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

TL;DR

This paper extends Iwasawa theory and the $ ext{Selmer}$ group analysis to supersingular primes, proving new corank results for elliptic curves over anticyclotomic extensions and deriving formulas for Selmer groups over finite layers.

Contribution

It develops a $ ext{Lambda}$-adic Kolyvagin method for supersingular primes, establishing the corank of plus/minus Selmer groups over $K_ ext{infty}$ and providing a supersingular analogue of known ordinary prime results.

Findings

Plus/minus Selmer groups have corank 1 over $ ext{Lambda}$.

Derived a $ ext{big O}$ formula for $ ext{Selmer}$ groups over finite layers.

Extended Kolyvagin method to supersingular setting.

Abstract

Let $E$ be an elliptic curve over $\mathbb Q$ and let $p\geq5$ 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 $\Lambda$ denote the Iwasawa algebra of $K_\infty/K$. By extending to the supersingular case the $\Lambda$-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 $\Lambda$. 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 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.