i start reading about modular forms of half-integral weight $k/2$ for $\Gamma' \subset \Gamma_0(4)$. As far as i understand these are holomorphic functions $f\colon \mathbb{H} \rightarrow \mathbb{C}$ which satisfies $$f(z)|[\zeta]_{k/2} = f(z)$$ for all $\zeta \in \widetilde{\Gamma}'$, where $\widetilde{\Gamma}'=\lbrace (\gamma,j(\gamma,z):\gamma \in \Gamma'\rbrace$. Here $j(\gamma,z)= \theta(\gamma z)/\theta(z)$ for the classical Jacobi thera series. And there is allways mentiont that these functions $f$ are holomorphic at the cusps, an easy condition therefor is that for a cusp $\gamma \tau$ the Fourier series of function $f(\gamma \tau)$ has no main part, and so is just a Taylor series.

My questions are: -Did i get it right so far? -Is there a way to write the holom. condition at the cusps by using the slash operator?

Thanks for the help!

p.s. Let the slash operator defined by $f(z)|[\zeta]_{k/2}=f(\alpha z)\phi(z)^{-k}$ for $\zeta \in G:=\lbrace (\alpha,\phi(z)):\alpha \in GL_2(\mathbb{Q}), \phi:\mathbb{H} \rightarrow \mathbb{C}\rbrace$.


1 Answer 1


There is a way to show that, analagous to the integer-weight case, if we write any cusp of $\tilde\Gamma'$ as $s=\alpha^{-1}\infty$, then the ramification index and type of the cusp depends only on the $\tilde\Gamma'$ equivalence class of $s$, and not on our choice of $\alpha$. In particular, this means that computing $f$ at a cusp "makes sense". Holomorphicity basically means that the coefficients in the q-series expansion of $f|[\tilde\alpha]_{k/2}$ is zero for all $n<0$, for all cusps.


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