjoint $L$-function of $f$ is then given by the Euler product :
\begin{equation}\label{Lnaive}
L^0(\Ad^0(f),s)=\prod_{v\nmid \gn}\left[
(1\!-\!\alpha_v\beta_v^{-1}\Nm(v)^{-s}) (1\!-\!\Nm(v)^{-s})
(1\!-\!\beta_v\alpha_v^{-1}\Nm(v)^{-s}) \right]^{-1}.
\end{equation}

We denote by $f^c$ the (external) complex conjugate of $f$. 
We  introduce  a twisted version  of the $L$-function  
associated to  the tensor product  $f\otimes f^c$  (see \cite{shimura45}) :
$$D(f,f^c,s)=\prod_{v}\left(1\!-\!
\alpha_v\beta_v\overline{\alpha_v\beta_v}\Nm(v)^{-2s}\right)
\left(1\!-\!{\alpha_v}\overline{\alpha_v}\Nm(v)^{-s}\right)^{-1}\cdot$$
$$\cdot\left(1\!-\!{\alpha_v}\overline{\beta_v}\Nm(v)^{-s}\right)^{-1}
\left(1\!-\!{\beta_v}\overline{\alpha_v}\Nm(v)^{-s}\right)^{-1}
\left(1\!-\!{\beta_v}\overline{\beta_v}\Nm(v)^{-s}\right)^{-1}$$

\noindent  as well as a  naive version $D^0(f,f^c,s)$ obtained by 
removing the  factors  for $v|\gn$.

Using that  for all ${v\nmid  \gn}$, 
  $\overline{c(f,v)}= \overline{\psi(v)} c(f,v)$, 
a direct computation, shows that 
\begin{equation}\label{Dnaive}
 \zeta_F^0(2s)D^0(f,f^c,s\!+\!k_0\!-\!1)=
\zeta_F^0(s) L^0(\Ad^0(f),s).
\end{equation}

The advantage to switch to $D(f,f^c,s)$ is that we  have the following 
formula, proved by Shimura  (see \cite{HiTi} lemma 7.2, for a proof that 
the $D(f,f^c,s)$ defined above equals the one  studied by Shimura) :


\begin{theo}\rm{(Shimura \cite{shimura45} (2.31), Prop. 4.13)}
Let $f\in S_k(\gn,\psi)$ be a newform. Then 
$$\Res_{s=1}D(f,f^c,s+k_0\!-\!1)=2^{d-1}(4\pi)^{|k|}\prod_{\tau\in
  J_F}\Gamma(k_\tau)^{-1}R_F[\go_{+}^\times:
\go^{\times 2}]\langle f,f\rangle \text{, where} $$
  $\langle f,f\rangle= \mu(\Gamma\bs \mathfrak{H}_F)^{-1}
(f,f)_{\gn}$, and  $\mu(\Gamma\bs \mathfrak{H}_F)=
\frac{2\Nm(\gd)^{3/2}\zeta_F(2)\Nm(\gn)}
{\pi^d[\go_{+}^\times:\go^{\times 2}]\prod_{v\mid\gn} (1+\Nm(v))^{-1}}.$
\end{theo}

We deduce the formula :
\begin{equation}\label{residu}
\zeta_F^0(2)\Res_{s=1}D(f,f^c,s+k_0\!-\!1)=
\frac{(4\pi)^{|k|}\pi^d\Res_{s=1}\zeta_F^0(s)}{2\Delta h_F\prod_{\tau\in
  J_F}\Gamma(k_\tau) } (f,f)_{\gn}.
\end{equation}

We define the imprimitive adjoint $L$-function    $L^*(\Ad^0(f),s)$
by  completing the naive adjoint $L$-function   $L^0(\Ad^0(f),s)$, defined 
in (\ref{Lnaive}), in order to have  the relation 
$$L^*(\Ad^0(f),s)D^0(f,f^c,s+k_0\!-\!1)=L^0(\Ad^0(f),s)D(f,f^c,s+k_0\!-\!1).$$

An    explicit computation  of  \cite{HiTi} (7.7) gives  
 $L^*(\Ad^0(f),s)=L^0(\Ad^0(f),s)
\prod_{v | \gn} L^*_v(\Ad^0(f),s) $, where for $v | \gn$ 
$$L^*_v(\Ad^0(f),s)=\begin{cases}1\!-\! \Nm(v)^{-s} & \text{, if }f 
\text{ is principal series  and minimal at  } v, \\ 
1\!-\! \Nm(v)^{-s-1} & \text{, if }f 
\text{ is special and minimal at  } v,
\\ 1 & \text{, otherwise. }\end{cases} $$


Following Deligne \cite{deligne33} we  associate to $L^*(\Ad^0(f),s)$
the  Euler factor at infinity    
$$\Gamma(\Ad^0(f),s)=\prod_{\tau\in J_F}
\pi^{-(s+1)/2}\Gamma((s+1)/2)
(2\pi)^{s+k_\tau-1}\Gamma(s+k_\tau-1).$$
Finally from (\ref{Dnaive}) and  (\ref{residu}) we obtain  
\begin{equation}\label{adjoint}
\Lambda^*(\Ad^0(f),1):=\Gamma(\Ad^0(f),s)L^*(\Ad^0(f),s)=
\frac{2^{|k|-1}}{\Delta h_F}(f,f)_{\gn}.
\end{equation}

\begin{rque}
Consider the adjoint $L$-function $L(\Ad^0(\rho),s)$ of the 
three dimensional $\G_F$-representation $\Ad^0(\rho)$ (on
trace zero matrices). By the compatibility between local and 
global Langlands correspondence $L(\Ad^0(\rho),s)$ equals 
the adjoint $L$-function  $L(\Ad^0(f),s)$ associated to the 
automorphic representation attached to $f$. 
Nevertheless $L(\Ad^0(f),s)$ may differ from  
$L^*(\Ad^0(f),s)$ at some  places $v$ dividing $\gn$ (see
\cite{HiTi} (7.3c)).
\end{rque}





\subsection{Construction of  congruences.}

\begin{lemma}\label{C0}
Let   $V_1$ and $V_2$ be two finite  dimensional $E$-vector spaces
and let $L$ be a $\cO$-lattice in  $V=V_1\oplus V_2$. 
For $j=1,2$, put $L_j=L\cap V_j$ and denote $L^j$  the projection of 
 $L$ in $V_j$ following the above direct sum decomposition. Then :

{\rm (i)} $L_j\!\subset\! L^j $ are two lattices of $V_j$, and
 $L_j$ is a direct factor  in $L$.

{\rm (ii)} we have isomorphisms of finite $\cO$-modules :
\begin{equation}\label{fusion}
L^1/L_1\overset{\sim}{\leftarrow} L/L_1\oplus
L_2\overset{\sim}{\rightarrow}L^2/L_2
\end{equation}
This finite $\cO$-module is called the congruence module, and 
is denoted by $C_0(L;V_1,V_2)$.
\end{lemma}

The following proposition follows from  Deligne-Serre 
(Lemma 6.11 of \cite{SeDe}) and will be used to construct congruences :

\begin{prop}\label{relevement}
Keep the notations of the  lemma \ref{C0}. 
Let $\cT$ be a commutative $\cO$-algebra consisting of 
endomorphisms of $V$, preserving  the lattice $L$ and the 
 direct sum decomposition  $V_1\oplus V_2$. Denote, for $j=1,2$, by 
  $\cT_j$   the image  of $\cT$ in $\End(V_j)$.

Assume that  $C_0(L;V_1,V_2)$ is non zero :
$\{\cP\}=\Ass(C_0(L;V_1,V_2))=\Supp(C_0(L;V_1,V_2))$.

Let $\gm_1$ be maximal ideal
$\cT_1$,  of residue  field  $\kappa_1$, such that  
$L^1/L_1\otimes_{\cT_1}\kappa_1$ is non zero, and denote by 
$\overline{\theta_1}:\cT_1\rightarrow \kappa_1$ the corresponding 
character.

Then  there exists a discrete valuation ring
$\cO'$ of maximal ideal   $\cP'$ (with
$\cP'\cap\cO=\cP$), of residue field  $\kappa'\supset \kappa_1$ 
and whose fraction field  $E'$ is a finite extension of $E$, 
and there exists  a 
character $\theta_2:\cT_2\rightarrow \cO'$ such that
for each $T\in\cT$,  $\enspace \overline{\theta_1}(T) \equiv
\theta_2(T) \pmod{\cP'}$.
\end{prop}

{\bf Proof : } Denote by   $\pi_j$ the projection of $\cT$  onto 
$\cT_j$, $j=1,2$. Then  $\gm=\pi_1^{-1}(\gm_1)$
is a maximal ideal of $\cT$ of residue field $\kappa_1$.
Put  $\gm_2=\pi_2(\gm)$.
As the isomorphism (\ref{fusion})  of the lemma $\ref{C0}$  is 
 $\cT$-equivariant, we get 
$$(L^1/L_1)\otimes_{\cT_1}(\cT_1/\gm_1)\cong
(L/(L_1\oplus L_2))\otimes_{\cT}(\cT/\gm) \cong
(L^2/L_2)\otimes_{\cT_2}(\cT_2/\gm_2)$$

By assumption $(L^1/L_1)\otimes_{\cT_1}(\cT_1/\gm_1)$
 is non zero. Therefore  $\gm_2$ is a 
maximal ideal of $\cT_2$ of residue field $\kappa_1$ and 
the corresponding  character  $\overline{\theta_2}:\cT_2\rightarrow \kappa_1$ 
fits in the  following commutative diagram :
$$\xymatrix@R=5pt{    &\cT_1\ar[rd]^{\overline{\theta_1}} & \\
\cT\ar[rd]\ar[ru] &   & \kappa_1    \\
  & \cT_2\ar[ru]_{\overline{\theta_2}}      & }$$

As $\cT_2$ is a (finite) flat $\cO$-algebra, there exist a
prime ideal $\cP_2$, contained in $\gm_2$ and such 
that $\cP_2\cap\cO=0$. The reduction modulo  $\cP_2$ gives 
a character $\theta_2$ of $\cT_2$  as in the statement. 

\begin{theo}{\rm(Theorem A)}\label{theo-A} 
Let  $f$ and  $p$ be such that {\bf (I)}, $\mathrm{\bf (Irr_{\overline{\rho}})}$ and {\bf (MW)} hold,
and $p-1>\max(1,\frac{5}{d})\sum_{\tau\in J_F}(k_\tau-1)$. If  
$\iota_p(\frac{W(f)\Lambda^*(\Ad^0(f),1)}{\Omega_f^+\Omega_f^-})\in\cP$, 
then  $\cP$  is a congruence prime for $f$.
\end{theo}

{\bf Proof : } We put $L=\rH_{!}^{d}(Y^{\an},\bV_n(\cO))_{\gm'}'
[\pm\widehat{\epsilon}_{J_0},\psi]\!\subset\! 
V=\rH_{!}^{d}(Y^{\an},\bV_n(E))_{\gm'} [\pm\widehat{\epsilon}_{J_0},\psi]$. 

We define   $V_1=\rH_{!}^{d}(Y^{\an},\bV_n(E))[\pm\widehat{\epsilon}_{J_0},f]$
(with the notations of {\S}\ref{periodes}
we have  $L_1=L\cap V_1=L_{f,J_0}\oplus L_{f,J_F\!\bs\!J_0}$). 

It follows from (\ref{discriminant}) that 
the bilinear  form  $[\enspace,\enspace ]$ is non-degenerate on $V_1$.
Let  $V_2$ be the orthogonal subspace of $V_1$ in $V$.

By  (\ref{adjoint}) and (\ref{discriminant}) and the assumption on $\cP$, we have
$0\neq \disc(L_1)\in \cP$. By  Thm.\ref{boundary}(ii) the $\cO$-lattice $L$ is 
autodual with respect to the twisted Poincar{\'e} pairing and  therefore, 
$\disc(L_1)=[L^1:L_1]$. Then by lemma \ref{C0}, the congruence module 
$C_0(L;V_1,V_2)$ is non-zero and $\cP$ belongs to its support. By  
Prop.\ref{relevement}  and the duality between   $\T(\C)$ and $S_k(\gn,\psi)$, we 
obtain another normalized eigenform $g\in S_k(\gn,\psi)$  congruent to $f$. 
Hence, $\cP$  is a congruence prime for $f$. \hfill $\square$


\section{Fontaine-Laffaille weights  of the Hilbert modular varieties.}


In this section all the objects are over $\cO$. 
The aim  is to establish a modulo $p$ version 
of Thm.\ref{hodge-pad}   under the   assumptions  that $p$ does not
divide $\Delta$ and $p-1>|n|+d$. 


\subsection{The BGG complex over $\cO$.} \label{BGG} ${ }$

\medskip
\noindent{\bf Koszul's complex.}
The  Koszul's complex of the trivial $G$-module   $\cO$ is given by     
$$...\rightarrow  U_{\cO}(\mathfrak{g})\otimes \wedge_{\cO}^2\mathfrak{g}
\rightarrow  U_{\cO}(\mathfrak{g})\otimes\mathfrak{g}
\rightarrow  U_{\cO}(\mathfrak{g})
\rightarrow \cO
\rightarrow 0$$

As  $\mathfrak{g}=\gb\oplus\gu^-$, the 
$\cO[\gb]$-module
$\mathfrak{g}/\gb$ is a direct factor  in 
$\mathfrak{g}$ and  we have a homomorphism of $B$-modules 
$ U_{\cO}(\mathfrak{g})\otimes \wedge_{\cO}^{\bullet}\mathfrak{g}
\rightarrow 
U_{\cO}(\mathfrak{g})\otimes_{U_{\cO}(\gb)}\wedge_{\cO}^{\bullet}(\mathfrak{g}/\gb)$. 
Thus, we   deduce another  complex  
$$ U_{\cO}(\mathfrak{g})\otimes_{U_{\cO}(\gb)}\wedge_{\cO}^{\bullet}
(\mathfrak{g}/\gb) \rightarrow \cO \rightarrow 0,$$
denoted by $S_{\cO}^{\bullet}(\mathfrak{g},\gb)$.

More generally,  for a free  $\cO$-module  $V$ endowed with an 
 action  of $U_{\cO}(\mathfrak{g})$, we consider  the complex  
$S_{\cO}^{\bullet}(\mathfrak{g},\gb)\otimes V$, endowed 
de the diagonal action  of $U_{\cO}(\mathfrak{g})$. 

For every   $U_{\cO}(\gb)$-module  $W$, free over $\cO$,
 we have a canonical isomorphism  of $U_{\cO}(\mathfrak{g})$-modules 
\begin{equation}
\left(U_{\cO}(\mathfrak{g})\otimes_{U_{\cO}(\gb)} W\right)\otimes V
\cong U_{\cO}(\mathfrak{g})\otimes_{U_{\cO}(\gb)} 
\left( W\otimes V|_{\gb}\right),
\end{equation}

So we get another complex 
$$ U_{\cO}(\mathfrak{g})\otimes_{U_{\cO}(\gb)}
\left(\wedge_{\cO}^{\bullet}(\mathfrak{g}/\gb)\otimes  V|_{\gb}\right)
\rightarrow V \rightarrow 0,$$
denoted $S_{\cO}^{\bullet}(\mathfrak{g},\gb,V)$.
In the case where $V=V_n$ we  denote  it 
$S_{\cO}^{\bullet}(\mathfrak{g},\gb,n)$.


\medskip
\noindent{\bf Verma modules.}
For each weight $\mu\in \Z[J_F]$, we define a  $U_{\cO}(\mathfrak{g})$-module 
$V_{\cO}(\mu):=U_{\cO}(\mathfrak{g}) \otimes_{U_{\cO}(\gb)}W_{\mu}(\cO)$, called the 
 {\it Verma  module} of weight $\mu$.

\begin{lemma}
Let $W$ be a $B$-module, free of finite rank  over $\cO$, whose 
weights are smaller than $ (p-1)t$. Then, there exists a filtration  of 
$B$-modules $0=W_0\!\subset\! W_1\!\subset\! ...\!\subset\! W_r=W$ such that 
for each $1\leq i\leq r$, $W_i/W_{i+1}\cong W_{\mu_i}(\cO)$, for some
 $\mu_i\in \Z[J_F]$. Moreover the  $W_{\mu_i}(\cO)$, $1\leq i\leq r$,
are  the irreducible factors  of the $T$-module $W$.

In  particular, if $U$ acts trivially on $W$, then  
$W\cong \oplus_{i=1}^{r}  W_{\mu_i}(\cO)$.
\end{lemma}

{\bf Proof : } Let $\mu_1$ be a maximal weight  of $W$ (for the partial   order  
given by the   positive roots  of $G$) and let  $v\in W$ be a $\cO$-primitive 
vector  of weight  $\mu_1$. Let $W'$ be the $U_{\cO}(\gb)$-submodule
 generated by $v$.  Then  $W'\cong W_{\mu_1}(\cO)$ and 
$W'\otimes \kappa$ is irreducible, because  $\mu_1$ is  smaller than $ (p-1)t$ 
(and  $W'$ is free of rank 1).
As $W$ is free over $\cO$ we have an  exact
 sequence  of $B$-modules  $$0\rightarrow \Tor_1^{\cO}(W/W',\kappa)
\rightarrow W'\otimes \kappa \rightarrow W\otimes \kappa.$$ As $W'\otimes \kappa$ is 
irreducible and $v$ is primitive, the last arrow is injective. Therefore 
$$\Tor_1^{\cO}(W/W',\kappa)=0,$$ that is  $W/W'$ is free over $\cO$.  
The lemma follows then  by induction. 
\hfill $\square$

\begin{lemma}\label{verma} The module  
$S^{i}_{\cO}(\mathfrak{g},\gb,n)$ has  a finite filtration  by    
$U_{ \cO}(\mathfrak{g})$-submodules whose graded pieces are of
 the form $V_{\cO}(\mu)$, $\mu\in \Omega^i(n)$,
where $\Omega^i(n)$ denoted  the set  of  weights of the 
$\gt$-module  $\wedge_{\cO}^{i}(\mathfrak{g}/\gb)\otimes  
V_{n}(\cO)|_{\gb}$.
\end{lemma}

{\bf Proof : } Since $p-1>|n|+d$ the previous lemma applies to 
 $\wedge_{\cO}^{\bullet}(\mathfrak{g}/\gb)\otimes
 V_{n}(\cO)|_{\gb}$. This gives  a filtration  $0\!=\!W_0\!\subset\! W_1\!\subset\! ...\!\subset\! W_r=
\wedge_{\cO}^{i} (\mathfrak{g}/\gb)\otimes  
V_{n}(\cO)|_{\gb}$ whose graded pieces are  $W_{\mu}(\cO)$, $\mu\in \Omega^i(n)$.
 As $U_{\cO}(\mathfrak{g})$ is 
  $U_{\cO}(\gb)$-free, the  functor $U_{\cO}(\mathfrak{g})
\otimes_{U_{\cO}(\gb)}\bullet$  is  exact. \hfill $\square$

\medskip
\noindent{\bf Central characters.}
Let $U_{\cO}(\mathfrak{g}) \rightarrow U_{\cO}(\gt)$
be the  projection coming from the  Poincar{\'e}-Birkoff-Witt decomposition
$U_{\cO}(\mathfrak{g})=U_{\cO}(\gt)\oplus
(\gu^-U_{\cO}(\mathfrak{g})+U_{\cO}(\mathfrak{g})\gu$). 
We take its restriction  to the invariants for the adjoint action  
$\theta : U_{\cO}(\mathfrak{g})^G\rightarrow U_{\cO}(\gt)$.
Note  that $U_{\overline{\F}_p}(\gt)$ 
identifies itself  with the algebra of regular  functions  on
$\Hom_{\cO}(\gt,\overline{\F}_p)\cong \overline{\F}_p[J_F]$
(a Laurent polynomial algebra).   The Weyl group $\{\pm 1\}^{J_F}$ 
of $G$ acts on it by  $(\epsilon_J\cdot P)(\mu)=P(\epsilon_J(\mu+t)-t)$.
The following result  is a analogous to a  the theorem of
Harish-Chandra :

\begin{theo}{\rm(Jantzen \cite{jantzen})}
$\theta_{\overline{\F}_p}$ induces  an algebra isomorphism  
$U_{\overline{\F}_p}(\mathfrak{g})^G\rightarrow 
U_{\overline{\F}_p}(\gt)^{\{\pm 1\}^{J_F}}$.
\end{theo}


For every $\mu\in \Z[J_F]$ and every $\cO$-algebra $R$, we denote by 
$d\mu_R:\gt_R\rightarrow R$ the corresponding
character  and by $\chi_{\mu,R}=d\mu_R\circ \theta_R$ the 
composed map 
$U_{R}(\mathfrak{g})^G\rightarrow U_{R}(\gt)\rightarrow R$. 
This  definition is compatible with the $\cO$-algebra homomorphisms.

If  $V$ is a $U_{R}(\mathfrak{g})$-module generated by a vector $v$ 
of  weight $\mu$, that  is annihilated by $\gu$, then 
$U_{R}(\mathfrak{g})^G$ acts over $V$ by $\chi_{\mu,R}$.
Put $\chi_{\mu,p}=\chi_{\mu,\cO}$
and  $\overline{\chi}_{\mu,p}=\chi_{\mu,\overline{\F}_p}$.

\begin{cor}
Let $\mu\in \Z[J_F]$. If 
$\overline{\chi}_{n,p}=\overline{\chi}_{\mu,p}$,
then  there exists $J\!\subset\! J_F$ such that 
$\mu-(\epsilon_J(n+t)-t)\in p\Z[J_F]$. In particular, 
if   $\mu$ is  smaller than $ (p-1)t$, then  we have $\mu=\epsilon_J(n+t)-t$.
\end{cor}

\begin{prop}\label{chandra}
Let  $\mu\in \Omega^{i}(n)$ (see  lemma \ref{verma}). 
Then  $\overline{\chi}_{n,p}=\overline{\chi}_{\mu,p}$,
if and only if there exists a subset 
$J\!\subset\! J_F$containing $i$ elements  and such that $\mu=\epsilon_J(n+t)-t$.
\end{prop}

 {\bf Proof : }
By  the corollary, it remains to show that for   $J\!\subset\! J_F$,
we have   $\epsilon_J(n+t)-t\in \Omega^{i}(n)$,
if and only if  $|J|=i$. By  the lemma \ref{verma}, we have to show that 
 $W_{\epsilon_J(n+t)-t}(E)$ occurs in 
$\wedge_{E}^{i}(\mathfrak{g}/\gb)\otimes  
V_{n}(E)|_{\gt}$ (with multiplicity one)
 if and only if  $|J|=i$. The weight of 
$\wedge_{E}^{i}(\mathfrak{g}/\gb)\otimes  
V_{n}(E)|_{\gt}$  are of the form 
$\epsilon_{J'}(n+t)-t+\nu$, where $J'\!\subset\! J_F$ is a subset containing 
$i$ elements   and $\nu$ is a weight of  $V_{n}(E)$. 
Therefore  $\epsilon_J(n+t)-t=\epsilon_{J'}(n+t)-t+\nu$ and so 
$n=\epsilon_J(\nu)+\epsilon_J(\epsilon_{J'}(t))-t$.
Because  $n$ is a maximal weight  of $V_{n}(E)$, 
we deduce that $J=J'$. 

\medskip
\noindent{\bf Decomposition with respect to the central characters.}
 By  the lemma \ref{verma} $S^{i}_{\cO}(\mathfrak{g},\gb,n)$
admits  a finite filtration  by    $U_{ \cO}(\mathfrak{g})$-submodules, whose graded are of the
 form $V_{\cO}(\mu)$, $\mu\in \Omega^i(n)$. Therefore $S^{\bullet}_{\cO}(\mathfrak{g},\gb,n)$
is  annihilated by a power of the ideal  $I:=\prod_{\mu\in \Omega^{\bullet}(n)}\ker(\chi_{\mu,p})$ 
of  the commutative  ring   $U_{\cO}(\mathfrak{g})^G$.  As  we will see at the end  of this section as 
a consequence of Prop.\ref{chandra},  $S^{\bullet}_{\cO}(\mathfrak{g},\gb,n)$ is 
annihilated by $I$ itself. We have the following  commutative algebra lemma : 

\begin{lemma}
Let $P_1,..,P_r$  be ideals of the commutative ring   $R$, such that 
 $P_1..P_r=0$ and  for all $i\neq j$,  $P_i+P_j=R$. 
Then   each $R$-module $W$ admits a direct sum decomposition 
 $W=\oplus_{1\leq i\leq r} W^{P_i}$, with  $W^{P_i}=\{m\in W|P_im=0\}$.
\end{lemma}

Consider the maximal ideals   $pR+\ker(\chi_{\mu,p})=
\ker(\overline{\chi}_{\mu,p})$ of $R$, where $\mu\in \Omega^{\bullet}(n)$. 
Let  $\overline{\chi}_1=\overline{\chi}_{n,p}$,
$\overline{\chi}_2$,...,$\overline{\chi}_r$ be the 
set of distinct characters among 
$\overline{\chi}_{\mu,p}$, $\mu\in \Omega^{\bullet}(n)$.
Put $P_i=\prod_{\overline{\chi}_{\mu,p}=\overline{\chi}_i} 
\ker(\chi_{\mu,p})$. By the above lemma we get a decomposition 
\begin{equation}
S^{\bullet}_{\cO}(\mathfrak{g},\gb,n)=
\oplus_{i=1}^{r}
S^{\bullet}_{\cO}(\mathfrak{g},\gb,n)^{P_i}
\end{equation}
which is a direct sum, because  the differentials  are 
$U_{ \cO}(\mathfrak{g})$-equivariant. Moreover,
$V_{\cO}(\mu)_{\overline{\chi}_{n,p}}=
V_{\cO}(\mu)$, if $\overline{\chi}_{\mu,p}=\overline{\chi}_{n,p}$,
and  $V_{\cO}(\mu)_{\overline{\chi}_{n,p}}=0$, otherwise. 
From here and from Prop.\ref{chandra} we get  :

\begin{theo} \label{decomposition-centrale}
The complex  $S^{\bullet}_{\cO}(\mathfrak{g},\gb,
n)_{\overline{\chi}_{n,p}}$ is a direct factor  in 
$S^{\bullet}_{\cO}(\mathfrak{g},\gb,n)$
and we have  $S^{0}_{\cO}(\mathfrak{g},\gb,
n)_{\overline{\chi}_{n,p}}= V_{n}(\cO)$. 
For each $i\geq 1$, 
$S^{i}_{\cO}(\mathfrak{g},\gb,
n)_{\overline{\chi}_{n,p}}$ has a filtration whose 
graded are   given by the $V_{\cO}(\epsilon_J(n+t)-t)$
where $J\!\subset\! J_F$, $|J|=i$ (with multiplicity one).
\end{theo}



\subsection{The  BGG complex for distributions  algebras.}


Let $\cU_{\cO}(G)$ be the distribution $\cO$-algebra  over $G$.
For each $G$-module $V$,  free over  $\cO$,  we define the complex 
$$  0 \leftarrow  V \leftarrow 
\cU_{\cO}(G)\otimes_{\cU_{\cO}(B)}
\left(\wedge_{\cO}^{\bullet}(\mathfrak{g}/\gb)\otimes  
V|_{\gb}\right)  ,$$

\noindent and denote it by  $\mathcal{S}_{\cO}^{\bullet}(G,B,V)$. 
In the case where $V=V_{n}(\cO)$ we denote this complex by  
$\mathcal{S}_{\cO}^{\bullet}(G,B,n)$.

\begin{rque}
The complex $\mathcal{S}_{\cO}^{\bullet}(G,B,V)$ 
is not exact. It will become exact after applying the
Grothendieck linearization functor to the associated complex
of vector bundles over the Hilbert modular variety.
\end{rque}

For all $\mu\in \Z[J_F]$,  we define the Verma module $\cV(\mu)=
\cU_{\cO}(G)\otimes_{\cU_{\cO}(B)} W_{\mu}(\cO)$ (see \S\ref{BGG}).
We recall  that,  since  $p-1> |n|+d$,  
$\Omega^i(n)$ is the set  of  $\mu\in \Z[J_F]$
such that $W_{\mu}(\cO)$ is a irreducible subquotient  of 
$\wedge_{\cO}^{i}(\mathfrak{g}/\gb)\otimes  
V_{n}(\cO)|_{\gb}$. The  lemma \ref{verma} translates as  :

\begin{lemma} The modules
$\mathcal{S}_{\cO}^{\bullet}(G,B,n)$ has a finite filtration by 
 $\cU_{\cO}(G)$-submodules whose  successive quotients are given by
$\cV_{\cO}(\mu)$, with $\mu\in \Omega^i(n)$.
\end{lemma}

As $U_{\cO}(\mathfrak{g})\!\subset\!  
\cU_{\cO}(G)\!\subset\! U_{E}(\mathfrak{g})$,
the  center 
$U_{\cO}(\mathfrak{g})^G$  of $U_{\cO}(\mathfrak{g})$ 
is  contained in the center of $\cU_{\cO}(G)$.
Consider the central  characters  
$\chi_{\mu,p}=\chi_{\mu,\cO}$
and  $\overline{\chi}_{\mu,p}=\chi_{\mu,\overline{\F}_p}$ (see \S\ref{BGG}).

If  $W$ is a $\cU_{\cO}(G)$-module  generated by a vector 
$v$ of weight $\mu$, that  is  annihilated by $\gu$,
then  $U_{\cO}(\mathfrak{g})^G$ acts  on $W$ by the character $\chi_{\mu,p}$. 
Put  $I=\prod_{\mu\in \Omega^{\bullet}(n)}\ker(\chi_{\mu,p})$. 
By  the  last lemma  the finite ${\cO}$-module  
$\mathcal{S}_{\cO}^{\bullet}(G,B,n)$ is  a  
$R:=U_{\cO}(\mathfrak{g})^G/I$-module. 
Let  $\overline{\chi}_1=\overline{\chi}_{n,p},\overline{\chi}_2,...
\overline{\chi}_r$ be the distinct  algebra homomorphisms
from  $R$ in $\overline{\F}_p$. For $1\leq j\leq r$, we put
$$\mathcal{S}_{\cO}^{\bullet}(G,B,n)_{\overline{\chi}_j}=
\left\{x\in\mathcal{S}_{\cO}^{\bullet}(G,B,n)
\Big{|} \left(\prod_{\mu\in \Omega^{\bullet}(n), \overline{\chi}_{\mu,p} 
=\overline{\chi}_j}  \ker(\chi_{\mu,p})\right)x=0\right\}.$$

The same way as in Thm.\ref{decomposition-centrale}  we obtain a decomposition 
\begin{equation}\label{decomposition}
\mathcal{S}_{\cO}^{\bullet}(G,B,n)= \oplus_{j=1}^{r}
\mathcal{S}_{\cO}^{\bullet}(G,B,n)_{\overline{\chi}_j}.
\end{equation}


The main theorem of this section states then :

\begin{theo}
$\displaystyle \mathcal{S}_{\cO}^{i}(G,B,n)_{\overline{\chi}_{n,p}}
\cong \bigoplus_{J\!\subset\! J_F, |J|=i}\cV_{\cO} (\epsilon_J(n+t)-t).$
\end{theo}

{\bf Proof : } We start with the case $n=0$.

As $\gu$ is abelian, $U$ acts trivially on  
$\wedge_{\cO}^{i}(\mathfrak{g}/\gb)$ and therefore, 
by  the lemma \ref{verma}, we have
$$\wedge_{\cO}^{i}(\mathfrak{g}/\gb)\cong
\bigoplus_{J\!\subset\! J_F, |J|=i} W_{\epsilon_J(t)-t}(\cO). $$ 

As  $\cU_{\cO}(G)$ is free over $\cU_{\cO}(B)$ we obtain :
\begin{equation}
\mathcal{S}_{\cO}^{i}(G,B,0)=
\mathcal{S}_{\cO}^{i}(G,B,0)_{\overline{\chi}_{0,p}}
\cong \bigoplus_{J\!\subset\! J_F, |J|=i}\cV_{\cO}
(\epsilon_J(t)-t).
\end{equation}

If $n\geq 0$,  we deduce from the $n=0$  case a  decomposition
\begin{equation}
\mathcal{S}_{\cO}^{i}(G,B,n)\cong \bigoplus_{J\!\subset\! J_F, |J|=i} 
\cU_{\cO}(G)\otimes_{\cU_{\cO}(B)}
\left( W_{\epsilon_J(t)-t}(\cO)\otimes V_{n}(\cO) \right).
\end{equation}

Using  (\ref{decomposition}) the theorem is a consequence of the 
following lemma, whose proof is a direct application of the proof 
 of Prop.\ref{chandra}.

\begin{lemma} 
$\left(\cU_{\cO}(G)\otimes_{\cU_{\cO}(B)}
\left( W_{\epsilon_J(t)-t}(\cO)\otimes
V_{n}(\cO) \right)\right)_{\overline{\chi}_{n,p}}
\cong \cV_{\cO}(\epsilon_J(n+t)-t).$
\end{lemma}



\subsection{BGG complex for crystals.}
Our reference is the section 4 of  \cite{MoTi}.
For every integer $r\geq 0$ we put  $S_r=\Spec(\Z/p^{r+1})$.
For a $\Z[\frac{1}{\Delta}]$-scheme $X$,   we put  $X_r=X \times S_r$. 

We have  an equivalence of categories between the category of crystals over 
$(\overline{X}_0 /S_r)_{\log}^{\crys}$ and the category of
$\cO_{\overline{X}_r}$-modules $\cM$ 
which are locally free and   endowed with integrable, quasi-unipotent
connection with logarithmic poles  $\nabla : \cM \rightarrow \cM
\otimes_{\cO_{\overline{X}_r}} \Omega^1_{\overline{X}_r/S_r}(\dlog(\infty_X))$.

\medskip
We have  a functor $\mathrm{L}$, called the  {\it linearization functor},  
from  the category of sheaves of  
 $\cO_{\overline{X}_r}$-modules to  the category of 
 crystals on  $(\overline{X}_0 /S_r)_{\log}^{\crys}$.

By the log-crystalline Poincar{\'e} lemma, we have a resolution :
\begin{equation}\label{poincare-crys}
0 \rightarrow \cM \rightarrow
\mathrm{L}(\cM\otimes_{\cO_{\overline{X}_r}} 
\Omega^{\bullet}_{\overline{X}_r/S_r}(\dlog \infty)).
\end{equation}


Let  $W_1$ and  $W_2$ be two $B$-modules with weights smaller than $ (p-1)t$. 
Put $\overline{\cW}_i=\cF_{B}(W_i)$, $i=1,2$ (see \S\ref{hodge-tate}).
By   \cite{MoTi}{\S}5.2.4   we have a homomorphism 
\begin{equation}
\Hom_{\cU_{\cO}(G)} (\cU_{\cO}(G) \otimes_{\cU_{\cO}(B)}W_1), \cU_{\cO}(G)
\otimes_{\cU_{\cO}(B)}W_2)\rightarrow 
\mathrm{Diff.Op.}(\overline{\cW}_{2,r},\overline{\cW}_{1,r}),
\end{equation}
which becomes  an isomorphism  after tensoring with $E$ (see (\ref{op-diff})).

\medskip
We apply now the above construction to the toroidal compactification 
over of  the Hilbert modular variety $\overline{M}{}'$ and 
the vector bundle $\overline{\cV}_n$. For every  $r\geq 0$ we have an injective 
 homomorphism  of complexes  of vector bundles  over $\overline{M}_r'$   
\begin{equation}\label{iota}
 \mathcal{K}^{\bullet}_n:=
\underset{J\!\subset\! J_F}{\oplus}\overline{\cW}_{\epsilon_J(n+t)-t}
\hookrightarrow \overline{\cV}_n \otimes_{\cO_{\overline{M}_r'}}
\Omega^{\bullet}_{\overline{M}_r'/S_r}(\dlog \infty).
\end{equation}

\begin{prop}\label{qis}
The map $(\ref{iota})$ is a strict  injective homomorphism  of filtered complexes.
\end{prop}

By the last  proposition   $\mathrm{L}( \mathcal{K}^{\bullet}_n)$
is a direct factor in $\mathrm{L}(\overline{\cV}_n \otimes_{\cO_{\overline{M}_r'}} 
\Omega^{\bullet}_{\overline{M}_r'/S_r}(\dlog \infty))$, which is 
 exact by  the Poincar{\'e}'s crystalline lemma. Therefore
 $\mathrm{L}( \mathcal{K}^{\bullet}_n)$ is also exact. 
As the functor $\mathrm{L}$ is exact, we deduce filtered  isomorphisms
$\cH_{\ldR}^j(\overline{M}_r'/S_r,  \overline{\cV}_n)
\cong \cH^j(\overline{M}_r'/S_r, \mathcal{K}^{\bullet}_n)$.

Recall that $p$ does not divide $\Delta$ and $p-1>|n|+d$. Under this 
assumptions we have 


\begin{theo}\label{bgg-modp}
The spectral sequence  given by the  Hodge filtration 
$$\rE_1^{i,j}=\bigoplus_{J \!\subset\! J_F, |p(J)|=i} 
\rH^{i+j-|J|} (\overline{M}_r, \overline{\cW}_{\epsilon_J(n+t)-t,n_0}) 
\Rightarrow\cH_{\ldR}^{i+j}(\overline{M}_r,\overline{\cV}_n)$$
degenerates at $\rE_1$  :
\begin{equation}\label{bgg-modp1}
\gr^i \cH_{\ldR}^r(\overline{M}_r, \overline{\cV}_n) 
=\bigoplus_{J \subset J_F, |J|\leq r, |p(J)|=i} \rH^{r-|J|}
(\overline{M}_r,  \overline{\cW}_{\epsilon_J(n+t)-t}).
\end{equation}
\end{theo}

{\bf Proof : } The proof  is formally  the same as the one 
of Thm.\ref{hodge-pad}(ii), once we have Prop.\ref{qis}.
The degeneration of the spectral sequence follows from a 
result of Illusie \cite{illusie2} Prop.4.13. applied to the
semi-stable morphism $\overline{\pi_s}:\overline{\cA^s}\rightarrow
\overline{M^1}$ of smooth $\Z_p$-schemes.
\hfill $\square$


\begin{rque} \label{hecke-equiv}
(i) It follows from the same arguments as in 
Cor.\ref{hodge-complex}(i), that the above decomposition is 
 Hecke equivariant, except for the $T_{\gp}$ operators, when 
 $\gp$ divides $p$. When $p$ is totally split in $F$,  
we could use Wedhorn's results \cite{Wed}  to write
$T_{\gp}$ as a sum of correspondences and try adapt to this case the method of 
\cite{FaJo}. Unfortunately, this approach is not available 
when $p$ is not  totally split in $F$. 

In the  proof of Thm.\ref{theo-B}, we will use  different method to 
prove the $T_{\gp}$-equivariance 
of the  above decomposition after a localization outside $p$. 

(ii) The commutativity of the  Hecke operators outside $p$ follows  from the  
degeneration at $\rE_1$ as in the proof of Cor.\ref{hodge-complex}(i). The last  
graded piece  $\rH^0(\overline{Y},\overline{\cW}_{\epsilon_{J_F}(n+t)-t,n_0})$ 
of the  filtration is   independent of the toroidal compactification
 by the Koecher principle (\ref{koecher}).
\end{rque}


\section{Integral cohomology  over  certain local  components  of 
the Hecke algebra.}


\subsection{The  key lemma.}
Let $q=p^r$ and denote by  $\sigma_1$,..,$\sigma_r$ the elements of
$\Gal(\F_q/\F_p)$.

\begin{theo} {\rm(Brauer-Nesbitt, Steinberg \cite{St})}
The group $\SL_2(\F_q)$ has exactly  $q$ irreducible representations 
 on finite dimensional  $\F_q$-vector spaces, namely  the
$\otimes_{j=1}^r(\Sym^{a_j})^{\sigma_j}$, for $0\leq a_j\leq p-1$.
\end{theo}


\begin{cor}\label{bn}
For every finite set  $I$, the group 
$ \prod_{i\in I}\SL_2(\F_q)$ has exactly  $q^{|I|}$ irreducible 
representations on finite dimensional  $\F_q$-vector spaces, namely the
$ \otimes_{i\in I}
\left(\otimes_{j=1}^r(\Sym_i^{a_{i,j}})^{\sigma_j}\right)$, 
for  $0\leq a_{i,j}\leq p-1$.
\end{cor}

In  \cite{Ma}  Mazur  states the following :

\begin{lemma} Let $\Phi$ be a group 
and $\rho_0$ be  a  representation of $\Phi$
on a  finite dimensional $\F_q$-vector space $W$. 
Let  $\rho: \Phi \rightarrow \GL_2(\F_q)$ be an absolutely irreducible 
representation such that  for all $g\in \Phi$, the characteristic polynomial
 of  $\rho(g)$ annihilates $\rho_0(g)$. Then,
$\rho_0^{\mathrm{s.s.}}=\rho\oplus..\oplus\rho$ and in particular  
$\rho\!\subset\!\rho_0$.
\end{lemma}

The  corresponding statement 
for a another group than $\GL_2$ is false  in  general.
Here is an example for $\GL_3$ : take 
$\rho=\Sym^2: \GL_2(\F_q)\rightarrow \GL_3(\F_q)$ and 
$\rho_0=\det:\GL_2(\F_q)\rightarrow \GL_1(\F_q)$.
Nevertheless, a we have a generalization for the special group :
$$H(\F_q)=\left(\prod_{i\in I}\GL_2(\F_q)\right)^\mathcal{D}:=
\left\{ (M_i)_{i\in I}\in  \prod_{i\in I}\GL_2(\F_q)\Big{|}\enspace\exists
 \delta\in\mathcal{D},\enspace \forall i\in I, \enspace  \det(M_i)=\delta\right\}$$

and the particular  representation 
$$\rho_1= \bigotimes_{i\in I,\tau\in J_F^i}
\mathrm{St}_i^{\sigma_{i,\tau}} : H(\F_q) \rightarrow
  \GL_{2^d}(\F_q) \text{ , } (M_i)_{i\in I}\mapsto 
\bigotimes_{i\in I,\tau\in J_F^i} M_i^{\sigma_{i,\tau}},$$ 
where $(J_F^i)_{i\in I}$ is a partition of $J_F$ and for all  $i\in I$,
 $(\sigma_{i,\tau})_{\tau\in J_F^i}$ are  two by two distinct elements 
of $\Gal(\F_q/\F_p)$ ($\St=\Sym^1$ is    
  the standard two-dimensional representation  of $\GL_2$).   
\begin{lemma}\label{variante-mazur}
Let $\rho_0$ be  a  representation of $H(\F_q)$
on a  finite dimensional $\F_q$-vector space $W$, such that 
 for all $g\in H(\F_q)$ the characteristic polynomial  
of $\rho_1(g)$ annihilates $\rho_0(g)$. Then 
$\rho_0^{\mathrm{s.s.}}={\rho_1}\oplus..\oplus{\rho_1}$ (each irreducible 
subquotient  of $\rho_0$ is isomorphic to $\rho_1$).
\end{lemma}

 {\bf Proof : } We can assume that $\rho_0$ is absolutely irreducible. 
Consider the exact 
sequence  $1\!\rightarrow\! H_1(\F_q)=\prod_{i\in I}\SL_2(\F_q)\!\rightarrow\!
 H(\F_q)\! \overset{\nu}{\rightarrow} \!\mathcal{D} \!\rightarrow\! 1$.
  By Cor.\ref{bn}, 
we know that  each irreducible  subquotient  of ${\rho_0}_{|H_1(\F_q)}$ is of 
the form   $\otimes_{i\in I}\left(\otimes_{j=1}^r(\Sym_i^{a_{i,j}})^{\sigma_j}
\right) \text{, with } 0\leq a_{i,j}\leq p-1$.

The subspace corresponding to the highest weight of the representation
${\rho_0}_{|H_1(\F_q)}$ is preserved by the  standard torus of $H(\F_q)$, and 
therefore contains an eigenvector $x$
for the action of this torus. Because $\rho_0$ is irreducible, it is 
generated by $x$, and therefore $\rho_0$ isomorphic to a  twist of 
$\otimes_{i\in I}\left(\otimes_{j=1}^r(\Sym_i^{a_{i,j}})^{\sigma_j}\right)$
 by some power  of the character $\nu$
(in particular, ${\rho_0}_{|H_1(\F_q)}$ is also irreducible).

 As the characteristic polynomial   of $\rho_1$ annihilates $\rho_0$, 
the set  of the  weights of $\rho_0$ is a subset of the set  of the  weights of 
$\rho_1$, and therefore $\rho_0=\rho_1$. \hfill$\square$

In {\S}\ref{ind-image} we proved  under the assumption
 $\mathrm{\bf (LI_{\Ind\overline{\rho}})}$, that 
$\Ind_F^{\Q}\overline{\rho}(\G_{\widehat{F}})$ contains the image of the map 
$\phi=(\phi^i)_{i\in I} : H(\F_q) \hookrightarrow  \GL_2(\F_q)^{J_F}$.

Denote by $\widehat{F}'$ the fixed field of 
$\overline{\rho}^{-1}(\phi(H(\F_q)))$.


\begin{lemma} {\rm(Key Lemma)} \label{key}
Let $\rho_0$ be  a  representation of $\G_{\widehat{F}'}$ on a 
finite dimensional $\kappa$-vector space $W$. 
Assume $\mathrm{\bf (LI_{\Ind\overline{\rho}})}$ and assume that, 
for every $g\in \G_{\widehat{F}'}$,  the characteristic polynomial  of 
$(\otimes\Ind_F^{\Q}\overline{\rho})(g)$ annihilates $\rho_0(g)$.
Then  each $\G_{\widehat{F}'}$-irreducible subquotient   of $\rho_0$ 
is isomorphic to  $\otimes\Ind_F^{\Q}\overline{\rho}$.
\end{lemma}

 {\bf Proof : } It is enough to treat the case where $\rho_0$ is irreducible. 
The idea is show that the action of $\G_{\widehat{F}'}$ on $W$ is through 
the  algebraic group $H(\F_q)$, and then  use the lemma 
$\ref{variante-mazur}$.

Put  $\overline{\rho}'= (\Ind_F^{\Q}\overline{\rho})_{|\G_{\widehat{F}'}}$. 
Because of the annihilation assumption, the group $\rho_0(\ker(\overline{\rho}'))$
is an unipotent $p$-group   and  therefore $W^{\ker(\overline{\rho}')}$
is non-zero.  Moreover  the subspace $W^{\ker(\overline{\rho}')}$ is 
preserved  by $\G_{\widehat{F}'}$. Because $W$ is irreducible, we get 
$W^{\ker(\overline{\rho}')}=W$ and therefore the action of $\G_{\widehat{F}'}$
 on  $W$ is through $H(\F_q)$.  Thus we get a homomorphism  $\rho_0'$  
fitting in the following commutative diagram :
$$\xymatrix@C=50pt{
 \G_{\Q}\ar[r]^{\otimes\Ind_F^{\Q}\overline{\rho}}&  \GL_{2^d}(\kappa)\\
  \G_{\widetilde{F}}\ar@{_{(}->}[u] \ar[r]^{\Ind_F^{\Q}\overline{\rho}}& 
 \GL_2(\kappa)^{J_F} \ar[u]^{\otimes} \\
  \G_{\widehat{F}'} \ar[d]_{\rho_0}
\ar@{_{(}->}[u]\ar@{->>}[r]^-{\phi^{-1}\circ\overline{\rho}'}& 
H(\F_q)\ar@{_{(}->}^{\phi}[u]
\ar@{-->}[dl]_{\rho_0'}\ar@{_{(}->}[u] \ar@/_3pc/[uu]_{\rho_1} \\
\GL(W)  & }$$
The characteristic polynomial  of   $\rho_1$ annihilates   the representation 
$\rho_0'$.  By the lemma \ref{variante-mazur} each $H$-irreducible 
subquotient of $W$ is  isomorphic to $\rho_1$, that is  to say
$W^{\mathrm{s.s.}}= \oplus\rho_1$ as  $H(\F_q)$-modules. As the action of 
$\G_{\widehat{F}'}$ on both sides is  through $H(\F_q)$, we're done. 
\hfill$\square$


\subsection{Localized cohomology of the  Hilbert modular variety.}${}$

Let $\T'\subset \T$ be the subalgebra generated by the Hecke operators
outside a finite set of places containing those dividing $\gn p$. Put
$\gm'=\gm\cap \T'$. 


\begin{theo}\label{coh-loc}
Assume  $f$ and  $p$ satisfy $\mathrm{{\bf (I)}}$, $\mathrm{{\bf (II)}}$ and 
$\mathrm{\bf (LI_{\Ind\overline{\rho}})}$. Then 

{\rm (i)}  $\rH^\bullet(Y,\bV_n(\kappa))_{\gm'}=
\rH^d(Y,\bV_n(\kappa))_{\gm'}$,

{\rm (ii)} 
$\rH^\bullet(Y,\bV_n(\cO)_{\gm'}
=\rH^d(Y,\bV_n(\cO))_{\gm'}$ is a free $\cO$-module 
of  finite rank and the $\cO$-module  $\rH^\bullet(Y,\bV_n(E/\!\cO))_{\gm'}
=\rH^d(Y,\bV_n(E/\!\cO))_{\gm'}$ is  divisible   of finite corank .


$\mathrm{(iii)}$ 
$\rH^d(Y,\bV_n(\cO))_{\gm'} 
\times \rH^d(Y,\bV_n(E/\!\cO))_{\gm'}\rightarrow \cO$ is a
perfect Pontryagin pairing.
\end{theo}

{\bf Proof : } 
(i) By Faltings' Comparison Theorem  \cite{Fa-jami} and Thm.\ref{bgg-modp}(i) the integer
$|p(J)|$ is not a Fontaine-Laffaille  weight 
of  $\rH^r(\kappa)$, when $r<d$.  Wedhorn \cite{Wed} has established the 
congruence  relations for  all totally split   primes  of  $F$.
By  the Cebotarev Density Theorem  the assumptions of the key lemma 
\ref{key} are fulfilled. We deduce  that  $\rH^r(\kappa)[\gm']=0$, and therefore 
$\rH^r(\kappa)_{\gm'}=0$
by Nakayama's lemma. The case  $n>d$ follows by  Poincar{\'e} duality.

(ii)(iii)  By  the long exact cohomology sequence
$$...\rightarrow \rH^{r-1}(\kappa)\longrightarrow
\rH^r(\cO)\overset{\varpi}{\longrightarrow}
\rH^r(\cO)\longrightarrow \rH^r(\kappa)\rightarrow  ... ,$$

\noindent and by the vanishing  of $\rH^r(\kappa)_{\gm'}$, 
for  $r \neq d$, we  deduce that (for $r \neq d$)
the  multiplication by an uniformizer $\varpi$ is a surjective endomorphism  of 
$\rH^r(\cO)_{\gm'}$, so this last vanishes.

The  same way, by the long exact sequence
$$...\rightarrow \rH^r(\varpi^{-1}\cO/\cO)\longrightarrow
\rH^r(E/\!\cO)\overset{\varpi}{\longrightarrow}
\rH^r(E/\!\cO)\longrightarrow
\rH^{r+1}(\varpi^{-1}\cO/\cO) \rightarrow  ... ,$$

\noindent we deduce a surjection $\rH^r(\kappa)_{\gm'}
\twoheadrightarrow  \rH^r(E/\!\cO)_{\gm'}[\varpi]$, when $r\neq d$. 
Since  $\rH^r(E/\!\cO)_{\gm'}$ is a torsion $\cO$-module, it
vanishes (for $r \neq d$).


\medskip 

The  localization at $\gm'$ of the  long exact  sequence  of $\cO$-modules :
$$...\rightarrow \rH^{r-1}(E/\!\cO)\longrightarrow
\rH^r(\cO)\longrightarrow \rH^r(E)\longrightarrow \rH^r(E/\!\cO)
\rightarrow... ,$$
 is concentrated  at the three  terms of degree  $r=d$.
From this we deduce the freeness. \hfill $\square$


\subsection{On the Gorensteiness of Hecke the algebra.}


\begin{theo}{\rm(Theorem B)}\label{theo-B}
Let  $f$ and  $p$ be such that $\mathrm{{\bf (I)}}$, $\mathrm{{\bf (II)}}$ 
and $\mathrm{\bf (LI_{\Ind\overline{\rho}})}$  hold. Then

{\rm(i)} $\rH^\bullet(Y,\bV_n(\kappa))[\gm]=
\rH^d(Y,\bV_n(\kappa))[\gm]$ is a  $\kappa$-vector 
space  of dimension $2^d$. 

{\rm(ii)} $\rH^\bullet(Y,\bV_n(\cO))_{\gm}=
\rH^d(Y,\bV_n(\cO))_{\gm}$ 
is free of rank  $2^d$ over  $\T_{\gm}$.  

{\rm(iii)} $\T_{\gm}$ is Gorenstein.
\end{theo}

{\bf Proof : } In this proof we put 
$W=\rH^d(Y_{\overline{\Q}},\bV_n(\kappa))_{\gm}$. 
By using an auxiliary level structure as in \cite{diamond2}, we
can assume that the condition {\bf (NT)} of \S\ref{hmv} is fullfilled. 

(i) As in the proof of Thm.\ref{coh-loc}(i), by  lemma \ref{key} 
we have an isomorphism of $\G_{\widehat{F}'}$-modules
$$ W[{\gm}]^{\mathrm{s.s.}}=
(\otimes \Ind_F^{\Q}\overline{\rho})^{\oplus r}.$$
It is crucial   to observe that  $I_p\subset \G_{\widehat{F}'}$.
By Thm.\ref{BL-thm} we have $r\geq 1$.
In order to show that  $r=1$  we consider  the restriction of these 
representations  to $I_p$. 
The multiplicity of the 
maximal Fontaine-Laffaille weight $|p(J_F)|$ in the right hand side  is $r$, 
by Thm.\ref{BL-thm}, Cor.\ref{hodge-complex}(ii)
and Fontaine-Laffaille's theory. 

On the other hand, the multiplicity of $|p(J_F)|$ in the left hand side 
is equal, by Thm.\ref{bgg-modp},  to the dimension of 
$\rH^0(\overline{Y}\otimes\kappa,\overline{\cW}_{\epsilon_{J_F}(n+t)-t,n_0})[\gm]$ :
In fact, by the remark \ref{hecke-equiv} it is sufficient to check 
the $T_{\gp}$-equivariance of the $\gm'$-localization of the projective limit over 
$r$ of (\ref{bgg-modp1}). By Thm.\ref{coh-loc}(ii) it is question of 
checking the $T_{\gp}$-equivariance of an isomorphism of {\it free} $\cO$-modules.
 Therefore, it is enough to be checked after  extending  the  scalars to 
$\C$. Then the  Strong Multiplicity One Theorem applies (recall
that $p$ is prime to the level $\gn$). We owe  this  idea to  Diamond  
(\cite{diamond2} proof of Prop.1).


\smallskip
We will now see that  $\dim_\kappa \rH^0(\overline{Y}\otimes \kappa, 
\overline{\cW}_{\epsilon_{J_F}(n+t)-t,n_0})[\gm]=1$.
We have $\overline{\cW}_{\epsilon_{J_F}(n+t)-t,n_0 }=
\underline{\omega}^k\otimes\underline{\nu}^{n_0t/2}$.
So we are led to show that two normalized Hilbert modular forms of weight $k$, 
level $\gn$ and coefficients in  $\kappa=\T_{\gm}/\gm$ 
 having  the same  eigenvalues for all the Hecke operators  are equal.
One should be careful to observe  that the Hecke operators 
permute the  connected components $M_1(\gc,\gn)$
 of the Shimura variety     $Y=Y_1(\gn)$ (here 
the ideal $\gc$ runs over a set of representatives of  $\Cl_F^+$).
We use then the Hecke relations  between Fourier coefficients 
and  eigenvalues  for the Hecke operators   and 
the   $q$-expansion principle (see \ref{qdev})  at the  $\infty$ cusp  
of each connected  component  $M_1(\gc,\gn)$.

\medskip

Even if we do not know the degeneration at 
$\rE_1$ of the   Hodge to  de Rham spectral sequence, we 
obtain by the same  arguments that $r\leq 1$ (instead of  $r=1$), 
because  we have always $\rH^0(\overline{Y}\otimes \kappa, 
\overline{\cW}_{\epsilon_{J_F}(n+t)-t,n_0})[\gm]
\supset  \gr^{|p(J_F)|}W[\gm]$. But 
$\rH^\bullet(Y,\bV_n(\cO))_{\gm}$ is non zero 
 as $\rH^\bullet(Y,\bV_n(\cO))_{\gm}\otimes \Q$
is  free of rank $2^d$ over $\T_{\gm}\otimes \Q$, and therefore $r=1$. 


(ii)(iii) Mazur's argument in the elliptic modular case remains valid.
By the theorem A, the twisted Poincar{\'e} pairing (\ref{accoupl-twisted}) on 
$\rH^d(Y,\bV_n(\cO))_{\gm}= \rH_c^d(Y,\bV_n(\cO))_{\gm}$ 
 is  a perfect duality of $\T_{\gm}$-modules,  
so it would be enough to show (ii).

Again using  the perfectness of the twisted Poincar{\'e} pairing
$W\times W\rightarrow \kappa$ 
we obtain  $W \cong \Hom_{\T_{\gm}} (W,\kappa)$, and so 
$W\otimes_{\T_{\gm}} \kappa  =W/\gm W \cong \Hom(W[\gm],\kappa)$,
and therefore 
$$\dim_\kappa(W\otimes_{\T_{\gm}} k)= \dim_\kappa(W[\gm]),$$
which equals  $2^d$, by (i). Then (ii) follows from the following

\begin{lemma}\label{freeness}
Let $\cT$ a torsion free local $\cO$-algebra 
($\cT\hookrightarrow \cT\otimes_{\cO}E$) of  maximal ideal 
 $\gm$ and  residue field $\kappa=\cT/\gm$.

Let $M$ be a finitely generated $\cT$-module such that  $M\otimes_{\cO}E$
is  free of rank $r$ over  $\cT\otimes_{\cO}E$. If
 $M\otimes_{\cT} \kappa$ is a $\kappa $-vector space  of dimension $\leq r$, 
then  $M$ is free of rank $r$ over $\cT$. 
\end{lemma}


{\bf Proof : } 
Since  $\cM\otimes_{\cT} k$ is of dimension $\leq r$, the  Nakayama's  lemma 
gives  a surjective homomorphism  of $\cT$-modules
$\cT^r\twoheadrightarrow M$. Denote by  $I$ its kernel. We have an 
 exact  sequence of $\cO$-modules 
$$0\rightarrow I \rightarrow \cT^r \rightarrow M \rightarrow 0.$$
By  tensoring it by  $\otimes_{\cO}E$ (or equivalently by  
$\otimes_{\cT}(\cT\otimes_{\cO}E)$) 
we obtain another  exact sequence   $$0\rightarrow I \otimes_{\cO}E
 \rightarrow (\cT\otimes_{\cO}E)^r \rightarrow M\otimes_{\cO}E\rightarrow 0.$$
By comparing the  dimensions over $E$ we get  $I\otimes_{\cO}E=0$. 
Since $I$ is torsion free, $I=0$. \hfill$\square$


\subsection{An application to $p$-adic ordinary families.}

For $r\geq 1$, consider the following open compact subgroups of $G(\A_f)$
$$K_0(p^r)=\left\{u\in K_1(\gn)| u\equiv 
\begin{pmatrix} *& * \\ 0 & *\end{pmatrix}\pmod{p^r}\right\},$$
$$K_{11}(p^r)=\left\{u\in K_1(\gn)| u\equiv 
\begin{pmatrix} 1 & * \\ 0 & 1\end{pmatrix}\pmod{p^r}\right\}.$$

Let $Y_0(p^r)$ (resp. $Y_{11}(p^r)$) be the Hilbert modular variety of 
level $K_0(p^r)$ (resp. $K_{11}(p^r)$).

\medskip
The cohomology group $\rH^{\bullet}(Y_{11}(p^r),\bV_n(E/\cO))^*$ has 
a natural action of $K_0(p^r)/K_{11}(p^r)\simeq
(\go/p^r)^\times\times (\go/p^r)^\times$
(we denote by ${}^*$ the Pontryagin dual). Therefore the group 
$T(\Z_p)/\overline{\go^\times}$ acts on 
the inductive  limit $\rH^{\bullet}(Y_{11}(p^{\infty}),\bV_n(E/\cO))^*:=
\limind \rH^{\bullet}(Y_{11}(p^r),\bV_n(E/\cO))^*$.


By Hida's stabilization lemma, the ordinary part of 
$\rH^{\bullet}(Y_{11}(p^{\infty}),\bV_n(E/\cO))^*$ (that is the part
where the Hecke operators $T_{0,\gp}$ of Def.\ref{ord-autom} 
are  invertible for all  $\gp$ dividing $p$) is independent on $n$. We
denote it by  $\cH_{\ord}^{\bullet}:=
\rH^{\bullet}_{\ord}(Y_{11}(p^{\infty}),E/\cO)^*$.

By the above discussion    $\cH_{\ord}^{\bullet}$ has a structure of a 
 $\Lambda:=\cO[[T(\Z_p)/\overline{\go^\times}]]$-module.
 It is of finite type, by a theorem of Hida. 

\medskip
We also define the $p$-adic ordinary Hecke $\Lambda$-algebra 
$\T_{k,\ord}^{\infty}:=\limproj \T_{k,\ord}(Y_{11}(p^r))$. 
As $\T_{k,\ord}^{\infty}$ is independant of $k$, we denote it by 
$\T^{\infty}_{\ord}$. 
Then  $\cH_{\ord}^{\bullet}$ is  a  $\T^{\infty}_{\ord}$-module. 


\medskip
An  arithmetic  character of $T(\Z_p)/\overline{\go^\times}$ is by definition 
a character whose restriction to an open subgroup is given by an 
algebraic character. It is immediate that such a character is a 
product of an algebraic character and a finite order character. An algebraic 
character of $T(\Z_p)\simeq D(\Z_p)\times D(\Z_p)$ trivial on $\go^\times$
is necessarily of the form $(u,\epsilon)\mapsto u^n\epsilon^{-m}$, 
where $m,n\in \Z[J_F]$ and
$m+2n\in \Z t$. Hence, the general form of an arithmetic character $\psi$
of  $T(\Z_p)/\overline{\go^\times}$ is   $(u,\epsilon)\mapsto 
u^n\epsilon^{-m}\psi_1(u)\psi_2(\epsilon)$, where $\psi_1,\psi_2$ are finite 
order characters.  Every such $\psi$ induces an $\cO$-algebra homomorphism 
$\Lambda\rightarrow \cO$, whose kernel is denoted by $P_\psi$. 

\medskip
Let $\gm$ be a maximal ordinary ideal of $\T=\T_k(\gn)$ and  $\gm_{\infty}$ be 
a maximal ideal of $\T^{\infty}_{\ord}$ above  $\gm$.
We denote by $\T^{\infty}_{\gm_{\infty}}$
(resp. $\cH_{\gm_{\infty}}^{\bullet}$) the localization of 
$\T^{\infty}_{\ord}$ (resp. of  $\cH_{\ord}^{\bullet}$) at
$\gm_{\infty}$. 


\begin{prop} \label{control} Let  $\gm$ be such  that $\mathrm{{\bf (I)}}$, $\mathrm{{\bf (II)}}$ and
 $\mathrm{\bf (LI_{\Ind\overline{\rho}})}$ hold. Then 

\rm{(i)} $\cH_{\gm_{\infty}}^d$ is free of finite rank over $\Lambda$ and
we have exact control :
 $$\cH_{\gm_{\infty}}^d/P_\psi\cH_{\gm_{\infty}}^d\simeq
\rH^{\bullet}(Y_{11}(p^r),\bV_\psi(E/\cO))^*_{\gm_r},$$

\rm{(ii)} $\cH_{\gm_{\infty}}^d$ is free of  rank $2^d$ over 
$\T^{\infty}_{\gm_{\infty}}$, and 

\rm{(iii)}  Hida's control theorem for the  Hecke algebra holds, that is 
$\T^{\infty}_{\gm_{\infty}}$ is a free $\Lambda$-algebra  of finite rank
and  for every $\psi$ we have
$\T^{\infty}_{\gm_{\infty}}/P_\psi\T^{\infty}_{\gm_{\infty}}\simeq 
\T_\psi(Y_{11}(p^r))_{\gm_r}$.
\end{prop}

{\bf Proof : } 
(i) The proof is very similar to the one of \cite{MoTi}Thm.9. It uses 
that a $\Lambda$-module is free, if it is free of constant rank over $\cO$ for
infinitely many specializations. In our case, it is enough to 
specialize at the weight of the form $k+(p-1)k'$, and use 
the exact control criterion and Thm.\ref{coh-loc}. 
We omit the details. Note that (i) follows from  (ii) and (iii).
 

(ii) Consider   $\Lambda\rightarrow \T^{\infty}_{\gm_{\infty}}
\rightarrow \End_{\cO}(\cH_{\gm_{\infty}}^d)$. The specialization 
 at $\psi=\psi_k$ gives 
$$\cO \rightarrow \T^{\infty}_{\gm_{\infty}}/P_k \T^{\infty}_{\gm_{\infty}}
\rightarrow \End_{\cO}(\cH_{\gm_{\infty}}^d/P_k \cH_{\gm_{\infty}}^d).$$

By Thm.\ref{coh-loc} we have 
$\rH^d(Y_0(p),\bV_n(E/\cO))^*_{\gm}\simeq \rH^d(Y,\bV_n(E/\cO))^*_{\gm}\simeq
\rH^d(Y,\bV_n(\cO))_{\gm}$ and an exact control : $\cH_{\gm_{\infty}}^d/P_k 
\cH_{\gm_{\infty}}^d\simeq  \rH^d(Y,\bV_n(\cO))_{\gm}$.

From here and from  Thm.B we obtain that $\cH_{\gm_{\infty}}^d\otimes_{\T^{\infty}_{\gm_{\infty}}}
(\T^{\infty}_{\gm_{\infty}}/P_\psi\T^{\infty}_{\gm_{\infty}})\simeq
\cH_{\gm_{\infty}}^d\otimes_{\Lambda} \Lambda/P_k$ 
is free  of rank $2^d$ over  $\T_{\gm}$. Hence 
$\cH_{\gm_{\infty}}^d\otimes_{\T^{\infty}_{\gm_{\infty}}}\kappa$
is free  of rank $2^d$ over  $\T_{\gm}
\otimes_{\T^{\infty}_{\gm_{\infty}}/P_\psi\T^{\infty}_{\gm_{\infty}}}\kappa=\kappa$.
Then lemma \ref{freeness}  applies to the $\T^{\infty}_{\gm_{\infty}}$-module
$\cH_{\gm_{\infty}}^d$ which is finitely generated  over the local algebra $\Lambda$. 


(iii) As  $\cH_{\gm_{\infty}}^d$ is a free  $\Lambda$-module, it 
admits a direct sum decomposition with respect to the Weyl group
action on the Betti cohomology :
$$\cH_{\gm_{\infty}}^d=\bigoplus_{J\subset J_F} 
\cH_{\gm_{\infty}}^d[\widehat{\epsilon_J}].$$

Every $\cH_{\gm_{\infty}}^d[\widehat{\epsilon_J}]$ is free 
of rank 1 over $\T^{\infty}_{\gm_{\infty}}$ and free over $\Lambda$.
Therefore   $\T^{\infty}_{\gm_{\infty}}$ is  free over $\Lambda$ and exact control 
holds.   \hfill$\square$

\begin{cor}  \label{thm-ord}
 Let $f\in S_{k+(p-1)k'}(Y_0(p^r))$ be a newform and $p$ be a prime not
dividing $\N_{F/\Q}(\gd)$, such that $p-1>\sum (k_\tau-1)$ and 
$\mathrm{\bf (LI_{\Ind\overline{\rho}})}$ holds. Then theorems A and B hold. 
\end{cor}

\newpage
\section*{List of symbols} 

\begin{multicols}{3}
\noindent $A$ \dotfill HBAV, Def.1.6

\noindent $A^t$ \dotfill dual HBAV 

\noindent $\cA$ \dotfill universal HBAV \S1.4

\noindent $\ga$ \dotfill ideal of $\go$

\noindent $B$ \dotfill standard Borel of $G$ 

\noindent $\gb$ \dotfill Lie algebra of  $B$

\noindent $\gc,\gc_+$ \dotfill  \S1.3

\noindent $c(f,\ga)$ \dotfill \S1.10

\noindent $\Cl_F$ \dotfill class group \S1.2

\noindent $\Cl_F^+$ \dotfill narrow class group \S1.1

\noindent $d$ \dotfill degree of $F$

\noindent $D$ \dotfill $\Res^{F}_{\Q} \Gm$ 

\noindent $D_{\gp}$ \dotfill decomposition group\S2.6

\noindent $\gd$ \dotfill different of $F$

\noindent $\mathcal{D}$ \dotfill \S3.4 

\noindent $E$ \dotfill large $p$-adic field

\noindent $f$ \dotfill Hilbert modular newform

\noindent $F$ \dotfill totally real number field

\noindent $\widetilde{F}$ \dotfill Galois closure of $F$

\noindent $\widetilde{F}'$ \dotfill before lemma 6.5

\noindent $\widehat{F}$ \dotfill \S3.4 

\noindent $\cF_D,\cF_B,\cF_G$ \dotfill functors \S2.3 

\noindent $g$ \dotfill Hilbert modular form

\noindent $G$ \dotfill $\Res^{F}_{\Q} \GL_2$ 

\noindent $\mathfrak{g}$ \dotfill Lie algebra of  $G$

\noindent $G^*$ \dotfill $G\times_{D}\Gm$ 

\noindent $g_J$ \dotfill \S1.11

\noindent $g_\tau$ \dotfill internal conjugate \S2.5

\noindent $\G_L$ \dotfill Galois group of $L$

\noindent $\mathfrak{G}$ \dotfill \S1.6

\noindent $H$ \dotfill above lemma 3.15

\noindent $\mathfrak{H}_F,\mathfrak{H}$ \dotfill  \S1.1

\noindent $h^+$ \dotfill  \S1.1

\noindent $\cH^1_{\dR}$ \dotfill  \S1.4

\noindent $I_{\gp}$ \dotfill inertia group at $\gp$ \S2.6

\noindent $J$ \dotfill subset of $J_F$

\noindent $J_F$ \dotfill set of infinite places of $F$

\noindent $J_F^i$ \dotfill \S3.5

\noindent $J_{F,\gp}$ \dotfill  \S2.6

\noindent $k=n+2t$ \dotfill  Def.\ref{weight} 

\noindent $k_0,n_0$ \dotfill Def.\ref{weight} 

\noindent $K$ \dotfill  CM field

\noindent $K_\infty,K_\infty^+$ \dotfill \S1.1

\noindent $K_1(\gn),K^1_1(\gn)$ \dotfill \S1.1

\noindent $K^\bullet_n$ \dotfill \S2.2

\noindent $\mathcal{K}^\bullet_n$ \dotfill \S2.3

\noindent $L$ \dotfill  field

\noindent $m=(k_0t-k)/2$ \dotfill  Def.\ref{weight} 

\noindent $M,M^1$ \dotfill connected HMV \S\ref{hmv}

\noindent $M'$ \dotfill \S1.4

\noindent $\overline{M},\overline{M^1}$ \dotfill \S1.6

\noindent $M^*,M^{1*}$ \dotfill \S1.8

\noindent $\gm$ \dotfill maximal ideal of $\T$ \S0.3

\noindent $\gm'$ \dotfill maximal ideal of $\T'$ \S6.2

\noindent $n$ \dotfill weight of $G$

\noindent $N$ \dotfill  normalizer of $T$ 

\noindent $\gn$ \dotfill level ideal

\noindent $\cO$ \dotfill integer ring of $E$

\noindent $\go$ \dotfill integer ring of $F$

\noindent $\go'$ \dotfill \S1.4

\noindent $\go_+^\times,\go_{\gn,1}^\times$ \dotfill \S0.4

\noindent $p$ \dotfill prime number 

\noindent $p(J)$ \dotfill \S2

\noindent $\cP$ \dotfill maximal ideal of $\cO$

\noindent $\gp$ \dotfill prime of $F$ dividing $p$

\noindent $q$ \dotfill power of $p$

\noindent $R$ \dotfill ring

\noindent $s$ \dotfill \S2.1

\noindent $S$ \dotfill base scheme

\noindent $S_k(\gn,\psi)$ \dotfill Def.1.3

\noindent $S_{\ga},T_{\ga}$ \dotfill Hecke operators \S1.10

\noindent $t=\sum_{\tau\in J_F}\tau$ \dotfill  Def.\ref{weight}

\noindent $T$ \dotfill standard torus of $G$ 

\noindent $\gt$ \dotfill Lie algebra of  $T$

\noindent $\T$ \dotfill Hecke algebra \S0.3

\noindent $\T'$ \dotfill reduced Hecke algebra 

\noindent $U$ \dotfill standard unipotent of $G$ 

\noindent $U(\gb),U(\mathfrak{g})$ \dotfill \S2.2
 
\noindent $\gu$ \dotfill Lie algebra of  $U$

\noindent $v$ \dotfill finite place of $F$

\noindent $V,V_n$ \dotfill  $G$-modules \S1.12

\noindent $\bV_n$ \dotfill local system \S1.12, \S2.1

\noindent $\overline{\cV},\overline{\cW}$ \dotfill \S2.3

\noindent $W,W_\mu$ \dotfill  $B$-modules 

\noindent $W_f$ \dotfill   \S2.1

\noindent $Y,Y^1$ \dotfill HMV \S\ref{hmv} 

\noindent $\alpha$  \dotfill $\mu_{\gn}$-level structure, Def.1.8

\noindent $\beta$  \dotfill \S2.4

\noindent $\gamma$  \dotfill element of $G(\R)$

\noindent $\Gamma_1(\gc,\gn)$  \dotfill \S1.1

\noindent $\Gamma^1_1(\gc,\gn)$ \dotfill \S1.1

\noindent $\delta,\delta_J$  \dotfill  \S1.12

\noindent $\Delta$  \dotfill $\Nm(\gn\gd)$

\noindent $\delta_{\gp},\varepsilon_{\gp}$  \dotfill tame characters of $I_{\gp}$ 

\noindent $\varepsilon$  \dotfill quadratic character

\noindent $\epsilon$  \dotfill unit of $F$

\noindent $\epsilon_J$  \dotfill \S1.11

\noindent $\widehat{\epsilon}_J$  \dotfill \S1.12

%\noindent $\zeta$  \dotfill

\noindent $\eta$  \dotfill id{\`e}le

%\noindent $\theta$  \dotfill 

\noindent $\iota$  \dotfill  Def.1.6

\noindent $\iota_p$  \dotfill embedding of $\overline{\Q}$ in  $\overline{\Q}_p$

\noindent $\kappa$  \dotfill residue field of $\cO$

\noindent $\lambda$  \dotfill $\gc$-polarization, Def.1.7

\noindent $\overline{\lambda}$  \dotfill $\gc$-polarization class,Def.1.7

\noindent $\mu$  \dotfill weight of $B$

\noindent $\nu$  \dotfill reduced norm  $G \rightarrow D$ \S\ref{hmfv}

\noindent $\underline{\nu}$  \dotfill \S1.5

\noindent $\xi$  \dotfill element of $F$

\noindent $\pi$  \dotfill \S1.4

\noindent $\varpi_v$  \dotfill uniformizer of $F_v$

\noindent $\overline{\pi}$  \dotfill \S1.6

\noindent $\rho=\rho_{f,p}$  \dotfill $p$-adic repr. \S0.1

\noindent $\overline{\rho}=\overline{\rho}_{f,p}$ \dotfill mod $p$ repr. \S0.1 

\noindent $\rho_1$  \dotfill \S6.1

\noindent $\sigma$  \dotfill cone \S1.6

\noindent $\Sigma$  \dotfill fan \S1.6

\noindent $\sigma_{i,\tau}$  \dotfill \S3.5

\noindent $\tau$  \dotfill  infinite place of $F$

\noindent $\phi$  \dotfill \S3.5

\noindent $\varphi$  \dotfill Hecke character

\noindent $\chi_n$  \dotfill \S2.2
  
\noindent $\psi$  \dotfill Hecke character, Def.1.3

\noindent $\omega$  \dotfill  mod $p$ cycl. char. 

\noindent $\Omega_f^{\pm}$  \dotfill  periods, Def.4.5

\noindent $\underline{\omega}$  \dotfill \S1.4

\noindent $\underline{\omega}^k$  \dotfill \S1.5

\noindent $(\enspace,\enspace)_{\gn}$  \dotfill  \S1.4
\end{multicols}



\newpage
\section*{Appendix}  
\renewcommand\thesection{A}
\renewcommand\thesubsection{A.\arabic{subsection}}
\setcounter{subsection}{0} 
\setcounter{theo}{0} 

The notations in this section are independent from the rest of the text.

Let $K$ be a local field  of characteristic zero  with a perfect
residue field  $\kappa$ of characteristic $p$.  Let
$W$ be the ring   of Witt vectors of $\kappa$ and $K_0$ 
be the fraction field of $W$. We  denote by $\sigma$ the 
Frobenius of $\kappa$, $W$ and $K_0$. 
Let $E$ be  a finite   extension  of $\Q_p$ in $\overline{\Q}_p$. 

%R{\'e}f{\'e}rences : \cite{FoLa}, \cite{Wach}, \cite{Mo}, \cite{Wi}, \cite{BrMe},\cite{Fo}

\subsection{ $p$-adic representations.} 

A $p$-adic representation of $\G_K$ is a finite dimensional 
$\Q_p$-vector space, endowed with continuous action of $\G_K$.
The $p$-adic representations form an abelian category, denoted
 $\Rep(\G_K)$. We denote by $\Rep_E(\G_K)$ the subcategory 
of $\Rep(\G_K)$ consisting of $E$-linear representations.

There are several interesting subcategories of $\Rep(\G_K)$, 
as the one of Hodge-Tate  representations $\Rep_{\mathrm{HT}}(\G_K)$,
the one of de Rham representations $\Rep_{\dR}(\G_K)$,
the one of semi-stables  representations $\Rep_{\mathrm{st}}(\G_K)$,
and the one of  crystalline representations $\Rep_{\crys}(\G_K)$
$$\Rep(\G_K)\supset \Rep_{\mathrm{HT}}(\G_K) \supset 
\Rep_{\dR}(\G_K)\supset  \Rep_{\mathrm{st}}(\G_K) \supset 
\Rep_{\crys}(\G_K).$$

Let  $\Rep_{\overline{\Q}_p}(\G_K)$ be the category of continuous
representations of $\G_K$ on  finite dimensional $\overline{\Q}_p$-vector spaces.
All these representations are obtained by scalar extension from $\Rep_E(\G_K)$ (for some $E$).
By an abuse of language will still  call them   $p$-adic representations.



\subsection{ Hodge-Tate weights.}
Denote by $C$ the $p$-adic  completion of the algebraic closure
$\overline{K}$ of $K$. The $\G_K$-action on $\overline{K}$ 
extends by continuity to an action on $C$. 
Put $B_{\mathrm{HT}}=\oplus_i C(i)$, where the $\G_K$-action on 
$C(i)$ is twisted by the $i$-th power  of the cyclotomic character . 


Let $V\in \Rep(\G_K)$. Then, by definition,  
$V\in \Rep_{\mathrm{HT}}(\G_K)$, if and only, if 
$\dim_K(V\otimes_{\Q_p}B_{\mathrm{HT}})^{\G_K}=\dim_{\Q_p}V$.
For $V\in \Rep_{\mathrm{HT}}(\G_K)$, we say  that $i$ is a  
Hodge-Tate weight of $V$, if  $V_i:=(V\otimes_{\Q_p}C(i))^{\G_K}\neq 0$ 
and we call  $h^i=\dim_K V_i$ its  multiplicity. 
We have a equality of $\G_K$-modules 
$V\otimes_{\Q_p}C=\bigoplus_i V_i\otimes_K C(-i)$.


If $V\in \Rep_E(\G_K)$, then   for all $i\in \Z$ 
$V_i=(V\otimes_{\Q_p}C(i))^{\G_K}$ is a $E\otimes_{\Q_p} K$-module 
in a natural way. It is not free in general. By decomposing the 
 $\Q_p$-algebra  $E\otimes_{\Q_p} K$ as a product of fields $\prod_j L_j$
(endowed with injections  $\sigma : E\hookrightarrow L_j$, 
$\tau : K\hookrightarrow L_j$), we obtain :
$$(V\otimes_{\Q_p}C(i))^{\G_K}
\underset{E\otimes K}{\otimes} L_j=
(V\otimes_{E}C(i))^{\G_{L_j}} $$

There is another way to index the  Hodge-Tate weights 
that is more appropriate to  the modular case. Consider the 
functor $\Rep_E(\G_K)\rightarrow \Rep_{\overline{\Q}_p}(\G_K)$
sending $V$ to $V_{\overline{\Q}_p}:=V\otimes_E \overline{\Q}_p$.


\begin{defin}\label{poids-ht} For all 
 $\tau: K\hookrightarrow \overline{\Q}_p$ 
we put $h_{\tau,i}=\dim_{\overline{\Q}_p}
(V_{\overline{\Q}_p}\otimes_{\tau,K} C(i))^{\G_K}$. The integer 
$h_{\tau,i}$ is called the multiplicity of $i$ as Hodge-Tate weight 
of $V$. For all $\tau$, we have $\sum_{i\in\Z} h_{\tau,i}= \dim_E V$. 
\end{defin}


\begin{ex}
Assume that $E=\Q_p$. Then   $V_i=(V\otimes_{\Q_p}C(i))^{\G_K}$
 is a $K$-vector space  and the $i$-th  Hodge-Tate  number is given by
 $h^i=\dim_K(V_i)$, $i\in \Z$. 

If we change the action of $\G_K$ on $C$ by an automorphism
$g\mapsto \tau^{-1} g\tau$, with $\tau\in \G_{\Q_p}$, 
then  we send $V_i$ onto $V_i^\tau$ by $v\otimes a\mapsto
v\otimes \tau(a)$ that does not  change  $h^i$ 
(since  the cyclotomic character  is invariant by  $g\mapsto \tau^{-1} g\tau$). 
\end{ex}

\subsection{Crystalline representations and filtered modules.}
The category  $\Rep_{\crys}(\G_K)$ of  crystalline representations 
is the $p$-adic  analogue of  the unramified  $l$-adic representations.

\begin{defin}
(i) A filtered $\phi$-module  over $K$ is  a $K_0$-vector space 
 $D$ of finite dimension, endowed with a $\sigma$-linear bijective map  
$\phi:D\rightarrow D$  and a  filtration $\Fil^i D$ of 
$D_K=D\otimes_{K_0} K$ which is decreasing ($\Fil^i D\supset \Fil^{i+1} D$), 
exhaustive ($\cup \Fil^i D=D_K$) and separated ($\cap \Fil^i D=0$).
We denote by $\MF_K$ the additive category  filtered $\phi$-module  over $K$.

(ii) A filtered $\phi$-module $D$   over $K$ is called  weakly admissible, if
it contains a $W$-lattice  $M$, such that 
$\sum p^{-i}\phi(\Fil^i D\cap M)=M$.
Such a lattice is called {\it strongly divisible} (or adapted to $D$). 
The weakly admissible filtered $\phi$-module  over $K$ form 
a full subcategory of $\MF_K$, denoted by $\MF_K^f$.
\end{defin} 


\begin{rque} 
To an object  $D\in \MF_K$ one can associate  Newton and Hodge
polygons and the notion of be  weakly admissible can be 
expressed in terms of inequalities between these two polygons.
\end{rque}


Fontaine's theory gives  an equivalence of categories  $D_{\crys}$ between 
$\Rep_{\crys}(\G_K)$ and a certain full subcategory of 
admissible objects $\MF_K^a$  of $\MF_K$. The Hodge-Tate weights of
$V\in \Rep_{\crys}(\G_K)$ are given by the jumps of the filtration on 
$D_{\crys}(V)$. 


It is known that  admissible implies weakly admissible, and
recently Colmez and Fontaine proved the converse, in the 
more general semi-stable case. When $K$ is an  unramified extension of
$\Q_p$ and the lenght of the filtration is $\leq p-1$, this has been 
established earlier by   Fontaine and Laffaille \cite{FoLa}. 


\subsection{Crystalline representations modulo $p$.} 
In the sequel, we assume  $K$ to be unramified ($K=K_0$).
 Fontaine and Laffaille \cite{FoLa} have introduced

\begin{defin}
(i) A filtered  $F$-module  over $W$ is defined by the following data 

$\bullet$  a  $W$-module $M$, 

$\bullet$  a  filtration $\Fil^iM$   by $W$-submodules which is decreasing
exhaustive and separated,

$\bullet$  $\sigma$-linear maps $\varphi_i: \Fil^iM\rightarrow M$ satisfying 
$\varphi_i|_{\Fil^{i+1}M}=p\varphi_{i+1}$.
 
We denote by $\MF_W$ the $\Z_p$-linear additive category of  filtered 
 $F$-modules  over $W$. 

(ii) We define two full abelian subcategories   
$\MF_{W,lf}\!\subset\! \MF_{W,tf}\!\subset\! \MF_W$, by  $M\in \MF_W$

$$M\in \MF_{W,tf}\iff \begin{cases} M\text{ is of finite  type  over } W,\\
\Fil^iM\text{ are  direct factors in }M, \\
\sum \varphi_i(\Fil^iM)=M.\end{cases}, $$


$$M\in \MF_{W,lf}\iff \begin{cases} M\text{ is of finite length  over } W,\\
\sum \varphi_i(\Fil^iM)=M.\end{cases}. $$
\end{defin}

\begin{rque}
Let $M$ be a strongly divisible lattice  for  $D\in \MF_K^f$, and put
$\Fil^iM=\Fil^iD\cap M$ and $\varphi_i=\phi/p^i$. Then we  have 
$M\in \MF_{W,tf}$. If moreover $M$ is a free $W$-module of  finite type,
then   we have $M/p^rM\in  \MF_{W,lf}$, for all $r \in \n$. 
\end{rque}



\subsection{The tame inertia.}
We choose to uniformize the Local Class Field Theory isomorphism by sending 
a uniformizer to the geometric Frobenius. 

 The tame inertia
$I_K^t$ is the quotient of the inertia group $I_K$ by its  maximal pro-$p$ subgroup
(called the wild inertia). There is a canonical isomorphism 
 $I_K^t\simeq \limproj \F_{p^h}^\times$ (see \cite{serre2}). 
A tame character of level $h\in \n$ is a character of
 $I_K^t$  factoring through  $\F_{p^h}^\times$. 

By the Local Class Field Theory, the inertia group of the maximal 
abelian extension of $K$ is isomorphic to $W^\times$. 
Thus we get a homomorphism $I_K\rightarrow W^\times \rightarrow
\kappa^\times$  equal to the  tame 
character $I_K\rightarrow I_K^t \rightarrow \kappa^\times$ of level $[\kappa:\F_p]$
(see \cite{serre2}).



\subsection{Fontaine-Laffaille's theory  and a theorem of Wintenberger.}
In  \cite{FoLa} Fontaine and Laffaille have introduced a contravariant functor
$$V_{\mathrm{FL}} :\MF_{W,tf}\longrightarrow \Rep_{\Z_p}(\G_K)\text{ , such that}$$

$\bullet$ the restriction to $\MF_{W,lf}^{<p-1}$ is exact and 
fully faithful,

$\bullet$ if $M\in \MF_{W,lf}^{<p-1}$, then $\mathrm{length}_W M=
\mathrm{length}_{\Z_p} V_{\mathrm{FL}}(M)$,

$\bullet$ if $M\in \MF_{W,tf}^{<p-1}$ is free, 
then  $V_{\mathrm{FL}}(M)$ is free and 
$\mathrm{rank}_W M= \mathrm{rank}_{\Z_p} V_{\mathrm{FL}}(M)$.

Let $X$ be the abelian  group of periodic map  $\xi:\Z\rightarrow \Z$.
By a result of Wintenberger \cite{Wint} we can decompose a 
filtered $F$-module  $M$ of  finite type over $W$  as a sum of 
isotypic components indexed by $X$,  $M=\bigoplus_{\xi\in X} M_{\xi}$.



\subsection{Hodge-Tate weights and Fontaine-Laffaille weights.}\label{ht-modere}
 The aim of this paragraph is to explain how the  theory of  Fontaine 
and Laffaille relates  the  Hodge-Tate weights of a crystalline 
 representation  to the weights of the tame inertia acting on the 
semi-simplification of its reduction modulo $p$. 
This formulation is due to Wintenberger \cite{Wint}.

Let $V$ be a  $p$-adic crystalline representation of   Hodge-Tate weights
 between $0$ and $p-1$ 
($V\in \Rep_{\mathrm{crys}}^{<p-1}(\G_K)$).
We can associate to it  $D=D_{\mathrm{crys}}(V)\in \MF_K^f$.
The  multiplicity of  $i\in \Z$ as  a  Hodge-Tate weight of $V$ 
is equal to $h^i=\dim_K(\Fil^i D)-\dim_K(\Fil^{i+1} D)$. 

Let $M\in \MF_{W,tf}^{<p-1}$ be a  $W$-lattice adapted to $D$.
By a theorem of Wintenberger we have two natural decompositions of  $M$ :
$$M=\bigoplus_{i\in \Z} M_i, \enspace 
M=\bigoplus_{\xi\in X} M_{\xi}.$$

Let's define $D_i=M_i\otimes_W K_0$ and $D_\xi=M_\xi\otimes_W K_0$.
Then we have $h^i=\dim D_i$ and 
$$D_i=\bigoplus_{\xi\in X, \xi(0)=i} D_{\xi}.$$


For each $r\in \n$, we have $M/p^rM\in \MF_{W,lf}^{<p-1}$. 
Put $L_r=V_{\mathrm{FL}}(M/p^rM)$ and $L=\limproj L_r$. 
Then  $L$ is a lattice of $V$ and  by construction we have $L/pL=L_1$. 

Moreover $M/pM=\oplus_{\xi\in X} M_{\xi}/pM_{\xi}$. 
But  $M_{\xi}/pM_{\xi}$ is a sum of copies  of a  simple object of
 $ \MF_{W,lf}^{<p-1}$ and  $V_{\mathrm{FL}}(M_{\xi}/pM_{\xi})$ 
is equal  to the sum of the same number of copies of the 
tame character $\theta(\xi)$. We deduce that the tame characters  occurring 
in $L/pL$ correspond exactly  to the $\xi$ occurring  in $M$. 
By the theorem of  Brauer-Nesbitt the semi-simplification of $L/pL$
does not depend  of the particular choice of a lattice.


\begin{theo} {\rm (Fontaine-Laffaille)} 
Assume  $V\in \Rep_{\crys}^{<p-1}(\G_K)$
has  Hodge-Tate weights $i_1,...,i_r$ (with multiplicities). 
Let $L$ be a stable lattice  of $V$ and consider  the tame inertia action
on  $(L/pL)$. Take a decomposition  
$(L/pL)^{\mathrm{ss}}=\oplus \overline{L}_j$, such that 
the tame inertia acts on $\overline{L}_j$ by a certain tame
character of level $h^j=\dim_{\F_p}\overline{L}_j$ and  weights
  $i_1^j,..,i_{h^j}^j$ ($r=\sum h^j$).  Then  the multisets $\{i_1,...,i_r\}$  and 
$\{i_k^j\enspace |\enspace 1\leq k\leq h^j\}$ are equal.
\end{theo}

The result remains  valid for  $V\in \Rep_{\crys,E}^{<p-1}(\G_K)$
(see  Def.\ref{poids-ht}).



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\bigskip

Universit{\'e} Paris 7,  UFR de Math{\'e}matiques,  Case 7012, 
2 place Jussieu,  75251 PARIS 


\medskip
 Email : \texttt{dimitrov@math.jussieu.fr}


\end{document}




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