Ranks of elliptic curves over Z_p^2-extensions

TL;DR

This paper provides an estimate for how the Mordell-Weil rank of an elliptic curve grows over certain infinite extensions of an imaginary quadratic field, specifically within the context of a $Z_p^2$-extension.

Contribution

It introduces a growth estimate for the Mordell-Weil rank of elliptic curves over $Z_p^2$-extensions of imaginary quadratic fields, extending previous rank growth results.

Findings

Provides explicit growth bounds for ranks in $Z_p^2$-extensions

Analyzes behavior under good reduction at an odd prime $p$

Extends rank growth theory to higher-dimensional $p$-adic Lie extensions

Abstract

Let $E$ be an elliptic curve with good reduction at a fixed odd prime $p$ and $K$ an imaginary quadratic field where $p$ splits. We give a growth estimate for the Mordell-Weil rank of $E$ over finite extensions inside the $\mathbb{Z}_p^2$-extension of $K$.