24차원 짝수 자기쌍대 격자

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Pythagoras0 (토론 | 기여)님의 2015년 6월 17일 (수) 17:08 판 (→‎$g=3$)
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개요

  • $\Gamma\subset \mathbb{R}^{24}$가 짝수 unimodular 격자라 하자
  • 루트로 생성되는 격자 $(\Gamma_2)_{\mathbb{Z}}$는 다음과 같은 24가지 경우만이 가능하다

$$ \begin{aligned} \emptyset &, & A_1^{24} &, & A_2^{12}&,& A_3^8&,& A_4^6&,& A_5^4D_4&,& D_4^6&,&A_6^4\\ A_7^2D_5^2&,&A_8^3&,&A_9^2D_6&,& D_6^4 &,& E_6^4&,&A_{11}D_7E_6&,&A_{12}^2&,&D_8^3 \\ A_{15}D_9 &,& A_{17}E_7&,&D_{10}E_7^2&,&D_{12}^2&,& A_{24}&,&D_{16}E_8&,& E_8^3&,&D_{24} \end{aligned} $$

  • 이 각각의 경우에 해당하는, 24차원 짝수 자기쌍대 격자를 찾을 수 있다

지겔 세타 함수

$g=1$

  • 세타함수는 다음과 같다

$$ \Theta^{(1)}_{\Lambda}=(E_4^{(1)})^3+(24h-720)\Delta $$

$g=2$

$$ \Theta^{(2)}_{\Lambda}=(E_4^{(2)})^3+(24h-720)Y_{12}^{(2)}+(48h^2-2880h+43200)X_{12}^{(2)} $$ 여기서 $$ X_{12}^{(2)}=a_1(E_4^{(2)})^3+a_2(E_6^{(2)})^2+a_3E_{12}^{(2)}\\ Y_{12}^{(2)}=b_1(E_4^{(2)})^3+b_2(E_6^{(2)})^2+b_3E_{12}^{(2)} $$

$g=3$

$$ \Theta^{(3)}_{\Lambda}=(E_4^{(3)})^3+(24h-720)Y_{12}^{(3)}+(48h^2-2880h+43200)X_{12}^{(3)}+(48h^2-288h^2+3144h-1131120)F_{12} $$ 여기서 $$ X_{12}^{(3)}=a_1(E_4^{(3)})^3+a_2(E_6^{(3)})^2+a_3E_{12}^{(3)}+\frac{4740}{337}F_{12}\\ Y_{12}^{(3)}=b_1(E_4^{(3)})^3+b_2(E_6^{(3)})^2+b_3E_{12}^{(3)}-\frac{356411}{337}F_{12} $$ 여기서 $$a_1=\frac{131\times 593}{2^{11} 3^4 5^3 337},a_2=\frac{131\times 593}{2^{10} 3^6 7^2 337}, a_3=-\frac{131\times 593\times 691}{2^{11} 3^6 5^3 7^2 337},$$ $$b_1=\frac{41\times 71\times 109}{2^7 3^3 5^3 337},b_2=\frac{1759}{2^2 3^4 7^2 337}, b_3=-\frac{13\times 593\times 691}{2^7 3^4 5^3 7^2 337}$$

테이블

\begin{array}{ccccc} \text{root system} & \text{Coxeter} & \text{num. of roots} & \text{order of Aut} & \text{factorization} \\ \hline \emptyset & 0 & 0 & 8315553613086720000 & 2^{22}\cdot 3^9\cdot 5^4\cdot 7^2\cdot 11^1\cdot 13^1\cdot 23^1 \\ A_1^{24} & 2 & 48 & 4107449023856640 & 2^{34}\cdot 3^3\cdot 5^1\cdot 7^1\cdot 11^1\cdot 23^1 \\ A_2^{12} & 3 & 72 & 413762786426880 & 2^{19}\cdot 3^{15}\cdot 5^1\cdot 11^1 \\ A_3^8 & 4 & 96 & 295882444505088 & 2^{31}\cdot 3^9\cdot 7^1 \\ A_4^6 & 5 & 120 & 716636160000000 & 2^{22}\cdot 3^7\cdot 5^7 \\ A_5^4 D_4 & 6 & 144 & 2476694568960000 & 2^{26}\cdot 3^{10}\cdot 5^4 \\ D_4^6 & 6 & 144 & 108208436847575040 & 2^{40}\cdot 3^9\cdot 5^1 \\ A_6^4 & 7 & 168 & 15485790781440000 & 2^{19}\cdot 3^9\cdot 5^4\cdot 7^4 \\ A_7^2 D_5^2 & 8 & 192 & 47943914618880000 & 2^{31}\cdot 3^6\cdot 5^4\cdot 7^2 \\ A_8^3 & 9 & 216 & 573416710078464000 & 2^{23}\cdot 3^{13}\cdot 5^3\cdot 7^3 \\ A_9^2 D_6 & 10 & 240 & 1213580338790400000 & 2^{27}\cdot 3^{10}\cdot 5^5\cdot 7^2 \\ D_6^4 & 10 & 240 & 6763027302973440000 & 2^{39}\cdot 3^9\cdot 5^4 \\ E_6^4 & 12 & 288 & 346657985428193280000 & 2^{32}\cdot 3^{17}\cdot 5^4 \\ A_{11} D_7 E_6 & 12 & 288 & 16019260472033280000 & 2^{28}\cdot 3^{11}\cdot 5^4\cdot 7^2\cdot 11^1 \\ A_{12}^2 & 13 & 312 & 155103152174530560000 & 2^{22}\cdot 3^{10}\cdot 5^4\cdot 7^2\cdot 11^2\cdot 13^2 \\ D_8^3 & 14 & 336 & 824788751971516416000 & 2^{43}\cdot 3^7\cdot 5^3\cdot 7^3 \\ A_{15} D_9 & 16 & 384 & 3887340541213409280000 & 2^{31}\cdot 3^{10}\cdot 5^4\cdot 7^3\cdot 11^1\cdot 13^1 \\ A_{17} E_7 & 18 & 432 & 37172693925353226240000 & 2^{27}\cdot 3^{12}\cdot 5^4\cdot 7^3\cdot 11^1\cdot 13^1\cdot 17^1 \\ D_{10} E_7^2 & 18 & 432 & 31316197926418513920000 & 2^{38}\cdot 3^{12}\cdot 5^4\cdot 7^3 \\ D_{12}^2 & 22 & 528 & 1924703466207817236480000 & 2^{43}\cdot 3^{10}\cdot 5^4\cdot 7^2\cdot 11^2 \\ A_{24} & 25 & 600 & 31022420086661971968000000 & 2^{23}\cdot 3^{10}\cdot 5^6\cdot 7^3\cdot 11^2\cdot 13^1\cdot 17^1\cdot 19^1\cdot 23^1 \\ D_{16} E_8 & 30 & 720 & 477676405704303732326400000 & 2^{44}\cdot 3^{11}\cdot 5^5\cdot 7^3\cdot 11^1\cdot 13^1 \\ E_8^3 & 30 & 720 & 2029289625631919702016000000 & 2^{43}\cdot 3^{16}\cdot 5^6\cdot 7^3 \\ D_{24} & 46 & 1104 & 5204698426366666226930810880000 & 2^{45}\cdot 3^{10}\cdot 5^4\cdot 7^3\cdot 11^2\cdot 13^1\cdot 17^1\cdot 19^1\cdot 23^1 \end{array}

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관련논문

  • 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. ‘On Theta Series Attached to the Leech Lattice’. arXiv:1412.7606 [math], 24 December 2014. http://arxiv.org/abs/1412.7606.
  • Nebe, Gabriele, and Boris Venkov. On Siegel Modular Forms of Weight 12. http://www.math.rwth-aachen.de/~Gabriele.Nebe/papers/Siemod12.pdf
  • Borcherds, Richard E., Eberhard Freitag, and Rainer Weissauer. "A Siegel cusp form of degree 12 and weight 12." Journal fur die Reine und Angewandte Mathematik (1998): 141-153.
  • Erokhin, V. A. 1984. “Automorphism Groups of 24-Dimensional Even Unimodular Lattices.” Journal of Soviet Mathematics 26 (3): 1876–79. doi:10.1007/BF01670573.
  • Erokhin, V. A. “Theta-Series of Even Unimodular Lattices.” Journal of Soviet Mathematics 25, no. 2 (April 1, 1984): 1012–20. doi:10.1007/BF01680824.
  • Erokhin, V. A. “Theta-Series of Even Unimodular 24-Dimensional Lattices.” Journal of Soviet Mathematics 17, no. 4 (November 1, 1981): 1999–2008. doi:10.1007/BF01465457.
  • Niemeier, Hans-Volker. 1973. “Definite Quadratische Formen Der Dimension 24 Und Diskriminante 1.” Journal of Number Theory 5 (2): 142–78. doi:10.1016/0022-314X(73)90068-1.
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