Arithmetic of elliptic curves and abelian varieties

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Elliptic curves and abelian varieties are increasingly important not only in number theory and arithmetic algebraic geometry, but also in cryptography and related applications. Some of the most interesting and important open theoretical questions about elliptic curves and abelian varieties are the Birch and Swinnerton-Dyer conjecture and other questions about ranks of Mordell-Weil groups, Selmer groups, and L-functions. On the applied side, a fundamental question is the difficulty of the discrete logarithm problem on elliptic curves over finite fields. 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.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. They also study other questions directly related to the cryptographic applications of elliptic curves, which come about by considering solutions in which the variables take values in finite fields.