Fuchsian groups, automorphic functions and Schwarzians. (English) Zbl 0958.11030
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.
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.
Reviewer: Jürgen Elstrodt (Münster)
MSC:
11F11 | Holomorphic modular forms of integral weight |
30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |
20H05 | Unimodular groups, congruence subgroups (group-theoretic aspects) |
11F03 | Modular and automorphic functions |
11F20 | Dedekind eta function, Dedekind sums |
11F27 | Theta series; Weil representation; theta correspondences |
Online Encyclopedia of Integer Sequences:
McKay-Thompson series of class 4C for the Monster group.McKay-Thompson series of class 6E for Monster (and, apart from signs, of class 12B).
Theta series of direct sum of 4 copies of hexagonal lattice.
Theta series of direct sum of 2 copies of D_4 lattice in powers of q^2.
Theta series of 8-d 6-modular lattice G_2 tensor F_4 (or A_2 tensor D_4) with det 1296 and minimal norm 4 in powers of q^2.
Expansion of square root of q times normalized Hauptmodul for Gamma(4) in powers of q^8.
McKay-Thompson series of class 16B for the Monster group.
McKay-Thompson series of class 8E for the Monster group.
McKay-Thompson series of class 9B for the Monster group.
McKay-Thompson series of class 12I for the Monster group.
Gamma_0(n) has genus 0 and does not contain any elliptic elements.
McKay-Thompson series of class 18D for the Monster group.
McKay-Thompson series of class 24c for the Monster group.
McKay-Thompson series of class 36B for the Monster group.
Expansion of Hauptmodul for group G’_{27|3}.
McKay-Thompson series of class 27c for the Monster group.
Expansion of a Schwarzian ({f_{27|3}, tau} / (4*Pi)^2) in powers of q^3.
McKay-Thompson series of class 32e for the Monster group.
McKay-Thompson series of class 16d for the Monster group.
Expansion of a Schwarzian ({f_{32|8}, tau} / (4*Pi)^2) in powers of q^8.
Expansion of (b(q) * c(q^3) / 3)^2 in powers of q where b(), c() are cubic AGM theta functions.