The ℓ-parity conjecture over the constant quadratic extension

Abstract

AbstractFor a prime ℓ and an abelian varietyAover a global fieldK, the ℓ-parity conjecture predicts that, in accordance with the ideas of Birch and Swinnerton–Dyer, the ℤℓ-corank of the ℓ∞-Selmer group and the analytic rank agree modulo 2. Assuming that charK> 0, we prove that the ℓ-parity conjecture holds for the base change ofAto the constant quadratic extension if ℓ is odd, coprime to charK, and does not divide the degree of every polarisation ofA. 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.