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.