A-expansions of Drinfeld modular forms

TL;DR

This paper introduces a new type of Drinfeld modular forms with $A$-expansions, enabling explicit Hecke action computations and revealing novel properties, congruences, and counterexamples in the theory of modular forms over function fields.

Contribution

It constructs an infinite family of eigenforms with $A$-expansions and explores their properties, including Hecke actions, congruences, and multiplicity phenomena, extending previous results to new groups.

Findings

Explicit computation of Hecke actions for $A$-expansion forms

Examples of congruences and multiplicity one failures

Construction of eigenforms as products of eigenforms

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 B\"{o}ckle 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 $\Gamma_0(T)$.