Defining equations of modular curves

Abstract

We obtain defining equations of modular curves X-0(N), X-1(N), and X(N) by explicitly constructing modular functions using generalized Dedekind eta functions. As applications, we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup. We also use our model of X-0(37) to find explicit modular parameterization of rational elliptic curves of conductor 37. (C) 2005 Elsevier Inc. All rights reserved.