Arithmetic and Geometry Around Relative Trace Formulae

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This project concerns research in number theory, which studies properties of the whole numbers and is at the core of modern cryptography. One central theme in number theory is to study the relationship between algebraic and geometric objects and special values of L-functions (generalizations of the Riemann zeta function introduced by Euler in the eighteenth century and studied extensively by Riemann in the nineteenth century). A motivating example is the conjecture of Birch and Swinnerton-Dyer, which relates rational points on elliptic curves (one of the simplest classes of polynomial equations) to the analytic property of L-functions. This research project explores several topics in number theory and aims to deepen understanding of these relationships.The project aims to study the Birch and Swinnerton-Dyer conjecture and its high dimensional generalizations, one of which is the arithmetic Gan-Gross-Prasad conjecture for unitary Shimura varieties. The investigator previously discovered a relative trace formula approach to study the first derivative of certain automorphic L-functions and intersection numbers on Shimura varieties, and recent work of the investigator and collaborator extends the idea to higher derivatives for L-functions for the general linear group of rank two over function fields. This research project aims to extend the relative trace formula approach to more general settings, for instance, to L-functions for the general linear groups of higher rank.