Algebraic cycles, L-functions and rational points on elliptic curves

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This proposal aims to investigate relations between special values of L-functions, cycles and periods with important applications to some open conjectures about L-functions such as those of Birch-Swinnerton-Dyer and Bloch-Kato-Beilinson. Specifically, the investigator and his collaborators will (i) Study the algebraic cycles associated to Rankin-Selberg L-functions and their images under Abel-Jacobi maps, and apply these results to give new constructions of rational points on CM elliptic curves; (ii) Study p-adic L-functions and the Iwasawa main conjecture for CM Hida deformations; (iii) Explore methods to prove a conjecture of his relating periods of quaternionic modular forms to adjoint L-values, with applications to some cases of the Bloch-Kato conjecture (iv) Study the problem of constructing and counting invariant linear forms on a triple product of representations of the metaplectic group, thus generalizing results on triple product L-functions associated to modular forms of integral weight to the setting of modular forms of half-integral weight. The general focus of this proposal is the area of number theory. Number theory has to do with such objects as prime numbers and diophantine equations. Other than being perhaps the oldest branch of mathematics, it is of great significance in today's world, since many cryptographic protocols (needed for secure transmissions over the internet) and error correcting codes (needed for compact discs, hard discs and the like) are based on number theoretic methods. These practical applications in fact involve rather sophisticated geometrical objects such as elliptic curves. Over the last half-century, we have realized that one can gain a better understanding of these geometric objects by studying certain functions, called L-functions. Conjecturally one expects that very interesting information about the geometric object is encoded in the behavior of the associated L-function at certain special points. The investigator hopes to deepen our understanding of this connection through the work to be done in the current proposal. One of the concrete consequences of this project will be a new method to find solutions in rational numbers to certain cubic equations, a central problem in number theory.