$N$-th root of modular forms

Question

In my research, I have encountered many $q$-series of functions that turn out to be the Fourier expansions of roots of modular forms. Examples are $n$-th roots of Eisenstein series and the $j$-invariant. These seem to be related in certain instances to noncongruence groups. I would be very grateful for recommendations on the literature. Thanks!

Answer 1

Functions that are roots of modular forms are pretty mysterious objects, but nonetheless are studied in a variety of different ways.

The major entrypoint into these sorts of functions are "modular forms of half-integral weight". These are morally "square roots" of modular forms. A modular form $\widetilde{f}$ of half-integral weight is a function such that $\widetilde{f}^2$ is a regular modular form. The classical example is the theta function $$\theta(z) = \sum_{n \in \mathbb{Z}} e^{2 \pi i n^2 z},$$ which is a modular form of weight $1/2$ on $\Gamma_0(4)$.

Actually defining these modular forms is a delicate matter, since one wants to make sure that the square root plays nicely. Specifically, such an $\widetilde{f}$ should still have locally nice properties, so at every point one might ask which square root to take (and there are right and wrong answers). This was the subject of Shimura's paper that I reference below. A substantial amount of work has been done in this field. I've also included a citation for Kohnen's seminar work below.

These are both the tip of an iceberg, as it is possible to consider more general fractional weight modular forms. These are usually thought of as modular forms on metaplectic covers of $\mathrm{GL}(n)$ (so, for instance, a modular form of half-integral weight is a metaplectic modular form on the double cover of $\mathrm{GL}(2)$). Unfortunately, I do not know of a readable reference for these general forms, but following citation trails beginning with either Kohnen's work or searches based on "metaplectic modular forms" or "modular forms on metaplectic covers" will get you to the edge very quickly.

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References

Shimura, Goro. "Modular forms of half integral weight." Modular Functions of One Variable I. Springer, Berlin, Heidelberg, 1973. 57-74.

Kohnen, Winfried. "Modular forms of half-integral weight on Γ 0 (4)." Mathematische Annalen 248.3 (1980): 249-266.