Research Interests

A fundamental problem in Algebraic Number Theory is to describe the Galois extensions of a number field K in a systematic manner so that properties such as ramification can be predicted naturally. The Langlands Program provides a set of conjectural correspondences between the representations of the Galois group of K and analytic objects associated to K called automorphic forms. Parts of this precise correspondence allow one to reduce the study of solutions to certain diophantine equations (like xⁿ + yⁿ = zⁿ) to the study of rational points on a class of curves called modular curves, which have a richer structure because they are moduli spaces for elliptic curves. The recent proof of Fermat's Last Theorem by Wiles, Taylor, et al. for instance was achieved by this method.

My research interests center around the above interplay between elliptic curves / modular curves, Galois representations, automorphic forms, and fall into two main areas with some computational interests.

Current Research

1. Galois representations
   

   The study of Galois representations has been central to much recent work in Number Theory. Given that all elliptic curves over the rationals are now known to be automorphic, a research interest of mine has been in transferring arithmetical questions about elliptic curves into automorphic questions. Elliptic curves have recently found applications in public key cryptography.

2. Rational points on modular curves
   

   The rational points on a curve can be analyzed through the arithmetic and geometry of its jacobian. With this motivation in mind, I have been involved in studying relationships between the jacobians of various kinds of modular curves. Such relationships can for instance extend the methods used in the proof of Fermat's Last Theorem to analyze other similar diophantine equations.