Iwasawa theory of elliptic curves at Eisenstein primes and applications

Abstract: In the study of Iwasawa theory of elliptic curves $E/\mathbb{Q}$, it is often assumed that $p$ is a non-Eisenstein prime, meaning that $E[p]$ is irreducible as a $G_{\mathbb{Q}}$-module. Because of this, most of the recent results on the $p$-converse to the theorem of Gross–Zagier and Kolyvagin (following Skinner and Wei Zhang) and on the $p$-part of the Birch–Swinnerton-Dyer formula in analytic rank 1 (following Jetchev–Skinner–Wan) were only known for non-Eisenstein primes $p$. In this talk, I’ll explain some of the ingredients in a joint work with Giada Grossi, Jaehoon Lee, and Christopher Skinner in which we study the (anticyclotomic) Iwasawa theory of elliptic curves over $\mathbb{Q}$ at Eisenstein primes. As a consequence of our study, we obtain an extension of the aforementioned results to the Eisenstein case. In particular, for $p=3$ this leads to an improvement on the best known results towards Goldfeld’s conjecture in the case of elliptic curves over $\mathbb{Q}$ with a rational $3$-isogeny.

number theory

Audience: researchers in the topic

University of Arizona Algebra and Number Theory Seminar