On Serre’s conjecture for 2-dimensional mod p representations of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$

Abstract

We prove the existence in many cases of minimally ramified $p$-adic lifts of 2-dimensional continuous, odd, absolutely irreducible, mod $p$ representations $\overline{\rho}$ of the absolute Galois group of $\mathbb{Q}$. It 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 $\mathbb{Z}$. We prove that a $\overline{\rho}$ 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 to proving generalisations of modularity lifting theorems of the type pioneered by Wiles.