Plus/minus Heegner Points and Iwasawa Theory of Elliptic Curves at Supersingular Primes

Let E be an elliptic curve over Q and let p ≥ 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∞ for the anticyclotomic Zp-extension of K and let Λ denote the Iwasawa algebra of K∞/K. By extending to the supersingular case the Λ-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∞ have corank 1 over Λ. 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 Zp-corank of the p-primary Selmer groups of E over the finite layers of K∞/K that represents the supersingular counterpart of a well-known result for ordinary primes.