Quadratic spaces and Selmer groups of abelian varieties with multiplication

Abstract

For certain symmetric isogeny $\lambda: A\rightarrow A^\vee$ of abelian varieties over a global field $F$, B. Poonen and E. Rains put an orthogonal quadratic structure on $\mathrm{H}^1(\mathbb{A}_F,A[\lambda])$ and realize the Selmer group $\mathrm{Sel}_\lambda(A)$ as an intersection of two maximal isotropic subspaces of $\mathrm{H}^1(\mathbb{A}_F,A[\lambda])$. With this understanding of Selmer groups, they expect to model the Selmer groups of elliptic curves and Jacobian varieties of hyperelliptic curves as the intersections of random maximal isotropic subspaces of orthogonal spaces. We extend this phenomenon to abelian varieties with multiplication and discuss the Shafarevich-Tate groups.