Math 2.0

Number theory Arithmetic geometry Theorem

Sato–Tate conjecture

Builds on Elliptic curveComplex multiplicationHasse's theorem on elliptic curvesFinite field

Statement

Let EE be an elliptic curve over ℚ\Q without complex multiplication. For each prime pp of good reduction for EE, let E~\tilde{E} be the reduction of EE modulo pp, a curve over the finite field 𝔽p\F_p. Let ap=p+1−#E~(𝔽p)a_p = p + 1 - \#\tilde{E}(\F_p). Let θp∈[0,π]\theta_p \in [0, \pi] be the angle with ap=2pcos⁡θpa_p = 2\sqrt{p} \cos \theta_p; it exists by Hasse's theorem. Then for all 0≤α≤β≤π0 \le \alpha \le \beta \le \pi, lim⁡x→∞#{p≤x:p good, α≤θp≤β}#{p≤x:p good}=2π∫αβsin⁡2θ dθ.\lim_{x \to \infty} \frac{\#\set{p \le x : p \text{ good}, \ \alpha \le \theta_p \le \beta}}{\#\set{p \le x : p \text{ good}}} = \frac{2}{\pi} \int_\alpha^\beta \sin^2 \theta \, d\theta.

Remarks

In words, the angles θp\theta_p are equidistributed with respect to the measure 2πsin⁡2θ dθ\frac{2}{\pi} \sin^2 \theta \, d\theta on [0,π][0, \pi]. This is the image of the Haar measure of SU(2)\mathrm{SU}(2) under the map that sends a matrix with eigenvalues e±iθe^{\pm i\theta} to θ\theta. The conjecture was formulated independently by Mikio Sato and John Tate around 1960. Tate showed that it follows from the meromorphic continuation and non-vanishing on Re(s)≥1\mathrm{Re}(s) \ge 1 of the symmetric power LL-functions L(SymmE,s)L(\mathrm{Sym}^m E, s) for all m≥1m \ge 1. Clozel, Harris, Shepherd-Barron and Taylor proved it for curves with multiplicative reduction at some prime, in work published in 2008. Barnet-Lamb, Geraghty, Harris and Taylor removed this condition in 2011, using potential automorphy of the symmetric powers. The hypothesis of no complex multiplication is necessary. For a CM curve, ap=0a_p = 0 for the good primes that are inert in the CM field, which are half of all primes. So θp=π/2\theta_p = \pi/2 for these primes, and the limit distribution is different.

Going deeper

For a student

The proof is a major application of automorphy lifting theorems in the Langlands program. The statement generalises, through the notion of Sato–Tate group, to other motives and to modular forms. For example, it is also known for non-CM holomorphic newforms of weight at least 22.

Reading path

To understand this page from the ground up, read these 5 pages in order. The foundations come first.

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Complex multiplication, concept, no evidence levelComplex multiplicationconceptElliptic curve, concept, no evidence levelElliptic curveconceptFinite field, concept, no evidence levelFinite fieldconceptHasse's theorem on elliptic curves, statement, no evidence levelHasse's theorem on el…statementSato–Tate conjecture, statement, no evidence levelSato–Tate conjecturestatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Complex multiplication concept
  2. Elliptic curve concept
  3. Finite field concept
  4. Hasse's theorem on elliptic curves statement
  5. Sato–Tate conjecture statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Sato%E2%80%93Tate_conjecture
About this page
TagTRGBJYPBXH
URIm2://TRGBJYPBXH
Kindstatement, theorem
WritingW0 stub
MSC11G05, 11F80, 11K06
StatusThis copy of the site has no build data. Run m2 status TRGBJYPBXH.
AboutElliptic curve
UsesElliptic curve, Complex multiplication, Hasse's theorem on elliptic curves, Finite field
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Changed2 November 2026

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