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arXiv · 2208.01474

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

Cédric Dion
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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 $Γ$-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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Cédric Dion. 2023-01-04. Functional equations for supersingular abelian varieties over $\mathbf{Z}_p^2$-extensions. https://arxiv.org/abs/2208.01474

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