Compactification of Drinfeld modular varieties and Drinfeld Modular Forms of Arbitrary Rank

TL;DR

This paper characterizes the Satake compactification of Drinfeld modular varieties, proves its existence and uniqueness, and constructs a sheaf whose sections correspond to Drinfeld modular forms, also exploring their behavior under morphisms.

Contribution

It provides an abstract, general construction of the Satake compactification for Drinfeld modular varieties and links it to modular forms via a natural sheaf.

Findings

Existence and uniqueness of the Satake compactification.

Construction of a sheaf whose sections are modular forms.

Analysis of modular forms and compactification under morphisms.

Abstract

We give an abstract characterization of the Satake compactification of a general Drinfeld modular variety. We prove that it exists and is unique up to unique isomorphism, though we do not give an explicit stratification by Drinfeld modular varieties of smaller rank which is also expected. We construct a natural ample invertible sheaf on it, such that the global sections of its $k$-th power form the space of (algebraic) Drinfeld modular forms of weight $k$. We show how the Satake compactification and modular forms behave under all natural morphisms between Drinfeld modular varieties; in particular we define Hecke operators. We give explicit results in some special cases.