The Analytic Theory of Vectorial Drinfeld Modular Forms

In this text we generalize the notion of Drinfeld modular form for the group Γ := GL2(Fq [θ]) to a vector-valued setting, where the target spaces are certain modules over positive characteristic Banach algebras over which are defined what we call the ’representations of the first kind’. Under quite reasonable restrictions, we show that the spaces of such modular forms are finite-dimensional, endowed with certain generalizations of Hecke operators, with differential operators ` a la Serre etc. The crucial point of this work is the introduction of a ’field of uniformizers’, a field extension of the valued field of formal Laurent series C∞((u)) where u is the usual uniformizer for Drinfeld modular forms, in which we can study the expansions at the cusp infinity of our modular forms and which is wildly ramified and not discretely valued. Examples of such modular forms are given through the construction of Poincar´ e and Eisenstein series. After the discussion of these fundamental properties, the paper continues with a more detailed analysis of the special case of modular forms associated to a re- stricted class of representations ρ∗ ! of Γ which has more importance in arithmetical applications. More structure results are given in this case, and a harmonic product formula is obtained which allows, with the help of conjectures on the structure of an Fp-algebra of A-periodic multiple sums, multiple Eisenstein series etc., to produce conjectural formulas for Eisenstein series. Other properties such as integrality of coe!cients of Eisenstein series, specialization at roots of unity etc. are included as well.