Elsevier

Advances in Mathematics

Volume 204, Issue 2, 20 August 2006, Pages 481-508
Advances in Mathematics

Defining equations of modular curves

https://doi.org/10.1016/j.aim.2005.05.019Get rights and content
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Abstract

We obtain defining equations of modular curves X0(N), X1(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 X0(37) to find explicit modular parameterization of rational elliptic curves of conductor 37.

MSC

primary 11F03
secondary 11G05
11G18
11G30

Keywords

Modular curves
Generalized Dedekind eta-functions
Cusp forms
Modular parameterization of rational elliptic curves

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