Fuchsian groups, automorphic functions and Schwarzians

Abstract

In this paper, a close connection is established between the geometry of certain genus zero Fuchsian groups and the analytic properties of the automorphic forms obtained by applying a certain differential operator to the Hauptmoduls of the groups.

The Schwarzian equation for completely replicable functions

“…This was used by Poincaré in his 1882 paper Mémoire sur les fonctions fuchsiennes [12]. It also appears in [5,8,10,11]. We give some properties of the Schwarz derivative (see [9]).…”

“…In [10], it was shown that the Schwarz derivative can have an important role in the theory of modular forms as well. The following proposition is an example.…”

“…Proposition 3.1 [10]. (i) If f is a modular function for a discrete subgroup G (of the first kind) of PSL 2 (R), then {f, z} is a weight four modular form for G.…”

On uniformization of Burnside’s curve y2=x5−x

“…On the other hand, explicit instances of all of these objects in the case of higher genera would enable one to construct nice applications wherever algebraic functions and Riemann surfaces appear. Such topics include algebraic-geometric integration, conformal field theories [1], integrable quantum/classical dynamical systems and nonlinear pde's, equations of Picard-Fuchs [10,14], second order linear ordinary differential equations (ode's) of Fuchsian type [21], number theory, "Monstrous Moonshine" [13], and many others. Yet another and deep application of uniformization originated in works of Takhtajan and collaborators in the late 1980's and for further applications of the general uniformization theory see work [1] and references therein.…”

Classification of torsion-free genus zero congruence groups

“…This study was motivated by previous work in collaboration with John McKay, [2,3], in which we studied the properties of Hauptmoduls originating from Moonshine under the action of the Schwarzian derivative.…”