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Moduli Stack Of Principal Bundles
In algebraic geometry, given a smooth projective curve ''X'' over a finite field \mathbf_q and a smooth affine group scheme ''G'' over it, the moduli stack of principal bundles over ''X'', denoted by \operatorname_G(X), is an algebraic stack given by: for any \mathbf_q-algebra ''R'', :\operatorname_G(X)(R) = the category of principal ''G''-bundles over the relative curve X \times_ \operatornameR. In particular, the category of \mathbf_q-points of \operatorname_G(X), that is, \operatorname_G(X)(\mathbf_q), is the category of ''G''-bundles over ''X''. Similarly, \operatorname_G(X) can also be defined when the curve ''X'' is over the field of complex numbers. Roughly, in the complex case, one can define \operatorname_G(X) as the quotient stack of the space of holomorphic connections on ''X'' by the gauge group. Replacing the quotient stack (which is not a topological space) by a homotopy quotient (which is a topological space) gives the homotopy type of \operatorname_G(X). In the fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight. Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that appeared more than 2000 years ago in Euclid's ''Elements'': "The urvedline is ¦the first species of quantity, which has only one dimension, namely length, without any width nor depth, and is nothing else than the flow or run of the point which ¦will leave from its imaginary moving some vestige in length, exempt of any width." This definition of a curve has been formalized in modern mathematics as: ''A curve is the image of an interval to a topological space by a continuous function''. In some contexts, the function that defines the curve is called a ''parametrization'', and the curve is a parametric curve. In this article, these curves are sometimes called ''topological curves'' to distinguish them from more constrained curves such a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hitchin Fibration
In mathematics, the Hitchin integrable system is an integrable system depending on the choice of a complex reductive group and a compact Riemann surface, introduced by Nigel Hitchin in 1987. It lies on the crossroads of algebraic geometry, the theory of Lie algebras and integrable system theory. It also plays an important role in the geometric Langlands correspondence over the field of complex numbers through conformal field theory. A genus zero analogue of the Hitchin system, the Garnier system, was discovered by RenĆ© Garnier somewhat earlier as a certain limit of the Schlesinger equations, and Garnier solved his system by defining spectral curves. (The Garnier system is the classical limit of the Gaudin model. In turn, the Schlesinger equations are the classical limit of the KnizhnikāZamolodchikov equations). Almost all integrable systems of classical mechanics can be obtained as particular cases of the Hitchin system or their common generalization defined by Bottacin an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tamagawa Number
In mathematics, the Tamagawa number \tau(G) of a semisimple algebraic group defined over a global field is the measure of G(\mathbb)/G(k), where \mathbb is the adele ring of . Tamagawa numbers were introduced by , and named after him by . Tsuneo Tamagawa's observation was that, starting from an invariant differential form Ļ on , defined over , the measure involved was well-defined: while could be replaced by with a non-zero element of k, the product formula for valuations in is reflected by the independence from of the measure of the quotient, for the product measure constructed from on each effective factor. The computation of Tamagawa numbers for semisimple groups contains important parts of classical quadratic form theory. Definition Let be a global field, its ring of adeles, and a semisimple algebraic group defined over . Choose Haar measures on the completions such that has volume 1 for all but finitely many places . These then induce a Haar measure on , which w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Mass Formula
Siegel (also Segal or Segel), is a German and Ashkenazi Jewish surname. it can be traced to 11th century Bavaria and was used by people who made wax seals for or sealed official documents (each such male being described as a ''Siegelbeamter''). Alternate spellings include Sigel, Sigl, Siegl, and others. "Siegel" is also the modern German word for seal. The name ultimately derives from the Latin ''sigillum,'' meaning "seal" as in the Seal of the City of New York: ''Sigillum Civitatis Novi Eboraci''. The Germanicized derivative of the name was given to professional seal makers and engravers. Some researchers have attributed the surname to Sigel, referring to SĆ³l (Sun), the goddess of the sun in Germanic mythology (SiČel or sigel in Old English / Anglo-Saxon), but that is highly speculative. Variants, and false cognates Other variants may routinely include Siegelman, Siegle, Sigl, and Sigel. Presumably, some bearers of these names are lineal descendants of ethnic Jews who chang ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Series (mathematics)
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graded Vector Space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces. Integer gradation Let \mathbb be the set of non-negative integers. An \mathbb-graded vector space, often called simply a graded vector space without the prefix \mathbb, is a vector space together with a decomposition into a direct sum of the form : V = \bigoplus_ V_n where each V_n is a vector space. For a given ''n'' the elements of V_n are then called homogeneous elements of degree ''n''. Graded vector spaces are common. For example the set of all polynomials in one or several variables forms a graded vector space, where the homogeneous elements of degree ''n'' are exactly the linear combinations of monomials of degree ''n''. General gradation The subspaces of a graded vector space need not be indexed by the set of natural numbers, and may be indexed by the elem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Torsor (algebraic Geometry)
In algebraic geometry, a torsor or a principal bundle is an analog of a principal bundle in algebraic topology. Because there are few open sets in Zariski topology, it is more common to consider torsors in Ć©tale topology or some other flat topologies. The notion also generalizes a Galois extension in abstract algebra. The category of torsors over a fixed base forms a stack. Conversely, a prestack can be stackified by taking the category of torsors (over the prestack). Definition Given a smooth algebraic group ''G'', a ''G''-torsor (or a principal ''G''-bundle) ''P'' over a scheme ''X'' is a scheme (or even algebraic space) with an action of ''G'' that is locally trivial in the given Grothendieck topology in the sense that the base change Y \times_X P along some covering map Y \to X is isomorphic to the trivial torsor Y \times G \to Y (''G'' acts only on the second factor). Equivalently, a ''G''-torsor ''P'' on ''X'' is a principal homogeneous space for the group scheme G_X = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Frobenius
In mathematics, the Frobenius endomorphism is defined in any commutative ring ''R'' that has characteristic ''p'', where ''p'' is a prime number. Namely, the mapping Ļ that takes ''r'' in ''R'' to ''r''''p'' is a ring endomorphism of ''R''. The image of Ļ is then ''R''''p'', the subring of ''R'' consisting of ''p''-th powers. In some important cases, for example finite fields, Ļ is surjective. Otherwise Ļ is an endomorphism but not a ring ''automorphism''. The terminology of geometric Frobenius arises by applying the spectrum of a ring construction to Ļ. This gives a mapping :Ļ*: Spec(''R''''p'') ā Spec(''R'') of affine schemes. Even in cases where ''R''''p'' = ''R'' this is not the identity, unless ''R'' is the prime field. Mappings created by fibre product with Ļ*, i.e. base changes, tend in scheme theory to be called ''geometric Frobenius''. The reason for a careful terminology is that the Frobenius automorphism in Galois groups, or defined by transport of structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-adic Integers
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory ā including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a discret ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Behrend's Trace Formula
In algebraic geometry, Behrend's trace formula is a generalization of the GrothendieckāLefschetz trace formula to a smooth algebraic stack over a finite field conjectured in 1993 and proven in 2003 by Kai Behrend. Unlike the classical one, the formula counts points in the " stacky way"; it takes into account the presence of nontrivial automorphisms. The desire for the formula comes from the fact that it applies to the moduli stack of principal bundles on a curve over a finite field (in some instances indirectly, via the HarderāNarasimhan stratification, as the moduli stack is not of finite type.) See the moduli stack of principal bundles and references therein for the precise formulation in this case. Pierre Deligne found an example that shows the formula may be interpreted as a sort of the Selberg trace formula. A proof of the formula in the context of the six operations formalism developed by Yves Laszlo and Martin Olsson is given by Shenghao Sun. Formulation By definiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Fiber
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension ''d'' is a point such that the field generated by its coordinates has transcendence degree ''d'' over the field generated by the coefficients of the equations of the variety. In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space ''X'' is a point whose closure is ''X''. Definition and motivation A generic point of the topological space ''X'' is a point ''P'' whose closure is all of ''X'', that is, a point that is dense in ''X''. T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Scheme
In algebraic geometry, a smooth scheme over a field is a scheme which is well approximated by affine space near any point. Smoothness is one way of making precise the notion of a scheme with no singular points. A special case is the notion of a smooth variety over a field. Smooth schemes play the role in algebraic geometry of manifolds in topology. Definition First, let ''X'' be an affine scheme of finite type over a field ''k''. Equivalently, ''X'' has a closed immersion into affine space ''An'' over ''k'' for some natural number ''n''. Then ''X'' is the closed subscheme defined by some equations ''g''1 = 0, ..., ''g''''r'' = 0, where each ''gi'' is in the polynomial ring ''k'' 'x''1,..., ''x''''n'' The affine scheme ''X'' is smooth of dimension ''m'' over ''k'' if ''X'' has dimension at least ''m'' in a neighborhood of each point, and the matrix of derivatives (ā''g''''i''/ā''x''''j'') has rank at least ''n''ā''m'' everywhere on ''X''. (It follows that ''X'' has dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
