## The main conjecture of Iwasawa theory for elliptic curves with complex multiplication over abelian extensions at supersingular primes

### Volume 181 / 2017

Acta Arithmetica 181 (2017), 209-238
MSC: Primary 11G05, 11R23; Secondary 11G40.
DOI: 10.4064/aa8606-8-2017
Published online: 1 December 2017

#### Abstract

Let $E$ be an elliptic curve over an abelian extension $F$ of an imaginary quadratic field $K$ with complex multiplication by $K$. Let $p$ be a prime number inert over $K/\mathbb Q$ (i.e. supersingular for $E$). We establish the main conjecture of Iwasawa theory under certain conditions on $p$. In other words, we prove that the characteristic ideal of the Pontryagin dual of the plus/minus Selmer group of $E$ over the cyclotomic $\mathbb Z_p$-extension of $F$ is generated by the plus/minus $p$-adic $L$-function of $E$.