UCLA

This dissertation focus on automorphy lifting theorems and related questions. There are two primary components.

The first deals with residually dihedral Galois representations. Namely, fix an odd prime $p$, and consider a continuous geometric representation $\rho: G_F \to \GL_n(\mathcal{O})$, where $F$ is either a totally real field if $n=2$, or a CM field if $n>2$, and $\mathcal{O}$ is the integer ring of a finite extension of $\mathbb{Q}_p$. The goal is to prove the automorphy of representations whose residual representation $\bar{\rho}$ has the property that the restriction to $G_{F(\zeta_p)}$ is reducible, where $\zeta_p$ denotes a primitive $p$-th root of unity. This means the classical Taylor-Wiles hypothesis fails and classical patching techniques do not suffice to prove the automorphy of $\rho$. Building off the work of Thorne, we prove an automorphy theorem in the $n=2$ case and apply the result to elliptic curves. The case $n>2$ is examined briefly as well.

The second component deals with the generic unobstructedness of compatible systems of adjoint representations. Namely, given a compatible system of representations, one can consider the adjoints of the residual representations and determine whether the second Galois cohomology group with the adjoints as coefficients vanishes for infinitely many primes. Such a question relates to classical problems such as Leopoldt's conjecture. While theorems are hard to prove, we discuss heuristics and provide computational evidence.