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

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])).