The investigator and his coworkers study elliptic curves, special values of L-functions, and related topics. Some of the most interesting open questions and conjectures in arithmetic algebraic geometry relate arithmetic invariants with special values of L-functions. Iwasawa theory and Euler systems have proved to be among the most successful tools for attacking these problems, by serving as a bridge between the algebraic world and the analytic world. The investigator continues to pursue this approach, by extracting additional information from Euler systems and their "derivative" Kolyvagin systems. In another direction, the investigator and his colleagues study ranks of elliptic curves, especially the asymptotic distribution of ranks in families of quadratic twists.
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.