Structure and bases of modular space sequences $(M_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$. Part III: Cuspidal spaces

TL;DR

This paper develops a structured approach to cuspidal modular form spaces, providing explicit bases for these spaces for levels up to 10, based on the notion of strong modular forms.

Contribution

It introduces a new framework for structuring cuspidal modular form spaces and explicitly determines bases for levels up to 10.

Findings

Explicit bases for cuspidal spaces when 1 ≤ N ≤ 10.

A theoretical structure for cuspidal modular form spaces.

Application of strong modular form concept to basis determination.

Abstract

Based on the notion of strong modular form developed in Part I, we propose to structure the family of cuspidal modular form spaces $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ and to determine bases for each of these spaces, once known bases for the first values of $k$. We then apply these theoretical results to explicitly determine bases for space families $(S_{2k}(\Gamma_0(N)))_{k\in \mathbb{N}^*}$ when $1\leq N \leq 10$.