The $\ell$-parity conjecture over the constant quadratic extension

Kestutis Cesnavicius

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arXiv · 1402.2939

The $\ell$-parity conjecture over the constant quadratic extension

Kestutis Cesnavicius

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Abstract

For a prime $\ell$ and an abelian variety $A$ over a global field $K$, the $\ell$-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton-Dyer, the $\mathbb{Z}_{\ell}$-corank of the $\ell^{\infty}$-Selmer group and the analytic rank agree modulo $2$. Assuming that $\mathrm{char} K > 0$, we prove that the $\ell$-parity conjecture holds for the base change of $A$ to the constant quadratic extension if $\ell$ is odd, coprime to $\mathrm{char} K$, and does not divide the degree of every polarization of $A$. The techniques involved in the proof include the \'{e}tale cohomological interpretation of Selmer groups, the Grothendieck-Ogg-Shafarevich formula, and the study of the behavior of local root numbers in unramified extensions.

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Kestutis Cesnavicius. 2014-02-12. The $\ell$-parity conjecture over the constant quadratic extension. https://arxiv.org/abs/1402.2939

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