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45 relations: Adele ring, Adelic algebraic group, Albrecht Fröhlich, Algebraic number, Algebraic number field, Algebraic variety, Artin L-function, Artin reciprocity law, Characteristic polynomial, Circle group, Class field theory, Complex multiplication, Dedekind zeta function, Dirichlet character, Dirichlet L-function, Dirichlet series, Distribution (mathematics), Elliptic curve, Emil Artin, Erich Hecke, Fractional ideal, Frobenius endomorphism, Functional equation (L-function), Gaussian rational, Global field, Glossary of arithmetic and diophantine geometry, Graduate Texts in Mathematics, Hasse–Weil zeta function, Ideal norm, J. W. S. Cassels, John Tate, Kenkichi Iwasawa, L-function, Modulus (algebraic number theory), Motive (algebraic geometry), Number theory, Polish Scientific Publishers PWN, Pontryagin duality, Positive real numbers, Ray class field, Riemann zeta function, Séminaire Nicolas Bourbaki, Schwartz–Bruhat function, Springer Science+Business Media, Theta function.
Adele ring
In mathematics, the adele ring (also adelic ring or ring of adeles) is defined in class field theory, a branch of algebraic number theory.
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Adelic algebraic group
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group G over a number field K, and the adele ring A.
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Albrecht Fröhlich
Albrecht Fröhlich FRS (22 May 1916 – 8 November 2001) was a mathematician famous for his major results and conjectures on Galois module theory in the Galois structure of rings of integers.
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Algebraic number
An algebraic number is any complex number (including real numbers) that is a root of a non-zero polynomial (that is, a value which causes the polynomial to equal 0) in one variable with rational coefficients (or equivalently – by clearing denominators – with integer coefficients).
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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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Algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry.
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Artin L-function
In mathematics, an Artin L-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group G. These functions were introduced in the 1923 by Emil Artin, in connection with his research into class field theory.
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Artin reciprocity law
The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory.
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Characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.
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Circle group
In mathematics, the circle group, denoted by T, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers The circle group forms a subgroup of C×, the multiplicative group of all nonzero complex numbers.
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Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of local fields (one-dimensional local fields) and "global fields" (one-dimensional global fields) such as number fields and function fields of curves over finite fields in terms of abelian topological groups associated to the fields.
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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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Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK(s), is a generalization of the Riemann zeta function (which is obtained in the case where K is the rational numbers Q).
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Dirichlet character
In number theory, Dirichlet characters are certain arithmetic functions which arise from completely multiplicative characters on the units of \mathbb Z / k \mathbb Z. Dirichlet characters are used to define Dirichlet ''L''-functions, which are meromorphic functions with a variety of interesting analytic properties.
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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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Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.
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Distribution (mathematics)
Distributions (or generalized functions) are objects that generalize the classical notion of functions in mathematical analysis.
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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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Emil Artin
Emil Artin (March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
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Erich Hecke
Erich Hecke (20 September 1887 – 13 February 1947) was a German mathematician.
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Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains.
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Frobenius endomorphism
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.
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Functional equation (L-function)
In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations.
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Gaussian rational
In mathematics, a Gaussian rational number is a complex number of the form p + qi, where p and q are both rational numbers.
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Global field
In mathematics, a global field is a field that is either.
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Glossary of arithmetic and diophantine geometry
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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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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Ideal norm
In commutative algebra, the norm of an ideal is a generalization of a norm of an element in the field extension.
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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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Kenkichi Iwasawa
Kenkichi Iwasawa (岩澤 健吉 Iwasawa Kenkichi, September 11, 1917 – October 26, 1998) was a Japanese mathematician who is known for his influence on algebraic number theory.
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L-function
In mathematics, an L-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects.
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Modulus (algebraic number theory)
In mathematics, in the field of algebraic number theory, a modulus (plural moduli) (or cycle, or extended ideal) is a formal product of places of a global field (i.e. an algebraic number field or a global function field).
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Motive (algebraic geometry)
In algebraic geometry, a motive (or sometimes motif, following French usage) denotes 'some essential part of an algebraic variety'.
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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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Polish Scientific Publishers PWN
Wydawnictwo Naukowe PWN (Polish Scientific Publishers PWN; until 1991 Państwowe Wydawnictwo Naukowe - National Scientific Publishers PWN, PWN) is a Polish book publisher, founded in 1951.
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Pontryagin duality
In mathematics, specifically in harmonic analysis and the theory of topological groups, Pontryagin duality explains the general properties of the Fourier transform on locally compact abelian groups, such as \R, the circle, or finite cyclic groups.
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Positive real numbers
In mathematics, the set of positive real numbers, \mathbb_.
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Ray class field
In mathematics, a ray class field is an abelian extension of a global field associated with a ray class group of ideal classes or idele classes.
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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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Séminaire Nicolas Bourbaki
The Séminaire Nicolas Bourbaki (Bourbaki Seminar) is a series of seminars (in fact public lectures with printed notes distributed) that has been held in Paris since 1948.
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Schwartz–Bruhat function
In mathematics, a Schwartz–Bruhat function, named after Laurent Schwartz and François Bruhat, is a complex valued function on a locally compact abelian group, such as the adeles, that generalizes a Schwartz function on a real vector space.
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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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Theta function
In mathematics, theta functions are special functions of several complex variables.
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References
[1] https://en.wikipedia.org/wiki/Hecke_character
