Plus/minus Heegner points and Iwasawa theory of elliptic curves at supersingular primes

Abstract

Let $E$ be an elliptic curve over $\mathbb{Q}$ and let $p\ge 5$ be a prime of good supersingular reduction for $E$. Let $K$ be an imaginary quadratic field satisfying a modified "Heegner hypothesis" in which $p$ splits, write $K_{\infty }$ for the anticyclotomic ${\mathbb{Z}}_p$-extension of $K$ and let $\hat{I}›$ denote the Iwasawa algebra of $K_{\infty }/K$. By extending to the supersingular case the $\hat{I}›$-adic Kolyvagin method originally developed by Bertolini in the ordinary setting, we prove that Kobayashi's plus/minus $p$-primary Selmer groups of $E$ over $K_{\infty }$ have corank $1$ over $\hat{I}›$. As an application, when all the primes dividing the conductor of $E$ split in $K$, we combine our main theorem with results of Çiperiani and of Iovita-Pollack and obtain a "big O" formula for the ${\mathbb{Z}}_p$-corank of the $p$-primary Selmer groups of $E$ over the finite layers of $K_{\infty }/K$ that represents the supersingular counterpart of a well-known result for ordinary primes.

26 pages