Sato–Tate Conjecture
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In mathematics, the Sato–Tate conjecture is a
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
statement about the family of
elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If ...
s ''Ep'' obtained from an elliptic curve ''E'' over the
rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all ra ...
s by reduction modulo almost all
prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only way ...
s ''p''.
Mikio Sato is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto. Education Sato studied at the Unive ...
and John Tate independently posed the conjecture around 1960. If ''Np'' denotes the number of points on the elliptic curve ''Ep'' defined over the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subt ...
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 Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below. If ''N'' is the number of points on the ellipt ...
, :N_p/p = 1 + \mathrm(1/\!\sqrt)\ as p\to\infty, and the point of the conjecture is to predict how the O-term varies. The original conjecture and its generalization to all
totally real field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...
s was proved by
Laurent Clozel Laurent Clozel (born October 23, 1953 in Gap) is a French mathematician. His mathematical work is in the area of automorphic forms, including major advances on the Langlands programme Career and distinctions Clozel was a student at the Écol ...
, 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.


Statement

Let ''E'' be an elliptic curve defined over the rational numbers without
complex multiplication In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...
. For a prime number ''p'', define ''θ''''p'' as the solution to the equation : p+1-N_p=2\sqrt\cos\theta_p ~~ (0\leq \theta_p \leq \pi). Then, for every two real numbers \alpha and \beta for which 0\leq \alpha < \beta \leq \pi, :\lim_\frac =\frac \int_\alpha^\beta \sin^2 \theta \, d\theta = \frac\left(\beta-\alpha+\sin(\alpha)\cos(\alpha)-\sin(\beta)\cos(\beta)\right)


Details

By
Hasse's theorem on elliptic curves Hasse's theorem on elliptic curves, also referred to as the Hasse bound, provides an estimate of the number of points on an elliptic curve over a finite field, bounding the value both above and below. If ''N'' is the number of points on the ellipt ...
, the ratio :\frac=\frac is between -1 and 1. Thus it can be expressed as cos ''θ'' for an angle ''θ''; in geometric terms there are two
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
accounting for the remainder and with the denominator as given they are
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
and of absolute value 1. The ''Sato–Tate conjecture'', when ''E'' doesn't have complex multiplication, states that the
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''. The difference between a probability measure and the more g ...
of ''θ'' is proportional to :\sin^2 \theta \, d\theta. This is due to
Mikio Sato is a Japanese mathematician known for founding the fields of algebraic analysis, hyperfunctions, and holonomic quantum fields. He is a professor at the Research Institute for Mathematical Sciences in Kyoto. Education Sato studied at the Unive ...
and John Tate (independently, and around 1960, published somewhat later).


Proof

In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over
totally real field In number theory, a number field ''F'' is called totally real if for each embedding of ''F'' into the complex numbers the image lies inside the real numbers. Equivalent conditions are that ''F'' is generated over Q by one root of an integer polyn ...
s satisfying a certain condition: of having multiplicative reduction at some prime, in a series of three joint papers. Further results are conditional on improved forms of the Arthur–Selberg trace formula. Harris has a
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. Overview The assumed antecedent of a conditional proof is called the condition ...
of a result for the product of two elliptic curves (not isogenous) following from such a hypothetical trace formula. 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, by improving the potential modularity results of previous papers. The prior issues involved with the trace formula were solved by Michael Harris, and Sug Woo Shin. In 2015, Richard Taylor was awarded the
Breakthrough Prize in Mathematics The Breakthrough Prize in Mathematics is an annual award of the Breakthrough Prize series announced in 2013. It is funded by Yuri Milner and Mark Zuckerberg and others. The annual award comes with a cash gift of $3 million. The Breakthrough Priz ...
"for numerous breakthrough results in (...) the Sato–Tate conjecture."


Generalisations

There are generalisations, involving the distribution of
Frobenius element In commutative algebra and field theory, the Frobenius endomorphism (after Ferdinand Georg Frobenius) is a special endomorphism of commutative rings with prime characteristic , an important class which includes finite fields. The endomorphis ...
s in
Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
s involved in the Galois representations on
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 conject ...
. In particular there is a conjectural theory for curves of genus ''n'' > 1. Under the random matrix model developed by
Nick Katz Nicholas Michael Katz (born December 7, 1943) is an American mathematician, working in arithmetic geometry, particularly on ''p''-adic methods, monodromy and moduli problems, and number theory. He is currently a professor of Mathematics at P ...
and
Peter Sarnak Peter Clive Sarnak (born 18 December 1953) is a South African-born mathematician with dual South-African and American nationalities. Sarnak has been a member of the permanent faculty of the School of Mathematics at the Institute for Advanced St ...
, there is a conjectural correspondence between (unitarized) characteristic polynomials of Frobenius elements and
conjugacy class In mathematics, especially group theory, two elements a and b of a group are conjugate if there is an element g in the group such that b = gag^. This is an equivalence relation whose equivalence classes are called conjugacy classes. In other w ...
es in the
compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
USp(2''n'') =  Sp(''n''). The Haar measure on USp(2''n'') then gives the conjectured distribution, and the classical case is USp(2) = 
SU(2) In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1. The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the speci ...
.


Refinements

There are also more refined statements. The Lang–Trotter conjecture (1976) of
Serge Lang Serge Lang (; May 19, 1927 – September 12, 2005) was a French-American mathematician and activist who taught at Yale University for most of his career. He is known for his work in number theory and for his mathematics textbooks, including the i ...
and Hale Trotter states 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 In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visibl ...
, trace ≠ 0) their formula states that the number of ''p'' up to ''X'' is asymptotically :c \sqrt/ \log X\ with a specified constant ''c''.
Neal Koblitz Neal I. Koblitz (born December 24, 1948) is a Professor of Mathematics at the University of Washington. He is also an adjunct professor with the Centre for Applied Cryptographic Research at the University of Waterloo. He is the creator of hypere ...
(1988) provided detailed conjectures for the case of a prime number ''q'' of points on ''E''''p'', motivated by
elliptic curve cryptography Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide e ...
.. In 1999,
Chantal David Chantal David (born 1964) is a French Canadian mathematician who works as a professor of mathematics at Concordia University. Her interests include analytic number theory, arithmetic statistics, and random matrix theory, and she has shown inter ...
and
Francesco Pappalardi Francesco, the Italian (and original) version of the personal name "Francis", is the most common given name among males in Italy. Notable persons with that name include: People with the given name Francesco * Francesco I (disambiguation), seve ...
proved an averaged version of the Lang–Trotter conjecture.


References


External links


Report on Barry Mazur giving contextMichael Harris notes, with statement (PDF)''La Conjecture de Sato–Tate'' [d'après Clozel, Harris, Shepherd-Barron, Taylor], Bourbaki seminar June 2007 by Henri Carayol (PDF)

Video introducing Elliptic curves and its relation to Sato-Tate conjecture, Imperial College London, 2014
(Last 15 minutes) {{DEFAULTSORT:Sato-Tate conjecture Elliptic curves Finite fields Conjectures Unsolved problems in geometry