The Cassels-Tate pairing for finite Galois modules

2021/03/15 by Adam Morgan, Alexander Smith, Morgan, Adam +1 · 1 citation
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2103.08530

Abstract

Given a global field F with absolute Galois group G_F, we define a category SMod_F whose objects are finite G_F-modules decorated with local conditions. We define this category so that `taking the Selmer group' defines a functor Sel from SMod_F to Ab. After defining a duality functor \vee on SMod_F, we show that every short exact sequence 0 → M₁ → M → M₂ → 0 in SMod_F gives rise to a natural bilinear pairing Sel (M₂) × Sel (M₁^\vee) → ℚ/ℤ whose left and right kernels are the images of Sel (M) and Sel (M^\vee), respectively. This generalizes the Cassels--Tate pairing defined on the Shafarevich--Tate group of an abelian variety over F and results in a flexible theory in which pairings associated to different exact sequences can be readily compared to one another. As an application, we give a new proof of Poonen and Stoll's results concerning the failure of the Cassels--Tate pairing to be alternating for principally polarized abelian varieties and extend this work to the setting of Bloch--Kato Selmer groups.

Citations

• Finding large Selmer rank via an arithmetic theory of local constants
• Modular Elliptic Curves and Fermat's Last Theorem

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• Polarizations, torsors and theta groups

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