On Serre's reciprocity conjecture for 2-dimensional mod p representations of the Galois group of Q

Abstract

We first prove the existence of minimally ramified p-adic lifts of 2-dimensional mod p representations, that are odd and irreducible, of the absolute Galois group of Q,in many cases. This is predicted by Serre's conjecture that such representations arise from newforms of optimal level and weight. Using these minimal lifts, and arguments using compatible systems, we prove some cases of Serre's conjectures in low levels and weights. For instance we prove that there are no irreducible (p,p) type group schemes over the rational integers. We prove that a Galois representation as above of Artin conductor 1 and Serre weight 12 arises from the Ramanujan Delta-function. In the last part of the paper we present arguments that reduce Serre's conjecture on modularity of Galois representations as above to proving modularity lifting theorems of the type pioneered by Wiles. While these modularity lifting results are not known as yet they might be relatively accessible because of the basic method of Wiles and Taylor-Wiles and recent developments in the p-adic Langlands programmeinitiated by Breuil.