Refined conjectures on Fitting ideals of Selmer groups over $\mathbf{Z}_p^2$-extensions

TL;DR

This paper investigates the structure of Selmer groups over $ extbf{Z}_p^2$-extensions of imaginary quadratic fields, showing under certain conditions that the Mazur-Tate element generates the Fitting ideal of the dual Selmer group.

Contribution

It refines conjectures on the Fitting ideals of Selmer groups over $ extbf{Z}_p^2$-extensions, establishing a link with Mazur-Tate elements under specific hypotheses.

Findings

Mazur-Tate element generates the Fitting ideal of the dual Selmer group.

Results hold for elliptic curves with good ordinary reduction at $p$.

Provides evidence supporting refined conjectures on Selmer groups.

Abstract

Let $p>3$ be a prime number and $K$ be an imaginary quadratic field where $p$ splits. Let $K_\infty$ be the $\mathbf{Z}_p^2$-extension of $K$ and let $K_n$ be a finite subextension of $K_\infty/K$. Let $E$ be an elliptic curve with good ordinary reduction at $p$. Under some hypotheses, we show that the Mazur-Tate element attached to $E$ over $K_n$ by S. Haran generates the Fitting ideal of the dual Selmer group of $E$ over $K_n$.