General Statement by

My research interests are in number theory where I use mostly analytic tools to study objects from algebraic number theory and arithmetic geometry. I am interested in topics such as modular forms, class numbers, quadratic forms, and finite fields. However, most of my current work is focused on elliptic curves, and in particular on their reductions. The theory of elliptic curves figured strongly in celebrated results such as the proof of Fermat’s Last Theorem and Tunnell’s characterization of the Congruent Number Problem. The theory continues to be relevant today. Indeed, one of the Clay Institute’s Millenium Prize Problems is the Birch and Swinnerton-Dyer Conjecture. Furthermore, elliptic curves occupy important roles in modern applications such as primality testing, integer factoring, cryptography, and digital signatures. An elliptic curve is a smooth curve of genus one having a specified base point. For simplicity, we may think of E as being given by a Weierstrass equation