Remarks on Serre's modularity conjecture

TL;DR

This paper proves Serre's modularity conjecture for odd level and arbitrary weight without relying on characteristic 2 generalizations, using Kisin's modularity lifting and previous results.

Contribution

It provides a new proof of Serre's conjecture for odd level and arbitrary weight, avoiding characteristic 2 complications and assuming GRH for some cases.

Findings

Proof of Serre's conjecture for odd level and arbitrary weight

Conditional results for even level assuming GRH

Utilization of Kisin's modularity lifting and prior methods

Abstract

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.