Coefficients of Drinfeld modular forms and Hecke operators

Abstract

Consider the space of Drinfeld modular forms of fixed weight and type for Γ 0 (n) ⊂ GL 2 (F q [T ]). It has a linear form b n , given by the coefficient of t m+n(q−1) in the power series expansion of a type m modular form at the cusp infinity, with respect to the uniformizer t. It also has an action of a Hecke algebra. Our aim is to study the Hecke module spanned by b 1 . We give elements in the Hecke annihilator of b 1 . Some of them are expected to be nontrivial and such a phenomenon does not occur for classical modular forms. Moreover, we show that the Hecke module considered is spanned by coefficients b n , where n runs through an infinite set of integers. As a consequence, for any Drinfeld Hecke eigenform, we can compute explicitly certain coefficients in terms of the eigenvalues. We give an application to coefficients of the Drinfeld Hecke eigenform h.

On certain coefficients of Drinfeld-Goss eigenforms with power eigenvalues

“…Theorem 2.3). This extends the level 1 case of an analogous result of Armana [1] for forms of types 0 and 1 on 0 (n) for n ∈ A, although the proof and the nature of the resulting formulas are rather different.…”

“…a n u(z) n , a n ∈ C ∞ , ( 1 ) and this expansion determines f uniquely. This expansion is called the u-expansion of f at 'infinity' .…”

On the Atkin Ut-operator for Γ1(t)-invariant Drinfeld cusp forms

“…Hecke operators. Hecke operators on Drinfeld modular forms are formally defined using a double coset decomposition as showed in [1]. Here we will use an equivalent definition employing a simplified notations which will involve just polynomials of A.…”

“…We would like to point out that some authors just declared U t to be the zero map (see, e.g., Böckle [3, Definition 6.5]), while others (like Armana [1] or Goss [15]) included a not trivial U t -operator in the Hecke algebra they worked with. The residue map allows us to define a Hecke action on harmonic cocycles in the following way:…”

Action of Hecke operators on two distinguished Drinfeld modular forms

“…In a recent paper, C. Armana has also obtained results relating eigenvalues to coefficients of Drinfeld modular forms (cf. [1]). (5) imposes a restrictive condition for an eigenform f to admit an expansion of the type f = c 0 + m monic c(m) t(mz) q−1 .…”