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

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