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

JOURNAL OF NUMBER THEORY

卷:232

Article

Gekeler, Ernst-Ulrich¹ 

[1] Univ Saarland, Fachrichtung Math, Campus E2 4, D-66123 Saarbrucken, Germany

关键词: Drinfeld modular forms;    Eisenstein series;    Compactification of moduli schemes;   

DOI  :  10.1016/j.jnt.2019.04.019

来源: Elsevier

【 摘 要 】

We construct and study a natural compactification (M) over bar (r) (N) of the moduli scheme M-r (N) for rank-r Drinfeld F-q [T]-modules with a structure of level N is an element of F-q [T]. Namely, (M) over bar (r) (N) = Proj Eis(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) over tilde (r) (N). Modular forms are characterized as those holomorphic functions on the Drinfeld space Omega(r) with the right transformation behavior under the congruence subgroup Gamma(N) of Gamma = GL(r, F-q[T]) (weak modular forms) which, along with all their conjugates under Gamma/Gamma(N), are bounded on the natural fundamental domain F for Gamma on Omega(r). (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

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