What is the space of weakly holomorphic modular forms of weight $\frac12$ for $\text{SL}(2,\mathbb Z)$?
My thoughts as of yet: The derivative $j_l:=(j^l)'$ of the $j$-invariant for $l\in\mathbb N$ is a weakly holomorphic modular form of weight $2$. So $j_l \, /\eta^{3}$ is a weakly holomorphic modular form of weight $\frac12$ for any $l\in\mathbb N$. Since
$$ (j^l)'=-2\pi i\, l \, j^l \frac{E_6}{E_4}$$
and $\Delta\sim \eta^{24}$, we can generalise this to any form $f_{klm}=E_4^k E_6^l \eta^m$, which has weight $\frac12$ if $4k+6l+\frac m2=\frac12$, for $k,l\geq 0$.
- Are there other such forms?
- Can we find such forms given more constraints, such as the asymptotics for $\tau\to i\infty$? I'm in particular interested in weakly holomorphic modular forms $f$ of weight $\frac12$ which behave as $f(\tau)=q^{-\frac18}+…$ as $\tau\to i\infty$. Does such a modular form even exist?
Best Answer
Let $G = [PSL_2(\Bbb{Z}),PSL_2(\Bbb{Z})]$ the commutator subgroup of the modular group, it is known to be a level $12$ congruence subgroup. Let $X_G=G\backslash\Bbb{H}^*$ its modular curve. If $f$ satisfies your definition of weight $1/2$ modular then $\forall \gamma\in PSL_2(\Bbb{Z}), f/\eta(\gamma z)= \psi(\gamma) f/\eta(z)$. Check that $\psi(\gamma\gamma') = \psi(\gamma)\psi(\gamma')$ so $\psi$ is a character $PSL_2(\Bbb{Z})\to \Bbb{C}^*$.
So $G\subset \ker(\psi)$ and $f/\eta\in \Bbb{C}(X_G)$.
Next $\Bbb{C}(X_G)/\Bbb{C}(j)$ is Galois with Galois group $ PSL_2(\Bbb{Z})^{ab} \cong \Bbb{Z}/12\Bbb{Z}$. Let $\chi$ be $\eta^2$'s order $12$ character given in example 2.5.
Being a degree $12$ cyclic extension by Kummer theory $\Bbb{C}(X_G) =\Bbb{C}(j)(h)$ with $h^{12}\in \Bbb{C}(j)$ which is such that $h(\gamma z) = \chi^k(\gamma) h( z)$ for all $\gamma\in PSL_2(\Bbb{Z})$ and for some $\gcd(k,12)=1$.
We get $$\Bbb{C}(X_G) = \bigoplus_{m=0}^{11} h^m\Bbb{C}(j)$$ a direct sum of eigenspaces of a generator of $PSL_2(\Bbb{Z})^{ab}$.
Let $I_m = \{ u(j) \in \Bbb{C}(j)$, $u(j) h^m$ is holomorphic on $\Bbb{H}\}$. It is a $\Bbb{C}[j]$ submodule of $\Bbb{C}(j)$ so $I_m=v_m(j) \Bbb{C}[j]$ for some $v_m(j)\in \Bbb{C}(j)$ that you'll find from the zeros/poles of $h$.
$f/\eta$ is in one of the eigenspaces $\Bbb{C}(j) h^m$, and being holomorphic on $\Bbb{H}$ we get $f/\eta \in h^m I_m$. ie. your space is $$\bigcup_{m=0}^{11} \eta h^m v_m(j) \Bbb{C}[j] $$