Computing actions on cusp forms

TL;DR

This paper presents a method to compute the action of SL_2(Z) on cusp forms by leveraging Atkin-Lehner operators, facilitating explicit models of modular curves from q-expansions.

Contribution

It introduces a computational approach to determine the action of SL_2(Z) on cusp forms using Atkin-Lehner operators, enabling explicit modular curve models.

Findings

Efficient algorithms for computing SL_2(Z) actions on cusp forms.

Reduction of the problem to Atkin-Lehner operator computations.

Application to explicit modeling of modular curves.

Abstract

For positive integers $k$ and $N$, we describe how to compute the natural action of $SL_2(\mathbb{Z})$ on the space of cusp forms $S_k(\Gamma(N))$, where a cusp form is given by sufficiently many terms of its $q$-expansion. This will reduce to computing the action of the Atkin--Lehner operator on $S_k(\Gamma)$ for a congruence subgroup $\Gamma_1(N)\subseteq \Gamma \subseteq \Gamma_0(N)$. Our motivating application of such fundamental computations is to compute explicit models of some modular curves $X_G$.