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

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 $K_n$.