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.
Publisher Copyright:
© 2019 Elsevier Inc.
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
UR - http://www.scopus.com/inward/record.url?scp=85072725346&partnerID=8YFLogxK
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 -