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

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 $K_{\infty }$.

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