The Cassels-Tate pairing for finite Galois modules

Given a global field $F$ with absolute Galois group $G_F$, we define a category $SMo{d}_F$ whose objects are finite $G_F$-modules decorated with local conditions. We define this category so that `taking the Selmer group’ defines a functor $Sel$ from $SMo{d}_F$ to $Ab$. After defining a duality functor $\vee$ on $SMo{d}_F$, we show that every short exact sequence $0\to M_1\to M\to M_2\to 0$ in $SMo{d}_F$ gives rise to a natural bilinear pairing $$Sel(M_2)\times Sel(M_1^{\vee })\to \mathbb{Q}\mathrm{/}\mathbb{Z}$$ 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.