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.

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    $\begingroup$ I believe that Weil pairing is in some natural sense the cup product in Tate cohomology. IIRC J.Silverman once tried to explain it to me in sci.math, but my recollection and understanding is a bit rusty :-) $\endgroup$ – Jyrki Lahtonen May 16 '14 at 13:26
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    $\begingroup$ The Weil pairing can be defined using the natural duality that occurs between the kernel of an isogeny and the kernel of its dual isogeny. This definition is very natural. It can then be shown that the Weil affords the geometric description given above. Unfortunately, I only know of a reference for this which deals with abelian varieties (here), perhaps it can be simplified for elliptic curves. $\endgroup$ – RghtHndSd May 16 '14 at 13:34
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    $\begingroup$ Silverman's Arithmetic of Elliptic Curves starts out by wanting to make the determinant Galois invariant, but I can't see how this property arises in constructing the pairing; only that it seems to appear magically afterwards. Can anyone explain this? $\endgroup$ – Matt B Jul 8 '15 at 18:27
  • $\begingroup$ @JyrkiLahtonen : you can have a look to this old but very interesting post :-) (same here). $\endgroup$ – Watson Feb 14 '17 at 17:18

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