The integrality and reduction of Drinfeld modular forms of arbitrary rank

We generalize Gekeler's results on the integrality and reductions of Drinfeld modular forms of rank 2. For an integer r≥2 and the polynomial ring Fq[T] over the finite field Fq of order q, we consider the Drinfeld modular forms for GLr(Fq[T]). First we show, for all ranks r, the integrality of certain Drinfeld modular forms called Drinfeld coefficient forms, and we determine relations between their reductions by nonzero prime ideals of Fq[T].