Euler Systems and Elliptic Curves

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The investigator and his coworkers study elliptic curves, special valuesof L-functions, and related topics. Some of the most interesting openquestions and conjectures in arithmetic algebraic geometry relatearithmetic invariants with special values of L-functions. Iwasawa theoryand Euler systems have proved to be among the most successful tools forattacking these problems, by serving as a bridge between the algebraicworld and the analytic world. The investigator continues to pursue thisapproach, by extracting additional information from Euler systems andtheir "derivative" Kolyvagin systems. In another direction, theinvestigator and his colleagues study ranks of elliptic curves,especially the asymptotic distribution of ranks in families of quadratictwists.Elliptic curves play a central role in many parts of mathematicsincluding its most applied areas. For example, elliptic curves are usedin algorithms to encrypt data for transmission, and for efficient digitalsignatures. In its most basic form, an elliptic curve is a special kindof polynomial equation in two variables. Historically number theoristsare interested in finding solutions of these equations in which thevariables take values which are either whole numbers, or fractions. Therank of an elliptic curve is a basic invariant which measures the size ofthe set of solutions. The investigator and his coworkers study ranks ofelliptic curves and their interrelations with other mathematical objectsand concepts. These questions are related to the cryptographicapplications of elliptic curves, which come about by consideringsolutions in which the variables take values in finite fields.