Sato–Tate conjecture

Statement

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

: 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

:\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)

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).

Proof

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, in a series of three joint papers.{{Cite journal | doi-access=free

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, 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,{{Cite journal | doi-access=free

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

Generalisations

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).

Refinements

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, 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 E**p, motivated by elliptic curve cryptography. In 1999, Chantal David and Francesco Pappalardi proved an averaged version of the Lang–Trotter conjecture.

References

References

1. In the case of an elliptic curve with complex multiplication, the [[Hasse–Weil L-function]] is expressed in terms of a [[Hecke character. Hecke L-function]] (a result of [[Max Deuring]]). The known analytic results on these answer even more precise questions.
2. To normalise, put 2/''π'' in front.
3. It is mentioned in J. Tate, ''Algebraic cycles and poles of zeta functions'' in the volume (O. F. G. Schilling, editor), ''Arithmetical Algebraic Geometry'', pages 93–110 (1965).
4. That is, for some ''p'' where ''E'' has [[bad reduction]] (and at least for elliptic curves over the rational numbers there are some such ''p''), the type in the singular fibre of the [[Néron model]] is multiplicative, rather than additive. In practice this is the typical case, so the condition can be thought of as mild. In more classical terms, the result applies where the [[j-invariant]] is not integral.
5. See Carayol's Bourbaki seminar of 17 June 2007 for details.
6. Theorem B of {{harvnb. Barnet-Lamb. Geraghty. Harris. Taylor. 2011
7. Harris, M.. (2011). "The stable trace formula, Shimura varieties, and arithmetic applications". International Press.
8. Shin, Sug Woo. (2011). "Galois representations arising from some compact Shimura varieties". [[Annals of Mathematics]].
9. See p. 71 and Corollary 8.9 of {{harvnb. Barnet-Lamb. Geraghty. Harris. Taylor. 2011
10. "Richard Taylor, Institute for Advanced Study: 2015 Breakthrough Prize in Mathematics".
11. (1999). "Random matrices, Frobenius Eigenvalues, and Monodromy". American Mathematical Society.
12. (1976). "Frobenius Distributions in GL₂ extensions". Springer-Verlag.
13. Koblitz, Neal. (1988). "Primality of the number of points on an elliptic curve over a finite field". Pacific Journal of Mathematics.
14. (2013-04-15). "Concordia Mathematician Recognized for Research Excellence". [[Canadian Mathematical Society]].
15. (1999-01-01). "Average Frobenius distributions of elliptic curves". International Mathematics Research Notices.