Sato–Tate conjecture

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

$N_{p}/p=1+{\mathcal {O}}\left(1/{\sqrt {p}}\right)\$ as p → ∞, and the point of the conjecture is to predict how the O-term varies.

Statement

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

$p+1-N_{p}=2{\sqrt {p}}\cos \theta _{p}~~(0\leq \theta _{p}\leq \pi ).$ Then, for every two real numbers $\alpha$ and $\beta$ for which $0\leq \alpha <\beta \leq \pi ,$ $\lim _{N\to \infty }{\frac {\#\{p\leq N:\alpha \leq \theta _{p}\leq \beta \}}{\#\{p\leq N\}}}={\frac {2}{\pi }}\int _{\alpha }^{\beta }\sin ^{2}\theta \,d\theta .$ Details

By Hasse's theorem on elliptic curves, the ratio

${\frac {((p+1)-N_{p})}{2{\sqrt {p}}}}=:{\frac {a_{p}}{2{\sqrt {p}}}}$ 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

$\sin ^{2}\theta \,d\theta .$ This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later). It is by now supported by very substantial evidence.[citation needed]

Proofs and claims in progress

In 2008, Richard Taylor, joint with Laurent Clozel, Michael Harris, and Nicholas Shepherd-Barron, 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, in a series of three papers.

Further results are conditional on improved forms of the Arthur–Selberg trace formula. Harris has a conditional proof of a result for the product of two elliptic curves (not isogenous) following from such a hypothetical trace formula. In 2011, Richard Taylor published joint work with Thomas Barnet-Lamb, David Geraghty, and Michael Harris, which claims to prove a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. They also assert that the prior issues involved with the trace formula have been solved by Michael Harris, and Sug Woo Shin.

In 2015, Richard Taylor was awarded the Breakthrough Prize in Mathematics "for numerous breakthrough results in (...) the Sato–Tate conjecture."

Generalisation

There are generalisations, involving the distribution of Frobenius elements in Galois groups involved in the Galois representations on étale cohomology. In particular there is a conjectural theory for curves of genus n > 1.

Under the random matrix model developed by Nick Katz and Peter Sarnak, 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 Haar measure on USp(2n) then gives the conjectured distribution, and the classical case is USp(2) = SU(2).

More precise questions

There are also more refined statements. The Lang–Trotter conjecture (1976) of Serge Lang and Hale Trotter states the asymptotic number of primes p with a given value of ap, the trace of Frobenius that appears in the formula. For the typical case (no complex multiplication, trace ≠ 0) their formula states that the number of p up to X is asymptotically

$c{\sqrt {X}}/\log X\$ with a specified constant c. Neal Koblitz (1988) provided detailed conjectures for the case of a prime number q of points on Ep, motivated by elliptic curve cryptography. In 1999, Chantal David and Francesco Pappalardi proved that the Lang–Trotter conjecture holds in most cases.