사토-테이트 추측 (Sato–Tate conjecture)

개요

• 유리수 체 위에 정의된 타원곡선 \(E\)를 생각하자
• 소수 \(p\)에 대하여 \(E(\mathbb{F}_p)\)의 원소의 개수를 \(M_p\)라 두고, \(\theta_p\)를 다음과 같이 정의하자

\[p+1-M_p=2\sqrt{p}\cos{\theta_p},\quad (0\leq \theta_p \leq \pi).\]

추측 (사토-테이트)

\(E\)가 complex multiplication을 갖지 않을 때, \(0\leq \alpha< \beta\leq \pi\)인 두 실수 \(\alpha, \beta\)에 대하여, 다음이 성립한다 \[\lim_{N\to\infty}\frac{\#\{p\leq N:\alpha\leq \theta_p \leq \beta\}} {\#\{p\leq N\}}=\frac{2}{\pi} \int_{\alpha}^{\beta} \sin^2 \theta \, d\theta. \]

예

• 타원곡선 \(y^2=x^3 + x + 1\)
• \(a_p=p+1-M_p\)라 두면, 다음과 같은 테이블을 얻는다

\[ \begin{array}{c|cc} p & a_p & \theta _p \\ \hline 2 & -2 & 2.35619 \\ 3 & 0 & 1.5708 \\ 5 & -3 & 2.30611 \\ 7 & 3 & 0.968002 \\ 11 & -2 & 1.87707 \\ 13 & -4 & 2.1588 \\ 17 & 0 & 1.5708 \\ 19 & -1 & 1.68576 \\ 23 & -4 & 2.00097 \\ 29 & -6 & 2.16167 \\ 31 & -1 & 1.66072 \\ 37 & -10 & 2.5357 \\ 41 & 7 & 0.992488 \\ 43 & 10 & 0.703639 \\ 47 & -12 & 2.63662 \\ 53 & -4 & 1.8491 \\ 59 & -3 & 1.76734 \\ 61 & 12 & 0.694738 \\ 67 & 12 & 0.74805 \\ 71 & 13 & 0.689745 \\ \end{array} \]

• 처음 5000개의 소수 \(p\)에 대하여, \(\theta_p\)의 분포는 다음과 같다
• 함께 그려진 곡선은 \(\frac{2}{\pi}\sin^2 \theta\)의 그래프

메모

• Lario, Joan-C., and Anna Somoza. “The Sato-Tate Conjecture for a Picard Curve with Complex Multiplication,” September 21, 2014. http://xxx.tau.ac.il/abs/1409.6020.

관련된 항목들

• 프로베니우스와 체보타레프 밀도(density) 정리

매스매티카 파일 및 계산 리소스

• https://docs.google.com/file/d/0B8XXo8Tve1cxN1F2dmVTMDE3LXc/edit

사전 형태의 자료

• http://en.wikipedia.org/wiki/Sato-Tate_conjecture

리뷰, 에세이, 강의노트

• Fité, Francesc. 2014. “Equidistribution, L-Functions, and Sato-Tate Groups.” arXiv:1405.5162 [math], May. http://arxiv.org/abs/1405.5162.

관련논문

• Yih-Dar Shieh, Character theory approach to Sato-Tate groups, arXiv:1605.07743 [math.NT], May 25 2016, http://arxiv.org/abs/1605.07743
• 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
• Jasmin Matz, Nicolas Templier, Sato-Tate equidistribution for families of Hecke-Maass forms on SL(n,R)/SO(n), http://arxiv.org/abs/1505.07285v3
• Sha, Min, and Igor E. Shparlinski. “The Sato--Tate Distribution in Families of Elliptic Curves with a Rational Parameter of Bounded Height.” arXiv:1512.07301 [math], December 22, 2015. http://arxiv.org/abs/1512.07301.
• Boyer, Pascal. “Lowering the Level and Ihara’s Lemma for Some Unitary Groups.” arXiv:1511.00144 [math], October 31, 2015. http://arxiv.org/abs/1511.00144.
• Sha, Min, Igor E. Shparlinski, and José Felipe Voloch. “The Sato-Tate Distribution in Thin Parametric Families of Elliptic Curves.” arXiv:1509.03009 [math], September 10, 2015. http://arxiv.org/abs/1509.03009.
• Chen, Evan, Peter S. Park, and Ashvin A. Swaminathan. ‘Linnik’s Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication’. arXiv:1506.09170 [math], 30 June 2015. http://arxiv.org/abs/1506.09170.
• Bucur, Alina, and Kiran S. Kedlaya. “An Application of the Effective Sato-Tate Conjecture.” arXiv:1301.0139 [math], January 1, 2013. http://arxiv.org/abs/1301.0139.
• Matz, Jasmin, and Nicolas Templier. ‘Sato-Tate Equidistribution for Families of Hecke-Maass Forms on SL(n,R)/SO(n)’. arXiv:1505.07285 [math], 27 May 2015. http://arxiv.org/abs/1505.07285.
• Thorner, Jesse. 2014. “The Error Term in the Sato-Tate Conjecture.” arXiv:1407.2656 [math], July. http://arxiv.org/abs/1407.2656.
• Barnet-Lamb, Tom, et al. "A family of Calabi-Yau varieties and potential automorphy II." Publ. Res. Inst. Math. Sci 47.1 (2011): 29-98.
• Harris, Michael, Nick Shepherd-Barron, and Richard Taylor. "A family of Calabi-Yau varieties and potential automorphy." Ann. of Math.(2) 171.2 (2010): 779-813.

메타데이터

위키데이터

• ID : Q2993331

Spacy 패턴 목록

• [{'LOWER': 'sato'}, {'OP': '*'}, {'LOWER': 'tate'}, {'LEMMA': 'conjecture'}]

노트

말뭉치

1. The Sato-Tate conjecture Suppose E does not have CM.
2. 13 / 33 A generalized Sato-Tate conjecture L-polynomials of algebraic varieties Let X be a smooth proper variety over K .
3. In order to build a strategy towards proving the Sato-Tate conjecture, we quickly review the method of moments from Probability Theory.
4. In any NOTES ON THE SATO-TATE CONJECTURE 5 case, cj = 1.
5. We prove the Sato-Tate conjecture for Hilbert modular forms.
6. In Section 2, the generalized Sato Tate conjecture for arithmetic curves will be reviewed as the main mathematical background for this paper.
7. To achieve this, under the same hypotheses, we prove an explicit version of the Sato–Tate conjecture for primes in arithmetic progressions.
8. This report nevertheless begins with an introduction to Langlands reciprocity conjectures, and their arithmetic variants, in the situation most relevant to the proof of the Sato-Tate conjecture.
9. The main goal of this talk is to present new results in the direction of the algebraic Sato-Tate conjecture, building on the previous work of Serre, Kedlaya and Banaszak.
10. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus.
11. The Sato-Tate conjecture, in its rst formulation, concerns elliptic curves.
12. Various people have asked me for an update on the status of the Sato-Tate conjecture for abelian surfaces in light of recent advances in modularity lifting theorems.
13. Case 2: The Sato-Tate conjecture is proved in this case.
14. Note that since the Sato-Tate conjecture is known in all other cases, one only has to consider cases (3) and (4), which behave slightly differently in this argument.
15. Bounded, Borel measurable R-valued central functions g on U Sp(4) are in one-one correspondence with bounded, Borel ON THE SATO-TATE CONJECTURE FOR GENUS TWO CURVES 3 Figure 1.