"16차원 짝수 자기쌍대 격자"의 두 판 사이의 차이

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(새 문서: ==개요== * $E_8^2$격자와 $D_{16}^{+}$격자 ==세타 급수== * $g=1$의 경우 * $E_8^2$격자와 $D_{16}^{+}$격자의 세타함수는 [[아이젠슈타인 급수(Eisenstein ...)
 
 
(같은 사용자의 중간 판 13개는 보이지 않습니다)
1번째 줄: 1번째 줄:
 
==개요==
 
==개요==
−
* $E_8^2$격자와 $D_{16}^{+}$격자
+
* <math>E_8^2</math>격자와 <math>D_{16}^{+}</math>격자
  
  
−
==세타 급수==
+
==격자의 지겔 세타함수==
−
* $g=1$의 경우
+
* 주어진 정수 <math>g\geq 1</math>와 <math>n</math>차원 격자 <math>L</math>에 대한 [[격자의 지겔 세타 급수|지겔 세타 급수]]
−
* $E_8^2$격자와 $D_{16}^{+}$격자의 세타함수는 [[아이젠슈타인 급수(Eisenstein series)]] $E_4^2=E_8$와 같으며 따라서 가중치평균도 이와 같다
+
* 지겔 상반 공간 <math>\mathcal{H}_g=\{Z\in {\rm Mat}(g,\C)\mid Z=Z^t,\ {\rm Im}(Z)>0\}</math>에서 다음과 같이 정의
 +
:<math>\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) }.</math>
 +
* <math>\Gamma_g:={\rm Sp}(2g,\Z)</math>의 합동부분군 <math>\Gamma</math>에 대해 weight이 <math>n/2</math>인 [[지겔 모듈라 형식]]
 +
===g가 1인 경우===
 +
* <math>E_8^2</math>격자와 <math>D_{16}^{+}</math>격자의 세타함수는 [[아이젠슈타인 급수(Eisenstein series)]] <math>(E_4(\tau))^2=E_8(\tau)</math>와 같으며 따라서 가중치평균도 이와 같다
 +
:<math>
 +
(E_4(\tau))^2=1+480 q+61920 q^2+1050240 q^3+7926240 q^4+37500480 q^5+\cdots
 +
</math>
 +
===g가 4인 경우===
 +
* <math>\Theta^{(4)}_{E_8^2}</math>, <math>\Theta^{(4)}_{D_{16}^{+}}</math>는 <math>\Gamma_4</math>에 대한 지겔 모듈라 형식
 +
* <math>\Theta^{(4)}_{E_8^2}-\Theta^{(4)}_{D_{16}^{+}}</math>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 <math>\mathcal{H}_4</math> 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 <math>T</math>-Duality of Heterotic Strings’. Physical Review B 88, no. 4 (26 July 2013). doi:10.1103/PhysRevB.88.045131.
  
 
==관련된 항목들==
 
==관련된 항목들==
 +
* [[스미스-민코프스키-지겔 질량 공식]]
 +
* [[24차원 짝수 자기쌍대 격자]]
 +
 +
==매스매티카 파일 및 계산 리소스==
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxUkhvXzBvVlZVZVE/edit
 
* https://docs.google.com/file/d/0B8XXo8Tve1cxUkhvXzBvVlZVZVE/edit
  
14번째 줄: 34번째 줄:
  
 
==관련논문==
 
==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. 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.
 
* 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.
  
 
[[분류:정수론]]
 
[[분류:정수론]]

2020년 11월 13일 (금) 03:47 기준 최신판

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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