"Half-integral weight modular forms"의 두 판 사이의 차이
imported>Pythagoras0 |
imported>Pythagoras0 |
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| 73번째 줄: | 73번째 줄: | ||
==articles== | ==articles== | ||
* http://arxiv.org/abs/1507.00518 | * 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 | ||
* [[3413025/attachments/1586151|serre-stark_1976.pdf]], Modular functions of one variable VI | * [[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] | * [http://www.springerlink.com/content/u5k773288424205q/ Fourier coefficients of modular forms of half-integral weight] | ||
2015년 7월 2일 (목) 21:25 판
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
\(f(z)|[\xi]_{k/2}:=f(\alpha z)\phi(z)^{-k}\)
unary theta functions of weight 1/2
theta functions of weight 3/2
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
- 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.