Abstract We determine the action of the Hecke operators T , i T p, i on the coefficient forms g₁, , gₑ-₁, gₑ = g 1, …, g r - 1, g r = Δ, and h, which together generate the ring of modular forms for {\, GL\, } (r, FₐT) 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 p p of FₐT 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 {\, GL\, } (r, FₐT) GL (r, F q T).
Ernst-Ulrich Gekeler (Wed,) studied this question.