On A-expansions of Drinfeld Modular Forms

Abstract

In this dissertation, we introduce the notion of Drinfeld modular forms with A-expansions, where instead of the usual Fourier expansion in tⁿ (t being the uniformizer at infinity), parametrized by n ∈ N, we look at expansions in tₐ, parametrized by a ∈ A = F(q)[T]. We construct an infinite family of such eigenforms. Drinfeld modular forms with A-expansions have many desirable properties that allow us to explicitly compute the Hecke action. The applications of our results include: (i) various congruences between Drinfeld eigenforms; (ii) interesting relations between the usual Fourier expansions and A-expansions, and resulting recursive relations for special families of forms with A-expansions; (iii) the computation of the eigensystems of Drinfeld modular forms with A-expansions; (iv) many examples of failure of multiplicity one result, as well as a restrictive multiplicity one result for Drinfeld modular forms with A-expansions; (v) the proof of diagonalizability of the Hecke action in 'non-trivial' cases; (vi) examples of eigenforms that can be represented as non-trivial' products of eigenforms; (vii) an extension of a result of Böckle and Pink concerning the Hecke properties of the space of cuspidal modulo double cuspidal forms for Γ₁(T) to the groups GL₂(F(q)[T]) and Γ₀(T).