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An Inverse Galois Deformation Problem

Author
Chen, Taoran
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
2018-08-30Subject
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
Type
dissertation or thesis