King's College London

Research portal

Serre weights and wild ramification in two-dimensional Galois representations

Research output: Contribution to journalArticlepeer-review

Lassina Dembélé, Fred Diamond, David P. Roberts

Original languageEnglish
Article numbere33
Pages (from-to)1-48
Early online date23 Dec 2016
Accepted/In press20 Oct 2016
E-pub ahead of print23 Dec 2016
PublishedDec 2016


King's Authors


A generalization of Serre’s Conjecture asserts that if F is a totally real field, then certain characteristic p representations of Galois groups over F arise from Hilbert modular forms. Moreover, it predicts the set of weights of such forms in terms of the local behaviour of the Galois representation at primes over p. This characterization of the weights, which is formulated using p-adic Hodge theory, is known under mild technical hypotheses if p > 2. In this paper we give, under the assumption that p is unramified in F, a conjectural alternative description for the set of weights. Our approach is to use the Artin–Hasse exponential and local class field theory to construct bases for local Galois cohomology spaces in terms of which we identify subspaces that should correspond to ones defined using p-adic Hodge theory. The resulting conjecture amounts to an explicit description of wild ramification in reductions of certain crystalline Galois representations. It enables the direct computation of the set of Serre weights of a Galois representation, which we illustrate with numerical examples. A proof of this conjecture has been announced by Calegari, Emerton, Gee and Mavrides.

Download statistics

No data available

View graph of relations

© 2020 King's College London | Strand | London WC2R 2LS | England | United Kingdom | Tel +44 (0)20 7836 5454