Trace Formulas, L-Functions, and Automorphic and Arithmetic Periods

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The main objective of this project is to study the relation between L-functions and various periods, both automorphic and arithmetic. The L-function is an important object in the modern study of questions in number theory, in particular, solving Diophantine equations like the famous Birch and Swinnerton-Dyer conjecture and its generalizations. L-functions are also crucial in the study of representations of transformation groups, which have an important role in physics. The work is expected to provide important new understanding of these relations in higher dimensional cases. The investigator plans to organize workshops related to the project, which will provide students, postdocs, and other researchers in the area substantial opportunities for instruction, discussion, and collaboration.The scope of this project consists of four parts. In the first part, the investigator will use some new techniques in the study of trace formulas to solve questions concerning the relation between L-functions and automorphic periods -- those periods obtained by integrating automorphic forms along certain subgroups. The second part aims to extend some explicit formulas of automorphic periods, known as the Ichino-Ikeda formula, to more general cases by removing certain assumptions that are usually hard to check. The third part aims to provide unconditional evidence toward the Beilinson-Bloch conjecture on the relation between L-functions and Chow groups, which will generalize parallel results in the Birch and Swinnerton-Dyer conjecture to higher dimensional cases. In the last part, the investigator plans to adopt a new approach via a relative trace formula to obtain variants of the arithmetic triple product formula concerning the height of certain cycles on a product of three modular elliptic curves.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.