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ON THE IRREDUCIBLE COMPONENTS OF SOME CRYSTALLINE DEFORMATION RINGS

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  24 April 2020

ROBIN BARTLETT
ROBIN BARTLETT*
Affiliation:
Max Planck Institute for Mathematics, Germany; robinbartlett18@mpim-bonn.mpg.de

Abstract

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We adapt a technique of Kisin to construct and study crystalline deformation rings of $G_{K}$ for a finite extension $K/\mathbb{Q}_{p}$. This is done by considering a moduli space of Breuil–Kisin modules, satisfying an additional Galois condition, over the unrestricted deformation ring. For $K$ unramified over $\mathbb{Q}_{p}$ and Hodge–Tate weights in $[0,p]$, we study the geometry of this space. As a consequence, we prove that, under a mild cyclotomic-freeness assumption, all crystalline representations of an unramified extension of $\mathbb{Q}_{p}$, with Hodge–Tate weights in $[0,p]$, are potentially diagonalizable.

MSC classification

Primary: 11F80: Galois representations
Secondary: 11F33: Congruences for modular and $p$-adic modular forms
Type
Number Theory
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e22
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author 2020

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