Editing Sato–Tate conjecture (section)

==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>