Ranks of $GL_2$ Iwasawa modules of elliptic curves

Abstract

Let $p \ge 5$ be a prime and $E$ an elliptic curve without complex multiplication and let $K_\infty=\mathbb{Q}(E[p^\infty])$ be a pro-$p$ Galois extension over a number field $K$. We consider $X(E/\K_\infty)$, the Pontryagin dual of the $p$-Selmer group $Sel_{p^\infty}(E/K_\infty)$. The size of this module is roughly measured by its rank $\tau$ over a $p$-adic Galois group algebra $\Lambda(H)$, which has been studied in the past decade. We prove $\tau \ge 2$ for almost every elliptic curve under standard assumptions. We find that $\tau = 1$ and $j \notin \mathbb{Z}$ is impossible, while $\tau = 1$ and $j \in \mathbb{Z}$ can occur in at most $8$ explicitly known elliptic curves. The rarity of $\tau=1$ was expected from Iwasawa theory, but the proof is essentially elementary. It follows from a result of Coates et al. that $\tau$ is odd if and only if $[\mathbb{Q}(E[p]) : \mathbb{Q}]/2$ is odd. We show that this is equivalent to $p=7$, $E$ having a $7$-isogeny, a simple condition on the discriminant and local conditions at $2$ and $3$. Up to isogeny, these curves are parametrised by two rational variables using recent work of Greenberg, Rubin, Silverberg and Stoll.