종수 2인 지겔 모듈라 형식

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Pythagoras0 (토론 | 기여)님의 2016년 8월 2일 (화) 02:19 판 (→‎테이블)
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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

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

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}

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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.
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