The Weil pairing for elliptic curves over the $\mathbb{C}$

Question

Let $E = \mathbb{C} / (\mathbb{Z} + \tau \mathbb{Z})$ be an elliptic curve over $\mathbb{C}$.
Then how can I show that $e_n(1/N, \tau/N) = \exp(2 \pi i / n)$?
If we can show it, then the fundamental property of the Weil pairing shows the complete description of it.

I know that this is the section 24 of Mumford's Abelian varieties.
But it's too hard for me.

For the case of $\dim = 1$, can I show it more easily?

Answer 1

So, here is the basic idea.

Consider the Weil pairing $e_N(P, Q)$ on an elliptic curve $E/K$ where $P, Q \in E[N]$. We start off by finding function $f \in \bar{K}(E)$ such that $$(f) = N(P) - N(O)$$

Now set some notation, let $p = 1/N$ and let $q = \tau/N$. Now in our case it is very convenient that the Weierstrass $\sigma$-function allows us to construct a function with this divisor (see e.g., Silverman AEC, VI Prop 3.4) and the construction found there gives $$f(z) = \frac{\sigma(z - 1/N)^N}{\sigma(z)^{N-1} \sigma(z - 1)}$$

Now we need to recall some facts about $\sigma(z)$, namely

Lemmas For $\omega \in \Lambda$ we have $$\frac{\sigma(z + \omega)}{\sigma(z)} = \pm e^{\eta(\omega)(z + \omega/2)}$$ where the sign is positive if and only if $\omega \in 2 \lambda$, and $\eta$ satisfies $$\tau\eta(1) - \eta(\tau) = 2\pi i$$

These facts are in Silverman Exercise 6.4.

With all that set up we can check $$ f(z) = \frac{-1}{e^{\eta(1)(z + 1/2)}} \left( \frac{\sigma(z - 1/N)}{\sigma(z)} \right)^N$$

So that in the Weil pairing we choose $$g(z) = \frac{\xi}{e^{\eta(1)(Nz + 1/2)/N}}\left( \frac{\sigma(Nz - 1/N)}{\sigma(Nz)} \right)$$ where $\xi^N = -1$, so that $g^N = f \circ [N]$.

Then we need to compute $$e_N(p, q) = \frac{g(q + z)}{g(z)}$$ (where $z$ is so that we're not doing anything illegal). This will give

$$e_N(p, q) = \frac{1}{e^{\eta(1)\tau/N}} \frac{\sigma(Nz - 1/N + \tau) / \sigma(Nz + \tau)}{\sigma(Nz - 1/N) / \sigma(Nz)} $$

Using the lemmas we get

$$e_N(p, q) = \frac{e^{\eta(\tau)/N}}{e^{\eta(1)\tau/N}} = e^{2\pi i /N}$$

Finally, you can actually work throught this mess with some arbitrary lattice $\mathbb{Z} \omega_1 + \mathbb{Z} \omega_2$ so the assumptions on $\Lambda$ are not nessicary.

Also, I am sure to have made a sign error and come out with the wrong way around in my skew-symmetry, so please edit the post if you find such an error.