Journal of Algebraic Geometry

Journal of Algebraic Geometry

Online ISSN 1534-7486; Print ISSN 1056-3911



Compactification of Drinfeld moduli spaces as moduli spaces of $ A$-reciprocal maps and consequences for Drinfeld modular forms

Author: Richard Pink
Journal: J. Algebraic Geom.
Published electronically: December 17, 2020
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Abstract: We construct a compactification of the moduli space of Drinfeld modules of rank $ r$ and level $ N$ as a moduli space of $ A$-reciprocal maps. This is closely related to the Satake compactification but not exactly the same. The construction involves some technical assumptions on $ N$ that are satisfied for a cofinal set of ideals $ N$. In the special case where $ A=\mathbb{F}_q[t]$ and $ N=(t^n)$, we obtain a presentation for the graded ideal of Drinfeld cusp forms of level $ N$ and all weights and can deduce a dimension formula for the space of cusp forms of any weight. We expect similar results in general, but the proof will require more ideas.

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Richard Pink
Affiliation: Department of Mathematics, ETH Zürich, 8092 Zürich, Switzerland

Received by editor(s): March 7, 2019
Received by editor(s) in revised form: January 30, 2020, and May 14, 2020
Published electronically: December 17, 2020
Dedicated: In memory of David Goss