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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] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting poin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Closure
In mathematics, particularly abstract algebra, an algebraic closure of a field ''K'' is an algebraic extension of ''K'' that is algebraically closed. It is one of many closures in mathematics. Using Zorn's lemmaMcCarthy (1991) p.21Kaplansky (1972) pp.74-76 or the weaker ultrafilter lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field ''K'' is unique up to an isomorphism that fixes every member of ''K''. Because of this essential uniqueness, we often speak of ''the'' algebraic closure of ''K'', rather than ''an'' algebraic closure of ''K''. The algebraic closure of a field ''K'' can be thought of as the largest algebraic extension of ''K''. To see this, note that if ''L'' is any algebraic extension of ''K'', then the algebraic closure of ''L'' is also an algebraic closure of ''K'', and so ''L'' is contained within the algebraic closure of ''K''. The algebraic closure of ''K'' is also the smallest algebraically closed ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mumford–Tate Group
In algebraic geometry, the Mumford–Tate group (or Hodge group) ''MT''(''F'') constructed from a Hodge structure ''F'' is a certain algebraic group ''G''. When ''F'' is given by a rational representation of an algebraic torus, the definition of ''G'' is as the Zariski closure of the image in the representation of the circle group, over the rational numbers. introduced Mumford–Tate groups over the complex numbers under the name of Hodge groups. introduced the ''p''-adic analogue of Mumford's construction for Hodge–Tate modules, using the work of on p-divisible groups, and named them Mumford–Tate groups. Formulation The algebraic torus ''T'' used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of ''a''+''bi'' on the basis of the complex numbers C over R: :\begin a & b \\ -b & a \end. The circle group inside this group of matrices is the unitary group ''U''(1). Ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Limit
In mathematics, the inverse limit (also called the projective limit) is a construction that allows one to "glue together" several related objects, the precise gluing process being specified by morphisms between the objects. Thus, inverse limits can be defined in any category although their existence depends on the category that is considered. They are a special case of the concept of limit in category theory. By working in the dual category, that is by reverting the arrows, an inverse limit becomes a direct limit or ''inductive limit'', and a ''limit'' becomes a colimit. Formal definition Algebraic objects We start with the definition of an inverse system (or projective system) of groups and homomorphisms. Let (I, \leq) be a directed poset (not all authors require ''I'' to be directed). Let (''A''''i'')''i''∈''I'' be a family of groups and suppose we have a family of homomorphisms f_: A_j \to A_i for all i \leq j (note the order) with the following properties: # f_ i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
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 field that contains \mathbb and has finite dimension when considered as a vector space over 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. This study reveals hidden structures behind usual rational numbers, by using algebraic methods. Definition Prerequisites The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplicat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ''F''. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory. A result of Emil Artin allows one to construct Galois extensions as follows: If ''E'' is a given field, and ''G'' is a finite group of automorphisms of ''E'' with fixed field ''F'', then ''E''/''F'' is a Galois extension. Characterization of Galois extensions An important theorem of Emil Artin states that for a finite extension E/F, each of the following statements is equivalent to the statement that E/F is Galois: *E/F is a normal extension and a separable extension. *E is a splitting field of a separable polynomial with coefficients in F. *, \!\operatorname(E/F), = :F that is, the numb ... [...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 theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
