Let $\mathfrak{M}(2)$ be the algberaic stack over $\mathbb{Z}[1/2]$ which classifies the elliptic curves with the two level structure and let $X(2)$ be the coarse moduli space of $\mathfrak{M}(2)$ ($X(2)$ exists since $\mathfrak{M}(2)$ is smooth and proper). It is easy to see that $X(2)(\mathbb{C})$ is the modular curve of level $2$ and it is known that the modular lambda function rises to an isomorphism $X(2)(\mathbb{C}) \simeq \mathbb{P}^1\_{\mathbb{C}}$. Is there a meromorphic function on $X(2)$ or on $X(2)\_{\mathbb{Q}}$ compatible with the modular lambda function on the $\mathbb{C}$-points of $X(2)$?



Modular curve X(2)
$y^2 =x (x-1)(x-\lambda)$ is a family of elliptic curves with level two structure on $\mathbb P^1\_{\mathbb Q}$. Hence by the definition of coarse moduli space, it defines a map from $\mathbb P^1$ to the coarse moduli space $X(2)$. It is sufficient to check that this map is an isomorphism and that it send $\lambda$ to the modular $\lambda$. The second claim implies the first, and the second claim can be checked easily from the definition of the modular $\lambda$ as the cross-ratio of the $x$-coordinates of the $2$-torsion points.


