"종수 3인 지겔 모듈라 형식"의 두 판 사이의 차이
Pythagoras0 (토론 | 기여) (→관련논문) |
Pythagoras0 (토론 | 기여) |
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1번째 줄: | 1번째 줄: | ||
==개요== | ==개요== | ||
− | * | + | * <math>\dim \left (M_k(\Gamma_3)\right )</math>의 생성함수 |
;정리 (Tsuyumine) | ;정리 (Tsuyumine) | ||
− | + | :<math> | |
\sum_{k=0}^{\infty}\dim \left (M_k(\Gamma_3)\right )T^k = \frac{f(T)}{(1-T^4) (1-T^{12})^2 (1-T^{14}) (1-T^{18}) (1-T^{20}) (1-T^{30})} | \sum_{k=0}^{\infty}\dim \left (M_k(\Gamma_3)\right )T^k = \frac{f(T)}{(1-T^4) (1-T^{12})^2 (1-T^{14}) (1-T^{18}) (1-T^{20}) (1-T^{30})} | ||
− | + | </math> | |
여기서 | 여기서 | ||
− | + | :<math> | |
\begin{align} | \begin{align} | ||
f(T) & =1 + T^6 + T^{10} + T^{12} + 3 T^{16} + 2 T^{18} + 2 T^{20} + 5 T^{22} + 4 T^{24} + 5 T^{26} \\ | f(T) & =1 + T^6 + T^{10} + T^{12} + 3 T^{16} + 2 T^{18} + 2 T^{20} + 5 T^{22} + 4 T^{24} + 5 T^{26} \\ | ||
14번째 줄: | 14번째 줄: | ||
& + 5 T^{88} + 4 T^{90} + 5 T^{92} + 2 T^{94} + 2 T^{96} + 3 T^{98} + T^{102} + T^{104} + T^{108} + T^{114} \\ | & + 5 T^{88} + 4 T^{90} + 5 T^{92} + 2 T^{94} + 2 T^{96} + 3 T^{98} + T^{102} + T^{104} + T^{108} + T^{114} \\ | ||
\end{align} | \end{align} | ||
− | + | </math> | |
===테이블=== | ===테이블=== | ||
* Miyawaki, 320p | * Miyawaki, 320p | ||
− | + | :<math> | |
\begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} | \begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} | ||
k & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & \cdots \\ | k & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & \cdots \\ | ||
26번째 줄: | 26번째 줄: | ||
\dim \left (S_k(\Gamma_3)\right ) & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 & 4 & 6 & \cdots \\ | \dim \left (S_k(\Gamma_3)\right ) & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 & 4 & 6 & \cdots \\ | ||
\end{array} | \end{array} | ||
− | + | </math> | |
46번째 줄: | 46번째 줄: | ||
* Nagaoka, Shoyu, and Sho Takemori. “Notes on Theta Series for Niemeier Lattices.” arXiv:1504.06715 [math], April 25, 2015. http://arxiv.org/abs/1504.06715. | * Nagaoka, Shoyu, and Sho Takemori. “Notes on Theta Series for Niemeier Lattices.” arXiv:1504.06715 [math], April 25, 2015. http://arxiv.org/abs/1504.06715. | ||
* Ikeda, Tamotsu. “Pullback of the Lifting of Elliptic Cusp Forms and Miyawaki’s Conjecture.” Duke Mathematical Journal 131, no. 3 (February 15, 2006): 469–97. doi:10.1215/S0012-7094-06-13133-2. | * Ikeda, Tamotsu. “Pullback of the Lifting of Elliptic Cusp Forms and Miyawaki’s Conjecture.” Duke Mathematical Journal 131, no. 3 (February 15, 2006): 469–97. doi:10.1215/S0012-7094-06-13133-2. | ||
− | * Katsurada, Hidenori. "An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree | + | * Katsurada, Hidenori. "An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree <math>3</math>." Nagoya Mathematical Journal 146 (1997): 199-223. |
* Miyawaki, Isao. “Numerical Examples of Siegel Cusp Forms of Degree 3 and Their Zeta-Functions.” Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics 46, no. 2 (1992): 307–39. doi:10.2206/kyushumfs.46.307. | * Miyawaki, Isao. “Numerical Examples of Siegel Cusp Forms of Degree 3 and Their Zeta-Functions.” Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics 46, no. 2 (1992): 307–39. doi:10.2206/kyushumfs.46.307. | ||
* Tsuyumine, Shigeaki. “On Siegel Modular Forms of Degree Three.” American Journal of Mathematics 108, no. 4 (August 1, 1986): 755–862. [http://doi.org/10.2206/kyushumfs.46.307 doi:10.2307/2374517]. | * Tsuyumine, Shigeaki. “On Siegel Modular Forms of Degree Three.” American Journal of Mathematics 108, no. 4 (August 1, 1986): 755–862. [http://doi.org/10.2206/kyushumfs.46.307 doi:10.2307/2374517]. | ||
* Ozeki, M., and T. Washio. "Further table of the Fourier Coefficients of Eisenstein Series of Degree 3." 長崎大学教養部紀要. 自然科学篇 24.2 (1984): 1-20. [https://www.researchgate.net/profile/Michio_Ozeki2/publication/29794462_Further_table_of_the_Fourier_Coefficients_of_Eisenstein_Series_of_Degree_3/links/5609d3c808ae1396914b8abc.pdf pdf] | * Ozeki, M., and T. Washio. "Further table of the Fourier Coefficients of Eisenstein Series of Degree 3." 長崎大学教養部紀要. 自然科学篇 24.2 (1984): 1-20. [https://www.researchgate.net/profile/Michio_Ozeki2/publication/29794462_Further_table_of_the_Fourier_Coefficients_of_Eisenstein_Series_of_Degree_3/links/5609d3c808ae1396914b8abc.pdf pdf] | ||
− | * Ozeki, Michio, and Tadashi Washio. “Table of the Fourier Coefficients of Eisenstein Series of Degree | + | * Ozeki, Michio, and Tadashi Washio. “Table of the Fourier Coefficients of Eisenstein Series of Degree <math>3</math>.” Proceedings of the Japan Academy, Series A, Mathematical Sciences 59, no. 6 (1983): 252–55. doi:10.3792/pjaa.59.252. |
[[분류:정수론]] | [[분류:정수론]] |
2020년 11월 16일 (월) 05:23 기준 최신판
개요
- \(\dim \left (M_k(\Gamma_3)\right )\)의 생성함수
- 정리 (Tsuyumine)
\[ \sum_{k=0}^{\infty}\dim \left (M_k(\Gamma_3)\right )T^k = \frac{f(T)}{(1-T^4) (1-T^{12})^2 (1-T^{14}) (1-T^{18}) (1-T^{20}) (1-T^{30})} \] 여기서 \[ \begin{align} f(T) & =1 + T^6 + T^{10} + T^{12} + 3 T^{16} + 2 T^{18} + 2 T^{20} + 5 T^{22} + 4 T^{24} + 5 T^{26} \\ & + 7 T^{28} + 6 T^{30} + 9 T^{32} + 10 T^{34} + 10 T^{36} + 12 T^{38} + 14 T^{40} + 15 T^{42} + 16 T^{44} + 18 T^{46} \\ & + 18 T^{48} + 19 T^{50} + 21 T^{52} + 19 T^{54} + 21 T^{56} + 21 T^{58} + 19 T^{60} + 21 T^{62} + 19 T^{64} + 18 T^{66} \\ & + 18 T^{68} + 16 T^{70} + 15 T^{72} + 14 T^{74} + 12 T^{76} + 10 T^{78} + 10 T^{80} + 9 T^{82} + 6 T^{84} + 7 T^{86} \\ & + 5 T^{88} + 4 T^{90} + 5 T^{92} + 2 T^{94} + 2 T^{96} + 3 T^{98} + T^{102} + T^{104} + T^{108} + T^{114} \\ \end{align} \]
테이블
- Miyawaki, 320p
\[ \begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} k & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & \cdots \\ \hline \dim \left (M_k(\Gamma_3)\right ) & 1 & 0 & 1 & 1 & 1 & 2 & 4 & 3 & 7 & 8 & 11 & \cdots \\ \hline \dim \left (S_k(\Gamma_3)\right ) & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 3 & 4 & 6 & \cdots \\ \end{array} \]
k=12인 경우
헤케 고유형식
관련된 항목들
매스매티카 파일 및 계산 리소스
관련논문
- Oliver D. King, Cris Poor, Jerry Shurman, David S. Yuen, Using Katsurada's Determination of the Eisenstein Series to Compute Siegel Eigenforms, arXiv:1604.07216 [math.NT], April 25 2016, http://arxiv.org/abs/1604.07216
- Nagaoka, Shoyu, and Sho Takemori. “Notes on Theta Series for Niemeier Lattices.” arXiv:1504.06715 [math], April 25, 2015. http://arxiv.org/abs/1504.06715.
- Ikeda, Tamotsu. “Pullback of the Lifting of Elliptic Cusp Forms and Miyawaki’s Conjecture.” Duke Mathematical Journal 131, no. 3 (February 15, 2006): 469–97. doi:10.1215/S0012-7094-06-13133-2.
- Katsurada, Hidenori. "An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree \(3\)." Nagoya Mathematical Journal 146 (1997): 199-223.
- Miyawaki, Isao. “Numerical Examples of Siegel Cusp Forms of Degree 3 and Their Zeta-Functions.” Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics 46, no. 2 (1992): 307–39. doi:10.2206/kyushumfs.46.307.
- Tsuyumine, Shigeaki. “On Siegel Modular Forms of Degree Three.” American Journal of Mathematics 108, no. 4 (August 1, 1986): 755–862. doi:10.2307/2374517.
- Ozeki, M., and T. Washio. "Further table of the Fourier Coefficients of Eisenstein Series of Degree 3." 長崎大学教養部紀要. 自然科学篇 24.2 (1984): 1-20. pdf
- Ozeki, Michio, and Tadashi Washio. “Table of the Fourier Coefficients of Eisenstein Series of Degree \(3\).” Proceedings of the Japan Academy, Series A, Mathematical Sciences 59, no. 6 (1983): 252–55. doi:10.3792/pjaa.59.252.