Finding solutions in integers or rational numbers of polynomial equations in several variables dates back to Diophantus in the 3rd century. This central subject in mathematics has seen contributions made by many well-known mathematicians including Fermat, Euler, and Gauss. Modern study of Diophantine equations in the early 20th century started with the theory of quadratic forms by Hilbert, followed by the remarkable discovery of the local-to-global principle of Hasse and Minkowski. Weil carried the idea of Hasse-Minkowski further, incorporating the idea of Riemann zeta function, to define what is now called Hasse-Weil zeta function, which is built up on the numbers of solutions of polynomial equations in the much simpler setting of modular arithmetic. Can one recover to a certain extent the integral or rational solutions from the Hasse-Weil zeta function (and hence from the solutions in modular arithmetic)? In the 1960s, based on computational experiment, Birch and Swinnerton-Dyer conjectured that, for a class of polynomial equations (corresponding to elliptic curves), the vanishing of the Hasse-Weil zeta function at the center of its symmetry reveals the existence of infinitely many solutions. This research project aims to deepen the understanding of Diophantine equations in the direction pioneered by the Birch and Swinnerton-Dyer (B-SD) conjecture.
The project is to study rational algebraic cycles (a natural high dimensional generalization of the concept of rational solutions to Diophantine equations) and their connection to the special values of L-functions over both functional fields and number fields. The theorems of Gross-Zagier and of Kolyvagin proved the B-SD conjecture when the analytic rank of the Hasse-Weil L-function of an elliptic curve is at most one. One of the PI?s goals is to establish the same type of results in certain high dimensional cases: the case of the Gan-Gross-Prasad cycles from the product of unitary Shimura varieties, and the new case arising from a symmetric pair of reductive groups. This would establish new cases of conjectures of Beilinson, Bloch, and Kato. Another goal of the project is to study special cycles on the moduli space of Shtukas, which is likely to shed light on the B-SD conjecture over function fields. The methods in the project are from the theory of automorphic L-functions and of relative trace formula, algebro-geometric technique for cycles and moduli spaces, and techniques from harmonic analysis and representation theory of reductive groups over local fields.
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.
