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

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