Automorphic lifts of prescribed types

TL;DR

This paper establishes new results on the existence and structure of automorphic Galois representations and their lifts, including modular lifts of specified types, and discusses conjectures on weights and level modifications.

Contribution

It introduces novel results on deformation rings, modular lifts of prescribed types, and level raising/lowering for automorphic Galois representations, advancing understanding in the field.

Findings

Proved structure results for deformation rings of local Galois representations.

Established existence of modular lifts of specified type for Hilbert modular forms.

Proved level raising and lowering results for automorphic Galois representations.

Abstract

We prove a variety of results on the existence of automorphic Galois representations lifting a residual automorphic Galois representation. We prove a result on the structure of deformation rings of local Galois representations, and deduce from this and the method of Khare and Wintenberger a result on the existence of modular lifts of specified type for Galois representations corresponding to Hilbert modular forms of parallel weight 2. We discuss some conjectures on the weights of $n$-dimensional mod $p$ Galois representations. Finally, we use recent work of Taylor to prove level raising and lowering results for $n$-dimensional automorphic Galois representations.