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The ℓ-parity conjecture over the constant quadratic extension

Published online by Cambridge University Press:  07 August 2017

KĘSTUTIS ČESNAVIČIUS
KĘSTUTIS ČESNAVIČIUS*
Affiliation:
Mathematisches Institut, Raum 4.025, Universität Bonn, Endenicher Allee 60, D-53115 Bonn, Deutschland e-mail: kestutis@berkeley.edu
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Abstract

For a prime ℓ and an abelian variety A over a global field K, 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 char K > 0, we prove that the ℓ-parity conjecture holds for the base change of A to the constant quadratic extension if ℓ is odd, coprime to char K, and does not divide the degree of every polarisation 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 385 - 409
Copyright
Copyright © Cambridge Philosophical Society 2017 

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