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Mathematical conjecture about elliptic curves From Wikipedia, the free encyclopedia
In mathematics, the Sato–Tate conjecture is a statistical statement about the family of elliptic curves Ep obtained from an elliptic curve E over the rational numbers by reduction modulo almost all prime numbers p. Mikio Sato and John Tate independently posed the conjecture around 1960.
Field | Arithmetic geometry |
---|---|
Conjectured by | Mikio Sato John Tate |
Conjectured in | c. 1960 |
First proof by | Laurent Clozel Thomas Barnet-Lamb David Geraghty Michael Harris Nicholas Shepherd-Barron Richard Taylor |
First proof in | 2011 |
If Np denotes the number of points on the elliptic curve Ep defined over the finite field with p elements, the conjecture gives an answer to the distribution of the second-order term for Np. By Hasse's theorem on elliptic curves,
as , and the point of the conjecture is to predict how the O-term varies.
The original conjecture and its generalization to all totally real fields was proved by Laurent Clozel, Michael Harris, Nicholas Shepherd-Barron, and Richard Taylor under mild assumptions in 2008, and completed by Thomas Barnet-Lamb, David Geraghty, Harris, and Taylor in 2011. Several generalizations to other algebraic varieties and fields are open.
Let E be an elliptic curve defined over the rational numbers without complex multiplication. For a prime number p, define θp as the solution to the equation
Then, for every two real numbers and for which
By Hasse's theorem on elliptic curves, the ratio
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,[1] states that the probability measure of θ is proportional to
This is due to Mikio Sato and John Tate (independently, and around 1960, published somewhat later).[3]
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,[4] in a series of three joint papers.[5][6][7]
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.[8] 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,[9] by improving the potential modularity results of previous papers.[10] The prior issues involved with the trace formula were solved by Michael Harris,[11] and Sug Woo Shin.[12][13]
In 2015, Richard Taylor was awarded the Breakthrough Prize in Mathematics "for numerous breakthrough results in (...) the Sato–Tate conjecture."[14]
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,[15] 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).
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 ap,[16] 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
with a specified constant c. Neal Koblitz (1988) provided detailed conjectures for the case of a prime number q of points on Ep, motivated by elliptic curve cryptography.[17] In 1999, Chantal David and Francesco Pappalardi proved an averaged version of the Lang–Trotter conjecture.[18][19]
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