Remarks on Serre’s modularity conjecture

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 use any modularity lifting theorem in 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 abelian varieties of small conductor. Assuming GRH, infinitely many cases of even level will also be proved.

Langlands base change for GL(2)

“…Swapping is the process used in [Die,§4], in order to transfer ramification from one set of primes to another. We have currently the set of primes: p 1 , .…”

Semistable abelian varieties with good reduction outside 15

“…Clearly the ring Z[η, ζ 3 , 3 ] is an extension of F that is unramified outside 3 and the infinite primes. Let π = Gal(H/K).…”

“…In his paper [3], Luis Dieulefait gives a proof of Serre's modularity conjecture for the case of odd level and arbitrary weight. By means of an intricate inductive procedure he reduces the issue to the case of Galois representations of level 3 and weight 2, 4 or 6.…”

“…By means of an intricate inductive procedure he reduces the issue to the case of Galois representations of level 3 and weight 2, 4 or 6. As explained in [3], these cases are taken care of by the following three theorems respectively. Theorem 1.1.…”

On modular forms and the inverse Galois problem

“…We finish this introduction by pointing out that our method of obtaining groups of the type PSL 2 (F n ) or PGL 2 (F n ) as Galois groups over Q via newforms is very general: if a group of this type occurs as the Galois group of a totally imaginary extension of Q, then it is the projective image of the Galois representation attached to a newform by Serre's modularity conjecture, which is now a theorem of KhareWintenberger ([KW1], [KW2]; see also [Ki] and [Di3] Proof. We interpret the number field K as a projective Galois representation ρ proj : Gal(Q/Q) → PGL 2 (F n ).…”