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THE WEIGHT PART OF SERRE’S CONJECTURE FOR $\text{GL}(2)$

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  05 February 2015

TOBY GEE ,
TONG LIU  and
DAVID SAVITT
TOBY GEE
Affiliation:
Department of Mathematics, Imperial College London SW7 2RH, UK; toby.gee@imperial.ac.uk
TONG LIU
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA; tongliu@math.purdue.edu
DAVID SAVITT
Affiliation:
Department of Mathematics, University of Arizona, Tucson, AZ 85721, USA; savitt@math.arizona.edu

Abstract

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Let $p>2$ be prime. We use purely local methods to determine the possible reductions of certain two-dimensional crystalline representations, which we call pseudo-Barsotti–Tate representations, over arbitrary finite extensions of $\mathbb{Q}_{p}$. As a consequence, we establish (under the usual Taylor–Wiles hypothesis) the weight part of Serre’s conjecture for $\text{GL}(2)$ over arbitrary totally real fields.

MSC classification

Primary: 11F33: Congruences for modular and $p$-adic modular forms
Secondary: 11F80: Galois representations
Type
Research Article
Information
Forum of Mathematics, Pi , Volume 3 , 2015 , e2
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/3.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2015

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