"Half-integral weight modular forms"의 두 판 사이의 차이
| 5번째 줄: | 5번째 줄: | ||
| − | <math>\Gamma_0(4)</math> | + | <math>\Gamma_0(4)</math> |
generated by <math>-I, T, ST^{-4}S</math> | generated by <math>-I, T, ST^{-4}S</math> | ||
| 35번째 줄: | 35번째 줄: | ||
| − | <h5> | + | <h5>action</h5> |
| − | + | For <math>\xi=(\alpha, \phi(z))</math> and function <math>f</math> on the upper half plane | |
| − | + | <math>f(z)|[\xi]_{k/2}:=f(\alpha z)\phi(z)^{-k}</math> | |
| − | |||
| − | |||
| − | + | <h5>references</h5> | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | [http://www.springerlink.com/content/u5k773288424205q/ Fourier coefficients of modular forms of half-integral weight] | + | * W. Kohnen, Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271 (1985), 237–268. |
| + | * [[3413025/attachments/1586151|serre-stark_1976.pdf]], Modular functions of one variable VI | ||
| + | * [http://www.springerlink.com/content/u5k773288424205q/ Fourier coefficients of modular forms of half-integral weight] | ||
Inventiones Mathematicae<br> Volume 87, Number 2 / 1987년 6월 | Inventiones Mathematicae<br> Volume 87, Number 2 / 1987년 6월 | ||
Henryk Iwaniec | Henryk Iwaniec | ||
2010년 1월 19일 (화) 16:05 판
\(\Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} {*} & {*} \\ 0 & {*} \end{pmatrix} \pmod{N} \right\}\)
\(\Gamma_0(4)\)
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