Drinfeld Quasi-Modular Forms of Higher Level

TL;DR

This work builds a comprehensive algebraic framework for Drinfeld quasi-modular forms on congruence subgroups by introducing two complementary structure theories: an $E$-expansion that expresses each form as a unique polynomial in the false Eisenstein series $E$ with modular coefficients, and a hyperderivative decomposition that represents forms as sums of derivatives of modular forms (under a non-vanishing binomial hypothesis). Central to the development is the associated polynomial $P_f$, which encodes depth and mediates between $E$-expansions and derivative expansions; the authors also define a robust double-slash operator and use it to formulate well-defined Hecke operators, along with explicit actions in the $ Gamma_0(rak m)$ case. The paper proves that hyperderivatives commute with Hecke operators, enabling a clear eigenform theory for quasi-modular forms in terms of modular forms of lower weight and type. These results yield a structurally rich parallel to the classical Kaneko–Zagier framework in positive characteristic and provide concrete tools for analyzing eigenforms, degeneracy maps, and Atkin–Lehner actions in the function-field setting.

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 $Γ_0(\mathfrak n)$, we give explicit formulas for the Hecke action on $E$-expansions.

Theorems & Definitions (85)

• Theorem 1.1: Theorem \ref{['t:StrutPolE']}
• Theorem 1.2: Theorem \ref{['Structure2']}
• Theorem 1.3: Corollary \ref{['c:EigenDer']}
• Theorem 1.4: Corollary \ref{['c:UpQuasiMod']}
• Definition 2.1
• Proposition 2.2
• proof
• Definition 2.3
• Remark 2.4
• Lemma 2.5