Let $A/k$ be an abelian variety with real multiplication by some ring of integers $\mathcal O \subset F$. Let $n$ be an integer prime to the characteristic of $k$.

We have the standart $e_n$ pairing $A[n] \times A^\vee[n] \to \mu_n$. It satisfies $e_n(a \cdot x, y) = e_n(x, a \cdot y)$ for any $a \in \mathcal O$.

It is desirable to "extend" this pairing to an $\mathcal O$-linear pairing with respect to the $\mathcal O$-action on $A[n] \times A^\vee[n]$.

In Rapoprt's "Compactifications de l'espace de modules de Hilbert-Blumenthal" 1.21 we find such an extension:

$e_{n_{\mathcal O}} \colon A[n] \times A^\vee[n] \to (\mathcal D^{-1}/n\mathcal D^{-1})(1)$

where $\mathcal D$ is the different of $F$ and with $(1)$ we denote the Tate-twist. Concerning the definition of the pairing it is only said that $Tr(e_{n_{\mathcal O}}) = e_n$.

Can someone make this definition more explicit?