My research interests include Iwasawa theory, Galois representations, and $p$-adic $L$-functions. For my senior thesis, I studied growth patterns of Selmer groups and fine Selmer groups in infinite towers of number fields. I enjoy problems that have a computational aspect, or where patterns can be discovered through examples.
Vanishing of Iwasawa Invariants of weight-$2$ newforms. Submitted for publication. This paper proves that if $p$ is a prime number satisfying mild conditions, there exists a weight $2$ newform $f$ of level $p$ whose Iwasawa $\lambda$- and $\mu$-invariants vanish for some branch of the $p$-adic $L$-series of $f$.
Non-vanishing of central $L$-values of weight-$2$ cuspforms. Submitted for publication. This paper proves the following result: if $N \geq 11$ is a prime number and $N \equiv 3$ modulo $4$, then there exists a weight $2$ newform $f$ of level $N$ whose $L$-series is non-vanishing at $s=1$.
The non-$p$-part of the Fine Selmer group in a $\mathbf{Z}_p$-extension. Accepted for publication at Acta Arithmetica. This paper studies how the $\ell$-part of the Fine Selmer group of an elliptic curve grows in a $\mathbf{Z}_p$-extension when $\ell \neq p$. It proves that there exist $\mathbf{Z}_p$-extensions where the $\ell$-part of the Fine Selmer group can grow arbitrarily quickly. This generalizes a theorem of Washington which proves an analogous statement for $\ell$-parts of class groups.
Crank equidistribution and $(k,j)$-overlined partitions (with Joshua Males and Shuyang Shen). Submitted for publication. This paper studies a variant of Ramanujan's partition function called the $(k,j)$-overlined partition function. We give an asymptotic for how this function grows, analogous to Ramanujan's celebrated asymptotic for the classical partition function.