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

Abstract

We prove the modularity of most reducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\) with k a finite field of characteristic an odd prime p. This is an analogue of Serre’s celebrated modularity conjecture (which concerned irreducible, odd representations \({\bar{\rho }}: \Gamma _{{\mathbb {Q}}} \rightarrow \mathrm {GL}_2(k)\)) for reducible, odd representations. Our proof lifts \({\bar{\rho }}\) to an irreducible geometric p-adic representation \(\rho \) which is known to arise from a newform by results of Skinner–Wiles and Pan. We likewise prove automorphy of many reducible representations \({\bar{\rho }}:\Gamma _{F} \rightarrow \mathrm {GL}_n(k)\) when F is a global function field of characteristic different from p, by establishing a p-adic lifting theorem and invoking the work of L. Lafforgue. Crucially, in both cases we show that the actual representation \({\bar{\rho }}\), rather than just its semisimplification, arises from reduction of the geometric representation attached to a cuspidal automorphic representation. Our main theorem establishes a geometric lifting result for mod p representations \({\bar{\rho }}:\Gamma _{F} \rightarrow G(k)\) of Galois groups of global fields F, valued in reductive groups G(k), and assumed to be odd when F is a number field. Thus we find that lifting theorems, combined with automorphy lifting results pioneered by Wiles in the number field case and the results in 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 both in reducible and irreducible cases.