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.
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We are grateful to Laurent Berger for the suggestion of considering the Artin- Hasse exponential in the context of a related question, and to David Savitt and Michael Schein for discussions that confirmed the compatibility of Conjecture 7.2 with the results of .We thank Victor Abrashkin for calling our attention to the paper , and Toby Gee for informing us of work on Conjecture 7.2 leading to its proof in . We would also like to thank the referee for numerous suggestions that improved the exposition of this paper. This research was partially supported by EPSRC Grant EP/J002658/1 (LD), Leverhulme Trust RPG-2012-530, EPSRC Grant EP/L025302/1 and the Heilbronn Institute for Mathematical Research (FD), and Simons Foundation Collaboration Grant #209472 (DPR).
© The Author(s) 2016.