# Weil pairing on abelian varieties and etale Chern classes

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 $$L$$?