Fake Congruence Modular Curves and Subgroups of the Modular Group

Abstract

We construct a special class of noncongruence modular subgroups and curves, analogous in some ways to the usual congruence ones. The subgroups are obtained via the Burau representation, and the associated quotient curves have a natural moduli space interpretation. In fact, they are reduced Hurwitz spaces corresponding to covers with 4 branch points and monodromy group equal to semi-direct products of a cyclic and an abelian group. Furthermore, they form a modular tower in the sense of Fried. We study representations on the cohomology of these fake congruence modular curves and also calculate the genera of certain quotient curves.