Proving Serre’s modularity conjecture via Sophie Germain primes

In this article we give a proof of Serre’s conjecture for the cases of odd conductor and even conductor semistable at 2, and arbitrary weight. Our proof in both cases will be unconditional: in particular, it 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 new key ingredient is the use of Sophie Germain primes to perform an efficient weight reduction (“Sophie Germain’s weight reduction”), which is combined with the methods and results of previous articles on Serre’s conjecture by Khare, Wintenberger, and myself. To conclude we present a second (simpler) proof which avoids the use of Khare’s weight reduction (thus, we give in particular an alternative proof of the level 1 case).