Kolyvagin's work and anticyclotomic tower fields: the supersingular case
Volume 201 / 2021
Abstract
Let be an elliptic curve, p a prime and K_{\infty }/K the anticyclotomic \mathbb Z_p -extension of a quadratic imaginary field K satisfying the Heegner hypothesis. Kolyvagin has shown under certain assumptions that if the basic Heegner point y_K \in E(K) is not divisible by p, then \operatorname{rank} (E(K))=1 and \Sha (E/K)[p^{\infty }]=0. Assuming that E has supersingular reduction at p and other conditions, we show using Kolyvagin’s result and Iwasawa theory that for all n we have \operatorname{rank} (E(K_n))=p^n and \Sha (E/K_n)[p^{\infty }]=0.