Modular forms for \(GL(r, \mathbbF_q[T])\): Hecke operators and growth of expansion coefficients

2025/11/03 by Ernst-Ulrich Gekeler, Gekeler, Ernst-Ulrich
Mathematics · #14 G 22 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT) #Primary 11F 52 #Secondary 11 G 09

paper · pdf · doi:10.48550/arxiv.2511.01712

openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

Abstract

We determine the action of the Hecke operators \(T_\mathfrakp,i\) on the coefficient forms \(g₁, …, g_r-1, g_r = Δ\), and \(h\), which together generate the ring of modular forms for \(GL(r, F_q[T])\). All these are eigenforms with powers of \(π\) as eigenvalues, where \(π\) is the monic generator of the prime ideal \(\mathfrakp\) of \(\mathbbF_q[T]\). We further describe the growth of the \(t\)-expansion coefficients of the discriminant function \(Δ\). It is such that the product expansion of \(Δ\) as well as the \(t\)-expansion of each modular form converges on the natural fundamental domain for \(GL(r, F_q[T])\).

Citations

• On Drinfeld modular forms of higher rank VII: Expansions at the boundary
• On Drinfeld modular forms of higher rank IV: Modular forms with level
• On Drinfeld modular forms of higher rank III: The analogue of the\n k/12-formula

Related

• Hecke operators on Hilbert-Siegel modular forms
• Multiplicative Hecke operators and their application II
• Hecke operators on differential modular forms mod p
• Hecke operators for higher rank Drinfeld modular forms
• Traces of Hecke operators on Drinfeld modular forms for \mathbbF_q[T]