Lifting and automorphy of reducible mod p Galois representations over global fields

TL;DR

This paper develops new methods to lift reducible mod p Galois representations over global fields to automorphic forms, establishing modularity results including a version of Serre's conjecture for reducible cases.

Contribution

It extends lifting techniques to reducible Galois representations valued in Chevalley groups, enabling the proof of modularity for a broad class of such representations.

Findings

Proves modularity of reducible mod p Galois representations over global fields.

Establishes a version of Serre's conjecture for reducible representations over Q.

Provides a method to realize Galois representations from automorphic forms.

Abstract

We extend the lifting methods of our previous paper to lift reducible odd representations $\bar{\rho}:\mathrm{Gal}(\overline{F}/F) \to G(k)$ of Galois groups of global fields $F$ valued in Chevalley groups $G(k)$. Lifting results, when combined with automorphy lifting results pioneered by Wiles in the number field case and the results on the global Langlands correspondence proved by Drinfeld and L. Lafforgue in the function field case, give the only known method to access modularity of mod $p$ Galois representations in both reducible and irreducible cases. In the reducible case this allows one to show that the actual representation, rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation on the dual group of $G$. As a particularly concrete application, we get a version of Serre's modularity conjecture for reducible, odd representations $\bar{\rho}: \mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q}) \to \mathrm{GL}_2(k)$. This extends earlier results of Hamblen and Ramakrishna in this classical case and proves modularity of infinitely many extensions of fixed characters that are not covered by loc. cit.