Compactification of Drinfeld Moduli Spaces as Moduli Spaces of $A$-Reciprocal Maps and Consequences for Drinfeld Modular Forms

Richard Pink

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arXiv · 1903.02432

Compactification of Drinfeld Moduli Spaces as Moduli Spaces of $A$-Reciprocal Maps and Consequences for Drinfeld Modular Forms

Richard Pink

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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 $A={\mathbb F}_q[t]$ 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 the same results in general, but the proof will require more ideas.

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BibTeX RIS

Richard Pink. 2019-03-06. Compactification of Drinfeld Moduli Spaces as Moduli Spaces of $A$-Reciprocal Maps and Consequences for Drinfeld Modular Forms. https://arxiv.org/abs/1903.02432

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