Half-integral weight modular forms

introduction

• modular forms of weight 1/2, which were classified by Serre & Stark (1977)

\(\Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} {*} & {*} \\ 0 & {*} \end{pmatrix} \pmod{N} \right\}\)

\(\Gamma_0(4)\)

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

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

related items

• Kohnen-Waldspurger formula
• Shimura correspondence

expositions

• Notes on modular forms of half-integral weight http://wwwf.imperial.ac.uk/~buzzard/maths/research/notes/modular_forms_of_half_integral_weight.pdf
• Funke, Jens. "CM points and weight 3/2 modular forms." Analytic Number Theory (2007): 107. https://www.maths.dur.ac.uk/~dma0jf/G-D-proceedings-funke.pdf

articles

• Chen, Bin, and Jie Wu. “Non-Vanishing and Sign Changes of Hecke Eigenvalues for Half-Integral Weight Cusp Forms.” arXiv:1512.08400 [math], December 28, 2015. http://arxiv.org/abs/1512.08400.
• Lau, Yuk-Kam, Emmanuel Royer, and Jie Wu. “Sign of Fourier Coefficients of Modular Forms of Half Integral Weight.” arXiv:1507.00518 [math], July 2, 2015. http://arxiv.org/abs/1507.00518.
• http://www.worldscientific.com/doi/abs/10.1142/S1793042110003484
• http://link.springer.com/article/10.1007%2Fs00013-013-0492-5
• serre-stark_1976.pdf, Modular functions of one variable VI
• Fourier coefficients of modular forms of half-integral weight
  • Henryk Iwaniec, Inventiones Mathematicae, Volume 87, Number 2 / 1987년 6월
• Fourier coefficients of modular forms of half-integral weight
  • W. Kohnen, Math. Ann. 271 (1985), 237–268.