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

2018/11/23 by Gekeler, Ernst-Ulrich · 2 citations
#11F52 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1811.09460

Abstract

We construct and study a natural compactification M^r(N) of the moduli scheme M^r(N) for rank-r Drinfeld \F_q[T]-modules with a structure of level N ∈ \F_q[T]. Namely, M^r(N) = \rm Proj \bf Eis(N), the projective variety associated with the graded ring \bf Eis(N) generated by the Eisenstein series of rank r and level N. We use this to define the ring \bf Mod(N) of all modular forms of rank r and level N. It equals the integral closure of \bf Eis(N) in their common quotient field \widetilde\MF_r(N). Modular forms are characterized as those holomorphic functions on the Drinfeld space \Om^r with the right transformation behavior under the congruence subgroup \Ga(N) of \Ga = \rm GL(r,\F_q[T]) ("weak modular forms") which, along with all their conjugates under \Ga/\Ga(N), are bounded on the natural fundamental domain \BF for \Ga on \Om^r.

Cited by

• On special values of meromorphic Drinfeld modular forms of arbitrary rank at CM points
• Modular forms for \(GL(r, \mathbbF_q[T])\): Hecke operators and growth of expansion coefficients

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