Editing Sato–Tate conjecture (section)

{{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.