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Given a line bundle $L$ on an abelian variety $A/k$, there is an associated Weil pairing $e_L\colon\bigwedge^2V_pA\to\mathbb Q_p(1)$, where $p$ is a prime different from the residue characteristic of the base field $k$ and $V_pA$ is the $\mathbb Q_p$-linear Tate module of $A$. This is usually constructed by explicitly constructing a pairing between $A[p^n]$ and $A^\vee[p^n]$ using the interpretation of the latter as classes of divisors, and then pulling back along the polarisation induced by $L$.

There is, however, another way to construct such a pairing, using the fact that $V_pA$ is dual to the etale cohomology $\mathrm H^1_{et}(A_{\bar k},\mathbb Q_p)$. Namely, the first etale Chern class $c^{et}_1(L)$ is an element of $\mathrm H^2_{et}(A_{\bar k},\mathbb Q_p)(1)=\mathrm{Hom}(\bigwedge^2V_pA,\mathbb Q(1))$, and we can just take the pairing corresponding to this element.

What I want to know (and ideally would like a reference for) is whether these two pairings are the same. In other words, is the element of $\mathrm H^2_{et}(A_{\bar k},\mathbb Q_p)(1)$ corresponding to the Weil pairing equal to the first etale Chern class of $L$?

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It seems that one of the pairings is the negative of the other (in char 0 this assertion is actually Lemma 2.6 of https://arxiv.org/pdf/1809.01440.pdf ).

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  • $\begingroup$ 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. $\endgroup$ May 15 '21 at 15:34
  • $\begingroup$ This is the same minus sign as that in Section 24, Theorem 1, of Mumford's book on Abelian Varieties (over the complex numbers). $\endgroup$
    – user166831
    May 16 '21 at 12:31

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