Schwarz maps for modular curves

Abstract

We solve a classical problem posed by F. Klein and studied by A. Hurwitz concerning the construction of linear ordinary differential equations associated with modular transformations of fixed degree. For every odd integer $N\ge 3$ (respectively, even integer $N\ge 4$), we construct a canonical invariant model of the modular curve $X(N)=\mathbb{H}/Γ(N)$ (respectively, $X_H(N)=\mathbb{H}/H(N)$ where $H(N)=Γ(N)\capΓ_0^0(2N)$), together with a linear ordinary differential equation with rational coefficients whose Schwarz map parametrizes this model and whose projective monodromy group is the finite quotient $PSL_2(\mathbb{Z})/\tildeΓ(N)$ (respectively, $PSL_2(\mathbb{Z})/\tilde{H}(N)$). The construction is expressed in terms of invariant projective geometry and Picard-Vessiot theory and yields equations that are canonical up to projective equivalence. In this framework, Hurwitz's classical equation for degree $7$ appears as a special case of a general mechanism. The results place Klein's question within the modern theory of algebraic linear ordinary differential equations and provide a uniform geometric realization of modular transformation groups as projective differential Galois groups. As an application, we construct an explicit example of a linear ordinary differential equation associated with $X(9)$.