Hecke polynomials for the mock modular form arising from the Delta-function

2025/06/20 by Gomez, Kevin, Ono, Ken · 1 citation
Mathematics · #11F25 #11F30 #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Mathematical functions and polynomials #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2506.17178

openalex publication_date 2025/06/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider a mock modular form M_Δ(τ) that arises naturally from Ramanujan's Delta-function. It is a weight -10 harmonic Maass form whose nonholomorphic part is the "period integral function'' of Δ(τ). The Hecke operator T_-10(m) acts on this mock modular form in terms of Ramanujan's τ(m) and a monic degree m polynomial F_m(x), evaluated at x=j(τ). In analogy with results by Asai, Kaneko, and Ninomiya on the zeros of Hecke polynomials for the j-function, we prove that the zeros of each F_m(x), including x=0 and x=1728, are distinct and lie in [0, 1728]. Additionally, as m → +∞, these zeros become equidistributed in [0, 1728].

Citations

• Dyson’s ranks and Maass forms
• Heegner divisors, L-functions and harmonic weak Maass forms
• On Two Geometric Theta Lifts
• On two geometric theta lifts

Cited by

• Zeros of Hecke polynomials arising from weak eigenforms

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