In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep 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.
The Sato–Tate conjecture, when E doesn't have complex multiplication,[1] states that the probability measure of θ is proportional to This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later).
[3] In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime,[4] in a series of three joint papers.
Harris has a conditional proof of a result for the product of two elliptic curves (not isogenous) following from such a hypothetical trace formula.
[10] The prior issues involved with the trace formula were solved by Michael Harris,[11] and Sug Woo Shin.
Under the random matrix model developed by Nick Katz and Peter Sarnak,[15] there is a conjectural correspondence between (unitarized) characteristic polynomials of Frobenius elements and conjugacy classes in the compact Lie group USp(2n) = Sp(n).
The Lang–Trotter conjecture (1976) of Serge Lang and Hale Trotter states the asymptotic number of primes p with a given value of ap,[16] the trace of Frobenius that appears in the formula.