TY - JOUR

T1 - Finiteness for crystalline representations of the absolute Galois group of a totally real field

AU - Choi, Dohoon

AU - Choi, Suh Hyun

N1 - Funding Information:
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.

PY - 2020/4

Y1 - 2020/4

N2 - Let K be a totally real field and GK:=Gal(K‾/K) its absolute Galois group, where K‾ is a fixed algebraic closure of K. Let ℓ be a prime and E a finite extension of Qℓ. 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:GK→GLn(E) unramified outside S∪{v:v|ℓ}, with fixed Hodge-Tate type h, such that r|GK′ ≃⊕ri ′ for some finite totally real field extension K′ of K unramified at all places of K over ℓ, where each representation ri ′ over E is an 1-dimensional representation of GK′ or a totally odd irreducible 2-dimensional representation of GK′ with distinct Hodge-Tate numbers.

AB - Let K be a totally real field and GK:=Gal(K‾/K) its absolute Galois group, where K‾ is a fixed algebraic closure of K. Let ℓ be a prime and E a finite extension of Qℓ. 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:GK→GLn(E) unramified outside S∪{v:v|ℓ}, with fixed Hodge-Tate type h, such that r|GK′ ≃⊕ri ′ for some finite totally real field extension K′ of K unramified at all places of K over ℓ, where each representation ri ′ over E is an 1-dimensional representation of GK′ or a totally odd irreducible 2-dimensional representation of GK′ with distinct Hodge-Tate numbers.

KW - Finiteness of Galois representations

KW - Potential automorphy

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U2 - 10.1016/j.jnt.2019.08.023

DO - 10.1016/j.jnt.2019.08.023

M3 - Article

AN - SCOPUS:85072725346

VL - 209

SP - 312

EP - 329

JO - Journal of Number Theory

JF - Journal of Number Theory

SN - 0022-314X

ER -