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

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 étale cohomological interpretation of Selmer groups, the Grothendieck-Ogg-Shafarevich formula, and the study of the behavior of local root numbers in unramified extensions.

22 pages; final version, to appear in Mathematical Proceedings of the Cambridge Philosophical Society

References in corpus (3)

• Twisted automorphisms of group algebras
• Selmer groups as flat cohomology groups
• 2-Selmer Parity for Hyperelliptic Curves in Quadratic Extensions