Birch and Swinnerton-Dyer conjecture

Birch–Swinnerton-Dyer conjectureBirch-Swinnerton-Dyer conjectureconjecture of Birch and Swinnerton-DyerBirch–Swinnerton–Dyer conjectureBSD conjectureconjecture of Birch and Swinnerton–Dyer
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve.wikipedia
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Millennium Prize Problems

Millennium ProblemsMillennium PrizeMillennium problem
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.
The problems are the Birch and Swinnerton-Dyer conjecture, Hodge conjecture, Navier–Stokes existence and smoothness, P versus NP problem, Poincaré conjecture, Riemann hypothesis, and Yang–Mills existence and mass gap.

Bryan John Birch

Bryan BirchB. J. BirchBirch, Bryan John
It is named after mathematicians Bryan Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation.
His name has been given to the Birch and Swinnerton-Dyer conjecture.

Peter Swinnerton-Dyer

Henry Peter Francis Swinnerton-DyerSir Peter Swinnerton-DyerH. P. F. Swinnerton-Dyer
It is named after mathematicians Bryan Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation.
As a mathematician he was best known for his part in the Birch and Swinnerton-Dyer conjecture relating algebraic properties of elliptic curves to special values of L-functions, which was developed with Bryan Birch during the first half of the 1960s with the help of machine computation, and for his work on the Titan operating system.

Elliptic curve

elliptic curveselliptic equationWeierstrass equation
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve.
The Birch and Swinnerton-Dyer conjecture is concerned with determining the rank.

Clay Mathematics Institute

Clay Research FellowClay InstituteClay Research Fellowship
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.

Rational point

rational pointsrationalk''-rational point
proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis.
That would follow from the conjecture that the Tate–Shafarevich group is finite, or from the related Birch–Swinnerton-Dyer conjecture.

Mordell–Weil theorem

Mordell's theoremMordell–Weil groupMordell-Weil group
proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis.

Néron–Tate height

canonical heightheight pairingheight
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.
(However, the height that naturally appears in the statement of the Birch and Swinnerton-Dyer conjecture is twice this height.) On abelian varieties of higher dimension, there need not be a particular choice of smallest ample line bundle to be used in defining the Néron–Tate height, and the height used in the statement of the Birch–Swinnerton-Dyer conjecture is the Néron–Tate height associated to the Poincaré line bundle on, the product of A with its dual.

Henri Darmon

Darmon, Henri
He works on Hilbert's 12th problem and its relation with the Birch-Swinnerton-Dyer conjecture.

Tunnell's theorem

theoremTunnell
. 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
In number theory, Tunnell's theorem gives a partial resolution to the congruent number problem, and under the Birch and Swinnerton-Dyer conjecture, a full resolution.

Congruent number

Congruent number problem
Assuming the Birch and Swinnerton-Dyer conjecture, n is the area of a right triangle with rational side lengths (a congruent number) if and only if the number of triplets of integers (x, y, z) satisfying
Tunnell's theorem provides an easily testable criterion for determining whether a number is congruent; but his result relies on the Birch and Swinnerton-Dyer conjecture, which is still unproven.

Riemann hypothesis

Critical line theoremcritical line1st
Much like the Riemann hypothesis, this conjecture has multiple consequences, including the following two:
Other examples of zeta functions with multiple zeros are the L-functions of some elliptic curves: these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.

Manjul Bhargava

BhargavaManjul BharğavaBhargava, Manjul
In 2015 Manjul Bhargava and Arul Shankar proved the Birch and Swinnerton-Dyer conjecture for a positive proportion of elliptic curves.

Tate–Shafarevich group

Tate-Shafarevich groupShafarevich groupShafarevich-Tate group
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.
*Birch and Swinnerton-Dyer conjecture

Heegner point

Gross–Zagier theoremGross-Zagier theoremGross–Zagier formula
Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic.

Mathematics

mathematicalmathmathematician
In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve.

Number theory

number theoristcombinatorial number theorytheory of numbers
It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.

Algebraic number field

number fieldnumber fieldsalgebraic number fields
The modern formulation of the conjecture relates arithmetic data associated with 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.

Hasse–Weil zeta function

Hasse–Weil L-functionHasse–Weil ''L''-functionHasse–Weil zeta-function
The modern formulation of the conjecture relates arithmetic data associated with 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.

Rank of an abelian group

rankrank-2rank of the abelian group
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. 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.

Abelian group

abelianabelian groupsadditive group
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.

Generating set of a group

generatedgeneratorsgenerator
proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis.

Infinite set

infiniteinfinitelyinfinitely many
If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order.

Invariant (mathematics)

invariantinvariantsinvariance
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.

Euler product

Euler factorEuler's product formulaproduct of Euler factors
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.