Iwasawa Theory

University Of Washington, Seattle WA

Investigators

Abstract

Professor Greenberg intends to study a diverse set of problems related to Selmer groups, elliptic curves, 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. In its essence, the idea is to study Selmer groups associated to a family of representations of the absolute Galois group of a number field. The formulation of these conjectures in a general setting leads to some fundamental problems. One problem is to find a simple way to measure how large the Selmer groups are. Their size should be measured by an element in a certain ring. But it is difficult to find a way to define that element. One of the objectives of this project is to tackle that question in two contrasting settings. In one setting, the ring is rather easy to describe, but is non-commutative. In the other setting, the ring is commutative, but we know very little about its actually structure. Another objective of this project is to better understand the behavior of certain quantities which indirectly reflect the structure of Selmer groups, especially in the non-commutative setting. These quantities are certain ``multiplicities.'' There are typically an infinite number of these quantities. Professor Greenberg hopes to show how to determine all of these quantities from just a finite number of them. 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 question 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 cryptography - 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 Swinnerton-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.