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.