Magnetic (quasi-)modular forms

Understand

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.