## Abstract

Let K be a totally real field and G_{K}:=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_{ℓ}. 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}→GL_{n}(E) unramified outside S∪{v:v|ℓ}, with fixed Hodge-Tate type h, such that r|_{GK′ }≃⊕r_{i} ^{′} for some finite totally real field extension K^{′} of K unramified at all places of K over ℓ, where each representation r_{i} ^{′} over E is an 1-dimensional representation of G_{K′ } or a totally odd irreducible 2-dimensional representation of G_{K′ } with distinct Hodge-Tate numbers.

Original language | English |
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Pages (from-to) | 312-329 |

Number of pages | 18 |

Journal | Journal of Number Theory |

Volume | 209 |

DOIs | |

Publication status | Published - 2020 Apr |

### Bibliographical note

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.

## Keywords

- Finiteness of Galois representations
- Potential automorphy

## ASJC Scopus subject areas

- Algebra and Number Theory