paper

The Cassels-Tate pairing for finite Galois modules

arXiv:2103.08530

Abstract

Given a global field $F$ with absolute Galois group $G_{F}$ , we define a category $S M o d_{F}$ whose objects are finite $G_{F}$ -modules decorated with local conditions. We define this category so that `taking the Selmer group' defines a functor $S e l$ from $S M o d_{F}$ to $A b$ . After defining a duality functor $\lor$ on $S M o d_{F}$ , we show that every short exact sequence $0 \to M_{1} \to M \to M_{2} \to 0$ in $S M o d_{F}$ gives rise to a natural bilinear pairing $S e l (M_{2}) \times S e l (M_{1}^{\lor}) \to Q / Z$ whose left and right kernels are the images of $S e l (M)$ and $S e l (M^{\lor})$ , 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.

45 pages, comments welcome!