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

Dohoon Choi, Suh Hyun Choi

Research output: Contribution to journalArticle

Abstract

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 Q. 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.

Original languageEnglish
Pages (from-to)312-329
Number of pages18
JournalJournal of Number Theory
Volume209
DOIs
Publication statusPublished - 2020 Apr

Keywords

  • Finiteness of Galois representations
  • Potential automorphy

ASJC Scopus subject areas

  • Algebra and Number Theory

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