Coefficients of some distinguished Drinfeld modular forms

Explicit formulas and vanishing conditions for certain coefficients of Drinfeld-Goss Hecke eigenforms

“…In [1] Armana had provided formulas for type 0 and type 1 cusp forms (with level) in terms of power sums of coefficients of the Carlitz module. Those results were extended by Baca and Lopez in [2] by explicitly evaluating those sums (together with some of Gekeler's results from [6,5]) to obtain explict formulas for some of the coefficients of the forms h, ∆ and g q k −1 . Work of the author and Petrov [4] provides another angle on Armana's work, as it was shown that for any type level 1 eigenform with power eigensystem, there is an infinite family of coefficients (coinciding with the ones given by Armana for the type 0 and type 1 cases) which are completely determined by the corresponding eigenvalues.…”

“…Such questions of vanishing and non-vanishing have been an important topic of study in the classical theory of modular forms (see [3,10,11] for example). They were addressed in the Drinfeld setting for certain special modular forms [6,5,1,2]. For instance, Gekeler [6] shows that if a i s i is the expansion of the forms ∆ or g q k −1 (where g q k −1 is the constant multiple of E q k −1 normalized to have leading coefficient 1) with respect to s = t q−1 then a i = 0 implies i ≡ 0, 1 (mod q).…”