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

Let E/ Q be an elliptic curve and p an odd 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 by a variation of the method of Greenberg–Vatsal. As a result of our study we obtain proofs (under relatively mild hypotheses) of Perrin-Riou’s Heegner point main conjecture, 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.