Poonen and Stoll had observed that the Cassels-Tate pairing on the Shafarevich-Tate group of a principally polarized abelian variety over a global field need not be alternating. Thus the order of $\Sha$ need not be a square.

For a jacobian of a curve there is an explicit criterion to decide when this happens in terms of the local points at primes of bad reduction. We will apply this to Shimura curves. The analysis of these local points uses the p-adic uniformization, and can be done in much greater generality.

The work described is joint with B. Jordan and Y. Varshavsky.