Rigidity of Automorphic Galois Representations over CM Fields

TL;DR

This paper proves the rigidity of automorphic Galois representations over CM fields by showing their adjoint Bloch-Kato Selmer groups vanish, implying they have no non-trivial deformations that are de Rham.

Contribution

It establishes the vanishing of Selmer groups for automorphic Galois representations over CM fields, demonstrating their rigidity and confirming they are de Rham.

Findings

Vanishing of adjoint Bloch-Kato Selmer groups

Automorphic Galois representations over CM fields are de Rham

Rigidity implies no non-trivial deformations

Abstract

We show the vanishing of adjoint Bloch-Kato Selmer groups of automorphic Galois representations over CM fields. This proves their rigidity in the sense that they have no deformations which are de Rham. In order for this to make sense we also prove that automorphic Galois representations over CM fields are de Rham themselves. Our methods draw heavily from the 10 author paper, where these Galois representations were studied extensively. Another crucial piece of inspiration comes from the work of P. Allen who used the smoothness of certain local deformation rings in characteristic 0 to obtain rigidity in the polarized case.