"지겔-아이젠슈타인 급수"의 두 판 사이의 차이

수학노트
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(같은 사용자의 중간 판 2개는 보이지 않습니다)
5번째 줄: 5번째 줄:
  
  
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==$g=2$인 경우==
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==<math>g=2</math>인 경우==
 
* 지겔-아이젠슈타인 시리즈의 푸리에 전개
 
* 지겔-아이젠슈타인 시리즈의 푸리에 전개
−
$$E_w=\sum_{T}a(T)\exp\left(2\pi i \operatorname{Tr}(T\tau)\right)$$
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:<math>E_w=\sum_{T}a(T)\exp\left(2\pi i \operatorname{Tr}(T\tau)\right)</math>
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* $T = \Bigl( {a \atop b/2} \thinspace {b/2 \atop c} \Bigr) \in
+
* <math>T = \Bigl( {a \atop b/2} \thinspace {b/2 \atop c} \Bigr) \in
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M_2({1 \over 2}\Z)$ 대각성분이 정수인 양의 준정부호행렬
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M_2({1 \over 2}\Z)</math> 대각성분이 정수인 양의 준정부호행렬
−
* $\tau=\left(
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* <math>\tau=\left(
 
\begin{array}{cc}
 
\begin{array}{cc}
 
  \tau _1 & z \\
 
  \tau _1 & z \\
 
  z & \tau _2
 
  z & \tau _2
 
\end{array}
 
\end{array}
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\right)$로 두면,
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\right)</math>로 두면,
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$$
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:<math>
 
\operatorname{Tr}(T\tau)=a \tau _1+b z+c \tau _2
 
\operatorname{Tr}(T\tau)=a \tau _1+b z+c \tau _2
−
$$
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</math>
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* $q_i=e^{2\pi i \tau_i}$, $\zeta=e^{2\pi i z}$로 두자
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* <math>q_i=e^{2\pi i \tau_i}</math>, <math>\zeta=e^{2\pi i z}</math>로 두자
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* 판별식 $D = b^2-4ac\leq 0$
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* 판별식 <math>D = b^2-4ac\leq 0</math>
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* 기본판별식 $D_0$, 체 $\Q(\sqrt{D})$의 판별식  
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* 기본판별식 <math>D_0</math>, 체 <math>\Q(\sqrt{D})</math>의 판별식  
 
;정리 ('''[BL2014]''' Thm 3.4)
 
;정리 ('''[BL2014]''' Thm 3.4)
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아이젠슈타인 급수 $E_w$의 푸리에계수 $a(T)$는 다음과 같이 주어진다
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아이젠슈타인 급수 <math>E_w</math>의 푸리에계수 <math>a(T)</math>는 다음과 같이 주어진다
−
$$
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:<math>
 
a(T)=
 
a(T)=
 
\begin{cases}
 
\begin{cases}
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  1, & \text{if $a=b=c=0$}\\  
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  1, & \text{if <math>a=b=c=0</math>}\\  
 
  {-2w \over B_{w}} \sum_{d | \gcd(a,b,c)} d^{w-1} \alpha(D/d^2), & \text{otherwise}
 
  {-2w \over B_{w}} \sum_{d | \gcd(a,b,c)} d^{w-1} \alpha(D/d^2), & \text{otherwise}
 
\end{cases}
 
\end{cases}
−
$$
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</math>
−
여기서 $B_{k}$는 [[베르누이 수]]이고 $\alpha$는  
+
여기서 <math>B_{k}</math>는 [[베르누이 수]]이고 <math>\alpha</math>는  
−
$$
+
:<math>
 
\alpha(D) =
 
\alpha(D) =
 
\begin{cases}  
 
\begin{cases}  
−
  1, & \text{if $D=0$}\\  
+
  1, & \text{if <math>D=0</math>}\\  
−
  0, & \text{if $D$ is not a discriminant} \\  
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  0, & \text{if <math>D</math> is not a discriminant} \\  
 
  {1 \over \zeta(3-2w)} C(w-1,D), & \text{otherwise}
 
  {1 \over \zeta(3-2w)} C(w-1,D), & \text{otherwise}
 
\end{cases}
 
\end{cases}
−
$$
+
</math>
−
여기서 $C$는 [[코헨 함수]]로 다음과 같이 주어진다
+
여기서 <math>C</math>는 [[코헨 함수]]로 다음과 같이 주어진다
−
$$
+
:<math>
 
C(s-1,D) = L_{D_0}(2-s) \sum_{d \mid f} \mu(d) \left(\frac{D_0}{d}\right) d^{s-2} \sigma_{2s-3}(f/d), \qquad\qquad D = D_0 f^2.
 
C(s-1,D) = L_{D_0}(2-s) \sum_{d \mid f} \mu(d) \left(\frac{D_0}{d}\right) d^{s-2} \sigma_{2s-3}(f/d), \qquad\qquad D = D_0 f^2.
−
$$
+
</math>
−
이 때, $\zeta$는 리만제타함수, $L_{D_0}$는 이차수체에 대한 디리클레 L-함수, $\mu$는 뫼비우스 함수, $\left(\frac{\cdot}{\cdot}\right)$는 자코비 부호,  
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이 때, <math>\zeta</math>는 리만제타함수, <math>L_{D_0}</math>는 이차수체에 대한 디리클레 L-함수, <math>\mu</math>는 뫼비우스 함수, <math>\left(\frac{\cdot}{\cdot}\right)</math>는 자코비 부호,  
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$\sigma_n(x)$는 $x$의 약수의 n차 거듭제곱의 합.
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<math>\sigma_n(x)</math>는 <math>x</math>의 약수의 n차 거듭제곱의 합.
  
  
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===$E_4$의 푸리에 계수===
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===<math>E_4</math>의 푸리에 계수===
−
$$
+
:<math>
 
\begin{array}{c|c|c|c|c|c|c|c|c|c|c}
 
\begin{array}{c|c|c|c|c|c|c|c|c|c|c}
 
  T & \left(
 
  T & \left(
107번째 줄: 107번째 줄:
 
  q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta^2} & \frac{q_1 q_2}{\zeta} & q_1 q_2 & q_1 q_2 \zeta & q_1 q_2 \zeta^2
 
  q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta^2} & \frac{q_1 q_2}{\zeta} & q_1 q_2 & q_1 q_2 \zeta & q_1 q_2 \zeta^2
 
\end{array}
 
\end{array}
−
$$
+
</math>
  
  
−
===$E_6$의 푸리에 계수===
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===<math>E_6</math>의 푸리에 계수===
−
$$
+
:<math>
 
\begin{array}{c|c|c|c|c|c|c|c|c|c|c}
 
\begin{array}{c|c|c|c|c|c|c|c|c|c|c}
 
  T & \left(
 
  T & \left(
169번째 줄: 169번째 줄:
 
  q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta^2} & \frac{q_1 q_2}{\zeta} & q_1 q_2 & q_1 q_2 \zeta & q_1 q_2 \zeta^2
 
  q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta^2} & \frac{q_1 q_2}{\zeta} & q_1 q_2 & q_1 q_2 \zeta & q_1 q_2 \zeta^2
 
\end{array}
 
\end{array}
−
$$
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</math>
  
  
−
===$E_8$의 푸리에 계수===
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===<math>E_8</math>의 푸리에 계수===
−
$$
+
:<math>
 
\begin{array}{c|c|c|c|c|c|c|c|c|c|c}
 
\begin{array}{c|c|c|c|c|c|c|c|c|c|c}
 
  T & \left(
 
  T & \left(
231번째 줄: 231번째 줄:
 
  q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta ^2} & \frac{q_1 q_2}{\zeta } & q_1 q_2 & \zeta  q_1 q_2 & \zeta ^2 q_1 q_2
 
  q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta ^2} & \frac{q_1 q_2}{\zeta } & q_1 q_2 & \zeta  q_1 q_2 & \zeta ^2 q_1 q_2
 
\end{array}
 
\end{array}
−
$$
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</math>
 
===테이블===
 
===테이블===
 
* [[종수 2인 지겔 모듈라 형식]] 항목 참조
 
* [[종수 2인 지겔 모듈라 형식]] 항목 참조
262번째 줄: 262번째 줄:
 
* Yang, Tonghai. “An Explicit Formula for Local Densities of Quadratic Forms.” Journal of Number Theory 72, no. 2 (1998): 309–56. doi:10.1006/jnth.1998.2258.
 
* Yang, Tonghai. “An Explicit Formula for Local Densities of Quadratic Forms.” Journal of Number Theory 72, no. 2 (1998): 309–56. doi:10.1006/jnth.1998.2258.
 
* Walling, Lynne H. “Explicit Siegel Theory: An Algebraic Approach.” Duke Mathematical Journal 89, no. 1 (1997): 37–74. doi:10.1215/S0012-7094-97-08903-1.
 
* Walling, Lynne H. “Explicit Siegel Theory: An Algebraic Approach.” Duke Mathematical Journal 89, no. 1 (1997): 37–74. doi:10.1215/S0012-7094-97-08903-1.
−
* Katsurada, Hidenori. "An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree $3$." Nagoya Mathematical Journal 146 (1997): 199-223.
+
* 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.
 
* Kitaoka, Yoshiyuki. 1986. “Local Densities of Quadratic Forms and Fourier Coefficients of Eisenstein Series.” Nagoya Mathematical Journal 103: 149–60.
 
* Kitaoka, Yoshiyuki. 1986. “Local Densities of Quadratic Forms and Fourier Coefficients of Eisenstein Series.” Nagoya Mathematical Journal 103: 149–60.
−
* 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.
+
* 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.
 
* Kudla, Stephen S. "Seesaw dual reductive pairs." Automorphic forms of several variables (Katata, 1983) 46 (1983): 244-268.
 
* Kudla, Stephen S. "Seesaw dual reductive pairs." Automorphic forms of several variables (Katata, 1983) 46 (1983): 244-268.
  
 
[[분류:정수론]]
 
[[분류:정수론]]
 
[[분류:목록]]
 
[[분류:목록]]

2020년 11월 16일 (월) 05:20 기준 최신판

개요


\(g=2\)인 경우

  • 지겔-아이젠슈타인 시리즈의 푸리에 전개

\[E_w=\sum_{T}a(T)\exp\left(2\pi i \operatorname{Tr}(T\tau)\right)\]

  • \(T = \Bigl( {a \atop b/2} \thinspace {b/2 \atop c} \Bigr) \in M_2({1 \over 2}\Z)\) 대각성분이 정수인 양의 준정부호행렬
  • \(\tau=\left( \begin{array}{cc} \tau _1 & z \\ z & \tau _2 \end{array} \right)\)로 두면,

\[ \operatorname{Tr}(T\tau)=a \tau _1+b z+c \tau _2 \]

  • \(q_i=e^{2\pi i \tau_i}\), \(\zeta=e^{2\pi i z}\)로 두자
  • 판별식 \(D = b^2-4ac\leq 0\)
  • 기본판별식 \(D_0\), 체 \(\Q(\sqrt{D})\)의 판별식
정리 ([BL2014] Thm 3.4)

아이젠슈타인 급수 \(E_w\)의 푸리에계수 \(a(T)\)는 다음과 같이 주어진다 \[ a(T)= \begin{cases} 1, & \text{if \(a=b=c=0\]}\\ 

{-2w \over B_{w}} \sum_{d | \gcd(a,b,c)} d^{w-1} \alpha(D/d^2), & \text{otherwise}

\end{cases} \) 여기서 \(B_{k}\)는 베르누이 수이고 \(\alpha\)는 \[ \alpha(D) = \begin{cases} 1, & \text{if \(D=0\]}\\ 

0, & \text{if <math>D\) is not a discriminant} \\ 
{1 \over \zeta(3-2w)} C(w-1,D), & \text{otherwise}

\end{cases} </math> 여기서 \(C\)는 코헨 함수로 다음과 같이 주어진다 \[ C(s-1,D) = L_{D_0}(2-s) \sum_{d \mid f} \mu(d) \left(\frac{D_0}{d}\right) d^{s-2} \sigma_{2s-3}(f/d), \qquad\qquad D = D_0 f^2. \] 이 때, \(\zeta\)는 리만제타함수, \(L_{D_0}\)는 이차수체에 대한 디리클레 L-함수, \(\mu\)는 뫼비우스 함수, \(\left(\frac{\cdot}{\cdot}\right)\)는 자코비 부호, \(\sigma_n(x)\)는 \(x\)의 약수의 n차 거듭제곱의 합.


\(E_4\)의 푸리에 계수

\[ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} T & \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 2 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 2 \end{array} \right) & \left( \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & -\frac{1}{2} \\ -\frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right) \\ \hline a(T) & 1 & 240 & 240 & 2160 & 2160 & 240 & 13440 & 30240 & 13440 & 240 \\ \hline q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta^2} & \frac{q_1 q_2}{\zeta} & q_1 q_2 & q_1 q_2 \zeta & q_1 q_2 \zeta^2 \end{array} \]


\(E_6\)의 푸리에 계수

\[ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} T & \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 2 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 2 \end{array} \right) & \left( \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & -\frac{1}{2} \\ -\frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right) \\ \hline a(T) & 1 & -504 & -504 & -16632 & -16632 & -504 & 44352 & 166320 & 44352 & -504 \\ \hline q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta^2} & \frac{q_1 q_2}{\zeta} & q_1 q_2 & q_1 q_2 \zeta & q_1 q_2 \zeta^2 \end{array} \]


\(E_8\)의 푸리에 계수

\[ \begin{array}{c|c|c|c|c|c|c|c|c|c|c} T & \left( \begin{array}{cc} 0 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 2 & 0 \\ 0 & 0 \end{array} \right) & \left( \begin{array}{cc} 0 & 0 \\ 0 & 2 \end{array} \right) & \left( \begin{array}{cc} 1 & -1 \\ -1 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & -\frac{1}{2} \\ -\frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & \frac{1}{2} \\ \frac{1}{2} & 1 \end{array} \right) & \left( \begin{array}{cc} 1 & 1 \\ 1 & 1 \end{array} \right) \\ \hline a(T) & 1 & 480 & 480 & 61920 & 61920 & 480 & 26880 & 175680 & 26880 & 480 \\ \hline q & 1 & q_1 & q_2 & q_1^2 & q_2^2 & \frac{q_1 q_2}{\zeta ^2} & \frac{q_1 q_2}{\zeta } & q_1 q_2 & \zeta q_1 q_2 & \zeta ^2 q_1 q_2 \end{array} \]

테이블

메모


관련된 항목들


매스매티카 파일 및 계산 리소스

관련논문

  • Ikeda, Tamotsu, and Hidenori Katsurada. “Explicit Formula for the Siegel Series of a Half-Integral Matrix over the Ring of Integers in a Non-Archimedian Local Field.” arXiv:1602.06617 [Math], February 21, 2016. http://arxiv.org/abs/1602.06617.
  • Walling, Lynne H. ‘Hecke Eigenvalues and Relations for Siegel Eisenstein Series of Arbitrary Degree, Level, and Character’. arXiv:1412.4588 [math], 15 December 2014. http://arxiv.org/abs/1412.4588.
  • [BL2014] Bröker, Reinier, and Kristin Lauter. 2014. “Evaluating Igusa Functions.” Mathematics of Computation. doi:10.1090/S0025-5718-2014-02816-0.
  • Pantchichkine, Alexei. 2012. “Analytic Constructions of P-Adic L-Functions and Eisenstein Series.” arXiv:1204.3878 [math], April. http://arxiv.org/abs/1204.3878.
  • Kudla, Stephen S. "Some extensions of the Siegel-Weil formula." Eisenstein series and applications. Birkhäuser Boston, 2008. 205-237.
  • King, Oliver. 2003. “A Mass Formula for Unimodular Lattices with No Roots.” Mathematics of Computation 72 (242): 839–63. doi:10.1090/S0025-5718-02-01455-2.
  • Katsurada, Hidenori. "An explicit formula for Siegel series." American journal of mathematics (1999): 415-452.
  • Shimura, Goro. “The Number of Representations of an Integer by a Quadratic Form.” Duke Mathematical Journal 100, no. 1 (1999): 59–92. doi:10.1215/S0012-7094-99-10002-0.
  • Yang, Tonghai. “An Explicit Formula for Local Densities of Quadratic Forms.” Journal of Number Theory 72, no. 2 (1998): 309–56. doi:10.1006/jnth.1998.2258.
  • Walling, Lynne H. “Explicit Siegel Theory: An Algebraic Approach.” Duke Mathematical Journal 89, no. 1 (1997): 37–74. doi:10.1215/S0012-7094-97-08903-1.
  • Katsurada, Hidenori. "An explicit formula for the Fourier coefficients of Siegel-Eisenstein series of degree \(3\)." Nagoya Mathematical Journal 146 (1997): 199-223.
  • Kitaoka, Yoshiyuki. 1986. “Local Densities of Quadratic Forms and Fourier Coefficients of Eisenstein Series.” Nagoya Mathematical Journal 103: 149–60.
  • 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.
  • Kudla, Stephen S. "Seesaw dual reductive pairs." Automorphic forms of several variables (Katata, 1983) 46 (1983): 244-268.
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