Kolyvagin's Conjecture and patched Euler systems in anticyclotomic Iwasawa theory
Abstract
Let E/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.