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We give an abstract characterization of the Satake compactification of ageneral Drinfeld modular variety. We prove that it exists and is unique up tounique isomorphism, though we do not give an explicit stratification byDrinfeld modular varieties of smaller rank which is also expected. We constructa 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 allnatural morphisms between Drinfeld modular varieties; in particular we defineHecke operators. We give explicit results in some special cases.

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