16차원 짝수 자기쌍대 격자

개요

주어진 정수 $g\geq 1$와 $n$차원 격자 $L$에 대한 지겔 세타함수
지겔 상반 공간 $\mathcal{H}_g=\{Z\in {\rm Mat}(g,\C)\mid Z=Z^t,\ {\rm Im}(Z)>0\}$에서 다음과 같이 정의

$$\Theta^{(g)}_L(Z)=\sum_{v_1,\,\ldots,\,v_g\in L}e^{2\pi i\,{\rm tr}((v_1,\ldots,v_g)(v_1,\ldots,v_g)^tZ) }.$$

$E_8^2$격자와 $D_{16}^{+}$격자의 세타함수는 아이젠슈타인 급수(Eisenstein series) $E_4^2=E_8$와 같으며 따라서 가중치평균도 이와 같다

$$ E_4^2(\tau)=1+480 q+61920 q^2+1050240 q^3+7926240 q^4+37500480 q^5+\cdots $$

$\Theta^{(4)}_{E_8^2}$, $\Theta^{(4)}_{D_{16}^{+}}$는 $\Gamma_4$에 대한 지겔 모듈라 형식
$\Theta^{(4)}_{E_8^2}-\Theta^{(4)}_{D_{16}^{+}}$는 weight 8인 지겔 cusp 형식으로 Schottky 형식이라 불림

The Schottky problem is the problem of characterizing Jacobians among all abelian varieties.
In 1888, for genus four, Schottky gave a homogeneous polynomial in the theta constants which vanishes on $\mathcal{H}_4$ precisely at the Jacobian points
a proof of this was finally published by Igusa in 1981

Poor, Cris, Nathan C. Ryan, and David S. Yuen. 2009. “LIFTING PUZZLES IN DEGREE FOUR.” Bulletin of the Australian Mathematical Society 80 (01): 65–82. doi:10.1017/S0004972708001317.
Poor, Cris. 1996. “Schottky’s Form and the Hyperelliptic Locus.” Proceedings of the American Mathematical Society 124 (7): 1987–91. doi:10.1090/S0002-9939-96-03312-6.
Poor, Cris. "The hyperelliptic locus." Duke Mathematical Journal 76.3 (1994): 809-884. http://faculty.fordham.edu/poor/files/Hyper93.pdf
Igusa, Jun-Ichi. “On the Irreducibility of Schottky’s Divisor.” Journal of the Faculty of Science, the University of Tokyo. Sect. 1 A, Mathematics 28, no. 3 (February 20, 1982): 531–45.
Igusa, Jun-ichi. 1981. “Schottky’s Invariant and Quadratic Forms.” In E. B. Christoffel, edited by P. L. Butzer and F. Fehér, 352–62. Birkhäuser Basel. http://link.springer.com/chapter/10.1007/978-3-0348-5452-8_24.