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In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep 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 Np denotes the number of points on Ep and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. That is, by Hasse's theorem on elliptic curves we have

$$N_{p}/p=1+\mathrm {O} (1/\!{\sqrt {p}})\ }$$

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

The original conjecture and its generalization to all totally real fields was proved by Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor under mild assumptions in 2008, and completed by Thomas Barnet-Lamb, David Geraghty, Harris, and Taylor in 2011. Several generalizations to other algebraic varieties and fields are open.

Statement

Let E be an elliptic curve defined over the rational numbers without complex multiplication. Define θp 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, states that the probability measure of θ is proportional to

$$\sin ^{2}\theta \,d\theta .}$$ 

This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later).
Proof

In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime, in a series of three joint papers.

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. In 2011, Barnet-Lamb, Geraghty, Harris, and Taylor proved a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. The prior issues involved with the trace formula were solved by Michael Harris, and Sug Woo Shin.

In 2015, Richard Taylor was awarded the Breakthrough Prize in Mathematics "for numerous breakthrough results in (...) the Sato–Tate conjecture."
Generalisations

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, 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).

Refinements

There are also more refined statements. The Lang–Trotter conjecture (1976) of Serge Lang and Hale Trotter states the asymptotic number of primes p with a given value of ap, 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 Ep, motivated by elliptic curve cryptography. In 1999, Chantal David and Francesco Pappalardi proved an averaged version of the Lang–Trotter conjecture.
References

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.
To normalise, put 2/π in front.
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).
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.
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. CiteSeerX 10.1.1.116.9791. doi:10.1007/s10240-008-0015-2. MR 2470688.
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. CiteSeerX 10.1.1.143.9755. 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
See Carayol's Bourbaki seminar of 17 June 2007 for details.
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.
Theorem B of Barnet-Lamb et al. 2009
Harris, M. (2011). "An introduction to the stable trace formula". In Clozel, L.; Harris, M.; Labesse, J.-P.; Ngô, B. C. (eds.). The stable trace formula, Shimura varieties, and arithmetic applications. Volume I: Stabilization of the trace formula. Boston: International Press. pp. 3–47. ISBN 978-1-57146-227-5.
Shin, Sug Woo (2011). "Galois representations arising from some compact Shimura varieties". Annals of Mathematics. 173 (3): 1645–1741. doi:10.4007/annals.2011.173.3.9.
See p. 71 and Corollary 8.9 of Barnet-Lamb et al. 2009
"Richard Taylor, Institute for Advanced Study: 2015 Breakthrough Prize in Mathematics".
Katz, Nicholas M. & Sarnak, Peter (1999), Random matrices, Frobenius Eigenvalues, and Monodromy, Providence, RI: American Mathematical Society, ISBN 978-0-8218-1017-0
Lang, Serge; Trotter, Hale F. (1976), Frobenius Distributions in GL2 extensions, Berlin: Springer-Verlag, ISBN 978-0-387-07550-1
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.
"Concordia Mathematician Recognized for Research Excellence". Canadian Mathematical Society. 2013-04-15. Archived from the original on 2017-02-01. Retrieved 2018-01-15.