Iwasawa theory of elliptic curves and the Birch--Swinnerton-Dyer conjecture

The study of rational number solutions of polynomial equations has a rich history. The case of elliptic curves, essentially cubic equations in two variables, continues to be fascinating. The Birch and Swinnerton-Dyer conjecture (BSD) connects the arithmetic of an elliptic curve to the associated Hasse-Weil L-function. Iwasawa theory seeks to shed light on the mysterious arithmetic incarnation of L-values. The PI aims to study cases of the BSD conjecture and variants, primarily based on Iwasawa theory. The PI also aims to train graduate students, and introduce undergraduates to research. The last decade has witnessed a key progress towards the BSD conjecture for elliptic curves of rank zero or one, as initiated by Skinner and Zhang. An aim of this project is to establish some of the key missing cases. For instance, the PI seeks the p-converse and p-part of the BSD formula for primes p of non-ordinary reduction. The project aims to develop two-variable Euler systems over an imaginary quadratic field for an elliptic curve as well as anticyclotomic Iwasawa theory at non-split primes. 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.