"Half-integral weight modular forms"의 두 판 사이의 차이
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imported>Pythagoras0 |
Pythagoras0 (토론 | 기여) (Pythagoras0 (토론)의 48551판 편집을 되돌림) 태그: 편집 취소 |
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| (사용자 2명의 중간 판 7개는 보이지 않습니다) | |||
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* modular forms of weight 1/2, which were classified by [http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFSerreStark1977 Serre & Stark (1977)] | * modular forms of weight 1/2, which were classified by [http://en.wikipedia.org/wiki/Mock_theta_function#CITEREFSerreStark1977 Serre & Stark (1977)] | ||
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<math>\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\}</math> | <math>\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\}</math> | ||
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<math>\Gamma_0(4)</math> | <math>\Gamma_0(4)</math> | ||
| − | generated | + | generated by <math>-I, T, ST^{-4}S</math> |
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Define | Define | ||
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<math>\epsilon_d = \begin{cases} 1 \mbox{ if }d\equiv 1 \pmod{4} \\i \mbox{ if } d\equiv 3 \pmod{4} \end{cases}</math> | <math>\epsilon_d = \begin{cases} 1 \mbox{ if }d\equiv 1 \pmod{4} \\i \mbox{ if } d\equiv 3 \pmod{4} \end{cases}</math> | ||
| − | <math>\sqrt z</math> | + | <math>\sqrt z</math> has branch in <math>(-\pi/2, \pi/2]</math> |
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Define | Define | ||
| − | <math>j(\gamma, z)=(\frac{c}{d})\epsilon_d^{-1}\sqrt{cz+d}</math> | + | <math>j(\gamma, z)=(\frac{c}{d})\epsilon_d^{-1}\sqrt{cz+d}</math> for <math>\gamma \in \Gamma_0(4)</math> |
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Check | Check | ||
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<math>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}</math> | <math>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}</math> | ||
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==action== | ==action== | ||
| − | + | 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> | <math>f(z)|[\xi]_{k/2}:=f(\alpha z)\phi(z)^{-k}</math> | ||
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==unary theta functions of weight 1/2== | ==unary theta functions of weight 1/2== | ||
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==theta functions of weight 3/2== | ==theta functions of weight 3/2== | ||
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| + | ==related items== | ||
| + | * [[Kohnen-Waldspurger formula]] | ||
| + | * [[Shimura correspondence]] | ||
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==expositions== | ==expositions== | ||
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==articles== | ==articles== | ||
| − | * [[3413025/attachments/1586151|serre-stark_1976.pdf]], | + | * 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 | ||
| + | * [[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] | ||
| − | ** Henryk Iwaniec, | + | ** Henryk Iwaniec, Inventiones Mathematicae, Volume 87, Number 2 / 1987년 6월 |
* [http://www.springerlink.com/content/p52527460724p36m/ Fourier coefficients of modular forms of half-integral weight] | * [http://www.springerlink.com/content/p52527460724p36m/ Fourier coefficients of modular forms of half-integral weight] | ||
| − | ** W. Kohnen, | + | ** W. Kohnen, Math. Ann. 271 (1985), 237–268. |
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[[분류:개인노트]] | [[분류:개인노트]] | ||
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[[분류:math and physics]] | [[분류:math and physics]] | ||
[[분류:math]] | [[분류:math]] | ||
[[분류:automorphic forms]] | [[분류:automorphic forms]] | ||
| + | [[분류:migrate]] | ||
2020년 12월 28일 (월) 04:49 기준 최신판
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
- 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.