The Cassels-Tate pairing for finite Galois modules

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.