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I have trouble in understanding Weil paring on $N$-torsion points on an elliptic curve. Please see Wikipedia for the definition of Weil paring. I would like to know what Weil paring is computing intuitively. I would also want to see some basic computation.

I will really appreciate your help.

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1 Answer 1

Imagine your elliptic curve is $\frac{\mathbb{C}}{\Lambda}$ for the lattice $\Lambda$ with basis $\langle 1, \tau \rangle$. Then the $N$ torsion is spanned (mod $\Lambda$) by $\frac{1}{N}$ and $\frac{\tau}{N}$ and the Weil pairing is the unique symplectic pairing with $$ \big\langle\frac{1}{N}, \frac{\tau}{N} \big\rangle = \exp\big(\frac{-2\pi i}{N}\big). $$ The point (I guess) is that this analytically "obvious" basis for the $N$-torsion doesn't make sense algebraically, but if you think of it as a basis for a symplectic pairing to $\mu_N$, then at least that pairing makes sense algebraically.

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