L-values, Galois representations, and elliptic curves

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This project focuses on one of the central problems in number theory: connecting special values of L-functions to associated algebraic quantities, such as class groups or Selmer groups. In the case of elliptic curves, which is addressed by the celebrated conjecture of Birch and Swinnerton-Dyer, the research described in this proposal aims to establish simple p-adic criteria for an elliptic curve to have both algebraic and analytic rank one, to use this criteria to prove a converse to the work of Kolyvagin and Gross-Zagier, and to show that this criteria holds for a positive proportion of all elliptic curves over the rationals (when ordered by height), thereby proving that both the average algebraic and analytic ranks are positive. It also aims to connect the vanishing of the central critical value of the L-function of an elliptic curve over the rationals or a totally real number field to the non-triviality of the associated p-adic Selmer group through methods involving the Iwasawa theory of unitary groups, extending some of the investigator's prior work on such problems. This project includes further development and generalization of these methods so as to apply to L-functions arising from more general unitary groups, especially groups of higher rank, and to connect values of p-adic L-functions to Abel-Jacobi maps of cycles of positive dimension on unitary Shimura varieties.The research described in this proposal aims to connect certain algebraic and analytic objects that are fundamental for many problems in number theory. The analytic objects are L-functions - a special class of functions built from number-theoretic data (this class includes the celebrated Riemann zeta function which is built from the prime numbers). For over a century and a half L-functions have been a crucial ingredient in efforts to tackle the most central problems in number theory (for example, understanding the distribution of prime numbers or the structure of the set of rational number solutions to cubic curves). An important feature of L-functions is that their values at certain special points are expected to encode information about the orders of algebraic quantities also associated with the number-theoretic data defining the L-function. The investigator aims to prove the existence of such relations, with an emphasis on the important case of elliptic curves. Understanding the structure of the set of rational points on an elliptic curve - essentially the rational number solutions to a cubic equation - has been an aim in number theory for over a century, with connections to open problems that go back even further (even to classical Greek geometry). The problem of connecting the structure of this set with the special values of the L-function of the elliptic curve is often listed as one of the most important unsolved problems in mathematics. It is expected that this project will especially further the understanding of this connection and that it will develop new methods that will be useful for making progress on more general problems of a similar nature. The methods will draw particularly on the theory of automorphic forms, which are closely connected to both analysis and algebra and a rich supply of L-functions.