Noncongruence modular curves as Hurwitz spaces

Abstract

In this survey article we give an overview of how noncongruence modular curves can be viewed as Hurwitz moduli spaces of covers of elliptic curves at most branched above the origin. We describe some natural questions that arise, and applications of these ideas to the Inverse Galois Problem, Markoff triples and the arithmetic of Fourier coefficients for noncongruence modular forms.