# Definition of modular curve associated to $\Gamma(N)$

For a positive integer $$N$$, we define $$\Gamma(N)=\big \{ \begin{bmatrix} a & b \newline c & d\end{bmatrix}\in \operatorname{SL}_2(\mathbb{Z}): \begin{bmatrix} a & b \newline c & d\end{bmatrix}\equiv \begin{bmatrix} 1 & 0 \newline 0 & 1\end{bmatrix}(\operatorname{mod}N)\big\}$$

Then, [Diamond, Shurman] defines the set of equivalence classes $$S(N)$$ of enhanced elliptic curves associated to $$\Gamma(N)$$ to be the set of equivalence classes of all triples $$(E,P,Q)$$, where $$E/\mathbb{C}$$ is an elliptic curve, and $$P,Q\in E(\mathbb{C})[N]$$ such that the Weil pairing $$e_N(P,Q)=e^{2\pi i/N}$$ .

Defining a Weil pairing:

Let $$P_1,P_2\in E(\mathbb{C})[N]$$ (possibly equal). Firstly, fix $$E(\mathbb{C})=\mathbb{C}/\Lambda$$ for some lattice $$\Lambda=\mathbb{Z}\omega_1+\mathbb{Z}\omega_2\subset \mathbb{C}$$. Now, there is a matrix $$\gamma\in M_{2\times 2}(\mathbb{Z}/N\mathbb{Z})$$ such that $$\begin{bmatrix} P_1 \\ P_2\end{bmatrix}=\gamma \begin{bmatrix} \omega_1/N+\Lambda \\ \omega_2/N+\Lambda\end{bmatrix}$$. We define $$e_N(P_1,P_2)=e^{2\pi i\operatorname{det}(\gamma)/N}$$.

Now, I know that there exists a smooth affine curve $$X_{\widetilde{\Gamma}(N)}/\operatorname{Spec}\mathbb{Z}[1/N]$$ which represents the moduli problem (assume $$N$$ is large enough): $$X_{\widetilde{\Gamma}(N)}(S)=\{(E,\alpha):E/S \operatorname{\ is\ an\ elliptic\ curve\ }, \alpha:\underline{(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}}_S\xrightarrow{\cong}E[N]\}.$$

At the face of it, it seems like $$X_{\widetilde{\Gamma}(N)}(\mathbb{C})=S(N)$$. But, the problem I am having is that an isomorphism $$\alpha:(\mathbb{Z}/N\mathbb{Z})^{\oplus 2}\xrightarrow{\cong}E(\mathbb{C})[N]$$ gives two points $$P,Q\in E(\mathbb{C})[N]$$ such that $$e_N(P,Q)=e^{2\pi i x/N}$$ for some $$x\in (\mathbb{Z}/N\mathbb{Z})^\times$$, which is not the same as a point in $$S(N)$$

Am I missing something? Are these two the same?

Reference:

Diamond, Fred; Shurman, Jerry, A first course in modular forms, Graduate Texts in Mathematics 228. Berlin: Springer (ISBN 0-387-23229-X/hbk). xv, 436 p. (2005). ZBL1062.11022.

Your $$S(N)$$ is naturally a scheme over $$\mathbb{Z}[1/N, \zeta_N]$$. Your $$X_{\widetilde{\Gamma}(N)}$$ is, as you have defined it, a scheme over $$\mathbb{Z}[1/N]$$. But there is actually a natural map $$\mathbb{Z}[1/N, \zeta_N] \hookrightarrow \mathcal{O}(X_{\widetilde{\Gamma}(N)})$$ which sends $$\zeta_N$$ to $$e_N(P, Q)$$; and if you regard $$X_{\widetilde{\Gamma}(N)}$$ as a $$\mathbb{Z}[1/N, \zeta_N]$$-scheme via this morphism, it coincides with $$S(N)$$.
• When you think of $e_N(P,Q)\in \mathcal{O}(X_{\widetilde{\Gamma}(N)})$, what is $P$ and $Q$? Are these some fixed generators of the $N$-torsion points of the universal elliptic curve? Commented May 9, 2023 at 21:26
• They are the canonical $N$-torsion points of the universal elliptic curve (which are part of what it means for the space to represent a functor). Commented May 10, 2023 at 0:05
• I had two follow-up question. Firstly, to summarise what you have said, would it be correct to say that over $\mathbb{Z}[1/N,\zeta_N]$, $X_{\widetilde{\Gamma}(N)}(\mathbb{C})=\Gamma(N)\backslash \mathcal{H}$, but over $\mathbb{Z}[1/N]$ this is not true anymore? Commented May 10, 2023 at 7:18
• Secondly, over $\mathbb{Z}[1/N$], will we get $X_{\widetilde{\Gamma}(N)}(\mathbb{C})=\sqcup_{(\mathbb{Z}/N\mathbb{Z})^{\times}}\Gamma(N)\backslash \mathcal{H}$ Commented May 10, 2023 at 7:21
• Perhaps the same thing in better language: this is about whether we study maximal ideals of $O(X) \otimes_{\mathbb{Z}[1/N, \zeta_N]}\mathbb{C}$ or of $O(X) \otimes_{\mathbb{Z}[1/N]} \mathbb{C}$. The $\mathbb{C}$-points of the first are $\Gamma(N) \backslash \mathcal{H}$. The $\mathbb{C}$-points of the second are $\varphi(N)$ copies of that. ($X = X_{\tilde{\Gamma}(N)}$) Commented May 10, 2023 at 14:23