Finiteness for crystalline representations of the absolute Galois group of a totally real field

Abstract

Let K be a totally real field and $G_K:=\mathrm{Gal}(\bar{K}/K)$ its absolute Galois group, where $\bar{K}$ is a fixed algebraic closure of K. Let ℓ be a prime and E a finite extension of ${\mathbb{Q}}_{\ell }$. Let S be a finite set of finite places of K not dividing ℓ. Assume that K, S, Hodge-Tate type h and a positive integer n are fixed. In this paper, we prove that if ℓ is sufficiently large, then, for any fixed E, there are only finitely many isomorphism classes of crystalline representations $r:G_K\to {\mathrm{GL}}_n(E)$ unramified outside $S\cup \{v:v|\ell \}$, with fixed Hodge-Tate type h, such that $r{|}_{G_{K^{\prime }}}\simeq \oplus r_i^{\prime }$ for some finite totally real field extension $K^{\prime }$ of K unramified at all places of K over ℓ, where each representation $r_i^{\prime }$ over E is an 1-dimensional representation of $G_{K^{\prime }}$ or a totally odd irreducible 2-dimensional representation of $G_{K^{\prime }}$ with distinct Hodge-Tate numbers.

Introduction

Let K be a finite extension of $\mathbb{Q}$ and $G_K:=\mathrm{Gal}(\bar{K}/K)$ its absolute Galois group, where $\bar{K}$ is a fixed algebraic closure of K. For each place v of K, let $K_v$ be the completion of K at v and $G_{K_v}:=\mathrm{Gal}({\bar{K}}_v/K_v)$. Let n be a positive integer and ℓ a rational prime. For a continuous representation of $G_K$

$$r:G_K\to {\mathrm{GL}}_n({\bar{\mathbb{Q}}}_{\ell }),$$

Fontaine and Mazur called r geometric if

• r is unramified outside a finite set of finite places of K,

• $r{|}_{G_{K_v}}$ is potentially semi-stable for all places v of K over ℓ.

In this paper, under the assumption that K is a totally real field, we investigate finiteness of isomorphism classes of geometric representations $r:G_K\to {\mathrm{GL}}_n({\bar{\mathbb{Q}}}_{\ell })$ satisfying certain conditions.

Let S be a finite set of finite places v of K satisfying $v\nmid \ell$. Fix ${\iota }^{\prime }:\bar{\mathbb{Q}}\hookrightarrow {\bar{\mathbb{Q}}}_{\ell }$, where $\bar{\mathbb{Q}}$ is a fixed algebraic closure of $\mathbb{Q}$. Let us note that there is 1-1 correspondence between the set of embeddings from K to ${\bar{\mathbb{Q}}}_{\ell }$ and the set of pairs $(\sigma ,v)$ such that $v|\ell$ is a place of K and that σ is an embedding from $K_v$ to ${\bar{\mathbb{Q}}}_{\ell }$. Thus, for each $\tau :K\hookrightarrow \bar{\mathbb{Q}}$, $H_{{\iota }^{\prime }\tau }$ denotes a multiset of Hodge-Tate numbers with respect to ${\iota }^{\prime }\tau$. Since ${\iota }^{\prime }\tau$ depends only on τ, let us simply denote $H_{{\iota }^{\prime }\tau }$ by $H_{\tau }$. We call it a Hodge-Tate type with respect to τ. Let ${\mathrm{HT}}_{\tau }(r)$ denote the Hodge-Tate type of r with respect to τ. Further, by a Hodge-Tate type h, we mean a set $\{H_{\tau }\}$. We say a Hodge-Tate type h is distinct if, for each τ, $H_{\tau }$ consists of distinct numbers. A representation r is totally odd if only if $\det r(c_v)=-1$ for all $v|\infty$ and $c_v\ne 1\in G_{K_v}$. For convenience, “r is crystalline” means that $r{|}_{G_{K_v}}$ is crystalline for all places v of K over ℓ.

The following is our main theorem.

Theorem 1.1

Assume that K, n, S and Hodge-Tate type h are fixed. Further, assume that ℓ is a sufficiently large prime. Then, for a fixed finite extension E of ${\mathbb{Q}}_{\ell }$, there are only finitely many isomorphism classes of crystalline representations

$$r:G_K\to {\mathrm{GL}}_n(E)$$

unramified outside $S\cup \{v:v|\ell \}$, with Hodge-Tate type h, such that $r{|}_{G_{K^{\prime }}}\simeq \oplus r_i^{\prime }$ over E for some finite totally real field extension $K^{\prime }$ of K unramified at all places of K over ℓ, where each $r_i^{\prime }$ is an 1-dimensional representation of $G_{K^{\prime }}$ or a totally odd irreducible 2-dimensional representation of $G_{K^{\prime }}$ with distinct Hodge-Tate type.

Remark 1.2

In Theorem 1.1, the assumption on the dimensions of $r_i$ can be removed by assuming other conditions on r (for details, see Remark 3.12).

This result is motivated by a conjecture of Fontaine and Mazur on finiteness of Galois representations. Shafarevich suggested the conjecture (proved by Faltings [6]) that, given a number field K, an integer d, and a finite set of primes S of K, there are only finitely many isomorphism classes of abelian varieties A over K of dimension d having good reduction at all primes outside S. In view of the Shafarevich conjecture, Fontaine and Mazur suggested the following conjecture.

Conjecture 1.3 [7], Conjecture 2c

For any finite extension E of ${\mathbb{Q}}_{\ell }$, there are only finitely many isomorphism classes of irreducible geometric representations

$$r:G_K\to {\mathrm{GL}}_n(E)$$

with fixed Hodge-Tate type such that r is unramified outside $S\cup \{v:v|\ell \}$.

Conjecture 1.3 is implied by another conjecture of Fontaine and Mazur. To state the conjecture, let us introduce some notation. Assume that r is geometric. For each finite place v of K, let $I_v$ be the inertia group of $G_{K_v}$. Let ${\mathfrak{L}}_v(r)$ be the unique open normal subgroup of $I_v$ such that if L is a finite extension of $K_v$ contained in ${\bar{K}}_v$, then $r{|}_{\mathrm{Gal}({\bar{K}}_v/L)}$ is semi-stable if and only if $\mathrm{Gal}({\bar{K}}_v/L)\cap I_v\subset {\mathfrak{L}}_v(r)$. If $v\notin S$, then ${\mathfrak{L}}_v(r)=I_v$. The inertial level of r for $S\cup \{v:v|\ell \}$ is defined by $\mathfrak{L}(r):={\prod }_{v\in S\cup \{v|\ell \}}{\mathfrak{L}}_v(r)$. By an inertial level for $S\cup \{v:v|\ell \}$, we mean an open normal subgroup $\mathfrak{L}$ of ${\prod }_{v\in S\cup \{v|\ell \}}{I}_v$. With this notation, we state another conjecture of Fontaine and Mazur.

Conjecture 1.4 [7], Conjecture 2a

There are only finitely many isomorphism classes of irreducible geometric representations

$$r:G_K\to {\mathrm{GL}}_n({\bar{\mathbb{Q}}}_{\ell })$$

with fixed Hodge-Tate type and fixed inertial level for $S\cup \{v:v|\ell \}$ such that r is unramified outside $S\cup \{v:v|\ell \}$.

Remark 1.5

We note some known results related to Conjecture 1.4.

• This conjecture was studied in [1] in terms of a compatible system of Galois representations for the case when the system is potentially abelian.

• This conjecture was proved in [17] for the case when r is potentially abelian.

• When $K=\mathbb{Q}$ and $n=2$, finiteness of the representations r in Conjecture 1.4 with certain conditions is immediately implied by the results of [11], [12], [13], and [14].

Let $G_{\mathrm{crys}}(K,\mathfrak{L},S,h;{\bar{\mathbb{Q}}}_{\ell })$ be the set of isomorphism classes of irreducible Galois representations

$$r:G_K\to {\mathrm{GL}}_2({\bar{\mathbb{Q}}}_{\ell })$$

with fixed distinct Hodge-Tate type h and fixed inertial level $\mathfrak{L}$ for S such that

• r is totally odd and unramified outside $S\cup \{v:v|\ell \}$,

• r is crystalline.

By potential automorphy results in [3], we prove the following theorem about the finiteness of crystalline representations $r:G_K\to {\mathrm{GL}}_2({\bar{\mathbb{Q}}}_{\ell })$.

Theorem 1.6

Assume that $K,\mathfrak{L},S$ and distinct Hodge-Tate type h are fixed. Further assume that, for some sufficiently large prime ${\ell }_0$, there are only finitely many isomorphism classes of the irreducible residual representations $\bar{r}$ of $r\in G_{\mathrm{crys}}(K,\mathfrak{L},S,h;{\bar{\mathbb{Q}}}_{{\ell }_0})$. Then, there exists a positive integer $m_0$ such that, for all but finitely many primes ℓ,

$$|G_{\mathrm{crys}}(K,\mathfrak{L},S,h;{\bar{\mathbb{Q}}}_{\ell })|<m_0.$$

Remark 1.7

Let ${\zeta }_{\ell }$ be a primitive ℓ-th root of the unity. Assume that $K,\mathfrak{L},S$ and h are fixed. Then, there exists a positive integer $n_0>3$ such that if $\ell >n_0$, then, for all $r\in G_{\mathrm{crys}}(K,\mathfrak{L},S,h;{\bar{\mathbb{Q}}}_{\ell })$, the representations $\bar{r}{|}_{G_{K({\zeta }_{\ell })}}$ is irreducible (see Lemma 3.2, Lemma 3.4). In Theorem 1.6, “sufficiently large” means that ${\ell }_0$ is larger than the smallest $n_0$.

If S, h and a finite extension E of ${\mathbb{Q}}_{\ell }$ are fixed, then the inertial level of geometric r is bounded for all representations

$$r:G_K\to {\mathrm{GL}}_n(E)$$

with Hodge-Tate type h such that r is unramified outside $S\cup \{v:v|\ell \}$ (see [7, §4.(a)]). Thus, by using the proof of Theorem 1.6, that of Theorem 1.1 reduces to showing the existence of $\alpha (n)$, independent of E and r, satisfying $[K^{\prime }:K]\le \alpha (n)$ for some $K^{\prime }$. The existence of such $\alpha (n)$ is proved by studying the number of connected components of the Zariski closure of $r(G_K)$ in ${\mathrm{GL}}_n$. In fact, we can put $\alpha (n)=n!$.

This paper is organized as follows. In section 2, we review results on potential automorphy of Galois representations. In section 3, we prove Theorem 1.1, Theorem 1.6.