Sato-Tate conjecture

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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.

1 Details
2 Taylor's announcement
3 Generalisation
4 More precise questions
5 Note
6 External links

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

sin² θ.dθ.^[2]

This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later).^[3] It is by now supported by very substantial evidence.

[edit] Taylor's announcement

On March 18, 2006, Richard Taylor of Harvard University announced on his web page 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. 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.

[edit] 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 > 1.

The form of the conjectured distribution is something that can be read from the Lie group geometry of a given case; so that in general terms there is a rationale for this particular distribution. In fact the distribution is the natural one, for conjugacy classes inside a compact Lie group, namely the pushforward of the probability Haar measure. The classical case is for SU(2) and a most natural parametrization of its conjugacy classes.

[edit] 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.

[edit] Note

^ 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.
^ To normalise, put 2/π in front.
^ 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).