16차원 짝수 자기쌍대 격자

수학노트
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개요

  • \(E_8^2\)격자와 \(D_{16}^{+}\)격자


격자의 지겔 세타함수

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

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

g가 1인 경우

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

g가 4인 경우

  • \(\Theta^{(4)}_{E_8^2}\), \(\Theta^{(4)}_{D_{16}^{+}}\)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

메모

  • 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
  • Piazza, Francesco Dalla, Davide Girola, and Sergio L. Cacciatori. “Classical Theta Constants vs. Lattice Theta Series, and Super String Partition Functions.” Journal of High Energy Physics 2010, no. 11 (November 1, 2010): 1–24. doi:10.1007/JHEP11(2010)082.
  • Plamadeala, Eugeniu, Michael Mulligan, and Chetan Nayak. ‘Short-Range Entangled Bosonic States with Chiral Edge Modes and \(T\)-Duality of Heterotic Strings’. Physical Review B 88, no. 4 (26 July 2013). doi:10.1103/PhysRevB.88.045131.

관련된 항목들

매스매티카 파일 및 계산 리소스


관련논문

  • Oura, Manabu, Cris Poor, Riccardo Salvati Manni, and David S. Yuen. 2010. “Modular Forms of Weight 8 for Γ G (1, 2).” Mathematische Annalen 346 (2): 477–98. doi:10.1007/s00208-009-0406-9.
  • 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.
  • Kneser, Martin. 1967. “Lineare Relationen zwischen Darstellungsanzahlen quadratischer Formen.” Mathematische Annalen 168 (1): 31–39. doi:10.1007/BF01361543.
  • Igusa, Jun-ichi. "Modular forms and projective invariants." American Journal of Mathematics (1967): 817-855.
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