Are arbitrary $\eta$-quotients weakly holomorphic half-integral weight modular forms?

Question

Let $\eta(z) = q^{1/24} \prod_{n=1}^\infty (1-q^n)$ be the usual Dedekind $\eta$-function, $q = e^{2\pi i z}$, and let an arbitrary $\eta$-quotient be $$f(q) = \prod_{\delta \in \mathbb{N}} \eta(\delta z)^{r_\delta}$$ with $r_\delta \in \mathbb{Z}$, all but finitely many entries of the sequence $(r_\delta)$ being zero.

A standard theorem of Gordon, Hughes, and Newman gives conditions on $(r_\delta)$ under which $f(q)$ is a weakly holomorphic modular form of weight $k$, level $N$, and character $\chi$, including giving $k$, $N$, and $\chi$.

After years of struggle (I don't know why the subject seems so impenetrable to me) I think I can claim some facility with using this theorem, along with the theorem of Ligozat (to bound the order at cusps), the standard fact that $M_k \cdot M_\ell = M_{k+\ell}$ for weakly holomorphic modular forms of integral weight (to get rid of cusps by appropriate multiplications), and Sturm's bounds on the resulting holomorphic forms (for congruences) to do the usual combinatorial work I need with these.

Currently, however, I find myself staring down some $\eta$-quotients which I believe are half-integral weight, which are of course an entirely different and much angrier kettle of fish. I have the following rather fundamental questions:

1.) Is it true that, for arbitrary $(r_\delta)$, $f(q)$ is a weakly holomorphic modular form of half-integral weight? I have heard this in non-rigorous settings but I'm not certain of it.

2.) If not, is there a half-integral weight equivalent to the theorem of Gordon, Hughes and Newman which will tell me conditions under which it is?

3.) Once it is, will that theorem, or another, tell me in what space $M_{k+1/2}(\Gamma_0(N),\chi)$ I might find $f(q)$?

These don't seem to be in the usual textbooks on modular forms which I have readily accessible to consult. Thanks for any help.

Answer 1

Let $$f(z) = \prod_{j=1}^{2k} \eta(a_j z), a_j \in \Bbb{Z}_{\ge 1}$$

$\eta(z)^2 \in M_1(\Gamma_1(24))$ thus $\eta(a_j z)^2 $ is weight $1$ modular for $\Gamma_1(a_j 24)$

so that $g(z)=f(z)^2$ is weight $2k$ modular for $\Gamma_1(N),N=24\prod_{j=1}^{2k} a_j$ ie. $(cz+d)^{-2k}g(\gamma(z))=g|_{2k}\gamma(z) = g(z)$ so that

$(cz+d)^{-k} f(\gamma(z))=f|_k\gamma(z) = \psi(\gamma) f(z)$ with $\psi(\gamma)= \pm 1$

Then $\psi(\beta \gamma) f(z)=f|_k\beta\gamma(z) =(f|_k\gamma)|_k\beta(z)= \psi(\beta)\psi(\gamma) f(z)$

thus $\psi$ is a character $\Gamma_1(N) \to \pm 1$,

$\Gamma=\ker(\psi)$ is of index $1$ or $2$,

the main point is to show $\Gamma$ is a congruence subgroup (not sure how, from $[\Gamma_1(N):\Gamma]=2,\Gamma \supset [\Gamma_1(N),\Gamma_1(N)]$ and the metaplectic double cover coming from $\eta$, or directly from the weight $2$ Eisenstein series expression for $\log f$)

once it is done $\Gamma$ contains $\alpha \Gamma_1(M)\alpha^{-1}$ for some $\alpha \in SL_2(\Bbb{Z}),M$

and $f$ is weight $k$ modular for $\Gamma$, analytic (with no zeros) on the upper half-plane and with poles at the cusps.

There is a formula for the order of the poles at the cusps but it involves complicated multiplicative sums.

From there it is clear in what space you'll find the weight $k+1/2$ versions $f(z)\eta(a_{2k+1}z)$ and the quotients of such eta products.