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I have a question on $\mathbb{H}/\Gamma(N)$, which parametrizes level $N$ structures on elliptic curves. Let $Y(N)$ be the set of isomorphism classes of such objects, then, according to Fact 2 on page 2 on this note, parametrization is given by $$ \mathbb{H}/\Gamma(N)\rightarrow Y(N): \tau \mapsto (\mathbb{C}/\mathbb{Z}+\tau \mathbb{Z}, \frac{1}{N},\frac{\tau}{N}). $$ My problem is that I don't see why all pair of $N$-torsion points are realized as above. What about $\mathbb{C}/\mathbb{Z}+\tau \mathbb{Z}, \frac{N-1}{N},\frac{\tau}{N})$? More precisely how can one get the canonical form above?

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up vote 2 down vote accepted

The pair of $N$-torsion points that you wrote down will not be obtained via your parametrization because its image under the Weil-pairing is not $\exp(\frac{2\pi i}{N})$.

(This is part of the moduli problem: we want a pair of torsion points that pair under the Weil pairing to something fixed. If we don't insist on this, the moduli space is disconnected.)

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I see. If one ignore Weil paring but want to parametrize simply an isomorphism classes with $(\mathbb{Z}/N\mathbb{Z})^2\cong E[N]$, do you know how one can solve the moduli problem? –  Pooya Dec 19 '12 at 22:46
It's just the disjoint union of the varying moduli spaces as one insists on different Weil pairings (so it has N components). So for instance your guy is in the connected component that pairs to $\zeta^{N-1}$. –  user29743 Dec 19 '12 at 22:55
Your claim is not obvious to me but I will think about it. Thank you for your answer. –  Pooya Dec 19 '12 at 23:30
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