This page contains a regularly-updated list of my published works, as well as works currently under review.  For search friendliness, I’ve included brief abstracts and links to the arXiv, where applicable.

Arithmetic Properties of Picard-Fuchs Equations and Holonomic Recurrences:

The coefficient series of the holomorphic Picard-Fuchs differential equation associated with the periods of elliptic curves often have surprising number-theoretic properties, whice have been widely studied for torsion-free, genus zero congruence subgroups of index 6 and 12 (e.g. the Beauville families). Here, we consider the Picard-Fuchs solutions associated to general elliptic families, with a particular focus on the index 24 congruence subgroups.

We prove that elliptic families over $\mathbb{Q}$ admit linear reparametrizations such that their associated Picard-Fuchs solutions lie in $\mathbb{Z}[[t]]$. A sufficient condition is given such that the same holds for holomorphic solutions at infinity. An Atkin-Swinnerton-Dyer congruence is proven for the coefficient series attached to $\Gamma_1(7)$. We conclude with a consideration of asymptotics, wherein we prove that many coefficient series satisfy asymptotic expressions of the form $u_n \sim \ell \lambda^n/n$. Certain arithmetic results extend to the study of general holonomic recurrences.

(Joint work with Zane Kun Li, to appear in the Journal of Number Theory.)