Ranks of GL2 Iwasawa modules of elliptic curves

TL;DR

This paper investigates the ranks of certain Iwasawa modules associated with elliptic curves over number fields, establishing lower bounds, characterizing when the rank is odd, and classifying cases where the rank equals one.

Contribution

It proves that the Iwasawa module rank is at least 2 for most elliptic curves, provides an alternative proof for the odd rank condition, and classifies elliptic curves with rank 1.

Findings

The rank $ au$ is at least 2 for almost all elliptic curves.

The rank $ au$ is odd if and only if $p=7$, $E$ has a 7-isogeny, and a certain discriminant condition holds.

Rank $ au=1$ with $j$-invariant in $Z$ occurs in at most 8 explicitly known curves.

Abstract

Let $p >= 5$ be a prime and $E$ an elliptic curve without complex multiplication and let $K_\infty=Q(E[p^\infty])$ 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 $\tau >= 2$ for almost every elliptic curve under standard assumptions. Following from a result of Coates et al, $\tau$ is odd if and only if $[Q(E[p]) \colon Q]/2$ is odd; we give an alternative proof. We show that this is equivalent to $p=7$, $E$ having a 7-isogeny and an easily verifiable condition on the discriminant. Up to isogeny, these curves are parametrised by two rational variables using recent work of Greenberg, Rubin, Silverberg and Stoll. We find that $\tau = 1$ and $j \notin Z$ is impossible, while $\tau = 1$ and $j \in Z$ 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.