Heegner Points, L-Functions of Elliptic Curves, and Generalizations

Columbia University, New York NY

Investigators

Abstract

This research project concerns one of the basic questions in mathematics: solving algebraic equations. Information about solutions is encoded in various mathematical objects: algebraic cycles, automorphic forms, and L-functions. The project aims to deepen the understanding of these mathematical objects and the connection between them. It will also advance the techniques for understanding the arithmetic of elliptic curves, particularly the Birch and Swinnerton-Dyer conjecture, one of the seven Millennium Prize Problems of the Clay Mathematics Institute. Elliptic curves and similar algebraic equations have wide application in other disciplines such as cryptography; results of the research are expected to advance understanding in these areas as well. The research comprises several projects in number theory: the arithmetic of Heegner points, L-functions of elliptic curves, and their higher dimensional generalizations. The work will investigate the congruences between Heegner points and their various applications, including Goldfeld's conjecture on elliptic curves in quadratic twists families and the rank of elliptic curves in Rubin-Silverberg families. The project will also investigate the Birch and Swinnerton-Dyer formula for elliptic curves at additive primes. For higher-dimensional generalizations, the investigator will explore arithmetic intersection problems on Rapoport-Zink spaces arising from arithmetic Gan-Gross-Prasad conjectures and the arithmetic fundamental lemma. It is also planned to initiate a new program for simultaneous generalization of the Waldspurger and Gross-Zagier formulas. 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.