On special values of generators of modular function fields (Q6085409)

Let \(N\) be a positive integer, \(\zeta_N\) a primitive \(N\)-th root of unity, \(X(N)\) the modular curve of level \(N\) and \({\mathcal F}_{X(N),{\mathbb Q}(\zeta_N)}\) its field of meromorphic functions with Fourier coefficients in \({\mathbb Q}(\zeta_N)\). Set \({\mathbb H}=\{\tau\in{\mathbb C}\mid \mathrm{Im}(\tau)>0\}\) and \({\mathbb H}^*={\mathbb H}\cup {\mathbb Q}\cup \{\infty\}\). Let \(X_1(N)=\Gamma_1(N)\setminus {\mathbb H}^*\) be the modular curve for the congruence subgroup \[ \Gamma_1=\Big\{\gamma\in\mathrm{SL}_2({\mathbb Z})\mid \gamma\equiv \Big[\begin{array}{cc} 1&*\\ 0& 1\end{array}\Big] (\bmod NM_2({\mathbb Z}))\Big\}. \] Let \(L\subseteq {\mathbb Q}(\zeta_N)\) and let \({\mathcal F}_{X,L}\) denote the field of meromorphic functions \(X\) whose Fourier coefficients belong to \(L\). We have \({\mathcal F}_{X(1),{\mathbb Q}} ={\mathbb Q}(j)\), where \(j\) is certain elliptic modular function. The authors consider the following Claim A: Let \(h\) be a nonconstant function on \({\mathbb H}\) satisfying \[ {\mathcal F}_{X(1),{\mathbb Q}(\zeta_N)}\subseteq {\mathbb Q}(\zeta_N, j,h)\subseteq {\mathcal F}_{X(N),{\mathbb Q}(\zeta_N)}. \] Let \(K\) be an imaginary quadratic field. If \(h\) is defined at \(z_K\), where \(z_K\in K\cap {\mathbb H}\) is such that \({\mathcal O}_K={\mathbb Z} z_K+{\mathbb Z}\). Then \[ K_{(N)}=K(\zeta_N, j(z_K), h(z_K))=H_K(\zeta_N, h(z_K)), \] where the Hilbert class field of \(K\) is \(H_K=K(j(z_K))\) and the ray class field of \(K\) modulo \(N{\mathcal O}_K\) is \(K_{(N)}=K(f(z_K)\mid f\in {\mathcal F}_{ X(N),{\mathbb Q}(\zeta_N)} \text{ is defined at }z_K)\). The paper consists of three parts. First, in Section 3, several counterexamples to Claim A are given. Next, in Section 4, the authors prove that Claim A holds true for all but finitely many imaginary quadratic fields \(K\) (Theorems 4.1 and 4.5). Finally, in Section 5, the authors present a practical way of specifying exceptions in Theorem 4.1 in terms of affine plane curves (Theorem 5.2).