Framed Deformation of Galois Representation

TL;DR

This paper investigates framed deformations of 2D Galois representations, establishing a modular lifting theorem, analyzing deformation rings, and applying results to p-adic L-functions.

Contribution

It introduces a modular lifting theorem for specific Galois deformations and explores the structure of deformation rings in characteristic zero fields.

Findings

Proved a modular lifting theorem for certain Galois representations.

Determined the structure of deformation rings over characteristic zero fields.

Applied results to study exceptional zeros of p-adic L-functions.

Abstract

We studied framed deformations of two dimensional Galois representation of which the residue representation restrict to decomposition groups are scalars, and established a modular lifting theorem for certain cases. We then proved a family version of the result, and used it to determine the structure of deformation rings over characteristic zero fields. These can be applied to the study of exceptional zero of p-adic L-function.