Sato–Tate conjecture

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

${\displaystyle p+1-N_{p}=2{\sqrt {p}}\cos \theta _{p}~~(0\leq \theta _{p}\leq \pi ).}$

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

${\displaystyle \lim _{N\to \infty }{\frac {\#\{p\leq N:\alpha \leq \theta _{p}\leq \beta \}}{\#\{p\leq N\}}}={\frac {2}{\pi }}\int _{\alpha }^{\beta }\sin ^{2}\theta \,d\theta ={\frac {1}{\pi }}\left(\beta -\alpha +\sin(\alpha )\cos(\alpha )-\sin(\beta )\cos(\beta )\right)}$

By Hasse's theorem on elliptic curves, the ratio

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

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,[1] states that the probability measure of θ is proportional to

${\displaystyle \sin ^{2}\theta \,d\theta .}$[2]

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.[5][6][7]

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.[8] In 2011, Barnet-Lamb, Geraghty, Harris, and Taylor proved a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two,[9] by improving the potential modularity results of previous papers.[10] The prior issues involved with the trace formula were solved by Michael Harris,[11] and Sug Woo Shin.[12][13]

In 2015, Richard Taylor was awarded the Breakthrough Prize in Mathematics "for numerous breakthrough results in (...) the Sato–Tate conjecture."[14]

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,[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 Haar measure on USp(2n) then gives the conjectured distribution, and the classical case is USp(2) = SU(2).

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 a_p,[16] 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

${\displaystyle c{\sqrt {X}}/\log X\ }$

with a specified constant c. Neal Koblitz (1988) provided detailed conjectures for the case of a prime number q of points on E_p, motivated by elliptic curve cryptography.[17] In 1999, Chantal David and Francesco Pappalardi proved an averaged version of the Lang–Trotter conjecture.[18][19]

• Report on Barry Mazur giving context
• Michael Harris notes, with statement (PDF)
• La Conjecture de Sato–Tate [d'après Clozel, Harris, Shepherd-Barron, Taylor], Bourbaki seminar June 2007 by Henri Carayol (PDF)
• Video introducing Elliptic curves and its relation to Sato-Tate conjecture, Imperial College London, 2014 (Last 15 minutes)