Sato–Tate distribution (for families of elliptic curves)

E753156

conjecture in number theory probability distribution theorem in arithmetic geometry

The Sato–Tate distribution (for families of elliptic curves) is a probabilistic law describing how the normalized Frobenius traces (or equivalently, the angles in the Hasse bound) of elliptic curves are distributed, typically following a specific sine-squared measure on the interval [0, π].

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Statements (49)

Predicate	Object
instanceOf	conjecture in number theory ⓘ probability distribution ⓘ theorem in arithmetic geometry ⓘ
angleDefinition	θ_p ∈ [0, π] with a_p(E) = 2√p cos θ_p ⓘ
appliesTo	elliptic curves over number fields ⓘ elliptic curves over ℚ ⓘ families of elliptic curves ordered by conductor ⓘ families of elliptic curves ordered by height ⓘ
associatedWith	Frobenius endomorphism NERFINISHED ⓘ Hasse–Weil zeta function NERFINISHED ⓘ L-functions of elliptic curves ⓘ
assumes	non-CM elliptic curves for the classical form ⓘ
concerns	primes of good reduction ⓘ
densityFunction	(2/π) sin²(θ) dθ on [0, π] ⓘ
describes	distribution of angles in the Hasse bound for elliptic curves ⓘ distribution of normalized Frobenius traces of elliptic curves ⓘ
excludes	primes of bad reduction ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ automorphic forms ⓘ number theory ⓘ
generalizedBy	Sato–Tate conjecture for motives NERFINISHED ⓘ Sato–Tate groups for abelian varieties NERFINISHED ⓘ
hasVariant	Sato–Tate distribution for CM elliptic curves NERFINISHED ⓘ generalized Sato–Tate distributions for higher-dimensional abelian varieties ⓘ
implies	moments of Frobenius traces match those of SU(2) ⓘ statistical regularity of point counts of elliptic curves modulo primes ⓘ
measureOn	conjugacy classes of SU(2) ⓘ
measureType	Haar measure on SU(2) pushed forward to [0, π] ⓘ
namedAfter	John Tate NERFINISHED ⓘ Mikio Sato NERFINISHED ⓘ
normalization	normalized Frobenius trace a_p(E)/(2√p) ⓘ
parameterization	a_p(E) = 2√p cos θ_p ⓘ
predicts	equidistribution of angles θ_p with respect to (2/π) sin²(θ) dθ ⓘ
proofYear	around 2006–2008 for elliptic curves over ℚ without CM ⓘ
provedBy	Laurent Clozel NERFINISHED ⓘ Michael Harris NERFINISHED ⓘ Nicholas Shepherd-Barron NERFINISHED ⓘ Richard Taylor NERFINISHED ⓘ
relatedConcept	Chebotarev density theorem NERFINISHED ⓘ Langlands program NERFINISHED ⓘ Sato–Tate conjecture NERFINISHED ⓘ equidistribution of Frobenius conjugacy classes ⓘ
relatedTo	Hasse bound \|a_p(E)\| ≤ 2√p NERFINISHED ⓘ
status	open in full generality for all motives ⓘ proved for many elliptic curves over totally real fields ⓘ
support	interval [0, π] ⓘ
uses	modularity of elliptic curves over ℚ ⓘ potential automorphy of symmetric power L-functions ⓘ

Referenced by (1)

Full triples — surface form annotated when it differs from this entity's canonical label.

Hasse bound for elliptic curves → relatedConcept → Sato–Tate distribution (for families of elliptic curves) ⓘ