Remarks on Serre's modularity conjecture

Details Remarks on Serre's modularity conjecture Remarks on Serre's modularity conjecture

2006-03-17

arxiv.org/abs/math/0603439v11

Mathematics

Number Theory

exposition improved, several Lemmas added to make the paper more readable

Scientific paper

In this article we give a proof of Serre's conjecture for the case of odd level and arbitrary weight. Our proof does not depend on any generalization of Kisin's modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations at any step of our proof). The key tool in the proof is a very general modularity lifting result of Kisin, which is combined with the methods and results of previous articles on Serre's conjecture by Khare, Wintenberger, and the author, and modularity results of Schoof for semistable abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved.

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