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

:<math>N_p/p = 1 + O(1/\sqrt{p})\ </math>

as ''p'' → ∞, and the point of the conjecture is to predict how the [[big-O notation|O-term]] varies.

==Statement==
Define ''θ''<sub>''p''</sub> as the solution to the equation

:<math> p+1-N_p=2\sqrt{p}\cos{\theta_p} ~~ (0\leq \theta_p \leq \pi).</math>

Let the elliptic curve ''E'' have no [[complex multiplication]]. Then, for every two real numbers &alpha; and &beta; for which 0&nbsp;&le;&nbsp;&alpha;&nbsp;&lt;&nbsp;&beta;&nbsp;&le;&nbsp;&pi;,

:<math>\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.  </math>

==Details==
It is easy to see that we can in fact choose the first ''M'' of the ''E''<sub>''p''</sub> as we like, as an application of the [[Chinese remainder theorem]], for any fixed integer ''M''.{{Clarify|date=December 2011}} In the case where ''E'' has [[complex multiplication]] the conjecture is replaced by another, simpler law.

By [[Hasse's theorem on elliptic curves]], the ratio

:<math>\frac{((p + 1)-N_p)}{2\sqrt{p}}=:\frac{a_p}{2\sqrt{p}} </math>

is between -1 and 1. Thus it can be expressed as cos&nbsp;''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]]&nbsp;1. The ''Sato–Tate conjecture'', when ''E'' doesn't have complex multiplication,<ref>In the case of an elliptic curve with complex multiplication, the [[Hasse–Weil L-function]] is expressed in terms of a [[Hecke character|Hecke L-function]] (a result of [[Max Deuring]]). The known analytic results on these answer even more precise questions.</ref> states that the [[probability measure]] of ''θ'' is proportional to

:<math>\sin^2 \theta \, d\theta.\ </math><ref>To normalise, put 2/''&pi;'' in front.</ref>

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

==Proofs and claims in progress==
On March 18, 2006,  [[Richard Taylor (mathematician)|Richard Taylor]] of [[Harvard University]]  announced on his web page the final step of a proof, joint with [[Laurent Clozel]], [[Michael Harris (mathematician)|Michael Harris]], and [[Nicholas Shepherd-Barron]], of the Sato–Tate conjecture for elliptic curves over [[totally real field]]s satisfying a certain condition: of having multiplicative reduction at some prime.<ref>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.</ref> Two of the three articles have since been published.<ref>{{harvnb|Clozel|Harris|Taylor|2008}} and {{harvnb|Taylor|2008}}, with the remaining one ({{harvnb|Harris|Shepherd-Barron|Taylor|2009}}) set to appear.</ref> 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 [[isogeny|isogenous]]) following from such a hypothetical trace formula.<ref>See Carayol's Bourbaki seminar of 17 June 2007 for details.</ref> {{As of|2008|07|08}}, Richard Taylor has posted on his website an article (joint work with [[Thomas Barnet-Lamb]], [[David Geraghty (mathematician)|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,<ref>Theorem B of {{harvnb|Barnet-Lamb|Geraghty|Harris|Taylor|2009}}</ref> 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"<ref>Some preprints available here [http://fa.institut.math.jussieu.fr/node/29] (retrieved July 8, 2009).</ref> and work of Sug Woo Shin.<ref>Preprint "Galois representations arising from some compact Shimura varieties" on author's website [http://math.mit.edu/~swshin/StableGal.pdf] (retrieved May 22, 2012).</ref><ref>See p. 71 and Corollary 8.9 of {{harvnb|Barnet-Lamb|Geraghty|Harris|Taylor|2009}}</ref>

==Generalisation==
There are generalisations, involving the distribution of [[Frobenius element]]s in [[Galois group]]s involved in the [[Galois representation]]s on [[étale cohomology]]. In particular there is a conjectural theory for curves of genus&nbsp;''n''&nbsp;>&nbsp;1.

Under the random matrix model developed by [[Nick Katz]] and [[Peter Sarnak]],<ref>{{Citation |title=Random matrices, Frobenius Eigenvalues, and Monodromy |first=Nicholas M. |last=Katz |lastauthoramp=yes |first2=Peter |last2=Sarnak |location=Providence, RI |publisher=American Mathematical Society |year=1999 |isbn=0-8218-1017-0 }}</ref> there is a conjectural correspondence between (unitarized) characteristic polynomials of Frobenius elements and [[conjugacy class]]es in the [[compact Lie group]] USp(2''n'')&nbsp;=&nbsp;[[Sp(n)|Sp(''n'')]].  The [[Haar measure]] on USp(2''n'') then gives the conjectured distribution, and the classical case is USp(2)&nbsp;=&nbsp;[[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''<sub>''p''</sub>,<ref>{{Citation |last=Lang |first=Serge |last2=Trotter |first2=Hale F. |year=1976 |title=Frobenius Distributions in GL<sub>2</sub> extensions |location=Berlin |publisher=Springer-Verlag |isbn=0-387-07550-X }}</ref> 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

:<math>c \sqrt{X}/ \log X\ </math>

with a specified constant ''c''. [[Neal Koblitz]] (1988) provided detailed conjectures for the case of a prime number ''q'' of points on ''E''<sub>''p''</sup>, motivated by [[elliptic curve cryptography]].<ref>{{Citation |last=Koblitz |first=Neal |year=1988 |title=Primality of the number of points on an elliptic curve over a finite field |journal=Pacific Journal of Mathematics |volume=131 |issue=1 |pages=157–165 |mr=89h:11023 }}.</ref>

Lang-Trotter conjecture is an analogue of Artin's conjecture on primitive roots, generated in 1977.

==Notes==
{{Reflist|colwidth=30em}}

==References==
*{{Citation
| last=Barnet-Lamb
| first=Thomas
| last2=Geraghty
| first2=David
| last3=Harris
| first3=Michael
| last4=Taylor
| first4=Richard
| title=A family of Calabi&ndash;Yau varieties and potential automorphy. II
| year=2009
| accessdate=July 8, 2009
}}, preprint (available [http://www.math.harvard.edu/~rtaylor/cy2.pdf here])
*{{Citation
| last=Clozel
| first=Laurent
| last2=Harris
| first2=Michael
| last3=Taylor
| first3=Richard
| title=Automorphy for some ''l''-adic lifts of automorphic mod ''l'' Galois representations
| journal=Publ. Math. Inst. Hautes Études Sci.
| volume=108
| year=2008
| pages=1&ndash;181
}}
*{{Citation
| last=Harris
| first=Michael
| last2=Shepherd-Barron
| first2=Nicholas
| last3=Taylor
| first3=Richard
| title=A family of Calabi&ndash;Yau varieties and potential automorphy
| year=2009
| accessdate=July 8, 2009
}}, preprint (available [http://www.math.harvard.edu/~rtaylor/cyfin.pdf here])
*{{Citation
| last=Taylor
| first=Richard
| title=Automorphy for some ''l''-adic lifts of automorphic mod ''l'' Galois representations. II
| journal=Publ. Math. Inst. Hautes Études Sci.
| volume=108
| year=2008
| pages=183&ndash;239
}}

==External links==
*[http://www.ams.org/mathmedia/archive/10-2006-media.html Report on Barry Mazur giving context]
*[http://www.cirm.univ-mrs.fr/videos/2006/exposes/17w2/Harris.pdf Michael Harris notes, with statement (PDF)]
*[http://www.mathematik.hu-berlin.de/gradkoll/Carayol_Exp.977.H.C4.pdf ''La Conjecture de Sato–Tate'' &#91;d'après Clozel, Harris, Shepherd-Barron, Taylor&#93;, Bourbaki seminar June 2007 by Henri Carayol (PDF)]

{{DEFAULTSORT:Sato-Tate conjecture}}
[[Category:Elliptic curves]]
[[Category:Finite fields]]
[[Category:Conjectures]]