Sato–Tate conjecture

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

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

Let E be an elliptic curve defined over the rationals numbers without complex multiplication. Define θ_p as the solution to the equation

Then, for every two real numbers ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ for which ${\displaystyle 0\leq \alpha <\beta \leq \pi }$,

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