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=J0(N), we can explicitly define a map Φ:∏Ei→J, where the Ei’s run through the 1-dimensional abelian subvarieties of J.
For my thesis, I leveraged the extra description of Φ and Galois cohomology to compute the group structure of J0(N)(Q) for as many totally split rank-0 J0(N) as possible. Moreover, I was able to provably enumerate the set of totally split J0(N).
Enumeration of Isogeny Classes of Prime Level Simple Modular Abelian Varieties
Let A⊆J0 be a simple abelian subvariety of J0(N) with N prime. For my thesis, I gave an algorithm for enumerating the odd-degree isogeny class of A when numN−112 is squarefree, Hecke algebra of A is integrally closed, and another technical condition.
Let G=Gal(¯Q/Q). I began by showing every finite odd-order G-submodule of A(¯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.