This monograph provides a foundation for the theory of Drinfeld modular forms
of arbitrary rank r and is subdivided into three chapters. In the first chapter, we
develop the analytic theory. Most of the work goes into defining and studying the
u-expansion of a weak Drinfeld modular form, whose coefficients are weak Drinfeld
modular forms of rank r − 1. Based on that we give a precise definition of when a
weak Drinfeld modular form is holomorphic at infinity and thus a Drinfeld modular
form in the proper sense.
In the second chapter, we compare the analytic theory with the algebraic one
that was begun in a paper of the third author. For any arithmetic congruence
subgroup and any integral weight we establish an isomorphism between the space
of analytic modular forms with the space of algebraic modular forms defined in
terms of the Satake compactification. From this we deduce the important result
that this space is finite dimensional.
In the third chapter, we construct and study some examples of Drinfeld modular
forms. In particular, we define Eisenstein series, as well as the action of Hecke
operators upon them, coefficient forms and discriminant forms. In the special case
A = Fq[t] we show that all modular forms for GLr(Γ(t)) are generated by certain
weight one Eisenstein series, and all modular forms for GLr(A) and SLr(A) are
generated by certain coefficient forms and discriminant forms. We also compute
the dimensions of the spaces of such modular forms

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Includes Index and Notes