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mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an
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 the ...
. It is an open problem in the field of
number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...
and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven. The modern formulation of the conjecture relates to arithmetic data associated with an elliptic curve ''E'' over a
number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a ...
''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 A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...
of the
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is commu ...
''E''(''K'') of points of ''E'' is the order of the zero of ''L''(''E'', ''s'') at ''s'' = 1. The first non-zero coefficient in the
Taylor expansion In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor ser ...
of ''L''(''E'', ''s'') at ''s'' = 1 is given by more refined arithmetic data attached to ''E'' over ''K'' . The conjecture was chosen as one of the seven
Millennium Prize Problems The Millennium Prize Problems are seven well-known complex mathematics, mathematical problems selected by the Clay Mathematics Institute in 2000. The Clay Institute has pledged a US $1 million prize for the first correct solution to each problem ...
listed by the Clay Mathematics Institute, which has offered a $1,000,000 prize for the first correct proof.


Background

proved Mordell's theorem: the group of
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
s on an elliptic curve has a finite basis. This means that for any elliptic curve there is a finite subset 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 Order, ORDER or Orders may refer to: * A socio-political or established or existing order, e.g. World order, Ancien Regime, Pax Britannica * Categorization, the process in which ideas and objects are recognized, differentiated, and understood ...
. The number of ''independent'' basis points with infinite order is called the
rank A rank is a position in a hierarchy. It can be formally recognized—for example, cardinal, chief executive officer, general, professor—or unofficial. People Formal ranks * Academic rank * Corporate title * Diplomatic rank * Hierarchy ...
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) it is unknown if these methods handle all curves. An ''L''-function ''L''(''E'', ''s'') can be defined for an elliptic curve ''E'' by constructing an
Euler product In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers. The original such product was given for the sum of all positive integers raised to a certain power as proven by Leonhard E ...
from the number of points on the curve modulo each
prime 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 ways ...
''p''. This ''L''-function is analogous to the
Riemann zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic c ...
and the Dirichlet L-series that is defined for a binary
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
. 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 (; 25 August 1898 – 26 December 1979) 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 class field theory and ...
conjectured that ''L''(''E'', ''s'') could be extended by
analytic continuation In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of definition of a given analytic function. Analytic continuation often succeeds in defining further values of a function, for example in a ne ...
to the whole complex plane. This conjecture was first proved by for elliptic curves with
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 visible wh ...
. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the
modularity theorem In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic c ...
in 2001. 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-Dyer used the EDSAC-2 computer at the
University of Cambridge Computer Laboratory The Department of Computer Science and Technology, formerly the Computer Laboratory, is the computer science department of the University of Cambridge. it employed 56 faculty members, 45 support staff, 105 research staff, and about 205 researc ...
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 conjectured that ''Np'' for a curve ''E'' with rank ''r'' obeys an asymptotic law :\prod_ \frac \approx C\log (x)^r \mbox x \rightarrow \infty where ''C'' is a constant. Initially, this was based on somewhat tenuous trends in graphical plots; this 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 behavior 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'') 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 :\frac = \frac where the quantities on the right-hand side are invariants of the curve, studied by Cassels,
Tate Tate is an institution that houses, in a network of four art galleries, the United Kingdom's national collection of British art, and international modern and contemporary art. It is not a government institution, but its main sponsor is the UK ...
, Shafarevich and others : \#E_ is the order of the
torsion group In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which every element has finite order. The exponent of such a group, if it exists, is the least common multiple of the orders of the elements. For exam ...
, \#\mathrm(E)= is the order of the
Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group \mathrm(A/K) = H^1(G_K, A), where G_K = \mathrm(K ...
, \Omega_E is the real period of ''E'' multiplied by the number of connected components of ''E'', R_E is the regulator of ''E'' which is defined via the canonical heights of a basis of rational points, c_p is the Tamagawa number of ''E'' at a prime ''p'' dividing the conductor ''N'' of ''E''. It can be found by Tate's algorithm. At the time of the inception of the conjecture little was known, not even the well-definedness of the left side (referred to as analytic) or the right side (referred to as algebraic) of this equation. John Tate expressed this in 1974 in a famous quote.
This remarkable conjecture relates the behavior of a function L at a point where it is not at present known to be defined to the order of a group which is not known to be finite!
By the
modularity theorem In number theory, the modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic c ...
proved in 2001 for elliptic curves over \mathbb the left side is now known to be well-defined and the finiteness of is known when additionally the analytic rank is at most 1, i.e., if L(E,s) vanishes at most to order 1 at s=1. Both parts remain open.


Current status

The Birch and Swinnerton-Dyer conjecture has been proved only in special cases: # 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 extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galoi ...
of ''K'' by . # showed that if a
modular elliptic curve A modular elliptic curve is an elliptic curve ''E'' that admits a parametrization ''X''0(''N'') → ''E'' by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called a ...
has a first-order zero at ''s'' = 1 then it has a rational point of infinite order; see Gross–Zagier theorem. # 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. # 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. # , extending work of , proved that all elliptic curves defined over the rational numbers are modular, 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. # proved 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 p-parity theorem of and and with the proof of the main conjecture of Iwasawa theory for GL(2) by , they conclude that a positive proportion of elliptic curves over Q have analytic rank zero, and hence, by , satisfy the Birch and Swinnerton-Dyer conjecture. There are currently no proofs involving curves with a rank greater than 1. There is extensive numerical evidence for the truth of the conjecture.


Consequences

Much like the
Riemann hypothesis In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part . Many consider it to be the most important unsolved problem in pure ...
, this conjecture has multiple consequences, including the following two: * Let be an odd
square-free {{no footnotes, date=December 2015 In mathematics, a square-free element is an element ''r'' of a unique factorization domain ''R'' that is not divisible by a non-trivial square. This means that every ''s'' such that s^2\mid r is a unit of ''R''. ...
integer. Assuming the Birch and Swinnerton-Dyer conjecture, is the area of a right triangle with rational side lengths (a
congruent number In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property. The sequence of (integer) cong ...
) if and only if the number of triplets of integers (, , ) satisfying is twice the number of triplets satisfying . This statement, due to Tunnell's theorem , is related to the fact that ''n'' is a congruent number if and only if the elliptic curve has a rational point of infinite order (thus, under the Birch and Swinnerton-Dyer conjecture, its -function has a zero at ). The interest in this statement is that the condition is easily verified. *In a different direction, certain analytic methods allow for an estimation of the order of zero in the center of the critical strip of families of ''L''-functions. Admitting the BSD conjecture, these estimations correspond to information about the rank of families of elliptic curves in question. For example: suppose the
generalized Riemann hypothesis The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whi ...
and the BSD conjecture, the average rank of curves given by is smaller than . *Because of the existence of the functional equation of the ''L''-function of an elliptic curve, BSD allows us to calculate the parity of the rank of an elliptic curve. This is a conjecture in its own right called the parity conjecture, and it relates the parity of the rank of an elliptic curve to its global root number. This leads to many explicit arithmetic phenomena which are yet to be proved unconditionally. For instance: **Every positive integer is a
congruent number In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property. The sequence of (integer) cong ...
. **The elliptic curve given by where has infinitely many solutions over \mathbb(\zeta_8). **Every positive rational number can be written in the form for and in \mathbb. **For every rational number , the elliptic curve given by has rank at least . **There are many more examples for elliptic curves over number fields.


Generalizations

There is a version of this conjecture for general
abelian varieties In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a smooth projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular f ...
over number fields. A version for abelian varieties over \mathbb is the following: : \lim_ \frac = \frac . All of the terms have the same meaning as for elliptic curves, except that the square of the order of the torsion needs to be replaced by the product \#A(\mathbb Q)_\cdot\#\hat A(\mathbb Q)_ involving the
dual abelian variety In mathematics, a dual abelian variety can be defined from an abelian variety ''A'', defined over a field (mathematics), field ''k''. A 1-dimensional abelian variety is an elliptic curve, and every elliptic curve is isomorphic to its dual, but this ...
\hat A. Elliptic curves as 1-dimensional abelian varieties are their own duals, i.e. \hat E = E, which simplifies the statement of the BSD conjecture. The regulator R_A needs to be understood for the pairing between a basis for the free parts of A(\mathbb Q) and \hat A(\mathbb Q) relative to the Poincare bundle on the product A\times\hat A. The rank-one Birch-Swinnerton-Dyer conjecture for modular elliptic curves and modular abelian varieties of GL(2)-type over totally real number fields was proved by Shou-Wu Zhang in 2001. Another generalization is given by the Bloch-Kato conjecture.


Notes


References

* * * * * * * * * * * * * * * * *


External links

* *
The Birch and Swinnerton-Dyer Conjecture
An Interview with Professor Henri Darmon by Agnes F. Beaudry
''What is the Birch and Swinnerton-Dyer Conjecture?''
lecture by
Manjul Bhargava Manjul Bhargava (born 8 August 1974) is a Canadian-American mathematician. He is the Brandon Fradd, Class of 1983, Professor of Mathematics at Princeton University, the Stieltjes Professor of Number Theory at Leiden University, and also holds A ...
(September 2016) given during the Clay Research Conference held at the University of Oxford {{DEFAULTSORT:Birch And Swinnerton-Dyer Conjecture Conjectures Diophantine geometry Millennium Prize Problems Number theory University of Cambridge Computer Laboratory Zeta and L-functions Unsolved problems in number theory