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.
This research project concerns a relationship between certain numbers that arise in two completely different ways. The project pursues a theme which first appeared in number theory in the 1840s in the work of the German mathematicians Ernst Kummer and Lejeune Dirichlet. On the one hand, there are certain numbers that arise as infinite sums. One sees such numbers in calculus courses. They arise as values of certain functions. One such function is often called the Riemann Zeta Function and is often studied in a calculus course. We will refer to the numbers that arise in this way by the term $L-values" because they occur as values of a type of function that has come to be called an "L-function." On the other hand, there are certain numbers that arise in a purely algebraic way and do not involve infinite sums. These numbers arise from certain objects which have come to be called "Selmer groups" (named after the Norwegian mathematician Ernst Selmer who introduced very special cases of this type of object in the early 1950s). The Selmer groups are usually finite sets and the numbers of interest arise by counting the number of elements in those finite sets. In the work of Kummer and Dirichlet mentioned above, the objects are called "ideal class groups" and measure the failure of a certain phenomenon occurring in number theory called "Unique Factorization." The objects called Selmer groups are a generalization of ideal class groups and also measure the failure of a certain phenomenon. And so they are interesting objects of study in number theory. Another instance of such a relationship was discovered experimentally by the British mathematicians Bryan Birch and Peter Swinnerton-Dyer in the early 1960s. Their study concerns the question of how many points exist on a certain type of curve (called an 'elliptic curve") which have coordinates in the rational numbers. Much progress has been made on the conjecture that they formulated, but it still remains open in general. In the early 1970s, the Japanese mathematician Kenkichi Iwasawa discovered a much more precise version of the relationship that Kummer had proved. His conjecture was proved in the 1980s by Barry Mazur (American) and Andrew Wiles (British) In the 1980s, it began to become apparent that the above instances were just the tip of the iceberg and that such relationships should exist in a vastly more general context. It was the PI for this project who found a precise formulation of such a relationship. The project just completed concerns another instance of such a relationship. The PI and his collaborator Vinayak Vatsal (Canadian) studied a situation concerning L-functions that were first defined in the 1920s by the German mathematician Emil Artin. In certain cases, the conjectured relationship has been already established completely by Wiles in the early 1990s. However, the PI and Vatsal considered a case where the L-values did not arise directly from the L-function, but could only be defined rather indirectly. And so this was somewhat different than the instances mentioned above. One outcome is that we discovered a surprising relationship which is an unexpected analogue of a conjecture of the American mathematician Harold Stark. Although the analogy with Stark's conjecture is quite convincing, the reason behind that analogy is still very mysterious.
