Interpretation of the Weil pairing in the complex torus The point addition on an elliptic curve corresponds to the vector addition on a complex torus (with suitable choice of the lattice and of the base point). Is there a similar interpretation for the Weil pairing? And for the Tate pairing?
Furthemore, the determinant of two vectors in $\mathbb{C}$ (considered as $\mathbb{R}^2$) is also an non-degenerate alternating form. Is there a corresponding pairing?
 A: Here are two possible answers:

*

*If you write $E=\mathbb{C}/\Lambda$ then the standard polarization
gives us an alternating map  $$\langle -,-\rangle:\Lambda\times
\Lambda\to \mathbb{Z}$$
One can obviously extend this to an alternating pairing
$$\Lambda_\mathbb{Q}\times \Lambda_\mathbb{Q}\to \mathbb{Q}$$
(where $\Lambda_\mathbb{Q}:=\Lambda\otimes_\mathbb{Z}\mathbb{Q}$).
Let us then note that
$$E[N]\subseteq \Lambda_\mathbb{Q}$$
and thus we can restrict to obtain a pairing
$$\langle -,-\rangle:E[N]\times E[N]\to \mathbb{Q}$$
We then can define
$$\langle \alpha,\beta\rangle_\text{Weil}:=\exp(2\pi i N \langle
\alpha,\beta\rangle)$$
One can then show, as the notation suggests, that $\langle
-,-\rangle_\text{Weil}$ is the Weil pairing.


*One can show that Weil pairing is nothing but the cup product in
cohomology under the identifications
$$H_1(E,\mathbb{Z}/N\mathbb{Z})=E[N],\qquad
H^2_\text{sing}(E,\mu_N)\cong \mathbb{Z}/N\mathbb{Z}$$
This perspective is nice since it also extends to etale cohomology.
Both of these discussion are contained, I'm pretty sure, in Mumford's book on abelian varieties.
