I'm wondering if one has a similiar result for modular forms of half-integral weight of a lemma for modular form of integral weight, we will get to a moment. The lemma and all defintions are taken from the book "Introduction to Elliptic Curves and Modular Forms" by Neal Koblitz.

First some definitions

Let $GL_2^{+}(\mathbb{Q})$ denote the subgroup of $GL_2(\mathbb{Q})$ consisting of matrices with positive determinant and $k \in \mathbb{Z}$. Then we define for $\gamma = \begin{pmatrix} a & b \\ c & d\end{pmatrix} \in GL_2^{+}(\mathbb{Q})$ \begin{equation} f(z)|[\gamma]_k = (\det \gamma)^{k/2} (cz+d)^{-k} f(\gamma z). \end{equation}

Let f(z) be a holomorphic function on the upper half-plane $\mathbb{H}$ and let $\Gamma' \subset \Gamma$ be a congruence subgroup of level $N$, i.e., $\Gamma(N) \subset \Gamma'$. Let $k \in \mathbb{Z}$. We call $f(z)$ a modular function of weight $k$ for $\Gamma'$ if $f|[\gamma]_k = f$ for all $\gamma \in \Gamma'$, and if for any $\gamma_0 \in SL_2(\mathbb{Z})$ \begin{equation} f(z)|[\gamma_0]_k \text{ has the form } \sum a(n)\exp((2\pi inz)/N)) \text{ with } a_n = 0 \text{ for } n < 0 \ (1) \end{equation}

Now we get to the lemma mentioned above.

Lemma: Suppose that $f(z)$ has the property $(1)$ for all $\gamma_0 \in SL_2(\mathbb{Z})$. Then $f(z)$ has the same property for all $\alpha \in GL_2^{+}(\mathbb{Q})$, i.e. $f(z)|[\alpha]_k = \sum_{n=0}^{\infty} b_n \exp{((2 \pi inz)/(ND))}$ for some positive integer $D$ which depend on $\alpha$.

Proof: Since $\alpha$ can be multiplied by a positive scalar without affecting $[\alpha]_k$, w.l.o.g we may suppose that $\alpha$ has integer entries. Then, there exists $\gamma_0 \in SL_2(\mathbb{Z})$ such that $\gamma_0^{-1} \alpha = \begin{pmatrix} a & b \\ 0 & d\end{pmatrix}$, where $a$ and $d$ are positive integers. Then

\begin{align}f(z)|[\alpha]_k &= (f(z)|[\gamma_0]_k) | \left[\begin{pmatrix} a & b \\ 0 & d\end{pmatrix}\right]_k = (ad)^{k/2}d^{-k} \sum_{n=0}^{\infty}a(n) \exp{(2\pi i n ((az+b)/d)/N)} \\ & = (a/d)^{k/2} \sum_{n=0}^{\infty} a(n) \exp{(2 \pi i b n /dN)} \exp(2\pi i anz /(Nd)) = (...) \end{align}

Question: Can one obtain a analogous result for modular forms of half-integral weight? The book implies that there is one and I fail to show it (starting to doubt hte claim, too).

If you need definitions regarding modular forms of half-integral weight, I'll add them to this post.


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