Potential Automorphy and E7 (pdf)
This project uses the potential automorphy machinary of Barnet-Lamb, Gee, Gereghty, and Taylor to produce compatible systems of Galois representations with exceptional monodromy group of type E7. Similar results were obtained for the exceptional group E6 by Boxer, Calegari, Emerton, Levin, Madapusi Pera, and Patrikis. As an application, we prove the analytic continuation of an L-function coming from Galois representations with E7-monodromy.
Galois deformation spaces with a sparsity of automorphic points (arXiv)
This project produces families of characteristic zero deformations of a residual Galois representation over a CM field, valued in a reducitive algebraic group, whose points which are conjecturally associated to automorphic forms must be nowhere dense. This generalizes a result of Calegair and Mazur for 2-dimensional Galois representations over a quadratic imaginary field. The main purpose is to show the constrast between the CM setting and the totally real setting, where the automorphic points are often provably dense.
Octahedral Extensions and Proofs of Two Conjectures of Wong (pdf)
My BYU M.S. thesis. It contains proofs of the results of the two published papers below.
Octahedral extensions of the rationals with a common cubic subfield (J. Number Theory 167 (2016), 141-146. MR 3504039) (pdf)
Joint with Darrin Doud. We prove a conjecture of Simon Wong about the number of octahedral extensions of the rational numbers with a common cubic subextension.
Proof of a conjecture of Wong concerning octahedral Galois representations of
prime power conductor (J. Number Theory 154 (2015), 101–104. MR 3339567) (pdf)
Joint with Darrin Doud. We prove a conjecture of Simon Wong relating the discriminant and Artin conductor for octahedral extensions of the rational numbers with prime power discriminant.