종수 2인 지겔 모듈라 형식

개요

• \(E_k\)는 weight \(k\)인 지겔-아이젠슈타인 급수
• cusp form \(X_{10},X_{12},X_{35}\)
• \(E_4, E_6, X_{10},X_{12}, X_{35}\)는 \(\mathbb{C}\)-algebra \(M(\Gamma_2)\)를 생성

테이블

• Igusa

\[ \begin{array}{c|c|c|c|c|c|c|c|c|c|c|c} k & 0 & 2 & 4 & 6 & 8 & 10 & 12 & 14 & 16 & 18 & 20 & \cdots \\ \hline \dim \left (M_k(\Gamma_2)\right ) & 1 & 0 & 1 & 1 & 1 & 2 & 3 & 2 & 4 & 4 & 5 & \cdots \\ \hline \dim \left (S_k(\Gamma_2)\right ) &0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 2 & 2 & 3 & \cdots \\ \end{array} \]

• https://oeis.org/A029143
• Miyawaki, 320p

생성원

• \(x_{10}=E_4E_6-E_{10}\), weight 10 cusp form
• \(x_{12}=441E_4^3+250E_6^2-691E_{12}\), weight 12 cusp form
• \(X_{10},X_{12}\)는 \(a(X_{k};(1,1,1))=1\)를 만족하는 \(x_{10},x_{12}\)의 상수배, 즉

\[ \begin{aligned} X_{10}&=\zeta q_1 q_2+\frac{q_1 q_2}{\zeta }-2 q_1 q_2+\cdots \\ X_{12}&=\zeta q_1 q_2+\frac{q_1 q_2}{\zeta }+10 q_1 q_2+\cdots \end{aligned} \]

• There exists a weight 35 cusp form \(X_{35}\);we normalize \(X_{35}\) so that \(a(X_{35};(2,-1,3))=1\)
• \(E_4, E_6, X_{10},X_{12}\) are algebraically independent over \(\mathbb{C}\)
• \(E_4, E_6, X_{10},X_{12}, X_{35}\) have integral Fourier coefficients

테이블

행렬 \(M_T=\left( \begin{array}{cc} a & \frac{b}{2} \\ \frac{b}{2} & c \\ \end{array} \right)\)를 \(T=\{a,b,c\}\)로 나타내었다

\begin{array}{c|ccccccc} T & a\left(E_4;T\right) & a\left(E_6;T\right) & a\left(E_8;T\right) & a\left(E_{10};T\right) & a\left(E_{12};T\right) & a\left(X_{10};T\right) & a\left(X_{12};T\right)\\ \hline \{0,0,0\} & 1 & 1 & 1 & 1 & 1 & 0 & 0 \\ \{0,0,1\} & 240 & -504 & 480 & -264 & \frac{65520}{691} & 0 & 0 \\ \{1,0,0\} & 240 & -504 & 480 & -264 & \frac{65520}{691} & 0 & 0 \\ \{0,0,2\} & 2160 & -16632 & 61920 & -135432 & \frac{134250480}{691} & 0 & 0 \\ \{1,-2,1\} & 240 & -504 & 480 & -264 & \frac{65520}{691} & 0 & 0 \\ \{1,-1,1\} & 13440 & 44352 & 26880 & \frac{227244864}{43867} & \frac{22266840960}{53678953} & 1 & 1 \\ \{1,0,1\} & 30240 & 166320 & 175680 & \frac{2626026480}{43867} & \frac{456798756960}{53678953} & -2 & 10 \\ \{1,1,1\} & 13440 & 44352 & 26880 & \frac{227244864}{43867} & \frac{22266840960}{53678953} & 1 & 1 \\ \{1,2,1\} & 240 & -504 & 480 & -264 & \frac{65520}{691} & 0 & 0 \\ \{2,0,0\} & 2160 & -16632 & 61920 & -135432 & \frac{134250480}{691} & 0 & 0 \\ \{1,-2,2\} & 30240 & 166320 & 175680 & \frac{2626026480}{43867} & \frac{456798756960}{53678953} & -2 & 10 \\ \{1,-1,2\} & 138240 & 2128896 & 6727680 & \frac{306175997952}{43867} & \frac{162868282536960}{53678953} & -16 & -88 \\ \{1,0,2\} & 181440 & 3792096 & 15914880 & \frac{950818774752}{43867} & \frac{661522702800960}{53678953} & 36 & -132 \\ \{1,1,2\} & 138240 & 2128896 & 6727680 & \frac{306175997952}{43867} & \frac{162868282536960}{53678953} & -16 & -88 \\ \{1,2,2\} & 30240 & 166320 & 175680 & \frac{2626026480}{43867} & \frac{456798756960}{53678953} & -2 & 10 \\ \{2,-2,1\} & 30240 & 166320 & 175680 & \frac{2626026480}{43867} & \frac{456798756960}{53678953} & -2 & 10 \\ \{2,-1,1\} & 138240 & 2128896 & 6727680 & \frac{306175997952}{43867} & \frac{162868282536960}{53678953} & -16 & -88 \\ \{2,0,1\} & 181440 & 3792096 & 15914880 & \frac{950818774752}{43867} & \frac{661522702800960}{53678953} & 36 & -132 \\ \{2,1,1\} & 138240 & 2128896 & 6727680 & \frac{306175997952}{43867} & \frac{162868282536960}{53678953} & -16 & -88 \\ \{2,2,1\} & 30240 & 166320 & 175680 & \frac{2626026480}{43867} & \frac{456798756960}{53678953} & -2 & 10 \\ \{2,-4,2\} & 2160 & -16632 & 61920 & -135432 & \frac{134250480}{691} & 0 & 0 \\ \{2,-3,2\} & 138240 & 2128896 & 6727680 & \frac{306175997952}{43867} & \frac{162868282536960}{53678953} & -16 & -88 \\ \{2,-2,2\} & 604800 & 24881472 & 225388800 & \frac{29960190114624}{43867} & \frac{46765376055216000}{53678953} & 240 & 2784 \\ \{2,-1,2\} & 967680 & 65995776 & 953948160 & \frac{199267181955072}{43867} & \frac{486707206711864320}{53678953} & -240 & -8040 \\ \{2,0,2\} & 1239840 & 90644400 & 1461833280 & \frac{345545694370800}{43867} & \frac{958912407409188960}{53678953} & 32 & 17600 \\ \{2,1,2\} & 967680 & 65995776 & 953948160 & \frac{199267181955072}{43867} & \frac{486707206711864320}{53678953} & -240 & -8040 \\ \{2,2,2\} & 604800 & 24881472 & 225388800 & \frac{29960190114624}{43867} & \frac{46765376055216000}{53678953} & 240 & 2784 \\ \{2,3,2\} & 138240 & 2128896 & 6727680 & \frac{306175997952}{43867} & \frac{162868282536960}{53678953} & -16 & -88 \\ \{2,4,2\} & 2160 & -16632 & 61920 & -135432 & \frac{134250480}{691} & 0 & 0 \\ \end{array}

weight 12

\begin{array}{c|ccc} T & a\left(E_4^3;T\right) & a\left(X_{12};T\right) & a\left(Y_{12};T\right) \\ \hline \{0,0,0\} & 1 & 0 & 0 \\ \{0,0,1\} & 720 & 0 & 1 \\ \{1,0,0\} & 720 & 0 & 1 \\ \{0,0,2\} & 179280 & 0 & -24 \\ \{1,-2,1\} & 720 & 0 & 1 \\ \{1,-1,1\} & 40320 & 1 & 116 \\ \{1,0,1\} & 436320 & 10 & 1206 \\ \{1,1,1\} & 40320 & 1 & 116 \\ \{1,2,1\} & 720 & 0 & 1 \\ \{2,0,0\} & 179280 & 0 & -24 \\ \{1,-2,2\} & 436320 & 10 & 1206 \\ \{1,-1,2\} & 19768320 & -88 & 22176 \\ \{1,0,2\} & 88672320 & -132 & 115236 \\ \{1,1,2\} & 19768320 & -88 & 22176 \\ \{1,2,2\} & 436320 & 10 & 1206 \\ \{2,-2,1\} & 436320 & 10 & 1206 \\ \{2,-1,1\} & 19768320 & -88 & 22176 \\ \{2,0,1\} & 88672320 & -132 & 115236 \\ \{2,1,1\} & 19768320 & -88 & 22176 \\ \{2,2,1\} & 436320 & 10 & 1206 \\ \{2,-4,2\} & 179280 & 0 & -24 \\ \{2,-3,2\} & 19768320 & -88 & 22176 \\ \{2,-2,2\} & 757296000 & 2784 & -36960 \\ \{2,-1,2\} & 7503805440 & -8040 & -2919840 \\ \{2,0,2\} & 15579220320 & 17600 & -2736144 \\ \{2,1,2\} & 7503805440 & -8040 & -2919840 \\ \{2,2,2\} & 757296000 & 2784 & -36960 \\ \{2,3,2\} & 19768320 & -88 & 22176 \\ \{2,4,2\} & 179280 & 0 & -24 \\ \end{array}

메모

• Heim, Bernhard, and Atsushi Murase. "Borcherds lifts on Sp2 (Z)." Geometry and Analysis of Automorphic Forms of Several Variables, Proceedings of the International Symposium in Honor of Takayuki Oda on the Occasion of his 60th Birthday. 2011. http://webdoc.sub.gwdg.de/ebook/serien/e/mpi_mathematik/2010/2010_074.pdf
• http://math.shinshu-u.ac.jp/~nu/html/sage/days/201310/doc/takemori/sagedays_slide.pdf

관련된 항목들

• 종수 3인 지겔 모듈라 형식
• 24차원 짝수 자기쌍대 격자
• 지겔-아이젠슈타인 급수

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxWUlELXdZTTNyS1U/edit

관련논문

• Dickson, Martin, Ameya Pitale, Abhishek Saha, and Ralf Schmidt. “Explicit Refinements of B"ocherer’s Conjecture for Siegel Modular Forms of Squarefree Level.” arXiv:1512.07204 [math], December 22, 2015. http://arxiv.org/abs/1512.07204.
• Kikuta, Toshiyuki, and Sho Takemori. “Sturm Bounds for Siegel Modular Forms of Degree 2 and Odd Weights.” arXiv:1508.01610 [math], August 7, 2015. http://arxiv.org/abs/1508.01610.
• McCarthy, Dermot. ‘Multiplicative Relations for Fourier Coefficients of Degree 2 Siegel Eigenforms’. arXiv:1505.07049 [math], 26 May 2015. http://arxiv.org/abs/1505.07049.
• 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.
• Pollack, Aaron, and Shrenik Shah. ‘On the Rankin-Selberg Integral of Kohnen and Skoruppa’. arXiv:1410.7870 [math], 28 October 2014. http://arxiv.org/abs/1410.7870.
• Vinberg, E. 2013 “On the Algebra of Siegel Modular Forms of Genus 2.” Transactions of the Moscow Mathematical Society 74: 1–13. doi:10.1090/S0077-1554-2014-00217-X.
• Ryan, Nathan, Nils-Peter Skoruppa, and Fredrik Strömberg. ‘Numerical Computation of a Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two’. Mathematics of Computation 81, no. 280 (2012): 2361–76. doi:10.1090/S0025-5718-2012-02584-1.
• Skoruppa, Nils-Peter. ‘Computations of Siegel Modular Forms of Genus Two’. Mathematics of Computation 58, no. 197 (1992): 381–98. doi:10.1090/S0025-5718-1992-1106982-0.
• Kohnen, W., and N.-P. Skoruppa. ‘A Certain Dirichlet Series Attached to Siegel Modular Forms of Degree Two’. Inventiones Mathematicae 95, no. 3 (1 October 1989): 541–58. doi:10.1007/BF01393889.
• Igusa, Jun-Ichi. 1962. “On Siegel Modular Forms of Genus Two.” American Journal of Mathematics 84 (1): 175. doi:10.2307/2372812.