Functiones et Approximatio Commentarii Mathematici

Ranks of $GL_2$ Iwasawa modules of elliptic curves

Tibor Backhausz

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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.

Article information

Funct. Approx. Comment. Math., Volume 52, Number 2 (2015), 283-298.

First available in Project Euclid: 18 June 2015

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11G05: Elliptic curves over global fields [See also 14H52]
Secondary: 11R23: Iwasawa theory

elliptic curve division field


Backhausz, Tibor. Ranks of $GL_2$ Iwasawa modules of elliptic curves. Funct. Approx. Comment. Math. 52 (2015), no. 2, 283--298. doi:10.7169/facm/2015.52.2.7.

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