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 $X(2)$?
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2$\begingroup$ The answer is yes, but a rigorous proof would be quite complicated. Roughly speaking, you have to use the fact the Tate curve over $\mathbb{Z}((q^{1/2}))[1/2]$ admits a $\Gamma(2)$-structure. $\endgroup$– Will ChenCommented Oct 1, 2016 at 17:42
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$\begingroup$ Can you give a reference please? $\endgroup$– Adel BETINACommented Oct 1, 2016 at 18:11
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3$\begingroup$ Corollary 5.3.4 of arxiv.org/pdf/1510.05687v4.pdf is basically what you want, where you can take $k = K = \mathbb{Q}$. Actually the corollary works over Dedekind domains (like $\mathbb{Z}[1/2]$) as well, though I didn't write that up in the paper. $\endgroup$– Will ChenCommented Oct 1, 2016 at 18:26
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$y^2 =x (x-1)(x-\lambda)$ 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.
answered Oct 1, 2016 at 21:41
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1$\begingroup$ Sorry I know modular forms but I get nothing of algebraic geometry. What is the map $\mathbb{P}^1 \to X(2)$ ? $\endgroup$– reunsCommented Oct 1, 2016 at 23:00
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1$\begingroup$ @user1952009 Objects of $\mathfrak{M}(2)$ are families of elliptic curves $E \to B$ with level 2 structure. That is, such a family with level two structure is "the same as" a map $B \to \mathfrak{M}(2)$. Take $B = \mathbb{P}^1_{\mathbb{Q}}$, and compose with the coarse moduli map $\mathfrak{M}(2) \to X(2)$. $\endgroup$– S. Carnahan ♦Commented Oct 2, 2016 at 2:46
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$\begingroup$ I have another question : If $E$ is a curve of genus one over a field $k$ and $E(k)$ is not empty ($E$ is an elliptic curve), by Riemann-Roch we can find an equation of $E$ given by $P(X,Y)$ such that the maximal degree of $X$ is three and the maximal degree of $Y$ is two, how we can find an Equation of the type $y^2=x(x-1)(x-\lambda)$ ? $\endgroup$ Commented Oct 2, 2016 at 19:42
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1$\begingroup$ @AdelBETINA One needs to know that the two-torsion points are defined over $k$. There $y$ coordinates are zero, and by assumption their $x$ coordinates are in $k$. Now apply a linear transformation to $x$ to send one point to $0$ and another to $1$. However, this only produces an equation of the form $D y^2 = x(x-1)(x-\lambda)$. The $D$ is related to the fact that it is a coarse moduli space and not a fine one. $\endgroup$ Commented Oct 3, 2016 at 7:04