Half-integral weight modular forms
\(\Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} {*} & {*} \\ 0 & {*} \end{pmatrix} \pmod{N} \right\}\)
\(\Gamma_0(4)\)
generated by \(-I, T, ST^{-4}S\)
Define
\(\epsilon_d = \begin{cases} 1 \mbox{ if }d\equiv 1 \pmod{4} \\i \mbox{ if } d\equiv 3 \pmod{4} \end{cases}\)
\(\sqrt z\) has branch in \((-\pi/2, \pi/2]\)
Define
\(j(\gamma, z)=(\frac{c}{d})\epsilon_d^{-1}\sqrt{cz+d}\) for \(\gamma \in \Gamma_0(4)\)
Check
\(j(\alpha\beta,z)=j(\alpha,\beta z)j(\beta,z)\)
\(j(\gamma, z)^2=\begin{cases} {cz+d} \mbox{ if }d\equiv 1 \pmod{4} \\ -(cz+d) \mbox{ if } d\equiv 3 \pmod{4} \end{cases}\)
W. Kohnen, Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271 (1985),
237–268.
Modular functions of one variable VI
Fourier coefficients of modular forms of half-integral weight
Inventiones Mathematicae
Volume 87, Number 2 / 1987년 6월
Henryk Iwaniec