Shafarevich-Tate groups of abelian varieties

Abstract

The Shafarevich-Tate group $W (\mathscr{A})$ measures the failure of the Hasse principle for an abelian variety $\mathscr{A}$. Using a correspondence between the abelian varieties and the higher dimensional non-commutative tori, we prove that $W (\mathscr{A})\cong Cl~(\Lambda)\oplus Cl~(\Lambda)$ or $W (\mathscr{A})\cong \left(\mathbf{Z}/2^k\mathbf{Z}\right) \oplus Cl_{~\mathbf{odd}}~(\Lambda)\oplus Cl_{~\mathbf{odd}}~(\Lambda)$, where $Cl~(\Lambda)$ is the ideal class group of a ring $\Lambda$ associated to the K-theory of the non-commutative tori and $2^k $ divides the order of $Cl~(\Lambda)$. The case of elliptic curves with complex multiplication is considered in detail.