Current Interests:

My thesis at U.C. Berkeley dealt with Galois representations associated to modular forms. Since then, however, my work has gravitated more toward the p-adic geometry of modular curves. In particular, I have been working on computing the stable reduction of the modular curve X0(pn) at the prime p (see research statement). Due to the work of various other people, this has been known when n<3 for over 15 years. Recently, I was able to compute the stable reduction of X0(125) by fairly explicit methods. Robert Coleman and I were then able to generalize that result to the case of X0(p3). Here are some reasonable pictures of these Stable Reduction Graphs.

I also have an enduring interest in explicit equations for modular curves. At some point I hope to publish a semi-expository paper on the various methods for obtaining explicit equations which I have come across and/or developed over the years. Here is a table with some Equations for X0(N) for a few small values of N (along with useful formulas for moduli-theoretic maps).


Research Papers:



Notes and Slides from Recent Talks: