Modular forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\): Hecke operators and growth of expansion coefficients

TL;DR

This paper analyzes the action of Hecke operators on modular forms for \\mathrm{GL}(r, \\mathbb{F}_{q}[T]) and describes the growth of expansion coefficients, providing insights into their eigenstructure and convergence properties.

Contribution

It determines the eigenvalues of Hecke operators on generators of the modular forms ring and describes the growth and convergence of their expansion coefficients.

Findings

All generators are eigenforms with eigenvalues as powers of \\pi.

The growth of the \\Delta function's expansion coefficients is characterized.

The product and \\t-expansions converge on the fundamental domain.

Abstract

We determine the action of the Hecke operators \(T_{\mathfrak{p},i}\) on the coefficient forms \(g_{1}, \dots, g_{r-1}, g_{r} = \Delta\), and \(h\), which together generate the ring of modular forms for \(\mathrm{GL}(r, \mathbf{F}_{q}[T])\). All these are eigenforms with powers of \(\pi\) as eigenvalues, where \(\pi\) is the monic generator of the prime ideal \(\mathfrak{p}\) of \(\mathbb{F}_{q}[T]\). We further describe the growth of the \(t\)-expansion coefficients of the discriminant function \(\Delta\). It is such that the product expansion of \(\Delta\) as well as the \(t\)-expansion of each modular form converges on the natural fundamental domain for \(\mathrm{GL}(r, \mathbf{F}_{q}[T])\).