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](https://zbmath.org/?q=an:1062.11022).



Definition of modular curve associated to $\Gamma(N)$
This is a subtle issue (which has come up before on this site several times, see e.g. [is the modular curve X(N) defined over Q?](https://mathoverflow.net/questions/192156/is-the-modular-curve-xn-defined-over-q?rq=1) for a related question).


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 $S(N)$.


