Fuchsian groups, automorphic functions and Schwarzians
Let \(G\) be a finitely generated Fuchsian group of the first kind acting on the upper half-plane such that the compactification of the associated Riemann surface has genus zero, and assume that the transformation \(\tau\mapsto \tau+1\) generates the stabilizer of \(\infty\) in \(G\). Moreover, let \(f\) be a Hauptmodul for \(G\) such that \(f\) has the Fourier expansion \[ f(\tau)= \tfrac 1q+ \sum_{n=1}^\infty a_nq^n \tag{1} \] \((q= \exp (2\pi i\tau))\) at \(\infty\). For such an \(f\) and any natural number \(n\) there exists a unique monic polynomial \(P_n\) of degree \(n\) such that \(P_n(f)- q^{-n}\) has a power series expansion in \(q\) with constant term zero. The authors show how to find \(P_n\), and they develop a relation between \(P_n(f)\) and the Schwarzian \(\{f,\tau\}\). The Schwarzian \(\{f,\tau\}\) is an automorphic form of weight 4 (precisely) for the normalizer of \(G\) in \(SL_2(\mathbb{R})\) with poles of order 2 in the elliptic fixed points which is holomorphic at the cusps. Example: \(\{\lambda,\tau\}= \pi^2 E_4(\tau)\) where \(\lambda\) denotes the Legendre modular form and \(E_4\) the normalized Eisenstein series of weight 4 on \(SL_2(\mathbb{Z})\). A major aim of the work under review is to determine all genus zero groups \(G\) containing some \(\Gamma_0(n)\) as a subgroup with finite index such that the stabilizer of \(\infty\) is generated by \(\tau\mapsto \tau+1\) and such that \(G\) contains no elliptic elements. The authors determine all natural numbers \(n\) such that \(\Gamma_0(n)\) or a conjugate satisfies these conditions. There are 14 such groups which have Hauptmoduls given by eta-products; the associated Schwarzians are evaluated in terms of classical theta or Eisenstein series. There are only 3 more groups with the above properties which are not equal to some \(\Gamma_0(n)\) or a conjugate. The corresponding Hauptmoduls and their Schwarzians are also determined.
scientific article; zbMATH DE number 1546881