The Cassels-Tate pairing for polarized abelian varieties, by Bjorn Poonen and Michael Stoll

This paper has now appeared in Annals of Mathematics 150 (1999), 1109-1149 and so the preprint has been removed. We develop criteria for determining when the Cassels-Tate pairing on a principally polarized abelian variety over a global field is alternating, and for determining when the order of the Shafarevich-Tate group (if finite) is a square. We give examples where the order is not square, and prove that for each even g>=2, a positive proportion (in a sense to be made precise) of the hyperelliptic curves of genus g over Q have a Jacobian whose Shafarevich-Tate group has order not the square of an integer.

Bjorn Poonen and Michael Stoll <poonen@math.berkeley.edu, stoll@math.uni-duesseldorf.de>