"지겔 모듈라 형식"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| (같은 사용자의 중간 판 19개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
==지겔 상반 공간== | ==지겔 상반 공간== | ||
| − | * 지겔 상반 공간 | + | * 지겔 상반 공간 <math>\mathcal{H}_g</math> |
| − | + | :<math> | |
\mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\mathrm{T}}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\} | \mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\mathrm{T}}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\} | ||
| − | + | </math> | |
| − | * 사교군 | + | * 사교군 <math>\Gamma_g:={\rm Sp}(2g,\Z)</math> |
| − | * | + | * <math>\mathcal{A}_g=\mathcal{H}_g/\Gamma_g</math> : moduli space of principally polarized abelian varieties |
==지겔 모듈라 형식== | ==지겔 모듈라 형식== | ||
;정의 | ;정의 | ||
| − | weight이 k이고 genus(또는 degree)가 | + | weight이 k이고 genus(또는 degree)가 <math>g</math>인 지겔 모듈라 형식은 다음 조건을 만족하는 해석함수 <math>f:\mathcal{H}_g\to \mathbb{C}</math>로 정의된다 |
| − | + | :<math> | |
f \left( (A\tau +B)(C\tau + D)^{-1}\right) = \det(C\tau +D)^{k} f(\tau),\, \forall \begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in {\rm Sp}(2g,\Z) | f \left( (A\tau +B)(C\tau + D)^{-1}\right) = \det(C\tau +D)^{k} f(\tau),\, \forall \begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in {\rm Sp}(2g,\Z) | ||
| − | + | </math> | |
| − | + | * [[종수 2인 지겔 모듈라 형식]] | |
| − | + | * [[종수 3인 지겔 모듈라 형식]] | |
==푸리에 전개== | ==푸리에 전개== | ||
| − | * 지겔 모듈라 형식 | + | * 지겔 모듈라 형식 <math>f\in M_k(\Gamma_g)</math>는 다음과 같은 형태의 푸리에 전개를 가진다 |
| − | + | :<math>f(\tau)=\sum_{T}a(T)\exp\left(2\pi i \operatorname{Tr}(T\tau)\right)</math> | |
| − | 여기서 | + | 여기서 <math>T\in \operatorname{Mat}_2(\frac{1}{2}\mathbb{Z})</math>는 대각성분이 정수인 대칭행렬. |
;Kocher 원리 | ;Kocher 원리 | ||
| − | 지겔 모듈라 형식 | + | 지겔 모듈라 형식 <math>f\in M_k(\Gamma_g)</math>의 푸리에 전개에서, <math>T</math>가 positive semi-definite 행렬이 아니면, <math>a(T)=0</math>이다 |
| 33번째 줄: | 33번째 줄: | ||
* [[리만 곡면의 주기 행렬과 겹선형 관계 (bilinear relation)]] | * [[리만 곡면의 주기 행렬과 겹선형 관계 (bilinear relation)]] | ||
* [[사교 행렬]] | * [[사교 행렬]] | ||
| − | * [[ | + | * [[지겔-베유 공식]] |
* [[스미스-민코프스키-지겔 질량 공식]] | * [[스미스-민코프스키-지겔 질량 공식]] | ||
| + | * [[자코비 세타함수와 자코비 형식]] | ||
| 40번째 줄: | 41번째 줄: | ||
* [http://math.berkeley.edu/~reb/papers/siegel/ Tables of coefficients of some Siegel automorphic forms] | * [http://math.berkeley.edu/~reb/papers/siegel/ Tables of coefficients of some Siegel automorphic forms] | ||
* [http://www.eg.bucknell.edu/~ncr006/papers/smf_in_sage.pdf Siegel modular forms in SAGE] | * [http://www.eg.bucknell.edu/~ncr006/papers/smf_in_sage.pdf Siegel modular forms in SAGE] | ||
| − | + | * http://math.shinshu-u.ac.jp/~nu/html/sage/days/201310/doc/takemori/demo.sage | |
==사전 형태의 자료== | ==사전 형태의 자료== | ||
| 51번째 줄: | 52번째 줄: | ||
* Chiera, [http://www.mat.uniroma2.it/ricerca/pre-print/aree/gavarini/yas01-02/3-CHIERA.pdf Some aspects of the theory of theta series] | * Chiera, [http://www.mat.uniroma2.it/ricerca/pre-print/aree/gavarini/yas01-02/3-CHIERA.pdf Some aspects of the theory of theta series] | ||
* Ghitza, 2004, [http://www.ms.unimelb.edu.au/~aghitza/research/An_elementary_introduction_to_Siegel_modular_forms.pdf An elementary introduction to Siegel modular forms] | * Ghitza, 2004, [http://www.ms.unimelb.edu.au/~aghitza/research/An_elementary_introduction_to_Siegel_modular_forms.pdf An elementary introduction to Siegel modular forms] | ||
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| 67번째 줄: | 60번째 줄: | ||
* Maass, [http://www.math.tifr.res.in/~publ/ln/tifr03.pdf Lectures on Siegel's Modular Functions] | * Maass, [http://www.math.tifr.res.in/~publ/ln/tifr03.pdf Lectures on Siegel's Modular Functions] | ||
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| + | ==관련논문== | ||
| + | * Andrew Knightly, Charles Li, On the distribution of Satake parameters for Siegel modular forms, arXiv:1605.03792 [math.NT], May 12 2016, http://arxiv.org/abs/1605.03792 | ||
| + | * Satoshi Wakatsuki, The dimensions of spaces of Siegel cusp forms of general degree, arXiv:1602.05676 [math.NT], February 18 2016, http://arxiv.org/abs/1602.05676 | ||
| + | * Gerard van der Geer, Exploring modular forms and the cohomology of local systems on moduli spaces by counting points, arXiv:1604.02654 [math.AG], April 10 2016, http://arxiv.org/abs/1604.02654 | ||
| + | * Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi, An equidistribution theorem for holomorphic Siegel modular forms for <math>GSp_4</math>, arXiv:1604.02036[math.NT], April 07 2016, http://arxiv.org/abs/1604.02036v1 | ||
| + | * Dickson, Martin J. “Hecke Eigenvalues of Klingen--Eisenstein Series of Squarefree Level.” arXiv:1512.09069 [math], December 30, 2015. http://arxiv.org/abs/1512.09069. | ||
| + | * Ichikawa, Takashi. “Integrality of Nearly (holomorphic) Siegel Modular Forms.” arXiv:1508.03138 [math], August 13, 2015. http://arxiv.org/abs/1508.03138. | ||
| + | * Schulze-Pillot, Rainer. “Averages of Fourier Coefficients of Siegel Modular Forms and Representation of Binary Quadratic Forms by Quadratic Forms in Four Variables.” arXiv:1202.4909 [math], February 22, 2012. http://arxiv.org/abs/1202.4909. | ||
| + | * Piazza, Francesco Dalla, Alessio Fiorentino, Samuel Grushevsky, Sara Perna, and Riccardo Salvati Manni. ‘Vector-Valued Modular Forms and the Gauss Map’. arXiv:1505.06370 [math], 23 May 2015. http://arxiv.org/abs/1505.06370. | ||
| + | * Heim, Bernhard, and Atsushi Murase. "On the Igusa modular form of weight 10." 数理解析研究所講究録 1767 (2011): 179-187. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1767-18.pdf | ||
| + | * Tsuyumine, Shigeaki. “Thetanullwerte on a Moduli Space of Curves and Hyperelliptic Loci.” Mathematische Zeitschrift 207, no. 1 (May 1, 1991): 539–68. doi:10.1007/BF02571407. | ||
| + | * Tsuyumine, Shigeaki. 1986. “On Siegel Modular Forms of Degree Three.” American Journal of Mathematics 108 (4): 755. doi:10.2307/2374517. | ||
| + | * Igusa, Jun-Ichi. 1962. “On Siegel Modular Forms of Genus Two.” American Journal of Mathematics 84 (1): 175. doi:10.2307/2372812. | ||
[[분류:정수론]] | [[분류:정수론]] | ||
| + | |||
| + | ==메타데이터== | ||
| + | ===위키데이터=== | ||
| + | * ID : [https://www.wikidata.org/wiki/Q7510567 Q7510567] | ||
| + | ===Spacy 패턴 목록=== | ||
| + | * [{'LOWER': 'siegel'}, {'LOWER': 'modular'}, {'LEMMA': 'form'}] | ||
2021년 2월 17일 (수) 03:21 기준 최신판
지겔 상반 공간
- 지겔 상반 공간 \(\mathcal{H}_g\)
\[ \mathcal{H}_g=\left\{\tau \in M_{g \times g}(\mathbb{C}) \ \big| \ \tau^{\mathrm{T}}=\tau, \textrm{Im}(\tau) \text{ positive definite} \right\} \]
- 사교군 \(\Gamma_g:={\rm Sp}(2g,\Z)\)
- \(\mathcal{A}_g=\mathcal{H}_g/\Gamma_g\) : moduli space of principally polarized abelian varieties
지겔 모듈라 형식
- 정의
weight이 k이고 genus(또는 degree)가 \(g\)인 지겔 모듈라 형식은 다음 조건을 만족하는 해석함수 \(f:\mathcal{H}_g\to \mathbb{C}\)로 정의된다 \[ f \left( (A\tau +B)(C\tau + D)^{-1}\right) = \det(C\tau +D)^{k} f(\tau),\, \forall \begin{pmatrix}A & B \\ C & D \\\end{pmatrix}\in {\rm Sp}(2g,\Z) \]
푸리에 전개
- 지겔 모듈라 형식 \(f\in M_k(\Gamma_g)\)는 다음과 같은 형태의 푸리에 전개를 가진다
\[f(\tau)=\sum_{T}a(T)\exp\left(2\pi i \operatorname{Tr}(T\tau)\right)\] 여기서 \(T\in \operatorname{Mat}_2(\frac{1}{2}\mathbb{Z})\)는 대각성분이 정수인 대칭행렬.
- Kocher 원리
지겔 모듈라 형식 \(f\in M_k(\Gamma_g)\)의 푸리에 전개에서, \(T\)가 positive semi-definite 행렬이 아니면, \(a(T)=0\)이다
지겔 모듈라 형식의 예
관련된 항목들
매스매티카 파일 및 계산 리소스
- Tables of coefficients of some Siegel automorphic forms
- Siegel modular forms in SAGE
- http://math.shinshu-u.ac.jp/~nu/html/sage/days/201310/doc/takemori/demo.sage
사전 형태의 자료
리뷰, 에세이, 강의노트
- Kohnen, A short course on Siegel modular forms
- Van der Geer, Gerard. 2006. “Siegel Modular Forms.” arXiv:math/0605346 (May 12). http://arxiv.org/abs/math/0605346.
- Chiera, Some aspects of the theory of theta series
- Ghitza, 2004, An elementary introduction to Siegel modular forms
관련도서
- Andrianov, Anatoli. Introduction to Siegel Modular Forms and Dirichlet Series Springer, 2010.
- Klingen, Helmut. Introductory Lectures on Siegel Modular Forms. Cambridge University Press, 1990.
- Maass, Lectures on Siegel's Modular Functions
관련논문
- Andrew Knightly, Charles Li, On the distribution of Satake parameters for Siegel modular forms, arXiv:1605.03792 [math.NT], May 12 2016, http://arxiv.org/abs/1605.03792
- Satoshi Wakatsuki, The dimensions of spaces of Siegel cusp forms of general degree, arXiv:1602.05676 [math.NT], February 18 2016, http://arxiv.org/abs/1602.05676
- Gerard van der Geer, Exploring modular forms and the cohomology of local systems on moduli spaces by counting points, arXiv:1604.02654 [math.AG], April 10 2016, http://arxiv.org/abs/1604.02654
- Henry H. Kim, Satoshi Wakatsuki, Takuya Yamauchi, An equidistribution theorem for holomorphic Siegel modular forms for \(GSp_4\), arXiv:1604.02036[math.NT], April 07 2016, http://arxiv.org/abs/1604.02036v1
- Dickson, Martin J. “Hecke Eigenvalues of Klingen--Eisenstein Series of Squarefree Level.” arXiv:1512.09069 [math], December 30, 2015. http://arxiv.org/abs/1512.09069.
- Ichikawa, Takashi. “Integrality of Nearly (holomorphic) Siegel Modular Forms.” arXiv:1508.03138 [math], August 13, 2015. http://arxiv.org/abs/1508.03138.
- Schulze-Pillot, Rainer. “Averages of Fourier Coefficients of Siegel Modular Forms and Representation of Binary Quadratic Forms by Quadratic Forms in Four Variables.” arXiv:1202.4909 [math], February 22, 2012. http://arxiv.org/abs/1202.4909.
- Piazza, Francesco Dalla, Alessio Fiorentino, Samuel Grushevsky, Sara Perna, and Riccardo Salvati Manni. ‘Vector-Valued Modular Forms and the Gauss Map’. arXiv:1505.06370 [math], 23 May 2015. http://arxiv.org/abs/1505.06370.
- Heim, Bernhard, and Atsushi Murase. "On the Igusa modular form of weight 10." 数理解析研究所講究録 1767 (2011): 179-187. http://www.kurims.kyoto-u.ac.jp/~kyodo/kokyuroku/contents/pdf/1767-18.pdf
- Tsuyumine, Shigeaki. “Thetanullwerte on a Moduli Space of Curves and Hyperelliptic Loci.” Mathematische Zeitschrift 207, no. 1 (May 1, 1991): 539–68. doi:10.1007/BF02571407.
- Tsuyumine, Shigeaki. 1986. “On Siegel Modular Forms of Degree Three.” American Journal of Mathematics 108 (4): 755. doi:10.2307/2374517.
- Igusa, Jun-Ichi. 1962. “On Siegel Modular Forms of Genus Two.” American Journal of Mathematics 84 (1): 175. doi:10.2307/2372812.
메타데이터
위키데이터
- ID : Q7510567
Spacy 패턴 목록
- [{'LOWER': 'siegel'}, {'LOWER': 'modular'}, {'LEMMA': 'form'}]