55 relations: Abelian extension, Abelian group, Algebraic number field, Analytic continuation, Annals of Mathematics, Basis (linear algebra), Bryan John Birch, Clay Mathematics Institute, Complex multiplication, Compositio Mathematica, Computer Laboratory, University of Cambridge, Congruent number, Crelle's Journal, Dirichlet L-function, EDSAC 2, Elliptic curve, Euler product, Generalized Riemann hypothesis, Hasse–Weil zeta function, Heegner point, Helmut Hasse, Henri Darmon, Ideal class group, Igor Shafarevich, Infinite set, Invariant (mathematics), Inventiones Mathematicae, J. W. S. Cassels, John Tate, Journal of the American Mathematical Society, Main conjecture of Iwasawa theory, Manjul Bhargava, Mathematical Proceedings of the Cambridge Philosophical Society, Mathematics, Millennium Prize Problems, Modular elliptic curve, Modularity theorem, Mordell–Weil theorem, Multiplicative inverse, Néron–Tate height, Number theory, Peter Swinnerton-Dyer, Prime number, Quadratic field, Quadratic form, Rank of an abelian group, Rational point, Riemann hypothesis, Riemann zeta function, Springer Science+Business Media, ..., Square-free element, Tate–Shafarevich group, Taylor series, Torsion group, Tunnell's theorem. Expand index (5 more) »

## Abelian extension

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian.

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## 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 the order in which they are written.

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## Algebraic number field

In mathematics, an algebraic number field (or simply number field) F is a finite degree (and hence algebraic) field extension of the field of rational numbers Q. Thus F is a field that contains Q and has finite dimension when considered as a vector space over Q. The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.

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## Analytic continuation

In complex analysis, a branch of mathematics, analytic continuation is a technique to extend the domain of a given analytic function.

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## Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

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## Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

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## Bryan John Birch

Bryan John Birch F.R.S. (born 25 September 1931) is a British mathematician.

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## Clay Mathematics Institute

The Clay Mathematics Institute (CMI) is a private, non-profit foundation, based in Peterborough, New Hampshire, United States.

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## Complex multiplication

In mathematics, complex multiplication (CM) 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 (it roughly means that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules).

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## Compositio Mathematica

Compositio Mathematica is a bimonthly peer-reviewed mathematics journal established by L.E.J. Brouwer in 1935.

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## Computer Laboratory, University of Cambridge

The Computer Laboratory is the computer science department of the University of Cambridge.

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## Congruent number

In mathematics, a congruent number is a positive integer that is the area of a right triangle with three rational number sides.

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## Crelle's Journal

Crelle's Journal, or just Crelle, is the common name for a mathematics journal, the Journal für die reine und angewandte Mathematik (in English: Journal for Pure and Applied Mathematics).

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## Dirichlet L-function

In mathematics, a Dirichlet L-series is a function of the form Here χ is a Dirichlet character and s a complex variable with real part greater than 1.

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## EDSAC 2

EDSAC 2 was an early computer (operational in 1958), the successor to the Electronic Delay Storage Automatic Calculator (EDSAC).

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## Elliptic curve

In mathematics, an elliptic curve is a plane algebraic curve defined by an equation of the form which is non-singular; that is, the curve has no cusps or self-intersections.

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## Euler product

In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.

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## Generalized Riemann hypothesis

The Riemann hypothesis is one of the most important conjectures in mathematics.

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## Hasse–Weil zeta function

In mathematics, the Hasse–Weil zeta function attached to an algebraic variety V defined over an algebraic number field K is one of the two most important types of L-function.

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## Heegner point

In mathematics, a Heegner point is a point on a modular curve that is the image of a quadratic imaginary point of the upper half-plane.

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## 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 diophantine geometry (Hasse principle), and to local zeta functions.

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## Henri Darmon

Henri Rene Darmon (born 22 October 1965) is a French Canadian mathematician specializing in number theory.

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## Ideal class group

In number theory, the ideal class group (or class group) of an algebraic number field is the quotient group where is the group of fractional ideals of the ring of integers of, and is its subgroup of principal ideals.

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## Igor Shafarevich

Igor Rostislavovich Shafarevich (И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Russian mathematician who contributed to algebraic number theory and algebraic geometry.

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## Infinite set

In set theory, an infinite set is a set that is not a finite set.

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## Invariant (mathematics)

In mathematics, an invariant is a property, held by a class of mathematical objects, which remains unchanged when transformations of a certain type are applied to the objects.

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## Inventiones Mathematicae

Inventiones Mathematicae is a mathematical journal published monthly by Springer Science+Business Media.

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## J. W. S. Cassels

John William Scott "Ian" Cassels, FRS (11 July 1922 – 27 July 2015) was a British mathematician.

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## John Tate

John Torrence Tate Jr. (born March 13, 1925) is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.

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## Journal of the American Mathematical Society

The Journal of the American Mathematical Society (JAMS), is a quarterly peer-reviewed mathematical journal published by the American Mathematical Society.

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## Main conjecture of Iwasawa theory

In mathematics, the main conjecture of Iwasawa theory is a deep relationship between ''p''-adic ''L''-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by.

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## Manjul Bhargava

Manjul Bhargava (born 8 August 1974) is a Canadian-American mathematician of Indian origins.

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## Mathematical Proceedings of the Cambridge Philosophical Society

Mathematical Proceedings of the Cambridge Philosophical Society is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society.

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## Mathematics

Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.

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## Millennium Prize Problems

The Millennium Prize Problems are seven problems in mathematics that were stated by the Clay Mathematics Institute in 2000.

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## Modular elliptic curve

A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve.

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## Modularity theorem

In mathematics, the modularity theorem (formerly called the Taniyama–Shimura conjecture or related names like Taniyama–Shimura–Weil conjecture due to rediscovery) states that elliptic curves over the field of rational numbers are related to modular forms.

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## Mordell–Weil theorem

In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell-Weil group.

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## Multiplicative inverse

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1.

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## Néron–Tate height

In number theory, the Néron–Tate height (or canonical height) is a quadratic form on the Mordell-Weil group of rational points of an abelian variety defined over a global field.

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## Number theory

Number theory, or in older usage arithmetic, is a branch of pure mathematics devoted primarily to the study of the integers.

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## Peter Swinnerton-Dyer

Sir Henry Peter Francis Swinnerton-Dyer, 16th Baronet (born 2 August 1927), commonly known as Peter Swinnerton-Dyer, is an English mathematician specialising in number theory at University of Cambridge.

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## Prime number

A prime number (or a prime) is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers.

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## Quadratic field

In algebraic number theory, a quadratic field is an algebraic number field K of degree two over Q, the rational numbers.

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## Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

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## 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.

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## Rational point

In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field.

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## Riemann hypothesis

In mathematics, the Riemann hypothesis is a conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part.

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## Riemann zeta function

The Riemann zeta function or Euler–Riemann zeta function,, is a function of a complex variable s that analytically continues the sum of the Dirichlet series which converges when the real part of is greater than 1.

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## Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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## Square-free element

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.

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## Tate–Shafarevich group

In arithmetic geometry, the Tate–Shafarevich group Ш(A/K), introduced by and, of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group WC(A/K).

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## Taylor series

In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point.

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## Torsion group

In group theory, a branch of mathematics, a torsion group or a periodic group is a group in which each element has finite order.

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## Tunnell's theorem

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.

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## Redirects here:

BSD conjecture, Birch Swinnerton-Dyer Conjecture, Birch-Swinnerton-Dyer conjecture, Birch-swinnerton-dyer conjecture, Birch–Swinnerton-Dyer conjecture, Birch–Swinnerton–Dyer conjecture, Conjecture of Birch and Swinnerton-Dyer, Rank conjecture.

## References

[1] https://en.wikipedia.org/wiki/Birch_and_Swinnerton-Dyer_conjecture