On Drinfeld modular forms of higher rank IV: Modular forms with level

Abstract

We construct and study a natural compactification M‾r(N) of the moduli scheme Mr(N) for rank-r Drinfeld Fq[T]-modules with a structure of level N∈Fq[T]. Namely, M‾r(N)=ProjEis(N), the projective variety associated with the graded ring Eis(N) generated by the Eisenstein series of rank r and level N. We use this to define the ring Mod(N) of all modular forms of rank r and level N. It equals the integral closure of Eis(N) in their common quotient field F˜r(N). Modular forms are characterized as those holomorphic functions on the Drinfeld space Ωr with the right transformation behavior under the congruence subgroup Γ(N) of Γ=GL(r,Fq[T]) (“weak modular forms”) which, along with all their conjugates under Γ/Γ(N), are bounded on the natural fundamental domain F for Γ on Ωr.