Variation of Selmer groups

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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.