Crystalline extensions and the weight part of Serre’s conjecture

Gee, Toby; Liu, Tong; Savitt, David

doi:10.2140/ant.2012.6.1537

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Crystalline extensions and the weight part of Serre's conjecture

Toby Gee, Tong Liu and David Savitt

Vol. 6 (2012), No. 7, 1537–1559

DOI: 10.2140/ant.2012.6.1537

Abstract

Let $p > 2$ be prime. We complete the proof of the weight part of Serre’s conjecture for rank-two unitary groups for mod $p$ representations in the totally ramified case by proving that any Serre weight which occurs is a predicted weight. This completes the analysis begun by Barnet-Lamb, Gee, and Geraghty, who proved that all predicted Serre weights occur. Our methods are a mixture of local and global techniques, and in the course of the proof we use global techniques (as well as local arguments) to establish some purely local results on crystalline extension classes. We also apply these local results to prove similar theorems for the weight part of Serre’s conjecture for Hilbert modular forms in the totally ramified case.

Keywords

Serre's conjecture, p-adic Hodge theory, automorphy lifting theorems

Mathematical Subject Classification 2010

Primary: 11F33

Milestones

Received: 2 July 2011

Revised: 4 October 2011

Accepted: 2 November 2011

Published: 4 December 2012

Authors

Toby Gee
Imperial College London London SW7 2AZ United Kingdom

Tong Liu
Department of Mathematics Purdue University 150 N. University Street West Lafayette, IN 47907 United States

David Savitt
Department of Mathematics University of Arizona 617 N. Santa Rita Ave. PO Box 210089 Tucson, AZ 85721-0089 United States

Vol. 6, No. 7, 2012