Instytut Matematyczny Polskiej Akademii Nauk / Institute of Mathematics / Publishing house / Journals and Serials / Acta Arithmetica / All issues

Abstract

Let $E/\mathbb Q $ 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 $\Sha (E/K_n)[p^{\infty }]=0$.