Analytic Number Theory on Groups

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9800048GoldfeldThis project involves the study of discrete groups with applications to analytic number theory. The focus is on two specific applications. Modular symbols for congruence subgroups are integrals over closed cycles of a differential one-form, which are, by a theorem of Shimura, integral linear combinations of periods of the Jacobian variety associated to the congruence subgroup. The PI has previously conjectured that such integral coefficients have a polynomial growth in the level and that this implies the abc-conjecure. New types of automorphic forms (twisted by modular symbols) are introduced to study the moments of modular symbols. In particular, the investigation of the second moment may lead to progress in the direction of the abc-conjecture. The second application involves the celebrated Gross-Zagier formula for critical values of L-functions associated to modular forms. Recently, the PI and S. Zhang have found a much simpler and very direct proof of the L-value computation of Gross-Zagier. The central idea is to compute the base change of a holomorphic Poincare series. It is proposed to generalize this proof to the case of Hilbert modular L-series over totally real fields. The ultimate goal is to obtain new results in the direction of the Birch-Swinnerton-Dyer conjecture.This research falls into the general mathematical field of number theory. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.