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

\(j(\alpha\beta,z)=j(\alpha,\beta z)j(\beta,z)\)

\(j(\gamma, z)^2=\begin{cases} {cz+d} \mbox{ if }d\equiv 1 \pmod{4} \\ -(cz+d) \mbox{ if } d\equiv 3 \pmod{4} \end{cases}\)

 

action

For \(\xi=(\alpha, \phi(z))\) and function \(f\) on the upper half plane

\(f(z)|[\xi]_{k/2}:=f(\alpha z)\phi(z)^{-k}\)

 

 

references

• W. Kohnen, Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271 (1985), 237–268.
• serre-stark_1976.pdf, 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