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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?

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Here are two possible answers:

  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.

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  • $\begingroup$ What is what you call "the standard polarization"? $\endgroup$
    – Weier
    Jun 27, 2023 at 9:16

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