Quasimodular forms and modular differential equations which are not apparent at cusps: I

2021/03/08 by Chang‐Shou Lin, Yifan Yang, Lin, Chang-Shou +1
Mathematics · #11F37 #34M03 #34M35 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Number Theory (math.NT) #Primary 11F11 #secondary 11F25

paper · pdf · doi:10.48550/arxiv.2103.04890

openalex publication_date 2021/03/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we explore a two-way connection between quasimodular forms of depth 1 and a class of second-order modular differential equations with regular singularities on the upper half-plane and the cusps. Here we consider the cases Γ=Γ₀⁺(N) generated by Γ₀(N) and the Atkin-Lehner involutions for N=1,2,3 (Γ₀⁺(1)=SL(2,\mathbb Z)). Firstly, we note that a quasimodular form of depth 1, after divided by some modular form with the same weight, is a solution of a modular differential equation. Our main results are the converse of the above statement for the groups Γ₀⁺(N), N=1,2,3.

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