Birch and Swinnerton-Dyer conjecture
Encyclopedia
In mathematics
, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory
. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems
listed by the Clay Mathematics Institute
, which has offered a $1,000,000 prize for the first correct proof. , only special cases of the conjecture have been proved correct.
The conjecture relates arithmetic data associated to an elliptic curve
E over a number field K to the behaviour of the Hasse-Weil L-function L(E, s) of E at s = 1. More specifically, it is conjectured that the rank
of the abelian group
E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K.
proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis
. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve, from which all further rational points may be generated.
If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is called the rank
of the curve, and is an important invariant
property of an elliptic curve.
If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points.
Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) these cannot be generalised to handle all curves.
An L-function L(E, s) can be defined for an elliptic curve E by constructing an Euler product
from the number of points on the curve modulo each prime
p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form
. It is a special case of a Hasse-Weil L-function.
The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2. Helmut Hasse
conjectured that L(E, s) could be extended by analytic continuation
to the whole complex plane. This conjecture was first proved by Max Deuring
for elliptic curves with complex multiplication
. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem.
Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
used the EDSAC
computer at the University of Cambridge Computer Laboratory
to calculate the number of points modulo p (denoted by Np) for a large number of primes p on elliptic curves whose rank was known. From these numerical results Bryan Birch
and Swinnerton-Dyer conjectured that Np for a curve E with rank r obeys an asymptotic law
where C is a constant.
Initially this was based on somewhat tenuous trends in graphical plots; which induced a measure of skepticism in J. W. S. Cassels
(Birch's Ph.D. advisor). Over time the numerical evidence stacked up.
This in turn led them to make a general conjecture about the behaviour of a curve's L-function L(E, s) at s = 1, namely that it would have a zero of order r at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of L(E, s) there was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the reciprocal of the L-function is from some points of view a more natural object of study; on occasion this means that one should consider poles rather than zeroes.)
The conjecture was subsequently extended to include the prediction of the precise leading Taylor coefficient of the L-function at s = 1. It is conjecturally given by
where the quantities on the right hand side are invariants of the curve, studied by Cassels, Tate
, Shafarevich
and others: these include the order of the torsion group, the order of the Tate–Shafarevich group, and the canonical heights of a basis of rational points.
Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.
, which is offering a prize of $1 million for the first proof or disproof of the whole conjecture.
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 Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems
Millennium Prize Problems
The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000. As of September 2011, six of the problems remain unsolved. A correct solution to any of the problems results in a US$1,000,000 prize being awarded by the institute...
listed by the Clay Mathematics Institute
Clay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
, which has offered a $1,000,000 prize for the first correct proof. , only special cases of the conjecture have been proved correct.
The conjecture relates arithmetic data associated to an 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...
E over a number field K to the behaviour of the Hasse-Weil L-function L(E, s) of E at s = 1. More specifically, it is conjectured that the rank
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
of the abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
E(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylor expansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K.
Background
In 1922 Louis MordellLouis Mordell
Louis Joel Mordell was a British mathematician, known for pioneering research in number theory. He was born in Philadelphia, USA, in a Jewish family of Lithuanian extraction...
proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis
Basis (linear algebra)
In linear algebra, a basis is a set of linearly independent vectors that, in a linear combination, can represent every vector in a given vector space or free module, or, more simply put, which define a "coordinate system"...
. This means that for any elliptic curve there is a finite sub-set of the rational points on the curve, from which all further rational points may be generated.
If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. The number of independent basis points with infinite order is called the rank
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
of the curve, and is an important invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
property of an elliptic curve.
If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points.
Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) these cannot be generalised to handle all curves.
An L-function L(E, s) can be defined for an elliptic curve E by constructing an Euler product
Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The name arose from the case of the Riemann zeta-function, where such a product representation was proved by Leonhard Euler.-Definition:...
from the number of points on the curve modulo each prime
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...
p. This L-function is analogous to the Riemann zeta function and the Dirichlet L-series that is defined for a binary quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
. It is a special case of a Hasse-Weil L-function.
The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2. Helmut Hasse
Helmut Hasse
Helmut Hasse was a German mathematician working in algebraic number theory, known for fundamental contributions to class field theory, the application of p-adic numbers to local classfield theory and diophantine geometry , and to local zeta functions.-Life:He was born in Kassel, and died in...
conjectured that L(E, s) could be extended by analytic continuation
Analytic continuation
In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an infinite series representation in terms of which...
to the whole complex plane. This conjecture was first proved by Max Deuring
Max Deuring
Max Deuring was a mathematician. He is known for his work in arithmetic geometry, in particular on elliptic curves in characteristic p...
for elliptic curves with 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...
. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem.
Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
History
In the early 1960s Peter Swinnerton-DyerPeter Swinnerton-Dyer
Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet KBE FRS , commonly known as Peter Swinnerton-Dyer, is an English mathematician specialising in number theory at University of Cambridge...
used the EDSAC
EDSAC
Electronic Delay Storage Automatic Calculator was an early British computer. The machine, having been inspired by John von Neumann's seminal First Draft of a Report on the EDVAC, was constructed by Maurice Wilkes and his team at the University of Cambridge Mathematical Laboratory in England...
computer at the University of Cambridge Computer Laboratory
University of Cambridge Computer Laboratory
The Computer Laboratory is the computer science department of the University of Cambridge. As of 2007, it employs 35 academic staff, 25 support staff, 35 affiliated research staff, and about 155 research students...
to calculate the number of points modulo p (denoted by Np) for a large number of primes p on elliptic curves whose rank was known. From these numerical results Bryan Birch
Bryan John Birch
Bryan John Birch F.R.S. is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture....
and Swinnerton-Dyer conjectured that Np for a curve E with rank r obeys an asymptotic law
where C is a constant.
Initially this was based on somewhat tenuous trends in graphical plots; which induced a measure of skepticism in J. W. S. Cassels
J. W. S. Cassels
John William Scott Cassels , FRS is a leading English mathematician.-Biography:Educated at Neville's Cross Council School in Durham and George Heriot's School in Edinburgh, Cassels graduated from the University of Edinburgh with an MA in 1943.His academic career was interrupted in World War II...
(Birch's Ph.D. advisor). Over time the numerical evidence stacked up.
This in turn led them to make a general conjecture about the behaviour of a curve's L-function L(E, s) at s = 1, namely that it would have a zero of order r at this point. This was a far-sighted conjecture for the time, given that the analytic continuation of L(E, s) there was only established for curves with complex multiplication, which were also the main source of numerical examples. (NB that the reciprocal of the L-function is from some points of view a more natural object of study; on occasion this means that one should consider poles rather than zeroes.)
The conjecture was subsequently extended to include the prediction of the precise leading Taylor coefficient of the L-function at s = 1. It is conjecturally given by
where the quantities on the right hand side are invariants of the curve, studied by Cassels, 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:...
, Shafarevich
Igor Shafarevich
Igor Rostislavovich Shafarevich is a Soviet and Russian mathematician, founder of a school of algebraic number theory and algebraic geometry in the USSR, and a political writer. He was also an important dissident figure under the Soviet regime, a public supporter of Andrei Sakharov's Human Rights...
and others: these include the order of the torsion group, the order of the Tate–Shafarevich group, and the canonical heights of a basis of rational points.
Current status
The Birch and Swinnerton-Dyer conjecture has been proved only in special cases:- In 1976, John Coates and Andrew WilesAndrew WilesSir Andrew John Wiles KBE FRS is a British mathematician and a Royal Society Research Professor at Oxford University, specializing in number theory...
proved that if E is a curve over a number field F with complex multiplication by an imaginary quadratic field K of class number 1, F = K or Q, and L(E, 1) is not 0 then E(F) is a finite group. This was extended to the case where F is any finite abelian extensionAbelian extensionIn abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
of K by Nicole Arthaud-Kuhman. - In 1983, Benedict GrossBenedict GrossBenedict Hyman Gross is an American mathematician, the George Vasmer Leverett Professor of Mathematics at Harvard University and former Dean of Harvard College....
and Don ZagierDon ZagierDon Bernard Zagier is an American mathematician whose main area of work is number theory. He is currently one of the directors of the Max Planck Institute for Mathematics in Bonn, Germany, and a professor at the Collège de France in Paris, France.He was born in Heidelberg, Germany...
showed that if a modular elliptic curveModular elliptic curveA modular elliptic curve is an elliptic curve E that admits a parametrisation X0 → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve...
has a first-order zero at s = 1 then it has a rational point of infinite order; see Gross–Zagier theorem. - In 1990, Victor KolyvaginVictor KolyvaginVictor Alexandrovich Kolyvagin is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerton-Dyer conjecture, and Iwasawa's conjecture for cyclotomic fields. His work also influenced Andrew Wiles's work on Fermat's Theorem.Kolyvagin...
showed that a modular elliptic curve E for which L(E,1) is not zero has rank 0, and a modular elliptic curve E for which L(E,1) has a first-order zero at s = 1 has rank 1. - In 1991, Karl RubinKarl RubinKarl Rubin is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. His research interest is in elliptic curves. He was the first mathematician to show that some elliptic curves over the rationals have finite Tate-Shafarevich groups...
showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s=1, then the p-part of the Tate–Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7. - In 2001, Christophe BreuilChristophe BreuilChristophe Breuil is a French mathematician, who works in algebraic geometry and number theory.-Academic life:Breuil attended schools in Brive-la-Gaillarde and Toulouse and studied from 1990 to 1992 at the Ecole Polytechnique....
, Brian ConradBrian ConradBrian Conrad , is an American mathematician and number theorist, working at Stanford University. Previously he was at the University of Michigan....
, Fred DiamondFred DiamondFred Diamond is an American mathematician, known for his role in proving the modularity theorem for elliptic curves. His research interest is in modular forms and Galois representations....
and Richard TaylorRichard Taylor (mathematician)-External links:**...
, extending work of Wiles, proved that all elliptic curves defined over the rational numbers are modular (the Taniyama–Shimura theorem), which extends results 2 and 3 to all elliptic curves over the rationals, and shows that the L-functions of all elliptic curves over Q are defined at s = 1. - In 2010, Manjul BhargavaManjul BhargavaManjul Bhargava is a Canadian-American mathematician of Indian origin. He is the R. Brandon Fradd Professor of Mathematics at Princeton University...
and Arul Shankar announced a proof that the average rank of the Mordell–Weil group of an elliptic curve over Q is bounded above by 7/6. Combining this with the announced proof of the main conjecture of Iwasawa theory for GL(2) by Chris Skinner and Éric Urban, they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by Kolyvagin's result, satisfy the Birch and Swinnerton-Dyer conjecture.
Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for the truth of the conjecture.
Clay Mathematics Institute Prize
The Birch and Swinnerton-Dyer conjecture is one of the seven Millennium Problems selected by the Clay Mathematics InstituteClay Mathematics Institute
The Clay Mathematics Institute is a private, non-profit foundation, based in Cambridge, Massachusetts. The Institute is dedicated to increasing and disseminating mathematical knowledge. It gives out various awards and sponsorships to promising mathematicians. The institute was founded in 1998...
, which is offering a prize of $1 million for the first proof or disproof of the whole conjecture.
External links
- The Birch and Swinnerton-Dyer Conjecture: An Interview with Professor Henri DarmonHenri DarmonHenri Rene Darmon is a French Canadian mathematician specializing in number theory. He works on Hilbert's 12th problem and its relation with the Birch-Swinnerton-Dyer conjecture...
by Agnes F. Beaudry