On the partial derivatives of Drinfeld modular forms of arbitrary rank

TL;DR

This paper develops a generalized differential operator for Drinfeld modular forms of arbitrary rank, extending previous work in rank two, and studies the algebraic structure of these forms under differentiation.

Contribution

It introduces an analogue of the Serre derivation for arbitrary rank Drinfeld modular forms and proves the stability of the associated algebra under partial derivatives.

Findings

Generalized Serre derivation for arbitrary rank Drinfeld modular forms

Construction of a finitely generated algebra containing all such forms

Proof of stability of this algebra under partial derivatives

Abstract

In this paper, we obtain an analogue of the Serre derivation acting on the product of spaces of Drinfeld modular forms which generalizes the differential operator introduced by Gekeler in the rank two case. We further introduce a finitely generated algebra $\mathcal{M}_r$ containing all the Drinfeld modular forms for the full modular group and show its stability under the partial derivatives.