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?



O-linear  Weil-pairing on abelian varieties with real multiplication
Yes, it's just a linear algebra stuff. Probably, it's more convenient to look at the Weil pairing between the $\ell$-adic Tate modules of $A$ and its dual, taking into account that these modules are free $O\otimes Z\_{\ell}$-modules of the same rank.


From the linear algebra point of view the situation is as follows.


Let $e: M \times N \to P$ be a perfect pairing of free $Z\_{\ell}$-modules $M$ and $N$ of the same rank that takes values in a free $Z\_{\ell}$-module $P$ of rank $1$. Assume also that $M$ and $N$ are free $O\otimes Z\_{\ell}$-modules of the same rank. Then we get a natural $Z\_{\ell}$-linear map


$M \times N \to Hom\_{Z\_{\ell}}(O\otimes Z\_{\ell}, P),$
$(m,n) \mapsto [a \mapsto e(ax,y)]$ for all $a \in O$. 
Taking into account that 
$$Hom\_{Z\_{\ell}}(O\otimes Z\_{\ell}, Z\_{\ell})=D^{-1}\otimes Z\_{\ell}$$ (via the trace map),
 we get the natural pairing


$$M \times N \to P \otimes\_{Z\_{\ell}}[D^{-1}\otimes Z\_{\ell}].$$
Here $M$ and $N$ are the Tate modules of $A$ and its dual while $P$ is $Z\_{\ell}(1)$ and $e$ is the Weil pairing.


