The Cassels-Tate pairing on polarized abelian varieties

Let (A, lambda) be a principally polarized abelian variety defined over a g lobal field k, and let TCC(A) be its Shafarevich-Tate group. Let III(A)(nd) denote the quotient of III(A) by its maximal divisible subgroup. Cassels a nd Tate constructed a nondegenerate pairing III(A)(nd) x III(A)(nd) --> Q/Z . If A is an elliptic curve, then by a result of Cassels the pairing is alt ernating. But in general it is only antisymmetric. Using some new but equivalent definitions of the pairing, we derive general criteria deciding whether it is alternating and whether there exists some alternating nondegenerate pairing on III(A)(nd). These criteria are express ed in terms of an element c is an element of III(A)(nd) that is canonically associated to the polarization lambda. In the case that A is the Jacobian of some curve, a down-to-earth version of the result allows us to determine effectively whether #III(A) (if finite) is a square or twice a square. We then apply this to prove that a positive proportion (in some precise sense) of all hyperelliptic curves of even genus g greater than or equal to 2 ove r Q have a Jacobian with nonsquare #III (if finite). For example, it appear s that this density is about 13% for curves of genus 2. The proof makes use of a general result relating global and local densities; this result can b e applied in other situations.