The Cassels-Tate pairing for finite Galois modules

Abstract

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 $$Sel(M_2)\times Sel(M_1^{\vee })\to \mathbb{Q}/\mathbb{Z}$$ 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.

arXiv comment45 pages, comments welcome!

References in corpus (5)

• The Cassels-Tate pairing on polarized abelian varieties
• $2^{\infty }$-Selmer groups, $2^{\infty }$-class groups, and Goldfeld's conjecture
• Self cup products and the theta characteristic torsor
• Governing fields and statistics for 4-Selmer groups and 8-class groups
• Cohen-Lenstra heuristics and bilinear pairings in the presence of roots of unity