  Let (A,\lambda) be a principally polarized abelian variety defined over a
global field k, and let \Sha(A) be its Shafarevich-Tate group. Let \Sha(A)_\nd
denote the quotient of \Sha(A) by its maximal divisible subgroup. Cassels and
Tate constructed a nondegenerate pairing \Sha(A)_\nd \times \Sha(A)_\nd
\rightarrow \Q/\Z. If A is an elliptic curve, then by a result of Cassels the
pairing is alternating. 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 \Sha(A)_\nd. These criteria are expressed
in terms of an element c \in \Sha(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
\#\Sha(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 \ge 2 over \Q have a Jacobian with nonsquare \#\Sha (if
finite). For example, it appears 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 be applied in other situations.
