Sato-Tate conjecture
Encyclopedia
In mathematics
, the Sato–Tate conjecture is a statistical statement about the family of elliptic curve
s 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 Hasse's theorem on elliptic curves we have
as p → ∞, and the point of the conjecture is to predict how the O-term varies.
. Let the elliptic curve E have no complex multiplication. Then, for real numbers α and β (0≦α<β≦π), :
.
, 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
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
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.
of Harvard University
announced on his web page the final step of a proof, joint with Laurent Clozel, Michael Harris
, and Nicholas 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. Two of the three articles have since been published. 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. , Richard Taylor has posted on his website an article (joint work with Thomas Barnet-Lamb, David Geraghty, and Michael Harris) which claims to prove a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. They also assert that the prior issues involved with the trace formula have been solved by Michael Harris' "Book project" and work of Sug Woo Shin.
s 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 class
es 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).
and Hale Trotter predicts the asymptotic number of primes p with a given value of ap, 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
.
Lang-Trotter conjecture is an analogue of Artin's conjecture on primitive roots, generated in 1977.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Sato–Tate conjecture is a statistical statement about the family of elliptic curve
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
s Ep over the finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
with p elements, with p a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, obtained from an elliptic curve E over the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
field, by the process of reduction modulo a prime for almost all
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
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 Hasse's theorem on elliptic curves we have
as p → ∞, and the point of the conjecture is to predict how the O-term varies.
Statement
Define θp by :. Let the elliptic curve E have no complex multiplication. Then, for real numbers α and β (0≦α<β≦π), :.Details
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 theoremChinese remainder theorem
The Chinese remainder theorem is a result about congruences in number theory and its generalizations in abstract algebra.In its most basic form it concerned with determining n, given the remainders generated by division of n by several numbers...
, for any fixed integer M. In the case where E has complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
the conjecture is replaced by another, simpler law.
It is known from the general theory that the remainder
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
Complex conjugate
In mathematics, complex conjugates are a pair of complex numbers, both having the same real part, but with imaginary parts of equal magnitude and opposite signs...
and of absolute value
Absolute value
In mathematics, the absolute value |a| of a real number a is the numerical value of a without regard to its sign. So, for example, the absolute value of 3 is 3, and the absolute value of -3 is also 3...
1. The Sato–Tate conjecture, when E doesn't have complex multiplication, states that the
probability measure
Probability measure
In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity...
of θ is proportional to
This is due to Mikio Sato
Mikio Sato
is a Japanese mathematician, who started the field of algebraic analysis. He studied at the University of Tokyo, and then did graduate study in physics as a student of Shin'ichiro Tomonaga...
and John Tate
John Tate
John Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
(independently, and around 1960, published somewhat later). It is by now supported by very substantial evidence.
Proofs and claims in progress
On March 18, 2006, Richard TaylorRichard Taylor (mathematician)
-External links:**...
of Harvard University
Harvard University
Harvard University is a private Ivy League university located in Cambridge, Massachusetts, United States, established in 1636 by the Massachusetts legislature. Harvard is the oldest institution of higher learning in the United States and the first corporation chartered in the country...
announced on his web page the final step of a proof, joint with Laurent Clozel, Michael Harris
Michael Harris (mathematician)
Michael Howard Harris is an American mathematician. He made notable contributions to the Langlands program, for which he won the 2007 Clay Research Award...
, and Nicholas Shepherd-Barron
Nicholas Shepherd-Barron
Nicholas Ian Shepherd-Barron is a British mathematician working in algebraic geometry. He is professor of mathematics at Cambridge University and a Fellow of Trinity College, Cambridge....
, of the Sato–Tate conjecture for elliptic curves over totally real fields satisfying a certain condition: of having multiplicative reduction at some prime. Two of the three articles have since been published. Further results are conditional on improved forms of the Arthur–Selberg trace formula
Arthur–Selberg trace formula
In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003....
. Harris has a conditional proof
Conditional proof
A conditional proof is a proof that takes the form of asserting a conditional, and proving that the antecedent of the conditional necessarily leads to the consequent....
of a result for the product of two elliptic curves (not isogenous
Isogeny
In mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....
) following from such a hypothetical trace formula. , Richard Taylor has posted on his website an article (joint work with Thomas Barnet-Lamb, David Geraghty, and Michael Harris) which claims to prove a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. They also assert that the prior issues involved with the trace formula have been solved by Michael Harris' "Book project" and work of Sug Woo Shin.
Generalisation
There are generalisations, involving the distribution of Frobenius elements in Galois groupGalois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
s involved in the Galois representations on étale cohomology
Étale cohomology
In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures...
. In particular there is a conjectural theory for curves of genus n > 1.
Under the random matrix model developed by Nick Katz
Nick Katz
Nicholas Michael Katz is an American mathematician, working in the fields of algebraic geometry, particularly on p-adic methods, monodromy and moduli problems, and number theory...
and Peter Sarnak
Peter Sarnak
Peter Clive Sarnak is a South African-born mathematician. He has been Eugene Higgins Professor of Mathematics at Princeton University since 2002, succeeding Andrew Wiles, and is an editor of the Annals of Mathematics...
, there is a conjectural correspondence between (unitarized) characteristic polynomials of Frobenius elements and conjugacy class
Conjugacy class
In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
es in the compact Lie group USp(2n)=Sp(n). The Haar measure
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
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 LangSerge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
and Hale Trotter predicts the asymptotic number of primes p with a given value of ap, the trace of Frobenius that appears in the formula. For the typical case (no complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
, trace ≠ 0) their formula states that the number of p up to X is asymptotically
with a specified constant c. Neal Koblitz
Neal Koblitz
Neal I. Koblitz is a Professor of Mathematics at the University of Washington in the Department of Mathematics. He is also an adjunct professor with the Centre for Applied Cryptographic Research at the University of Waterloo. He is the creator of hyperelliptic curve cryptography and the...
(1988) provided detailed conjectures for the case of a prime number q of points on Ep, motivated by elliptic curve cryptography
Elliptic curve cryptography
Elliptic curve cryptography is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S...
.
Lang-Trotter conjecture is an analogue of Artin's conjecture on primitive roots, generated in 1977.