Shafarevich-Tate groups of abelian varieties

Abstract

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