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Before today, every source on the subject talks about algebraic models of the modular curve $X(N)$ over $\mathbb{Q}(\zeta_N)$, but in Ogg's paper "Rational Points on Certain Elliptic Modular Curves", he says that there is a canonical $\mathbb{Q}$-rational model of $X(N)$, where the cusps are rational over $\mathbb{Q}(\zeta_N)$, and identifying the cusps with pairs $(x,y)\in(\mathbb{Z}/N\mathbb{Z})^2/(\pm 1)$ for which $\gcd(x,y,N) = 1$, there is an action of the Galois group $(\mathbb{Z}/N\mathbb{Z})^*$ on the cusps acting via multiplication on the first coordinate (ie, $x$).

Ogg calls it a "model of Shimura, as communicated to me by Casselman", and cites a preprint: "W. Casselman, The Arithmetic of the cusps of the classical modular curves (to appear)", which I can't seem to find anywhere.

Can someone point out a good reference for this model?

thanks,

  • will
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  • $\begingroup$ This $Y(N)$ represents the moduli problem of pairs $(E,i)$ consisting of elliptic curves $E \rightarrow S$ (with $S$ a $\mathbf{Z}[1/N]$-scheme) and $i:(\mathbf{Z}/N\mathbf{Z})_S \times \mu_N \simeq E[N]$ is an isomorphism of $S$-group schemes. It is a smooth affine curve over $\mathbf{Z}[1/N]$ with geometrically connected fibers. The "classical" moduli problem uses an isomorphism between $E[N]$ and the constant $S$-group scheme $(\mathbf{Z}/N\mathbf{Z})_S^2$, so the relative Weil pairing of the standard basis specifies a map from $S$ to ${\rm{Spec}}(\mathbf{Z}[1/N][X]/(\Phi_N))$. $\endgroup$
    – user29283
    Commented Apr 23, 2013 at 14:18

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This is done by Shimura in Introduction to the Arithmetic Theory of Automorphic Functions, Chapter 6.

By definition, the field $\mathbf{Q}(X(N))$ of rational functions with respect to this model is the field of modular functions for the congruence subgroup $\Gamma(N)$ whose Fourier expansions belong to $\mathbf{Q}(\zeta_N)((q^{1/N}))$.

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