Some of the most interesting and important open questions in number theory and arithmetic algebraic geometry involve special values of L-functions and their connections with arithmetic. These questions include Stark's conjecture in the case of L-functions of number fields, and the Birch and Swinnerton-Dyer conjecture in the case of elliptic curves and abelian varieties. In this project the investigator and his colleagues plan to use many different techniques, including algebraic, p-adic, and analytic tools, to study various aspects of these questions. Particular questions to be studied include the distribution of Selmer ranks in families of quadratic twists of elliptic curves, refined class number formulas over number fields, and higher rank Kolyvagin systems. In previous work of the investigator Kolyvagin systems have proved to be a very useful tool for relating L-values and arithmetic.
Elliptic curves and abelian varieties 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.
Elliptic curves are curves defined by polynomials of a special type, and a basic problem in number theory is to find all rational points (i.e., rational solutions of the polynomials). Having "genus one", elliptic curves lie at the borderline between curves that are relatively simple and well-understood (genus zero), and those that are much more complicated and mysterious (genus at least two). Elliptic curves are increasingly important not only in number theory, but also in cryptography and related applications. In recent years many aspects of the theory of elliptic curves have turned out to have surprising uses. One way to determine the size of the set of solutions, but not the solutions themselves, is to compute the so-called Selmer groups of the elliptic curve. In joint work with Barry Mazur and Zev Klagsbrun, we study the distribution of the sizes of Selmer groups in certain special families of elliptic curves. One of the main tools for studying a Selmer group is a Kolyvagin system. In other joint work with Mazur, we generalize the definition of a Kolyvagin system to more general objects than elliptic curves. An an application of this work we develop, and obtain evidence for, a generalization of a "refined class number formula" of Darmon. In another direction, in joint work with Ralph Greenberg, Alice Silverberg, and Michael Stoll we study the properties of special points (torsion points) on elliptic curves. Our main result shows that, outside of two exceptional elliptic curves that we identify, these special points are as "generic" as possible.
