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The ℓ-parity conjecture over the constant quadratic extension

  • KĘSTUTIS ČESNAVIČIUS (a1)
Abstract
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 ℤ-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.

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