Drinfeld Quasi-Modular Forms of Higher Level

TL;DR

This paper explores the structure of Drinfeld quasi-modular forms at higher levels, introducing new representations, operators, and explicit Hecke action formulas to advance understanding in the field.

Contribution

It provides polynomial and hyperderivative representations of Drinfeld quasi-modular forms, introduces the double-slash operator, and characterizes Hecke eigenforms at higher levels.

Findings

Representation of forms as polynomials in false Eisenstein series

Introduction of the double-slash operator for Hecke actions

Explicit formulas for Hecke operators on E-expansions

Abstract

We study the structure of the vector space of Drinfeld quasi-modular forms for congruence subgroups. We provide representations as polynomials in the false Eisenstein series with coefficients in the space of Drinfeld modular forms (the $E$-expansion), and, whenever possible, as sums of hyperderivatives of Drinfeld modular forms. \\ Moreover, we introduce and study the double-slash operator, and use it to provide a well-posed definition for Hecke operators on Drinfeld quasi-modular forms. We characterize eigenforms and, for the special case of Hecke congruence subgroups $\Gamma_0(\mathfrak n)$, we give explicit formulas for the Hecke action on $E$-expansions.