On the anticyclotomic Iwasawa theory of rational elliptic curves at Eisenstein primes

TL;DR

This paper advances the understanding of the anticyclotomic Iwasawa theory for rational elliptic curves at Eisenstein primes, proving key conjectures and formulas under mild hypotheses.

Contribution

It proves Perrin-Riou's Heegner point main conjecture and the $p$-part of the Birch--Swinnerton-Dyer formula for elliptic curves with Eisenstein primes.

Findings

Proof of Perrin-Riou's Heegner point main conjecture

Establishment of a $p$-converse to Gross--Zagier and Kolyvagin's theorem

Verification of the $p$-part of BSD formula in rank 1

Abstract

Let $E/\mathbb{Q}$ be an elliptic curve, and $p$ a prime where $E$ has good reduction, and assume that $E$ admits a rational $p$-isogeny. In this paper, we study the anticyclotomic Iwasawa theory of $E$ over an imaginary quadratic field in which $p$ splits, which we relate to the anticyclotomic Iwasawa theory of characters following the method of Greenberg--Vatsal. As a result of our study, we obtain a proof, under mild hypotheses, of Perrin-Riou's Heegner point main conjecture, as well as a $p$-converse to the theorem of Gross--Zagier and Kolyvagin and the $p$-part of the Birch--Swinnerton-Dyer formula in analytic rank $1$ for Eisenstein primes $p$.