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?

1. 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.

2. 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.

• What is what you call "the standard polarization"? Jun 27, 2023 at 9:16