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May 16, 2021 at 12:31 comment added user166831 This is the same minus sign as that in Section 24, Theorem 1, of Mumford's book on Abelian Varieties (over the complex numbers).
May 15, 2021 at 15:34 vote accept Alexander Betts
May 15, 2021 at 15:34 comment added Alexander Betts Great! The minus sign that intervenes here is a good cautionary tale: not all naturally-defined pairings are the same as one another. In any case, the instance I care about is in characteristic zero, so I will accept this answer.
May 15, 2021 at 1:35 history answered Yuri Zarhin CC BY-SA 4.0