Sato–Tate conjecture

In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves E_p over the finite field with p elements, with p a prime number, obtained from an elliptic curve E over the rational number field, by the process of reduction modulo a prime for almost all p. If N_p denotes the number of points on E_p and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for N_p. That is, by Hasse's theorem on elliptic curves we have

$N_p/p = 1 + O(1/\sqrt{p})\$

as p → ∞, and the point of the conjecture is to predict how the O-term varies.

Statement

Let E be an elliptic curve defined over the rationals numbers without complex multiplication. Define θ_p as the solution to the equation

$p+1-N_p=2\sqrt{p}\cos{\theta_p} ~~ (0\leq \theta_p \leq \pi).$

Then, for every two real numbers $\alpha$ and $\beta$ for which $0\leq \alpha < \beta \leq \pi$,

$\lim_{N\to\infty}\frac{\#\{p\leq N:\alpha\leq \theta_p \leq \beta\}} {\#\{p\leq N\}}=\frac{2}{\pi} \int_{\alpha}^{\beta} \sin^2 \theta \, d\theta.$

Details

By Hasse's theorem on elliptic curves, the ratio

$\frac{((p + 1)-N_p)}{2\sqrt{p}}=:\frac{a_p}{2\sqrt{p}}$

is between -1 and 1. Thus it can be expressed as cos θ for an angle θ; in geometric terms there are two eigenvalues accounting for the remainder and with the denominator as given they are complex conjugate and of absolute value 1. The Sato–Tate conjecture, when E doesn't have complex multiplication,^[1] states that the probability measure of θ is proportional to

$\sin^2 \theta \, d\theta.\$^[2]

This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later).^[3] It is by now supported by very substantial evidence.

Proofs and claims in progress

On March 18, 2006, Richard Taylor of Harvard University announced on his web page the final step of a proof, joint with Laurent Clozel, Michael Harris, and Nicholas Shepherd-Barron, of the Sato–Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime.^[4] Two of the three articles have since been published.^[5] Further results are conditional on improved forms of the Arthur–Selberg trace formula. Harris has a conditional proof of a result for the product of two elliptic curves (not isogenous) following from such a hypothetical trace formula.^[6] As of 8 July 2008, Richard Taylor has posted on his website an article (joint work with Thomas Barnet-Lamb, David Geraghty, and Michael Harris) which claims to prove a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two,^[7] by improving the potential modularity results of previous papers. They also assert that the prior issues involved with the trace formula have been solved by Michael Harris' "Book project"^[8] and work of Sug Woo Shin.^[9][10] In 2013 Taylor was awarded the Breakthrough Prize in Mathematics "for numerous breakthrough results in (...) the Sato–Tate conjecture."^[11]

Generalisation

There are generalisations, involving the distribution of Frobenius elements in Galois groups involved in the Galois representations on étale cohomology. In particular there is a conjectural theory for curves of genus n > 1.

Under the random matrix model developed by Nick Katz and Peter Sarnak,^[12] there is a conjectural correspondence between (unitarized) characteristic polynomials of Frobenius elements and conjugacy classes in the compact Lie group USp(2n) = Sp(n). The Haar measure on USp(2n) then gives the conjectured distribution, and the classical case is USp(2) = SU(2).

More precise questions

There are also more refined statements. The Lang–Trotter conjecture (1976) of Serge Lang and Hale Trotter predicts the asymptotic number of primes p with a given value of a_p,^[13] the trace of Frobenius that appears in the formula. For the typical case (no complex multiplication, trace ≠ 0) their formula states that the number of p up to X is asymptotically

$c \sqrt{X}/ \log X\$

with a specified constant c. Neal Koblitz (1988) provided detailed conjectures for the case of a prime number q of points on E_p, motivated by elliptic curve cryptography.^[14]

Notes

1. In the case of an elliptic curve with complex multiplication, the Hasse–Weil L-function is expressed in terms of a Hecke L-function (a result of Max Deuring). The known analytic results on these answer even more precise questions.
2. To normalise, put 2/π in front.
3. It is mentioned in J. Tate, Algebraic cycles and poles of zeta functions in the volume (O. F. G. Schilling, editor), Arithmetical Algebraic Geometry, pages 93–110 (1965).
4. That is, for some p where E has bad reduction (and at least for elliptic curves over the rational numbers there are some such p), the type in the singular fibre of the Néron model is multiplicative, rather than additive. In practice this is the typical case, so the condition can be thought of as mild. In more classical terms, the result applies where the j-invariant is not integral.
5. Clozel, Harris & Taylor 2008 and Taylor 2008, with the remaining one (Harris, Shepherd-Barron & Taylor 2009) set to appear.
6. See Carayol's Bourbaki seminar of 17 June 2007 for details.
7. Theorem B of Barnet-Lamb et al. 2009
8. Some preprints available here (retrieved July 8, 2009). Archived July 13, 2009, at the Wayback Machine.
9. Preprint "Galois representations arising from some compact Shimura varieties" on author's website (retrieved May 22, 2012).
10. See p. 71 and Corollary 8.9 of Barnet-Lamb et al. 2009
11. https://breakthroughprize.org/?controller=Page&action=laureates&p=3&laureate_id=59
12. Katz, Nicholas M. & Sarnak, Peter (1999), Random matrices, Frobenius Eigenvalues, and Monodromy, Providence, RI: American Mathematical Society, ISBN 0-8218-1017-0
13. Lang, Serge; Trotter, Hale F. (1976), Frobenius Distributions in GL₂ extensions, Berlin: Springer-Verlag, ISBN 0-387-07550-X
14. Koblitz, Neal (1988), "Primality of the number of points on an elliptic curve over a finite field", Pacific Journal of Mathematics 131 (1): 157–165, doi:10.2140/pjm.1988.131.157, MR 0917870.

References

• Barnet-Lamb, Thomas; Geraghty, David; Harris, Michael; Taylor, Richard (2011), "A family of Calabi–Yau varieties and potential automorphy. II", Publ. Res. Inst. Math. Sci 47 (1): 29–98, doi:10.2977/PRIMS/31, MR 2827723, preprint (available here)
• Clozel, Laurent; Harris, Michael; Taylor, Richard (2008), "Automorphy for some l-adic lifts of automorphic mod l Galois representations", Publ. Math. Inst. Hautes Études Sci. 108: 1–181, doi:10.1007/s10240-008-0016-1, MR 2470687
• Harris, Michael; Shepherd-Barron, Nicholas; Taylor, Richard (2010), "A family of Calabi–Yau varieties and potential automorphy", Annals of Mathematics 171 (2): 779–813, doi:10.4007/annals.2010.171.779, MR 2630056, (preprint available here)
• Taylor, Richard (2008), "Automorphy for some l-adic lifts of automorphic mod l Galois representations. II", Publ. Math. Inst. Hautes Études Sci. 108: 183–239, doi:10.1007/s10240-008-0015-2, MR 2470688, (preprint available here)

External links

• Report on Barry Mazur giving context
• Michael Harris notes, with statement (PDF)
• La Conjecture de Sato–Tate [d'après Clozel, Harris, Shepherd-Barron, Taylor], Bourbaki seminar June 2007 by Henri Carayol (PDF)