Sato–Tate conjecture

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,

${\displaystyle N_{p}/p=1+\mathrm {O} (1/\!{\sqrt {p}})\ }$

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.