Let E/Q be an elliptic curve, and p>2 a prime where E has good reduction. In the study of Iwasawa theory of E, it is common to assume that p is a non-Eisenstein prime, meaning that E[p] is irreducible as a Galois module.
As a result, the recent spectacular results on the p-converse to the Gross-Zagier-Kolyvagin theorem (following Skinner and Wei Zhang) and the p-part of the Birch-Swinnerton-Dyer formula in analytic rank 1 (following Jetchev-Skinner-Wan) have left untouched the cases where p is Eisenstein.
In this talk, I will explain a joint work in progress with Giada Grossi, Jaehoon Lee, and Chris Skinner in which we study the anticyclotomic Iwasawa theory of E at Eisenstein primes by relating it to the anticyclotomic Iwasawa theory of characters. As a result of our study, we can obtain an extension of the aforementioned results to the cases where p is Eisenstein, as well as a proof of Perrin-Riou’s Heegner point main conjecture in this setting.