종수 3인 지겔 모듈라 형식

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

• http://siegelmodularforms.org/cgi-bin/degree3/show.pl?k=12&v=versionb&stf=1&spf=1&showMoreCoeffs=1#more

관련된 항목들

• 종수 2인 지겔 모듈라 형식
• 지겔-아이젠슈타인 급수

매스매티카 파일 및 계산 리소스

• http://siegelmodularforms.org/pages/degree3/

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.