Open Access
June 2015 Ranks of $GL_2$ Iwasawa modules of elliptic curves
Tibor Backhausz
Funct. Approx. Comment. Math. 52(2): 283-298 (June 2015). DOI: 10.7169/facm/2015.52.2.7


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.


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Tibor Backhausz. "Ranks of $GL_2$ Iwasawa modules of elliptic curves." Funct. Approx. Comment. Math. 52 (2) 283 - 298, June 2015.


Published: June 2015
First available in Project Euclid: 18 June 2015

zbMATH: 06862263
MathSciNet: MR3358321
Digital Object Identifier: 10.7169/facm/2015.52.2.7

Primary: 11G05
Secondary: 11R23

Keywords: division field , Elliptic curve

Rights: Copyright © 2015 Adam Mickiewicz University

Vol.52 • No. 2 • June 2015
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