General SectionFiniteness for crystalline representations of the absolute Galois group of a totally real field☆
Introduction
Let K be a finite extension of and its absolute Galois group, where is a fixed algebraic closure of K. For each place v of K, let be the completion of K at v and . Let n be a positive integer and ℓ a rational prime. For a continuous representation of Fontaine and Mazur called r geometric if
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r is unramified outside a finite set of finite places of K,
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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 satisfying certain conditions.
Let S be a finite set of finite places v of K satisfying . Fix , where is a fixed algebraic closure of . Let us note that there is 1-1 correspondence between the set of embeddings from K to and the set of pairs such that is a place of K and that σ is an embedding from to . Thus, for each , denotes a multiset of Hodge-Tate numbers with respect to . Since depends only on τ, let us simply denote by . We call it a Hodge-Tate type with respect to τ. Let denote the Hodge-Tate type of r with respect to τ. Further, by a Hodge-Tate type h, we mean a set . We say a Hodge-Tate type h is distinct if, for each τ, consists of distinct numbers. A representation r is totally odd if only if for all and . For convenience, “r is crystalline” means that 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 , there are only finitely many isomorphism classes of crystalline representations unramified outside , with Hodge-Tate type h, such that over E for some finite totally real field extension of K unramified at all places of K over ℓ, where each is an 1-dimensional representation of or a totally odd irreducible 2-dimensional representation of with distinct Hodge-Tate type.
Remark 1.2 In Theorem 1.1, the assumption on the dimensions of 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 , there are only finitely many isomorphism classes of irreducible geometric representations with fixed Hodge-Tate type such that r is unramified outside .
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 be the inertia group of . Let be the unique open normal subgroup of such that if L is a finite extension of contained in , then is semi-stable if and only if . If , then . The inertial level of r for is defined by . By an inertial level for , we mean an open normal subgroup of . 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 with fixed Hodge-Tate type and fixed inertial level for such that r is unramified outside .
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 and , 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 be the set of isomorphism classes of irreducible Galois representations with fixed distinct Hodge-Tate type h and fixed inertial level for S such that
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r is totally odd and unramified outside ,
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r is crystalline.
Theorem 1.6 Assume that and distinct Hodge-Tate type h are fixed. Further assume that, for some sufficiently large prime , there are only finitely many isomorphism classes of the irreducible residual representations of . Then, there exists a positive integer such that, for all but finitely many primes ℓ,
Remark 1.7 Let be a primitive ℓ-th root of the unity. Assume that and h are fixed. Then, there exists a positive integer such that if , then, for all , the representations is irreducible (see Lemma 3.2, Lemma 3.4). In Theorem 1.6, “sufficiently large” means that is larger than the smallest .
If S, h and a finite extension E of are fixed, then the inertial level of geometric r is bounded for all representations with Hodge-Tate type h such that r is unramified outside (see [7, §4.(a)]). Thus, by using the proof of Theorem 1.6, that of Theorem 1.1 reduces to showing the existence of , independent of E and r, satisfying for some . The existence of such is proved by studying the number of connected components of the Zariski closure of in . In fact, we can put .
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.
Section snippets
Potential automorphy
In this section, we review, with some observations, some potential automorphy results in [3]. For our purpose, we need only the case when the dimension of a given representation is 2. Assume that r is a 2-dimensional totally odd crystalline representation of . It is known that the pair is a polarized 2-dimensional representation of if and only if r factors through with multiplier μ (satisfying certain conditions on the signs of for all and ) (see [3, §2.1]
Proofs of Theorem 1.1 and 1.6
In this section, we prove Theorem 1.1, Theorem 1.6. We keep the notation of Section 1 and 2 except the notation and ρ. Especially, let us recall the following notation. Let K be a totally real field, and S a set of places of K. Let ℓ be a rational prime. Let be the set of isomorphism classes of irreducible Galois representations with fixed distinct Hodge-Tate type h and fixed inertial level for such that
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r is totally odd and unramified outside
Acknowledgments
The authors are grateful to the anonymous referee for helpful and kind comments.
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The first author was partially supported by the National Research Foundation of Korea (NRF) grant (NRF-2019R1A2C1007517). The second author was supported in part by the Basic Science Research Program through the National Research Foundation of Korea (NRF) grant 2011-0013981.