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Special Values Of L-functions
In mathematics, the study of special values of -functions is a subfield of number theory devoted to generalising formulae such as the Leibniz formula for , namely 1 \,-\, \frac \,+\, \frac \,-\, \frac \,+\, \frac \,-\, \cdots \;=\; \frac,\! by the recognition that expression on the left-hand side is also L(1) where L(s) is the Dirichlet -function for the field of Gaussian rational numbers. This formula is a special case of the analytic class number formula, and in those terms reads that the Gaussian field has class number 1. The factor \tfrac14 on the right hand side of the formula corresponds to the fact that this field contains four roots of unity. Conjectures There are two families of conjectures, formulated for general classes of -functions (the very general setting being for -functions associated to Chow motives over number fields), the division into two reflecting the questions of: how to replace \pi in the Leibniz formula by some other "transcendental" number (rega ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Beilinson Regulator
In mathematics, especially in algebraic geometry, the Beilinson regulator is the Chern class map from algebraic K-theory to Deligne cohomology: :K_n (X) \rightarrow \oplus_ H_D^ (X, \mathbf Q(p)). Here, ''X'' is a complex smooth projective variety, for example. It is named after Alexander Beilinson. The Beilinson regulator features in Beilinson's conjecture on special values of L-functions. The ''Dirichlet regulator'' map (used in the proof of Dirichlet's unit theorem) for the ring of integers \mathcal O_F of a 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'' :\mathcal O_F^\times \rightarrow \mathbf R^, \ \ x \mapsto (\log , \sigma (x), )_\sigma is a particular case of the Beilinson regulator. (As usual, \sigma: F \subset \mathbf C runs over all complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Main Conjecture Of Iwasawa Theory
In mathematics, the main conjecture of Iwasawa theory is a deep relationship between ''p''-adic ''L''-functions and ideal class groups of cyclotomic fields, proved by Kenkichi Iwasawa for primes satisfying the Kummer–Vandiver conjecture and proved for all primes by . The Herbrand–Ribet theorem and the Gras conjecture are both easy consequences of the main conjecture. There are several generalizations of the main conjecture, to totally real fields,, CM fields, elliptic curves, and so on. Motivation was partly motivated by an analogy with Weil's description of the zeta function of an algebraic curve over a finite field in terms of eigenvalues of the Frobenius endomorphism on its Jacobian variety. In this analogy, * The action of the Frobenius corresponds to the action of the group Γ. * The Jacobian of a curve corresponds to a module ''X'' over Γ defined in terms of ideal class groups. * The zeta function of a curve over a finite field corresponds to a ''p''-adic ''L' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Iwasawa Theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite Tower of fields, towers of number fields. It began as a Galois module theory of ideal class groups, initiated by (), as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur considered generalizations of Iwasawa theory to abelian variety, abelian varieties. More recently (early 1990s), Ralph Greenberg has proposed an Iwasawa theory for motive (algebraic geometry), motives. Formulation Iwasawa worked with so-called \Z_p-extensions: infinite extensions of a number field F with Galois group \Gamma isomorphic to the additive group of p-adic integers for some prime ''p''. (These were called \Gamma-extensions in early papers.) Every closed subgroup of \Gamma is of the form \Gamma^, so by Galois theory, a \Z_p-extension F_\infty/F is the same thing as a tower of fields :F=F_0 \subset F_1 \subset F_2 \subset \cdots \subset F_\infty such that \operatorname(F_n/F)\cong \Z/p^n\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Algebraic Group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n\times n matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation M^TM = I_n where M^T is the transpose of M. Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(''n'',R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and . In the 1950s, Armand Borel constructed much of the theory of algebraic groups ... [...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 ω 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 , whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Curve
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birch–Swinnerton-Dyer Conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. Only special cases of the conjecture have been proven. The modern formulation of the conjecture relates to arithmetic data associated with an elliptic curve ''E'' over a number field ''K'' to the behaviour of the Hasse–Weil ''L''-function ''L''(''E'', ''s'') of ''E'' at ''s'' = 1. More specifically, it is conjectured that the rank of the abelian group ''E''(''K'') of points of ''E'' is the order of the zero of ''L''(''E'', ''s'') at ''s'' = 1. The first non-zero c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Milnor Conjecture (K-theory)
In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' with coefficients in Z/2Z. It was proved by . Statement Let ''F'' be a field of characteristic different from 2. Then there is an isomorphism :K_n^M(F)/2 \cong H_^n(F, \mathbb/2\mathbb) for all ''n'' ≥ 0, where ''KM'' denotes the Milnor ring. About the proof The proof of this theorem by Vladimir Voevodsky uses several ideas developed by Voevodsky, Alexander Merkurjev, Andrei Suslin, Markus Rost, Fabien Morel, Eric Friedlander, and others, including the newly minted theory of motivic cohomology (a kind of substitute for singular cohomology for algebraic varieties) and the motivic Steenrod algebra. Generalizations The analogue of this result for primes other than 2 was known as the Bloch–Kato conjecture. Work ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kazuya Kato
is a Japanese mathematician who works at the University of Chicago and specializes in number theory and arithmetic geometry. Early life and education Kazuya Kato grew up in the prefecture of Wakayama in Japan. He attended college at the University of Tokyo, from which he also obtained his master's degree in 1975, and his PhD in 1980. Career Kato was a professor at Tokyo University, Tokyo Institute of Technology and Kyoto University. He joined the faculty of the University of Chicago in 2009. A special volume of '' Documenta Mathematica'' was published in honor of his 50th birthday, together with research papers written by leading number theorists and former students it contains Kato's song on Prime Numbers. Research Kato's first work was in the higher-dimensional generalisations of local class field theory using algebraic K-theory. His theory was then extended to higher global class field theory in which several of his papers were written jointly with Shuji Saito. He contri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spencer Bloch
Spencer Janney Bloch (born May 22, 1944; New York City) is an American mathematician known for his contributions to algebraic geometry and algebraic ''K''-theory. Bloch is a R. M. Hutchins Distinguished Service Professor Emeritus in the Department of Mathematics of the University of Chicago. Research Bloch introduced the Bloch group in 1978. He introduced Bloch's higher Chow group, a generalization of Chow groups, in 1986. He also introduced Bloch's formula in Algebraic K-theory. Bloch and Kazuya Kato formulated the motivic Bloch–Kato conjecture relating Milnor K-theory and Galois cohomology in 1986 and the Bloch–Kato conjectures for special values of ''L''-functions in 1990. Awards and honors Bloch is a member of the U.S. National Academy of Sciences and a Fellow of the American Academy of Arts and Sciences [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
