Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory

Abstract: Let $E/\mathbb{Q}$ be an elliptic curve and let $K$ be an imaginary quadratic field. Under a certain Heegner hypothesis, Kolyvagin constructed cohomology classes for $E$ using $K$-CM points and conjectured they did not all vanish. Conditional on this conjecture, he described the Selmer rank of $E$ using his system of classes. We extend work of Wei Zhang to prove new cases of Kolyvagin's conjecture by considering congruences of modular forms modulo large powers of $p $. Additionally, we prove an analogous result, and give a description of the Selmer rank, in a complementary "definite" case (using certain modified $L$-values rather than CM points). Similar methods are also used to improve known results on the Heegner point main conjecture of Perrin-Riou. One consequence of our results is a new converse theorem, that $p$-Selmer rank one implies analytic rank one, when the residual representation has dihedral image.