On pseudo-nullity of fine Mordell-Weil group

TL;DR

This paper investigates the structure of the fine Mordell-Weil group of an elliptic curve over a specific extension of an imaginary quadratic field, showing it is pseudo-null under certain conditions, contributing to Iwasawa theory.

Contribution

It demonstrates that the Pontryagin dual of the fine Mordell-Weil group is pseudo-null over the Iwasawa algebra in a new setting involving $ ext{Z}_p^2$-extensions.

Findings

The Pontryagin dual of the fine Mordell-Weil group is pseudo-null.

Results apply to elliptic curves with good ordinary reduction at prime p.

Provides new insights into Iwasawa modules over $ ext{Z}_p^2$-extensions.

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p\geq 5$, and let $F$ be an imaginary quadratic field. Under appropriate assumptions, we show that the Pontryagin dual of the fine Mordell-Weil group of $E$ over the $\mathbb{Z}_p^2$-extension of $F$ is pseudo-null as a module over the Iwasawa algebra of the group $\mathbb{Z}_p^2$.