In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves E_p obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960.

If N_p denotes the number of points on the elliptic curve E_p defined over the finite field with p elements, the conjecture gives an answer to the distribution of the second-order term for N_p. By Hasse's theorem on elliptic curves,

as ${\displaystyle p\to \infty }$, and the point of the conjecture is to predict how the O-term varies.

The original conjecture and its generalization to all totally real fields was proved by Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor under mild assumptions in 2008, and completed by Thomas Barnet-Lamb, David Geraghty, Harris, and Taylor in 2011. Several generalizations to other algebraic varieties and fields are open.

Let E be an elliptic curve defined over the rational numbers without complex multiplication. For a prime number p, define θ_p as the solution to the equation

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

By Hasse's theorem on elliptic curves, the ratio

is between -1 and 1. Thus it can be expressed as cos 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 1. The Sato–Tate conjecture, when E doesn't have complex multiplication, states that the probability measure of θ is proportional to