Modular forms for \(\mathrm{GL}(r, \mathbb{F}_{q}[T])\): Hecke operators and growth of expansion coefficients
Ernst-Ulrich Gekeler
TL;DR
The work analyzes higher-rank Drinfeld modular forms for GL$(r,A)$, showing the ring of modular forms is generated by the coefficient forms $g_1,\\dots,g_{r-1},\\Delta$ and the root $h$ of $\\Delta$, with all these forms being eigenforms for the Hecke operators $T_{\\mathfrak{p},i}$ whose eigenvalues are powers of the uniformizer $\\pi$. It develops a lattice-based Hecke-action framework, including oriented lattices and a boundary splitting that yields explicit $t$-expansion and $A$-expansion formulas, and proves these expansions converge uniformly on the fundamental domain. The paper provides detailed growth bounds for $t$-expansion coefficients of $\\Delta$ (and related forms), and establishes that the Hecke action preserves modular-type data and can be computed inductively from rank $r-1$. These results illuminate the structure of coefficient forms in higher rank and connect Drinfeld-module data to explicit analytic expansions with uniform convergence on the natural domain, enriching the arithmetic of function-field modular forms. The findings have implications for extending $A$-expansions and for understanding Hecke eigenstructure in the nonclassical, higher-rank setting.
Abstract
We determine the action of the Hecke operators \(T_{\mathfrak{p},i}\) on the coefficient forms \(g_{1}, \dots, g_{r-1}, g_{r} = Δ\), 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 \(π\) as eigenvalues, where \(π\) 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 \(Δ\). 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 \(\mathrm{GL}(r, \mathbf{F}_{q}[T])\).