Models for quotients of modular curves

TL;DR

This paper presents an algorithm to compute rational models of quotient modular curves using q-expansions and provides a moduli interpretation for morphisms between modular curves.

Contribution

It introduces a new algorithm for constructing rational models of quotient modular curves and offers a moduli interpretation for general morphisms between these curves.

Findings

Algorithm successfully computes Q-rational models of quotient modular curves.

Provides a moduli interpretation for morphisms between modular curves.

Enhances understanding of the structure of modular curves and their quotients.

Abstract

We describe an algorithm for computing a $\Q$-rational model for the quotient of a modular curve by an automorphism group, under mild assumptions on the curve and the automorphisms, by determining $q$-expansions for a basis of the corresponding space of cusp forms. We also give a moduli interpretation for general morphisms between modular curves.