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The Cassels-Tate pairing for finite Galois modules

Adam Morgan, Alexander Smith

TL;DR

The paper develops a general framework for the Cassels--Tate pairing for finite Galois modules with local conditions by introducing the Selmer-category $ extup{SMod}_F$ and a duality functor. It constructs a natural bilinear pairing from any exact sequence in $ extup{SMod}_F$, analyzes its left and right kernels, and proves a duality identity that governs symmetry. A central theme is the interplay between local Tate duality, Poitou–Tate dualities, and theta groups, which clarifies when the pairing is alternating and how obstructions to alternation arise. The results unify and extend classical Cassels--Tate theory to Bloch–Kato Selmer settings, provide algebraic proofs of Poonen–Stoll phenomena, and furnish tools for comparing pairings across different exact sequences, with concrete applications to isogeny Selmer groups and sums of local conditions.

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 $0 \to M_1 \to M \to M_2 \to 0$ in $SMod_F$ gives rise to a natural bilinear pairing $$Sel (M_2) \times Sel (M_1^{\vee}) \to \mathbb{Q}/\mathbb{Z}$$ 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.

The Cassels-Tate pairing for finite Galois modules

TL;DR

The paper develops a general framework for the Cassels--Tate pairing for finite Galois modules with local conditions by introducing the Selmer-category and a duality functor. It constructs a natural bilinear pairing from any exact sequence in , analyzes its left and right kernels, and proves a duality identity that governs symmetry. A central theme is the interplay between local Tate duality, Poitou–Tate dualities, and theta groups, which clarifies when the pairing is alternating and how obstructions to alternation arise. The results unify and extend classical Cassels--Tate theory to Bloch–Kato Selmer settings, provide algebraic proofs of Poonen–Stoll phenomena, and furnish tools for comparing pairings across different exact sequences, with concrete applications to isogeny Selmer groups and sums of local conditions.

Abstract

Given a global field with absolute Galois group , we define a category whose objects are finite -modules decorated with local conditions. We define this category so that `taking the Selmer group' defines a functor from to . After defining a duality functor on , we show that every short exact sequence in gives rise to a natural bilinear pairing whose left and right kernels are the images of and , respectively. This generalizes the Cassels--Tate pairing defined on the Shafarevich--Tate group of an abelian variety over 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.

Paper Structure

This paper contains 23 sections, 23 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Selmer groups and the category $\textup{SMod}_F$
  3. The pairing and its main properties
  4. The Poonen--Stoll class for modules with alternating structure
  5. Acknowledgments
  6. Notation
  7. The pairing and its main symmetry
  8. The left kernel of the Cassels--Tate pairing
  9. The duality identity and the right kernel of the Cassels--Tate pairing
  10. The category $\textup{SMod}_F$
  11. The quasi-abelian property
  12. The pairing for infinite modules
  13. More general local conditions
  14. Theta groups and the Poonen--Stoll class
  15. Proving Theorem \ref{['thm:theta_main']}
  16. ...and 8 more sections

Key Result

Theorem 1.3

Take an exact sequence $E$ in $\textup{SMod}_F$ as in eq:example_exact_intro. There is then a natural bilinear pairing with left kernel $\pi(\textup{Sel}\, M)$ and right kernel $\iota^{\vee}(\textup{Sel}\, M^{\vee})$. Moreover:

Theorems & Definitions (79)

  • Definition 1.1
  • Remark 1.2
  • Theorem 1.3
  • Example 1.4
  • Definition 1.6
  • Theorem 1.7
  • Remark 1.8
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.3
  • ...and 69 more