지겔-아이젠슈타인 급수

개요

아이젠슈타인 급수(Eisenstein series)의 일반화
지겔 모듈라 형식의 예

$g=2$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 $$ \alpha(D) = \begin{cases} 1, & \text{if $D=0$}\\ 0, & \text{if $D$ is not a discriminant} \\ {1 \over \zeta(3-2w)} C(w-1,D), & \text{otherwise} \end{cases} $$ 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

종수 2인 지겔 모듈라 형식 항목 참조

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

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.