Uniformizations of modular curves
This long and subtle paper stands within a series of notes devoted to the intimate relations between theta constants with rational characteristics, meromorphic modular forms for the principal congruence subgroup \(\Gamma(k)\) of the modular group \(\Gamma = \mathbb{P}\text{SL} (2,\mathbb{Z})\) and the geometry of the punctured Riemann surface \(\mathbb{H}^2/ \Gamma(k)\). These connections are established by associating the theta characteristics to the cusps of the group which leads in turn to new theta constant identities and explicit constructions of covering maps. In the present paper this approach is refined and extended to the larger group \(\Gamma_0(k)\). For this let \(V(k)\) be the finite-dimensional Hilbert space of holomorphic functions on the upper half plane \(\mathbb{H}^2\) spanned by the modified theta constants \(\varphi_\ell : \tau\mapsto \theta [\chi_\ell] (0,k. \tau)\) where \(\chi_\ell\) varies over the classes of rational characteristics. Then for \(k\) odd the full modular group operates on \(\mathbb{P} V(k) \simeq \mathbb{P} \mathbb{C}^{(k-3)/2}\) and induces a holomorphic map of the compactification \(\overline{\mathbb{H}^2/ \Gamma(k)}\) into \(\mathbb{P}\mathbb{C}^{(k-3)/2}\). Moreover for \(k\) prime this map reveals the automorphism group \(\Gamma/ \Gamma(k)\) of the surface as a group of projective transformations hence of isometries. It is this fact which yields a lot of geometric insight into the surface. Since the \({k-1 \over 2}\) ``distinguished'' punctures of the surface are permuted by the group \(\Gamma_0 (k)\) and support the divisor of \(\varphi_\ell/ \varphi_\ell\), two explicit generators of the function field \(K\overline {(\mathbb{H}^2/ \Gamma(k)})\) are constructed in terms of \(\varphi_1/ \varphi_0\) and \(\varphi_2/ \varphi_1\) for \(k\) prime \(> 7\). In general this field is a Galois extension of \(K\overline {(\mathbb{H}^2/ \Gamma)}\) generated by \(\varphi_\ell/ \varphi_{k-3 \over 2}\), \(\ell= 0,1, \dots, {k-5 \over 2}\) with Galois group \(\Gamma/ \Gamma(k)\). Finally quartic relations on the theta constants are given. It seems to be unknown whether these relations already determine the image of the curve \(\overline {\mathbb{H}^2/ \Gamma (k)}\) in \(\mathbb{P}\mathbb{C}^{(k-3)/2}\). Nevertheless for certain \(k\)'s the defining equations are found.
theta constants with rational characteristics
geometry of the punctured Riemann surface
scientific article; zbMATH DE number 929683