Modular solutions to equations of generalized Halphen type

Abstract

Solutions to a class of differential systems that generalize the Halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group. These functions uniformize Riemann surfaces of genus zero and have q-series with integral coefficients. Rational maps between these functions are derived, implying subgroup relations between their automorphism groups, as well as symmetrization maps between the associated differential systems.