Sato-Tate conjecture

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

:"N_p/p" = 1 + O(1/√"p")

as "p" → ∞, and the point of the conjecture is to predict how the O-term varies.

Details

It is easy to see that we can in fact choose the first "M" of the "E_p" as we like, as an application of the Chinese remainder theorem, for any fixed integer "M". In the case where "E" has complex multiplication the conjecture is replaced by another, simpler law.

It is known from the general theory that the remainder

:−½("N_p" − ("p" + 1))/√"p"

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, [In the case of an elliptic curve with complex multiplication, the Hasse-Weil L-function is expressed in terms of a Hecke L-function (result of Max Deuring. The known analytic results on these answer even more precise questions.] states that the
probability measure of θ is proportional to

:sin² θ.dθ. [To normalise, put 2/π in front.]

This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later). [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).] It is by now supported by very substantial evidence.

Taylor's announcement

On March 18, 2006, Richard Taylor of Harvard University announced on his web page the final step ofa proof, joint with L. Clozel, M. Harris, and N. Shepherd-Barron, of the Sato-Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime. 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.

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, ["Random matrices, Frobenius Eigenvalues, and Monodromy", Nicholas M. Katz and Peter Sarnak, AMS, 1999.] 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 predicts 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

:constant × √"X"/ log "X"

with a specified constant. Neal Koblitz (1988) provided detailed conjectures for the case of a prime number "q" of points on "E"_{"p", motivated by elliptic curve cryptography.}

Notes

External links

* [http://modular.math.washington.edu/sage/apps/2006-04-20-sato-tate Diagrams illustrating the Sato-Tate conjecture] , made with the computer algebra system SAGE.
* [http://www.ams.org/mathmedia/archive/10-2006-media.html Report on Barry Mazur giving context]
* [http://www.cirm.univ-mrs.fr/videos/2006/exposes/17w2/Harris.pdf Michael Harris notes, with statement (PDF)]