Variation of Selmer Groups of Elliptic Curves

University Of California-Irvine, Irvine CA

Investigators

Abstract

Elliptic curves are increasingly important not only in number theory and arithmetic algebraic geometry, but also in cryptography and related applications. Some of the most interesting and important open questions about elliptic curves are the Birch and Swinnerton-Dyer conjecture and other questions about ranks, Selmer groups, and L-functions. In this project the investigator and his colleagues plan to study how the rank of an elliptic curve varies in certain families of number fields, especially the subfields of Iwasawa towers and the family of all quadratic fields. The investigation will make use of many different techniques, including algebraic, p-adic, and analytic tools. Elliptic curves play a central role in many parts of mathematics including its most applied areas. For example, elliptic curves are used in algorithms to encrypt data for transmission, and for efficient digital signatures. In its most basic form, an elliptic curve is a special kind of polynomial equation in two variables. Historically number theorists are interested in finding solutions of these equations in which the variables take values which are either whole numbers, or fractions. The rank of an elliptic curve is a basic invariant which measures the size of the set of solutions. The investigator and his coworkers study ranks of elliptic curves and their interrelations with other mathematical objects and concepts. These questions are related to the cryptographic applications of elliptic curves, which come about by considering solutions in which the variables take values in finite fields.