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 $n\in \mathbb{N}$, we look at expansions in $t_a$, parametrized by $a \in A = \Fq [T]$. We construct an infinite family of eigenforms with $A$-expansions. 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 ${\Gamma }_1(T)$ to the groups $\text{GL}_2 (\Fq [T])$ and ${\Gamma }_0(T)$.

This version does not use normalized Goss polynomials, which fixes various inaccuracies in the previous version

References in corpus (3)

• A-expansions of Drinfeld modular forms
• tau-recurrent sequences and modular forms
• Relations between multizeta values in characteristic p

Cited by in corpus (11)

• A-expansions of Drinfeld modular forms
• Drinfeld modular forms of arbitrary rank, Part III: Examples
• $\wp$-adic continuous families of Drinfeld eigenforms of finite slope
• On the structure and slopes of Drinfeld cusp forms
• Vectorial Drinfeld modular forms over Tate algebras
• Notes on Atkin-Lehner theory for Drinfeld modular forms
• On the action of Hecke operators on Drinfeld modular forms
• A Hecke-equivariant decomposition of spaces of Drinfeld cusp forms via representation theory, and an investigation of its subfactors
• The analytic theory of vectorial Drinfeld modular forms
• Theta operators, Goss polynomials, and v-adic modular forms
• A construction of $\mathfrak{v}$-adic modular forms