Selmer Groups

University Of Washington, Seattle WA

Investigators

Abstract

The investigator intends to pursue a diverse set of projects related to Selmer groups, elliptic curves, modular forms, p-adic L-functions and Iwasawa theory. Selmer groups have traditionally been an important tool for studying the Mordell-Weil group of an elliptic curve over a number field. They have also played an important role in proving special cases of the Birch and Swinnerton-Dyer conjecture which provides a relationship between arithmetic properties of an elliptic curve E and the behavior of the Hasse-Weil L-function for E. In recent years it has become clear that certain natural generalizations of the Selmer group should provide similar conjectural relationships to the behavior of much more general kinds of L-functions. Iwasawa theory provides a framework for studying these conjectures by allowing the L-functions to vary in families defined by congruences modulo powers of a prime p. This leads to the notion of a p-adic L-function and to the formulation of analogous conjectural relationships relating the behavior of such p-adic L-functions to the corresponding Selmer groups. The investigator, together with several collaborators, intends to study this kind of conjectural relationship in some new settings: (1) p-adic analogues of classical Artin L-functions, (2) families of L-functions associated to certain families of Galois representations of varying dimension. In addition, the investigator intends to study relationships between Selmer groups associated to modular forms and ideal class groups of certain number fields and to study some unresolved questions about the derivatives of p-adic L-functions. One of the fundamental questions in the theory of numbers is the study of solutions of an algebraic equation. The difficulty of this question depends on the degree of the equation and the number of variables. It has been understood since antiquity how to study this question when the degree is one or two and the number of variables is also one or two. However, the question becomes much more subtle when one considers equations of degree three, even if the number of variables is just two. A fundamental conjecture concerning this kind of equation was formulated in the 1960s by Birch and Swinnerton-Dyer. Although considerable progress has been made since then, the conjecture remains unresolved. Such equations define a class of curves known as "elliptic curves." The study of their properties has proved to be of importance in crytography - designing codes for the secure transmission of information. Professor Greenberg intends to continue his study of "Selmer groups" which have been a traditional tool in understanding the arithmetic properties of elliptic curves and in studying the conjecture of Birch and Swinnnerton-Dyer. The ultimate goal is to achieve a deeper understanding of the solutions to the algebraic equations that define an elliptic curve, and to develop a more general point of view concerning conjectures analogous to the Birch and Swinnerton-Dyer conjecture.