Magnetic (quasi-)modular forms

TL;DR

This paper investigates the existence of holomorphic modular forms with integral coefficients whose anti-derivatives also have integral coefficients, exploring recent counterexamples and extending the discussion to transcendental extensions of quasi-modular forms.

Contribution

It provides a systematic study of the arithmetic properties of modular forms with integral coefficients and discusses related transcendental extensions.

Findings

Counterexamples of meromorphic modular forms with integral anti-derivatives

Discussion of transcendental extensions of quasi-modular forms

Analysis of the conjecture on holomorphic modular forms with integral coefficients

Abstract

A (folklore?) conjecture states that no holomorphic modular form $F(\tau)=\sum_{n=1}^\infty a_nq^n\in q\mathbb Z[[q]]$ exists, where $q=e^{2\pi i\tau}$, such that its anti-derivative $\sum_{n=1}^\infty a_nq^n/n$ has integral coefficients in the $q$-expansion. A recent observation of Broadhurst and Zudilin, rigorously accomplished by Li and Neururer, led to examples of meromorphic modular forms possessing the integrality property. In this note we investigate the arithmetic phenomenon from a systematic perspective and discuss related transcendental extensions of the differentially closed ring of quasi-modular forms.