Vol. 6, No. 7, 2012

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ISSN: 1944-7833 (e-only)
ISSN: 1937-0652 (print)
Crystalline extensions and the weight part of Serre's conjecture

Toby Gee, Tong Liu and David Savitt

Vol. 6 (2012), No. 7, 1537–1559
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