Analytic Number Theory on Groups

Columbia University, New York NY

Investigators

Abstract

This award supports the research of Dorian Goldfeld in the study of discrete groups with applications to analytic number theory. The proposed research concerns various open problems related to: the distribution of additive characters on Fuchsian groups, the ABC-conjecture, higher rank Eisenstein series twisted by Ash-Borel modular symbols, the Waldspurger correspondence for cubic covers of GL(2), and the geometric periods associated to derivatives of L-series. Number theory has its historical roots in the study of whole numbers, and is among the oldest branches of mathematics. Diophantus, of the third century, proposed many problems in his arithmetic, requiring the solutions to be whole numbers. The study of integer solutions to equations is now referred to as Diophantine analysis, and has many applications to computational complexity, data transmission, signal processing, cryptography, etc. Professor Goldfeld introduces and studies new types of geometrical L-series (infinite sums of geometric objects depending on a complex parameter) in order to develop a novel general method to attack a large class of hitherto still unsolved Diophantine problems.