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Let $E$ be an elliptic curve over $\mathbb Q$ and let $p\geq5$ be a prime ofgood supersingular reduction for $E$. Let $K$ be an imaginary quadratic fieldsatisfying a modified "Heegner hypothesis" in which $p$ splits, write$K_\infty$ for the anticyclotomic $\mathbb Z_p$-extension of $K$ and let$\Lambda$ denote the Iwasawa algebra of $K_\infty/K$. By extending to thesupersingular case the $\Lambda$-adic Kolyvagin method originally developed byBertolini in the ordinary setting, we prove that Kobayashi's plus/minus$p$-primary Selmer groups of $E$ over $K_\infty$ have corank $1$ over$\Lambda$. As an application, when all the primes dividing the conductor of $E$split in $K$, we combine our main theorem with results of \c{C}iperiani and ofIovita-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 $K_\infty/K$ thatrepresents the supersingular counterpart of a well-known result for ordinaryprimes.

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