An Inverse Galois Deformation Problem

Chen, Taoran

An Inverse Galois Deformation Problem

Abstract

Suppose $\bar{\rho}: \Gal({\bar{F}/F}) \rightarrow \GL_2(\mathbf{k})$ is a residual Galois representation satisfying several mild conditions, where $F$ is a number field and $\mathbf{k}$ is a finite field with characteristics $p \geq 7$. In this work, we show that for any finite flat reduced complete intersection over $W(\mathbf{k})$, $\mathcal{R}$, we can construct a deformation problem defined by local conditions imposed on some finite set of places in $F$, such that the corresponding universal deformation ring of $\bar{\rho}$ is $\mathcal{R}$. It's a theorem of Wiles that if the local conditions are chosen to express restriction to deformations coming from modular forms, then the corresponding universal deformation ring is a finite flat reduced complete intersection, so our work can be regarded as a converse to Wiles' result.

Date Issued

2018-08-30

Keywords

Galois representation; number theory; universal deformation ring; Mathematics; deformation theory

Committee Chair

Ramakrishna, Ravi Kumar

Committee Member

Zywina, David J.
Templier, Nicolas P.

Degree Discipline

Mathematics

Degree Name

Ph. D., Mathematics

Degree Level

Doctor of Philosophy

Types

dissertation or thesis