The Cassels-Tate pairing is typically defined for elliptic curves (or abelian varieties) over number fields, but it seems like it should be defined for elliptic curves over function fields as well. Does anybody know if this is not true or if there is a reference?



Is the Cassels-Tate pairing defined for elliptic curves over function fields?
You can find it for Abelian varieties over global fields in [Milne, Arithmetic Duality Theorems] <http://jmilne.org/math/Books/ADTnot.pdf>, p. 202, Theorem 5.6. (See also Remark 5.7.)


See also Poonen/Stoll: <http://www.mathe2.uni-bayreuth.de/stoll/papers/sha.pdf>


Edit: I also constructed a Cassels-Tate pairing for Abelian schemes over *higher dimensional* bases over finite fields under certain conditions, see <http://arxiv.org/abs/1410.5293>


