Drinfeld Quasi-Modular Forms of Higher Level

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 $\hat{I}{“}_0(\mathfrak{n})$, we give explicit formulas for the Hecke action on $E$-expansions.

Additional results concerning associated polynomials, as well as Hecke operators and their eigenvalues, have been included. Furthermore, the notion of being holomorphic at infinity for a quasi-modular function is discussed in detail