How do modular forms of half-integral weight relate to the (quasi-modular) Eisenstein series?

Question

The Eisenstein series $$ G_{2k} = \sum_{(m,n) \neq (0,0)} \frac{1}{(m + n\tau)^{2k}} $$ are modular forms (if $k>1$) of weight $2k$ and quasi-modular if $k=1$. It is clear that given modular forms $f,g$ of weight $2k$ and $2\ell$ that $f\cdot g$ is a modular form of weight $2(k + \ell)$.

We can also define modular forms of half-integral weight if we are a little more careful. However, the functional equation $$ f\Big(\frac{az + b}{cz+d}\Big) = (cz+d)^{2k}f(z) $$ must be replaced with something more subtle.

In particular, the Dedekind $\eta$-function is a modular form of weight 1/2; it satisfies $$ \eta(z + 1) = e^{\frac{\pi i}{12}}\eta(z) \qquad \eta\Big(-\frac{1}{z}\Big) = \sqrt{-iz}\ \eta(z) $$ Now, it would be nice if $\eta^4$ were a modular form of weight 2; however, an easy check using the above relations shows that this is only the case up to roots of unity, and so $\eta^4$ is not a multiple of $G_2$.

My question is then the following: What is the relation between modular forms of half-integral weight and (quasi-)modular forms of even integer weight? I know that $\eta^{24}$ is an honest modular form of weight 12, so I'm more curious about the general setting, or even what can be said about things like $\eta^4$.

Edit: As was pointed out in the comments, $\eta^4$ is a modular form for a congruence subgroup of $SL_2(\mathbb{Z})$, but not for the full modular group. The space of modular forms for the full modular group is generated by $G_4$ and $G_6$ (with $G_2$ thrown in if we are looking at the space of quasi-modular forms); is there a corresponding statement for modular forms on congruence subgroups?

Answer 1

One of the best-known relations between modular forms of half integral weight and even weight is the Shimura correspondence taking certain weight k+1/2 forms to weight 2k forms. See Koblitz's book or Shimura's paper for details.

By the way, η⁴ is indeed a modular form of weight 2 for the whole modular group SL₂(Z), for a nontrivial character of this group. For half-integral weight forms one needs to use a character of the metaplectic double cover in the functional equation; this is what the funny roots of unity in the functional equation of η are doing.

Answer 2

To answer Scott; yes, every modular form of integer weight $k$ for a congruence group can be expressed rationally in terms of $\eta(r\tau)$ for rational $r$. For a start, $g=f/\eta^{2k}$ is a modular function for some $\Gamma(N)$, so it suffices to consider these. Replacing $g$ by $g(N\tau)$ we can reduce to the case where $g$ is a modular function for some $\Gamma_0(M)$. The field of modular functions for $\Gamma_0(M)$ is generated by $j(\tau)$ and $j(M\tau)$. So all we need is that $j(\tau)$ expressible in terms of the $\eta(r\tau)$. This is well-known; one can express $j$ in terms of the Weber functions, and them in terms of $\eta(\tau)$, $\eta(2\tau)$ and $\eta(\tau/2)$.

Note that this argument is essentially the same as the proof for a very similar theorem in this paper by Kilford.

Answer 3

Regarding the question you added later, the ring of modular forms for a congruence group is finitely generated, but for large level it can be computationally difficult to find explicit generators. I believe you can always write the generators as rational functions in $\eta(r\tau)$ for $r$ ranging over rational numbers, but it is not clear to me that this is a useful method of presentation.