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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?

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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

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    $\begingroup$ There are though known problems with the proof that the kernels are the maximal divisible subgroup. See Harari-Szamuely and Gonzalez-Aviles for the corrections, both in Crelle 632 in 2009 with a title containing "arithmetic duality theorems". $\endgroup$ – Chris Wuthrich Oct 16 '14 at 20:56

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