Remarks on Serre’s modularity conjecture

In this article we give a proof of Serre’s conjecture for the case of odd level and arbitrary weight. Our proof will not depend on any yet unproved generalization of Kisin’s modularity lifting results to characteristic 2 (moreover, we will not consider at all characteristic 2 representations in any step of our proof). The key tool in the proof is Kisin’s recent modularity lifting result, 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