Functional equations for supersingular abelian varieties over $\mathbf{Z}_p^2$-extensions

TL;DR

This paper proves an algebraic functional equation for the Pontryagin dual of the Selmer group of supersingular abelian varieties over $ extbf{Z}_p^2$-extensions, extending to elliptic curves and using advanced $ extbf{Gamma}$-system theory.

Contribution

It establishes the first algebraic functional equation for Selmer groups in this setting, answering a question of Ahmed and Lim, and applies to supersingular elliptic curves.

Findings

Proves algebraic functional equation for supersingular abelian varieties.

Extends results to Sprung's chromatic Selmer groups of elliptic curves.

Uses $ extbf{Gamma}$-system theory in the proof.

Abstract

Let $K$ be an imaginary quadratic field and $K_\infty$ be the $\mathbf{Z}_p^2$-extension of $K$. Answering a question of Ahmed and Lim, we show that the Pontryagin dual of the Selmer group associated to a supersingular polarized abelian variety admits an algebraic functional equation. The proof uses the theory of $\Gamma$-system developed by Lai, Longhi, Tan and Trihan. We also show the algebraic functional equation holds for Sprung's chromatic Selmer groups of supersingular elliptic curves along $K_\infty$.