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

TL;DR

This paper advances the understanding of Kolyvagin's conjecture in anticyclotomic Iwasawa theory by proving new cases using congruences of modular forms and explores implications for Selmer ranks and the Heegner point main conjecture.

Contribution

It extends work on Kolyvagin's conjecture by proving new cases via modular form congruences and introduces a converse theorem linking Selmer rank and analytic rank.

Findings

Proved new cases of Kolyvagin's conjecture using congruences of modular forms.

Provided a description of Selmer ranks in a definite case using modified L-values.

Established a converse theorem relating p-Selmer rank one to analytic rank one.

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.