On the real zeros of depth 1 quasimodular forms

2024/01/02 by Bo‐Hae Im, Wonwoong Lee, Im, Bo-Hae +1
Mathematics · #11F11 #11F99 #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometry and complex manifolds #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2401.01000

openalex publication_date 2024/01/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We discuss the critical points of modular forms, or more generally the zeros of quasimodular forms of depth 1 for PSL₂(\mathbb Z). In particular, we consider the derivatives of the unique weight k modular forms f_k with the maximal number of consecutive zero Fourier coefficients following the constant 1. Our main results state that (1) every zero of a depth 1 quasimodular form near the derivative of the Eisenstein series in the standard fundamental domain lies on the geodesic segment \z ∈ \mathbb H: \Re(z)=1/2\, and (2) more than half of zeros of f_k in the standard fundamental domain lie on the geodesic segment \z ∈ \mathbb H: \Re(z)=1/2\ for large enough k with k≡ 0 \pmod12.

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