Thesis
I did my Ph.D. thesis with William Stein. It was a combination of two projects.
Arithmetic of Totally Split Modular Jacobians
A modular Jacobian variety, $J$, is said to be said to be totally split if it is isogenous to a product of elliptic curves. When $J=J_0(N)$, we can explicitly define a map $\Phi:\prod E_i \to J$, where the $E_i$’s run through the 1-dimensional abelian subvarieties of $J$.
For my thesis, I leveraged the extra description of $\Phi$ and Galois cohomology to compute the group structure of $J_0(N)(\mathbf{Q})$ for as many totally split rank-0 $J_0(N)$ as possible. Moreover, I was able to provably enumerate the set of totally split $J_0(N)$.
Enumeration of Isogeny Classes of Prime Level Simple Modular Abelian Varieties
Let $A\subseteq J_0$ be a simple abelian subvariety of $J_0(N)$ with $N$ prime. For my thesis, I gave an algorithm for enumerating the odd-degree isogeny class of $A$ when $\mathrm{num} \frac{N-1}{12}$ is squarefree, Hecke algebra of $A$ is integrally closed, and another technical condition.
Let $G = \mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$. I began by showing every finite odd-order $G$-submodule of $A(\overline{\mathbf{Q}})$ is a Hecke module. This allows me to split into the case of Eisenstein and non-Eisenstein isogenies. In the Eisenstein case, I used an idea of Klosin and Papikan. In the non-Eisenstein case, I followed an idea of Frank Calegari and was able to bound the isogeny class by the class group of the Hecke algebra of $A$.