Selmer Groups, Euler Systems, and Rational Points on Curves

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This project concerns work in the general area of number theory, which is studied in this proposal using methods from geometry. Algebraic varieties -- geometric objects defined by equations -- play a central role in many parts of mathematics, including its most applied areas. For example, special algebraic varieties called 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 curve defined by a certain type of polynomial equation in two variables. Historically number theorists have been interested in finding solutions of these equations in which the variables take values that are either whole numbers or fractions. The investigator will study some new questions about points on algebraic varieties, and the connections between these points and other mathematical objects and concepts.Some of the most basic and important questions in number theory are about rational points on varieties. These questions include connections with L-functions, such as the Birch and Swinnerton-Dyer conjecture. In this project the investigator plans to use many different techniques, including algebraic, p-adic, and analytic tools, to study various aspects of these questions. One set of questions to be studied includes 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. In another direction, the investigator plans to use Selmer groups of twists of abelian varieties to study how the set of rational points on a curve changes when the base field is increased.