Deformation of rigid conjugate self-dual Galois representations

TL;DR

This paper investigates the deformation theory of conjugate self-dual Galois representations, establishing an R=T theorem for rigid cases and analyzing rigidity properties for families related to elliptic curves and automorphic forms.

Contribution

It proves an R=T theorem for rigid conjugate self-dual Galois representations and explores rigidity in families linked to elliptic curves and automorphic representations.

Findings

Proved an R=T theorem for certain rigid conjugate self-dual Galois representations.

Analyzed rigidity properties for residue Galois representations from elliptic curves.

Established conditions under which these Galois representations exhibit rigidity.

Abstract

In this article, we study deformations of conjugate self-dual Galois representations. The study has two folds. First, we prove an R=T type theorem for a conjugate self-dual Galois representation with coefficients in a finite field, satisfying a certain property called rigid. Second, we study the rigidity property for the family of residue Galois representations attached to a symmetric power of an elliptic curve, as well as to a regular algebraic conjugate self-dual cuspidal representation.