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Class Formation
In mathematics, a class formation is a topological group acting on a module satisfying certain conditions. Class formations were introduced by Emil Artin and John Tate to organize the various Galois groups and modules that appear in class field theory. Definitions A formation is a topological group ''G'' together with a topological ''G''-module ''A'' on which ''G'' acts continuously. A layer ''E''/''F'' of a formation is a pair of open subgroups ''E'', ''F'' of ''G'' such that ''F'' is a finite index subgroup of ''E''. It is called a normal layer if ''F'' is a normal subgroup of ''E'', and a cyclic layer if in addition the quotient group is cyclic. If ''E'' is a subgroup of ''G'', then ''A''''E'' is defined to be the elements of ''A'' fixed by ''E''. We write :''H''''n''(''E''/''F'') for the Tate cohomology group ''H''''n''(''E''/''F'', ''A''''F'') whenever ''E''/''F'' is a normal layer. (Some authors think of ''E'' and ''F'' as fixed fields rather than subgroup of ''G'', ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Module (mathematics)
In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a (not necessarily commutative) ring. The concept of a ''module'' also generalizes the notion of an abelian group, since the abelian groups are exactly the modules over the ring of integers. Like a vector space, a module is an additive abelian group, and scalar multiplication is distributive over the operations of addition between elements of the ring or module and is compatible with the ring multiplication. Modules are very closely related to the representation theory of groups. They are also one of the central notions of commutative algebra and homological algebra, and are used widely in algebraic geometry and algebraic topology. Introduction and definition Motivation In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a founding member of the Bourbaki group. Life His father, Abel Chevalley, was a French diplomat who, jointly with his wife Marguerite Chevalley née Sabatier, wrote ''The Concise Oxford French Dictionary''. Chevalley graduated from the École Normale Supérieure in 1929, where he studied under Émile Picard. He then spent time at the University of Hamburg, studying under Emil Artin and at the University of Marburg, studying under Helmut Hasse. In Germany, Chevalley discovered Japanese mathematics in the person of Shokichi Iyanaga. Chevalley was awarded a doctorate in 1933 from the University of Paris for a thesis on class field theory. When World War II broke out, Chevalley was at Princeton University. After reporting to the French Embass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Extension
In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian group, abelian. When the Galois group is also cyclic group, cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable if its Galois group is solvable group, solvable, i.e., if the group can be decomposed into a series of normal Group extension, extensions of an abelian group. Every finite extension of a finite field is a cyclic extension. Description Class field theory provides detailed information about the abelian extensions of number fields, function field of an algebraic variety, function fields of algebraic curves over finite fields, and local fields. There are two slightly different definitions of the term cyclotomic extension. It can mean either an extension formed by adjoining roots of unity to a field, or a subextension of such an extension. The cyclotomic fields are examples. A cyclotomic extension, under either d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Langlands Program
In mathematics, the Langlands program is a set of conjectures about connections between number theory, the theory of automorphic forms, and geometry. It was proposed by . It seeks to relate the structure of Galois groups in algebraic number theory to automorphic forms and, more generally, the representation theory of algebraic groups over local fields and adeles. It was described by Edward Frenkel as the " grand unified theory of mathematics." Background The Langlands program is built on existing ideas: the philosophy of cusp forms formulated a few years earlier by Harish-Chandra and , the work and Harish-Chandra's approach on semisimple Lie groups, and in technical terms the trace formula of Selberg and others. What was new in Langlands' work, besides technical depth, was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-called functoriality). Harish-Chandra's work exploited the principle that what can be d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mordell–Weil Group
In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety A defined over a number field K. It is an arithmetic invariant of the Abelian variety. It is simply the group of K-points of A, so A(K) is the Mordell–Weil grouppg 207. The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of A(K) to the zero of the associated L-function at a special point. Examples Constructing explicit examples of the Mordell–Weil group of an abelian variety is a non-trivial process which is not always guaranteed to be successful, so we instead specialize to the case of a specific elliptic curve E/\mathbb. Let E be defined by the Weierstrass equationy^2 = x(x-6)(x+6)over the rational numbers. It has discriminant \Delta_E = 2^\cdot 3^6 (and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weil–Châtelet Group
In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety ''A'' defined over a field ''K'' is the abelian group of principal homogeneous spaces for ''A'', defined over ''K''. named it for who introduced it for elliptic curves, and , who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent. It can be defined directly from Galois cohomology, as H^1(G_K,A), where G_K is the absolute Galois group of ''K''. It is of particular interest for local fields and global fields, such as algebraic number fields. For ''K'' a finite field, proved that the Weil–Châtelet group is trivial for elliptic curves, and proved that it is trivial for any connected algebraic group. See also The Tate–Shafarevich group In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Group
In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections through the hyperplanes orthogonal to at least one of the roots, and as such is a finite reflection group. In fact it turns out that ''most'' finite reflection groups are Weyl groups. Abstractly, Weyl groups are finite Coxeter groups, and are important examples of these. The Weyl group of a semisimple Lie group, a semisimple Lie algebra, a semisimple linear algebraic group, etc. is the Weyl group of the root system of that group or algebra. Definition and examples Let \Phi be a root system in a Euclidean space V. For each root \alpha\in\Phi, let s_\alpha denote the reflection about the hyperplane perpendicular to \alpha, which is given explicitly as :s_\alpha(v)=v-2\frac\alpha, where (\cdot,\cdot) is the inner product on V. The Weyl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lubin–Tate Formal Group Law
In mathematics, the Lubin–Tate formal group law is a formal group law introduced by to isolate the local field part of the classical theory of complex multiplication of elliptic functions. In particular it can be used to construct the totally ramified abelian extensions of a local field. It does this by considering the (formal) endomorphisms of the formal group, emulating the way in which elliptic curves with extra endomorphisms are used to give abelian extensions of global fields. Definition of formal groups Let Z''p'' be the ring of ''p''-adic integers. The Lubin–Tate formal group law is the unique (1-dimensional) formal group law ''F'' such that ''e''(''x'') = ''px'' + ''x''''p'' is an endomorphism of ''F'', in other words :e(F(x,y)) = F(e(x), e(y)).\ More generally, the choice for ''e'' may be any power series such that :''e''(''x'') = ''px'' + higher-degree terms and :''e''(''x'') = ''x''''p'' mod ''p''. All such group ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Extension
Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist * Clare Kummer (1873–1958), American composer, lyricist and playwright * Clarence Kummer (1899–1930), American jockey * Christopher Kummer (born 1975), German economist * Corby Kummer (born 1957), American journalist * Dirk Kummer (born 1966), German actor, director, and screenwriter * Eberhard Kummer (1940–2019), Austrian concert singer, lawyer, and medieval music expert * Eduard Kummer, also known as the following Ernst Kummer * Eloise Kummer (1916–2008), American actress *Ernst Kummer (1810–1893), German mathematician ** Kummer configuration, a mathematical structure discovered by Ernst Kummer ** Kummer surface, a related geometrical structure discovered by Ernst Kummer * Ferdinand von Kummer (1816–1900), German general * Frederic Arnold Kummer (1873–1943), American author, playwright, and screenwriter * Friedrich August Kummer (1797–1879), German ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Takagi Existence Theorem
{{short description, Correspondence between finite abelian extensions and generalized ideal class groups In class field theory, the Takagi existence theorem states that for any number field ''K'' there is a one-to-one inclusion reversing correspondence between the finite abelian extensions of ''K'' (in a fixed algebraic closure of ''K'') and the generalized ideal class groups defined via a modulus of ''K''. It is called an existence theorem because a main burden of the proof is to show the existence of enough abelian extensions of ''K''. Formulation Here a modulus (or ''ray divisor'') is a formal finite product of the valuations (also called primes or places) of ''K'' with positive integer exponents. The archimedean valuations that might appear in a modulus include only those whose completions are the real numbers (not the complex numbers); they may be identified with orderings on ''K'' and occur only to exponent one. The modulus ''m'' is a product of a non-archimedean (finite) p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate's Theorem
In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence. They were introduced by , and are used in class field theory. Definition If ''G'' is a finite group and ''A'' a ''G''-module, then there is a natural map ''N'' from H_0(G,A) to H^0(G,A) taking a representative ''a'' to \sum_ ga (the sum over all ''G''-conjugates of ''a''). The Tate cohomology groups \hat H^n(G,A) are defined by *\hat H^n(G,A) = H^n(G,A) for n\ge 1, *\hat H^0(G,A)=\operatorname N= quotient of H^0(G,A) by norms of elements of ''A'', *\hat H^(G,A)=\ker N= quotient of norm 0 elements of ''A'' by principal elements of ''A'', *\hat H^(G,A) = H_(G,A) for n\le -2. Properties * If :: 0 \longrightarrow A \longrightarrow B \longrightarrow C \longrightarrow 0 :is a short exact sequence of ''G''-modules, then we get the usual long exact sequence of Tate cohomology groups: ::\cdots \longrightarro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
