A New $\pm$ Iwasawa Theory and Converse of Gross-Zagier and Kolyvagin Theorem (with an Appendix by Yangyu Fan)

Abstract: Let $p>3$ be a prime. In this paper we develop a new kind of anticyclotomic local $\pm$-Iwasawa theory at $p$ for Hecke characters of quadratic imaginary fields which is valid for all ramification types of $p$ (split, inert and ramified). As an application we deduce the converse of Gross-Zagier-Kolyvagin theorem for these CM forms, which states that Selmer rank one implies analytic rank one. To carry out the Iwasawa theory argument we employ a recent construction of a new type of $p$-adic $L$-function by Andreatta-Iovita, and generalized by Yangyu Fan to Shimura curves in the Appendix, and a ``virtual Heenger family'' made via a limiting procedure from a Heegner family along Coleman-Mazur eigencurve constructed by Jetchev-Loeffler-Zerbes.