Nearly holomorphic Drinfeld modular forms and their special values at CM points

TL;DR

This paper introduces nearly holomorphic Drinfeld modular forms, explores their special values at CM points, and investigates their algebraic independence and structure, extending the theory of Drinfeld modular forms.

Contribution

It defines nearly holomorphic Drinfeld modular forms, studies their special values at CM points, and analyzes the structure of related quasi-modular forms and their algebraic properties.

Findings

Special values at CM points are algebraically independent for distinct endomorphism algebras.

The structure of vector spaces and algebras generated by Drinfeld quasi-modular forms is characterized.

Introduction of nearly holomorphic Drinfeld modular forms and Maass-Shimura operators in this setting.

Abstract

In the present paper, we introduce the notion of nearly holomorphic Drinfeld modular forms and study an analogue of Maass-Shimura operators in this context. Furthermore, for a given nearly holomorphic Drinfeld modular form, we show that its special values at CM points are algebraically independent whenever the associated endomorphism algebras are distinct. As an application of our results on nearly holomorphic Drinfeld modular forms, we study Drinfeld quasi-modular forms for arbitrary congruence subgroups and investigate the structure of the vector spaces and the algebras generated by them.