A -expansions of Drinfeld modular forms

Abstract

We introduce the notion of Drinfeld modular forms with A-expansions, where instead of the usual Fourier expansion in t n (t being the uniformizer at 'infinity'), parametrized by n ∈ N, we look at expansions in t a , parametrized by a ∈ A = F q [T ]. We construct an infinite family of eigenforms with Aexpansions. 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) the computation of the eigensystems of Drinfeld modular forms with A-expansions; (iii) examples of failure of multiplicity one result, as well as a restrictive multiplicity one result for Drinfeld modular forms with A-expansions; (iv) examples of eigenforms that can be represented as 'non-trivial' products of eigenforms; (v) an extension of a result of Böckle and Pink concerning the Hecke properties of the space of cuspidal modulo double-cuspidal forms for Γ 1 (T ) to the groups GL 2 (F q [T ]) and Γ 0 (T ).

Vectorial Drinfeld modular forms over Tate algebras

“…The connection of VMF with Petrov's special family. Petrov, building on the work of B. López [21], discovered in his thesis (see [33] for the published version) a family of Drinfeld modular forms with special expansions at the cusp at infinity, similar to those discovered by Goss for his Eisenstein series [9, (6.3)], which Petrov dubbed A-expansions ( …”

On hyperderivatives of single-cuspidal Drinfeld modular forms with A-expansions

“…1.3] are precisely the hyperderivatives of the subfamily of single-cuspidal Drinfeld modular forms with A-expansions that remain modular after differentiation. This is a consequence of results scattered in [1] and [8], but has not appeared in a single and concise form.…”

Theta operators, Goss polynomials, and v-adic modular forms

“…In [21], Serre defined p-adic modular forms as p-adic limits of Fourier series of classical modular forms and determined their properties, in particular their behavior under the ϑ-operator. For a fixed finite place v of K, Goss and Vincent recently transferred Serre's definition to the function field setting of v-adic modular forms, and Goss produced families of examples based on work of Petrov [19] (see Theorem 6.3). In §6, we show that v-adic modular forms are invariant under the operators Θ r .…”

“…Using this as a starting point, Goss [14] and Vincent [24] defined v-adic modular forms in the sense of Serre by taking v-adic limits of u-expansions and thus defining v-adic forms as power series in K ⊗ A A v [[u]] (see §5). Goss [14] constructed a family of v-adic forms based on forms with A-expansions due to Petrov [19] (see Theorem 6.3), and Vincent [24] showed that forms for the group Γ 0 (v) ⊆ GL 2 (A) with v-integral u-expansions are also v-adic modular forms. It is natural to ask how Drinfeld modular forms and v-adic forms behave under differentiation, and since we are in positive characteristic it is favorable to use hyperdifferential operators ∂ r z , rather than straight iteration…”

“…For more information and examples on A-expansions the reader is directed to Gekeler [7], López [16,17], and Petrov [19,20].…”