Analytic Number Theory on Groups

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This award supports the research of Dorian Goldfeld in thestudy of discrete groups with applications to analytic numbertheory. The proposed research concerns various open problemsrelated to: the distribution of additive characters onFuchsian groups, the ABC-conjecture, higher rank Eisensteinseries twisted by Ash-Borel modular symbols, the Waldspurgercorrespondence for cubic covers of GL(2), and the geometricperiods associated to derivatives of L-series.Number theory has its historical roots in the study of wholenumbers, and is among the oldest branches of mathematics. Diophantus, of the third century, proposed many problems inhis arithmetic, requiring the solutions to be whole numbers.The study of integer solutions to equations is now referredto as Diophantine analysis, and has many applications tocomputational complexity, data transmission, signalprocessing, cryptography, etc. Professor Goldfeld introducesand studies new types of geometrical L-series (infinite sumsof geometric objects depending on a complex parameter) inorder to develop a novel general method to attack a largeclass of hitherto still unsolved Diophantine problems.