The Weil pairing $$e_\phi:E[\phi]\times E'[\hat{\phi}]\to \mu_n$$ for an elliptic curve is defined as follows. Let $\phi:E\to E'$ be an isogeny of degree $n$ and $\hat\phi:E'\to E$ be the dual isogeny. Given $(S,T)\in E[\phi]\times E'[\hat{\phi}]$, pick $g$ such that $\text{div}(g)=\phi^*((T)-(O))$, and define $e_\phi(S,T)=\frac{g(X+S)}{g(X)}$ (choose any $X$). Of course, one has to check this is well-defined, in particular that it doesn't depend on $X$, i.e., $\frac{\tau_S^*g}{g}$ is a constant where $\tau_S$ is translation by $S$.
This definition is not at all transparent to me. I know that we want the Weil pairing because we want a nondegenerate, bilinear, alternating (when $\phi=[n]$), Galois invariant, compatible map (see Silverman III.8.1). But given that we want a map satisfying these conditions, how might one come up with the right definition?
For instance, in the case of elliptic curves over $\mathbb C$ one could map to a lattice and define the map using the determinant---this seems a motivated construction since we get a canonical way of identifying $E[n]$ with $(\mathbb Z/n\mathbb Z)^2$ (I don't know the details of this, I believe it's in Lang's Elliptic Functions). However, I don't know how this would transfer over to the definition given above.
