|
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 law ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Group Law
In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one of several generalizations. Formal groups are intermediate between Lie groups (or algebraic groups) and Lie algebras. They are used in algebraic number theory and algebraic topology. Definitions A one-dimensional formal group law over a commutative ring ''R'' is a power series ''F''(''x'',''y'') with coefficients in ''R'', such that # ''F''(''x'',''y'') = ''x'' + ''y'' + terms of higher degree # ''F''(''x'', ''F''(''y'',''z'')) = ''F''(''F''(''x'',''y''), ''z'') (associativity). The simplest example is the additive formal group law ''F''(''x'', ''y'') = ''x'' + ''y''. The idea of the definition is that ''F'' should be something like the formal power series expansion of the product of a Lie group, where we choose coordinates so that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing ''CF' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacob Lurie
Jacob Alexander Lurie (born December 7, 1977) is an American mathematician who is a professor at the Institute for Advanced Study. Lurie is a 2014 MacArthur Fellow. Life When he was a student in the Science, Mathematics, and Computer Science Magnet Program at Montgomery Blair High School, Lurie took part in the International Mathematical Olympiad, where he won a gold medal with a perfect score in 1994. In 1996 he took first place in the Westinghouse Science Talent Search and was featured in a front-page story in the ''Washington Times''. Lurie earned his bachelor's degree in mathematics from Harvard College in 2000 and was awarded in the same year the Morgan Prize for his undergraduate thesis on Lie algebras. He earned his Ph.D. from the Massachusetts Institute of Technology under supervision of Michael J. Hopkins, in 2004 with a thesis on derived algebraic geometry. In 2007, he became associate professor at MIT, and in 2009 he became professor at Harvard University. In 20 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Johnson–Wilson Theory
In algebraic topology, Johnson–Wilson theory ''E''(''n'') is a generalized cohomology theory In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ... introduced by . Real Johnson–Wilson theory ''ER''(''n'') was introduced by . References * * Homotopy theory {{Topology-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum (homotopy Theory)
In algebraic topology, a branch of mathematics, a spectrum is an object representing a generalized cohomology theory. Every such cohomology theory is representable, as follows from Brown's representability theorem. This means that, given a cohomology theory\mathcal^*:\text^ \to \text,there exist spaces E^k such that evaluating the cohomology theory in degree k on a space X is equivalent to computing the homotopy classes of maps to the space E^k, that is\mathcal^k(X) \cong \left , E^k\right/math>.Note there are several different categories of spectra leading to many technical difficulties, but they all determine the same homotopy category, known as the stable homotopy category. This is one of the key points for introducing spectra because they form a natural home for stable homotopy theory. The definition of a spectrum There are many variations of the definition: in general, a ''spectrum'' is any sequence X_n of pointed topological spaces or pointed simplicial sets together with th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Extraordinary Cohomology Theory
In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. Some versions of cohomology arise by dualizing the construction of homology. In other words, cochains are functions on the group of chains in homology theory. From its beginning in topology, this idea became a dominant method in the mathematics of the second half of the twentieth century. From the initial idea of homology as a method of constructing algebraic invariants of topological spaces, the range of applications of homology and cohomology theories has spread throughout geometry and algebra. The terminology tends to hide the fact that cohomology, a contravariant theory, is more natural than homology in many applications. At a basic level, this has to do wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stable Homotopy Theory
In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the Freudenthal suspension theorem, which states that given any pointed space X, the homotopy groups \pi_(\Sigma^n X) stabilize for n sufficiently large. In particular, the homotopy groups of spheres \pi_(S^n) stabilize for n\ge k + 2. For example, :\langle \text_\rangle = \Z = \pi_1(S^1)\cong \pi_2(S^2)\cong \pi_3(S^3)\cong\cdots :\langle \eta \rangle = \Z = \pi_3(S^2)\to \pi_4(S^3)\cong \pi_5(S^4)\cong\cdots In the two examples above all the maps between homotopy groups are applications of the suspension functor. The first example is a standard corollary of the Hurewicz theorem, that \pi_n(S^n)\cong \Z. In the second example the Hopf map, \eta, is mapped to its suspension \Sigma\eta, which generates \pi_4(S^3)\cong \Z/2. One of the most i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deformation Theory
In mathematics, deformation theory is the study of infinitesimal conditions associated with varying a solution ''P'' of a problem to slightly different solutions ''P''ε, where ε is a small number, or a vector of small quantities. The infinitesimal conditions are the result of applying the approach of differential calculus to solving a problem with constraints. The name is an analogy to non-rigid structures that deform slightly to accommodate external forces. Some characteristic phenomena are: the derivation of first-order equations by treating the ε quantities as having negligible squares; the possibility of ''isolated solutions'', in that varying a solution may not be possible, ''or'' does not bring anything new; and the question of whether the infinitesimal constraints actually 'integrate', so that their solution does provide small variations. In some form these considerations have a history of centuries in mathematics, but also in physics and engineering. For example, in th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eisenstein Polynomial
In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. Criterion Suppose we have the following polynomial with integer coefficients. : Q(x)=a_nx^n+a_x^+\cdots+a_1x+a_0 If there exists a prime number such that the following three conditions all apply: * divides each f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unramified Extension
In geometry, ramification is 'branching out', in the way that the square root function, for complex numbers, can be seen to have two ''branches'' differing in sign. The term is also used from the opposite perspective (branches coming together) as when a covering map degenerates at a point of a space, with some collapsing of the fibers of the mapping. In complex analysis In complex analysis, the basic model can be taken as the ''z'' → ''z''''n'' mapping in the complex plane, near ''z'' = 0. This is the standard local picture in Riemann surface theory, of ramification of order ''n''. It occurs for example in the Riemann–Hurwitz formula for the effect of mappings on the genus. See also branch point. In algebraic topology In a covering map the Euler–Poincaré characteristic should multiply by the number of sheets; ramification can therefore be detected by some dropping from that. The ''z'' → ''z''''n'' mapping shows this as a local p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Class Field Theory
In mathematics, local class field theory, introduced by Helmut Hasse, is the study of abelian extensions of local fields; here, "local field" means a field which is complete with respect to an absolute value or a discrete valuation with a finite residue field: hence every local field is isomorphic (as a topological field) to the real numbers R, the complex numbers C, a finite extension of the ''p''-adic numbers Q''p'' (where ''p'' is any prime number), or a finite extension of the field of formal Laurent series F''q''((''T'')) over a finite field F''q''. Approaches to local class field theory Local class field theory gives a description of the Galois group ''G'' of the maximal abelian extension of a local field ''K'' via the reciprocity map which acts from the multiplicative group ''K''×=''K''\. For a finite abelian extension ''L'' of ''K'' the reciprocity map induces an isomorphism of the quotient group ''K''×/''N''(''L''×) of ''K''× by the norm group ''N'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Class Field
In algebraic number theory, the Hilbert class field ''E'' of a number field ''K'' is the maximal abelian unramified extension of ''K''. Its degree over ''K'' equals the class number of ''K'' and the Galois group of ''E'' over ''K'' is canonically isomorphic to the ideal class group of ''K'' using Frobenius elements for prime ideals in ''K''. In this context, the Hilbert class field of ''K'' is not just unramified at the finite places (the classical ideal theoretic interpretation) but also at the infinite places of ''K''. That is, every real embedding of ''K'' extends to a real embedding of ''E'' (rather than to a complex embedding of ''E''). Examples *If the ring of integers of ''K'' is a unique factorization domain, in particular if K = \mathbb , then ''K'' is its own Hilbert class field. *Let K = \mathbb(\sqrt) of discriminant -15. The field L = \mathbb(\sqrt, \sqrt) has discriminant 225=(-15)^2 and so is an everywhere unramified extension of ''K'', and it is abelian. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
