A note on transcendence of special values of functions related to modularity

TL;DR

This paper investigates the algebraic and transcendental properties of special values of modular and quasi-modular functions with algebraic Fourier coefficients, unifying and extending classical results in the field.

Contribution

It provides a unified framework for understanding the arithmetic nature of these special values and generalizes key theorems by Schneider, Nesterenko, and others.

Findings

Unified approach to values of modular functions with algebraic Fourier coefficients

Generalizations of classical theorems on transcendence and algebraic independence

Broader applicability to arbitrary congruence subgroups

Abstract

In this note, we study the arithmetic nature of values of modular functions, meromorphic modular forms and meromorphic quasi-modular forms with respect to arbitrary congruence subgroups, that have algebraic Fourier coefficients. This approach unifies many of the known results, and leads to generalizations of the theorems of Schneider, Nesterenko and others.