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Taniyama Group
In mathematics, the Taniyama group is a group that is an extension of the absolute Galois group of the rationals by the Serre group. It was introduced by using an observation by Deligne, and named after Yutaka Taniyama. It was intended to be the group scheme In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in ... whose representations correspond to the (hypothetical) CM motives over the field Q of rational numbers. References * * Algebraic groups Langlands program {{math-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group (mathematics)
In mathematics, a group is a Set (mathematics), set and an Binary operation, operation that combines any two Element (mathematics), elements of the set to produce a third element of the set, in such a way that the operation is Associative property, associative, an identity element exists and every element has an Inverse element, inverse. These three axioms hold for Number#Main classification, number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and the axioms that define it were elaborated for handling, in a unified way, essential structural properties of very different mathematical entities such as numbers, geometric shapes and polynomial roots. Because the concept of groups is ubiquitous in numerous areas both within and outside mathematics, some authors consider it as a central organizing principle of contemporary mathematics. In geometry groups arise naturally in the study of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Extension
In mathematics, a group extension is a general means of describing a group in terms of a particular normal subgroup and quotient group. If Q and N are two groups, then G is an extension of Q by N if there is a short exact sequence :1\to N\;\overset\;G\;\overset\;Q \to 1. If G is an extension of Q by N, then G is a group, \iota(N) is a normal subgroup of G and the quotient group G/\iota(N) is isomorphic to the group Q. Group extensions arise in the context of the extension problem, where the groups Q and N are known and the properties of G are to be determined. Note that the phrasing "G is an extension of N by Q" is also used by some. Since any finite group G possesses a maximal normal subgroup N with simple factor group G/N, all finite groups may be constructed as a series of extensions with finite simple groups. This fact was a motivation for completing the classification of finite simple groups. An extension is called a central extension if the subgroup N lies in the center o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Absolute Galois Group
In mathematics, the absolute Galois group ''GK'' of a field ''K'' is the Galois group of ''K''sep over ''K'', where ''K''sep is a separable closure of ''K''. Alternatively it is the group of all automorphisms of the algebraic closure of ''K'' that fix ''K''. The absolute Galois group is well-defined up to inner automorphism. It is a profinite group. (When ''K'' is a perfect field, ''K''sep is the same as an algebraic closure ''K''alg of ''K''. This holds e.g. for ''K'' of characteristic zero, or ''K'' a finite field.) Examples * The absolute Galois group of an algebraically closed field is trivial. * The absolute Galois group of the real numbers is a cyclic group of two elements (complex conjugation and the identity map), since C is the separable closure of R and ''C:Rnbsp;= 2. * The absolute Galois group of a finite field ''K'' is isomorphic to the group :: \hat = \varprojlim \mathbf/n\mathbf. (For the notation, see Inverse limit.) :The Frobenius automorphism Fr is a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rational Number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (e.g. ). The set of all rational numbers, also referred to as "the rationals", the field of rationals or the field of rational numbers is usually denoted by boldface , or blackboard bold \mathbb. A rational number is a real number. The real numbers that are rational are those whose decimal expansion either terminates after a finite number of digits (example: ), or eventually begins to repeat the same finite sequence of digits over and over (example: ). This statement is true not only in base 10, but also in every other integer base, such as the binary and hexadecimal ones (see ). A real number that is not rational is called irrational. Irrational numbers include , , , and . Since the set of rational numbers is countable, and the set of real numbers is uncountable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Serre Group
In mathematics, the Serre group ''S'' is the pro-algebraic group whose representations correspond to CM-motives over the algebraic closure of the rationals, or to polarizable rational Hodge structures with abelian Mumford–Tate groups. It is a projective limit of finite dimensional tori, so in particular is abelian. It was introduced by . It is a subgroup of the Taniyama group. There are two different but related groups called the Serre group, one the connected component of the identity in the other. This article is mainly about the connected group, usually called the Serre group but sometimes called the connected Serre group. In addition one can define Serre groups of algebraic 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 f ...s, and the Serre group is the inverse lim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne
Pierre René, Viscount Deligne (; born 3 October 1944) is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal. Early life and education Deligne was born in Etterbeek, attended school at Athénée Adolphe Max and studied at the Université libre de Bruxelles (ULB), writing a dissertation titled ''Théorème de Lefschetz et critères de dégénérescence de suites spectrales'' (Theorem of Lefschetz and criteria of degeneration of spectral sequences). He completed his doctorate at the University of Paris-Sud in Orsay 1972 under the supervision of Alexander Grothendieck, with a thesis titled ''Théorie de Hodge''. Career Starting in 1972, Deligne worked with Grothendieck at the Institut des Hautes Études Scientifiques (IHÉS) near Paris, initially on the generalization within scheme theory of Zariski's main theorem. In 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yutaka Taniyama
was a Japanese mathematician known for the Taniyama–Shimura conjecture. Contribution Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and refined case of this conjecture for elliptic curves over rationals is called the Taniyama–Shimura conjecture or the modularity theorem whose statement he subsequently refined in collaboration with Goro Shimura. The names Taniyama, Shimura and Weil have all been attached to this conjecture, but the idea is essentially due to Taniyama. “Taniyama's interests were in algebraic number theory and his fame is mainly due to two problems posed by him at the symposium on Algebraic Number Theory held in Tokyo and Nikko in 1955. His meeting with André Weil at this symposium was to have a major influence on Taniyama's work. These problems form the basis of a conjecture: every elliptic curve defined over the rational field is a factor of the Jacob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Scheme
In mathematics, a group scheme is a type of object from Algebraic geometry, algebraic geometry equipped with a composition law. Group schemes arise naturally as symmetries of Scheme (mathematics), schemes, and they generalize algebraic groups, in the sense that all algebraic groups have group scheme structure, but group schemes are not necessarily connected, smooth, or defined over a field. This extra generality allows one to study richer infinitesimal structures, and this can help one to understand and answer questions of arithmetic significance. The Category (mathematics), category of group schemes is somewhat better behaved than that of Group variety, group varieties, since all homomorphisms have Kernel (category theory), kernels, and there is a well-behaved deformation theory. Group schemes that are not algebraic groups play a significant role in arithmetic geometry and algebraic topology, since they come up in contexts of Galois representations and moduli problems. The ini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application. David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science. There is also the higher-dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Motive (algebraic Geometry)
In algebraic geometry, motives (or sometimes motifs, following French usage) is a theory proposed by Alexander Grothendieck in the 1960s to unify the vast array of similarly behaved cohomology theories such as singular cohomology, de Rham cohomology, etale cohomology, and crystalline cohomology. Philosophically, a "motif" is the "cohomology essence" of a variety. In the formulation of Grothendieck for smooth projective varieties, a motive is a triple (X, p, m), where ''X'' is a smooth projective variety, p: X \vdash X is an idempotent correspondence, and ''m'' an integer, however, such a triple contains almost no information outside the context of Grothendieck's category of pure motives, where a morphism from (X, p, m) to (Y, q, n) is given by a correspondence of degree n-m. A more object-focused approach is taken by Pierre Deligne in ''Le Groupe Fondamental de la Droite Projective Moins Trois Points''. In that article, a motive is a "system of realisations" – that is, a tupl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Groups
In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Many groups of geometric transformations are algebraic groups; for example, orthogonal groups, general linear groups, projective groups, Euclidean groups, etc. Many matrix groups are also algebraic. Other algebraic groups occur naturally in algebraic geometry, such as elliptic curves and Jacobian varieties. An important class of algebraic groups is given by the affine algebraic groups, those whose underlying algebraic variety is an affine variety; they are exactly the algebraic subgroups of the general linear group, and are therefore also called ''linear algebraic groups''. Another class is formed by the abelian varieties, which are the algebraic groups whose underlying variety is a projective variety. Chevalley's structure theorem s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
