Structure and bases of modular spaces sequences (M 2k (Γ 0 (N))) k∈ℕ *

Jean-Christophe Feauveau

doi:10.5802/ambp.395

Abstract

Structure and bases of modular spaces sequences <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>M</mml:mi> <mml:mrow><mml:mn>2</mml:mn><mml:mi>k</mml:mi></mml:mrow> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>Γ</mml:mi> <mml:mn>0</mml:mn> </mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>N</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow> <mml:mrow><mml:mi>k</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>ℕ</mml:mi> <mml:mo>*</mml:mo> </mml:msup></mml:mrow> </mml:msub></mml:math>

Highlights

When studying modular forms, an important result relates to the structure of the sequence (M2k (SL2 (Z)))k ∈N∗ obtained using the Δ function, and the opportunity to obtain an explicit basis for each subspace [11, p. 143–144].Such a result appears to be missing for the sequences (M2k (Γ0 (N)))k ∈N∗, whenever N 2
We propose in this paper an explicit decomposition of modular form spaces (M2k (Γ0 (N)))(k,N ) ∈N∗2
The following result provides the structure of the sequence of modular forms spaces (M2k (Γ0 (N)))k ∈N∗ under the assumption that a strong modular unit exists

Summary

Introduction

When studying modular forms, an important result relates to the structure of the sequence (M2k (SL2 (Z)))k ∈N∗ obtained using the Δ function, and the opportunity to obtain an explicit basis for each subspace [11, p. 143–144]. When studying modular forms, an important result relates to the structure of the sequence (M2k (SL2 (Z)))k ∈N∗ obtained using the Δ function, and the opportunity to obtain an explicit basis for each subspace [11, p. The structure of families (M2k (Γ0 (N)))k ∈N∗ will be studied in Section 2 under the assumption of the existence of a strong modular unit ΔN.

Primer on modular forms

Primer on Dedekind η function

Putting theory into practice

Conclusion