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 $L$?



Weil pairing on abelian varieties and etale Chern classes
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> ).


