Faber polynomials and Poincaré series

2010/10/11 by Ben Kane, Kane, Ben
Mathematics · #11F11 #11F37 #11J91 #11N37 #26A33 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11F11 #msc:11F37 #msc:11J91 #msc:11N37 #msc:26A33

paper · pdf · doi:10.48550/arxiv.1010.2176

openalex publication_date 2010/10/11 · arxiv created 2011/04/18 · arxiv updated 2011/04/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider weakly holomorphic modular forms (i.e. those meromorphic modular forms for which poles only possibly occur at the cusps) of weight 2-k∈ 2\Z for the full modular group \SL₂(\Z). The space has a distinguished set of generators f_m,2-k. Such weakly holomorphic modular forms have been classified in terms of finitely many Eisenstein series, the unique weight 12 newform Δ, and certain Faber polynomials in the modular invariant j(z), the Hauptmodul for \SL₂(\Z). We employ the theory of harmonic weak Maass forms and (non-holomorphic) Maass-Poincaré series in order to obtain the asymptotic growth of the coefficients of these Faber polynomials. Along the way, we obtain an asymptotic formula for the partial derivatives of the Maass-Poincaré series with respect to y as well as extending an asymptotic for the growth of the ℓ-th repeated integral of the Gauss error function at x to include ℓ∈ \R and a wider range of x.

Citations

• The f(q) mock theta function conjecture and partition ranks
• On two geometric theta lifts

Related

• Weak Maass-Poincare series and weight 3/2 mock modular forms
• A class of non-holomorphic modular forms III: real analytic cusp forms for SL₂(ℤ)
• Zeros of modular forms and Faber polynomials
• A polyharmonic Maass form of depth 3/2 for SL₂(Z)
• Singular invariants and coefficients of weak harmonic Maass forms of weight 5/2