Sato-Tate conjecture
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In mathematics, the Sato-Tate conjecture is a statistical statement about the family of elliptic curves
- Ep
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
- Np
denotes the number of points on Ep and defined over the field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. That is, by a theorem of Helmut Hasse we have
- Np/p = 1 + O(1/√p)
as p → ∞, and the point of the conjecture is to predict how the O-term varies. It is easy to see that we can in fact choose the first M of the Ep 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(Np − (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, states that probability measure of θ is proportional to
- sin2 θ.dθ.
This is due to Mikio Sato and John Tate (independently?, and around 1960, published somewhat later). It is by now supported by very substantial evidence, but is not proved in general (2004).
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.