Editing Sato–Tate conjecture

{{Short description|Mathematical conjecture about elliptic curves}}
{{Infobox mathematical statement
| name = Sato–Tate conjecture
| image = 
| caption = 
| field = [[Arithmetic geometry]]
| conjectured by = [[Mikio Sato]]{{br}}[[John Tate (mathematician)|John Tate]]
| conjecture date = {{circa|1960}}
| first proof by = [[Laurent Clozel]]{{br}}Thomas Barnet-Lamb{{br}}David Geraghty{{br}}[[Michael Harris (mathematician)|Michael Harris]]{{br}}[[Nicholas Shepherd-Barron]]{{br}}[[Richard Taylor (mathematician)|Richard Taylor]]
| first proof date = 2011
}}
In [[mathematics]], the '''Sato–Tate conjecture''' is a [[statistical]] statement about the family of [[elliptic curve]]s ''E<sub>p</sub>'' obtained from an elliptic curve ''E'' over the [[rational number]]s by [[reduction modulo a prime|reduction modulo]] almost all [[prime number]]s ''p''. [[Mikio Sato]] and [[John Tate (mathematician)|John Tate]] independently posed the conjecture around 1960.

If ''N<sub>p</sub>'' denotes the number of points on the elliptic curve ''E<sub>p</sub>'' defined over the [[finite field]] with ''p'' elements, the conjecture gives an answer to the distribution of the second-order term for ''N<sub>p</sub>''. By [[Hasse's theorem on elliptic curves]],

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

as <math>p\to\infty</math>, and the point of the conjecture is to predict how the [[big-O notation|O-term]] varies.

The original conjecture and its generalization to all [[totally real field]]s was proved by [[Laurent Clozel]], [[Michael Harris (mathematician)|Michael Harris]], [[Nicholas Shepherd-Barron]], and [[Richard Taylor (mathematician)|Richard Taylor]] under mild assumptions in 2008, and completed by [[Thomas Barnet-Lamb]], [[David Geraghty (mathematician)|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]].  For a prime number ''p'', 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>

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

:<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 =  \frac{1}{\pi}\left(\beta-\alpha+\sin(\alpha)\cos(\alpha)-\sin(\beta)\cos(\beta)\right)</math>

==Details==
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 (mathematician)|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>

==Proof==
In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof 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> in a series of three joint papers.<ref>{{Cite journal
| 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
| doi=10.1007/s10240-008-0015-2
| mr=2470688
| citeseerx=10.1.1.116.9791
}}</ref><ref>{{Cite journal
| last1=Clozel
| first1=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
| doi=10.1007/s10240-008-0016-1
| mr=2470687
| citeseerx=10.1.1.143.9755
}}</ref><ref>{{Citation
| last1=Harris
| first1=Michael
| last2=Shepherd-Barron
| first2=Nicholas
| last3=Taylor
| first3=Richard
| title=A family of Calabi&ndash;Yau varieties and potential automorphy
| journal=[[Annals of Mathematics]]
| year=2010
| volume=171
| issue=2
| pages=779&ndash;813
| doi=10.4007/annals.2010.171.779
| mr=2630056
| doi-access=free
}}</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> 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,<ref>{{Cite journal
| last1=Barnet-Lamb
| first1=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=2011
| pages=29&ndash;98
| journal=Publ. Res. Inst. Math. Sci.
| volume=47
| issue=1
| mr=2827723
| doi=10.2977/PRIMS/31
| doi-access=free
}}</ref> by improving the potential modularity results of previous papers.<ref>Theorem B of {{harvnb|Barnet-Lamb|Geraghty|Harris|Taylor|2011}}</ref> The prior issues involved with the trace formula were solved by [[Michael Harris (mathematician)|Michael Harris]],<ref>{{cite book |last=Harris |first=M. |chapter=An introduction to the stable trace formula |editor-first=L. |editor-last=Clozel |editor2-first=M. |editor2-last=Harris |editor3-first=J.-P. |editor3-last=Labesse |editor4-first=B. C. |editor4-last=Ngô |title=The stable trace formula, Shimura varieties, and arithmetic applications |volume=I: Stabilization of the trace formula |location=Boston |publisher=International Press |year=2011 |pages=3–47 |isbn=978-1-57146-227-5 }}</ref> and [[Sug Woo Shin]].<ref>{{cite journal |first=Sug Woo |last=Shin |title=Galois representations arising from some compact Shimura varieties |journal=[[Annals of Mathematics]] |volume=173 |issue=3 |pages=1645–1741 |year=2011 |doi=10.4007/annals.2011.173.3.9 |doi-access=free }}</ref><ref>See p. 71 and Corollary 8.9 of {{harvnb|Barnet-Lamb|Geraghty|Harris|Taylor|2011}}</ref>

In 2015, Richard Taylor was awarded the [[Breakthrough Prize in Mathematics]] "for numerous breakthrough results in (...) the Sato–Tate conjecture."<ref>{{cite web |title=Richard Taylor, Institute for Advanced Study: 2015 Breakthrough Prize in Mathematics |url=https://breakthroughprize.org/?controller=Page&action=laureates&p=3&laureate_id=59 }}</ref>

==Generalisations==
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 |first1=Nicholas M. |last1=Katz |name-list-style=amp |first2=Peter |last2=Sarnak |location=Providence, RI |publisher=American Mathematical Society |year=1999 |isbn=978-0-8218-1017-0 }}</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)]].

==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 ''a''<sub>''p''</sub>,<ref>{{Citation |last1=Lang |first1=Serge |last2=Trotter |first2=Hale F. |year=1976 |title=Frobenius Distributions in GL<sub>2</sub> extensions |location=Berlin |publisher=Springer-Verlag |isbn=978-0-387-07550-1 }}</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''</sub>, 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=0917870 |doi=10.2140/pjm.1988.131.157|doi-access=free }}.</ref>
In 1999, [[Chantal David]] and [[Francesco Pappalardi]] proved an averaged version of the Lang–Trotter conjecture.<ref name="david-pappalardi">{{cite news |url=https://cms.math.ca/MediaReleases/2013/kn-prize.html |archiveurl=https://web.archive.org/web/20170201044745/https://cms.math.ca/MediaReleases/2013/kn-prize.html |archivedate=2017-02-01 |date=2013-04-15 |title=Concordia Mathematician Recognized for Research Excellence |access-date=2018-01-15 |work=[[Canadian Mathematical Society]] }}</ref><ref name="David Pappalardi distributions of elliptic curves">{{cite journal | last1=David | first1=Chantal | last2=Pappalardi | first2=Francesco | title=Average Frobenius distributions of elliptic curves | journal=International Mathematics Research Notices | volume=1999 | issue=4 | date=1999-01-01 | pages=165–183| doi=10.1155/S1073792899000082 }}</ref>

== See also ==

* [[Wigner semicircle distribution]]

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

==External links==
*[http://www.ams.org/mathmedia/archive/10-2006-media.html Report on Barry Mazur giving context]
*[https://web.archive.org/web/20070930010424/http://www.cirm.univ-mrs.fr/videos/2006/exposes/17w2/Harris.pdf Michael Harris notes, with statement (PDF)]
*[https://web.archive.org/web/20110719043348/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)]
* [https://www.youtube.com/watch?v=6eZQu120A80 Video introducing Elliptic curves and its relation to Sato-Tate conjecture, Imperial College London, 2014]  (Last 15 minutes)

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