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Motivic L-function
In 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 ar ..., motivic ''L''-functions are a generalization of Hasse–Weil ''L''-functions to general motives over global fields. The local ''L''-factor at a finite place ''v'' is similarly given by the characteristic polynomial of a Frobenius element at ''v'' acting on the ''v''-inertial invariants of the ''v''-adic realization of the motive. For infinite places, Jean-Pierre Serre gave a recipe in for the so-called Gamma factors in terms of the Hodge realization of the motive. It is conjectured that, like other ''L''-functions, that each motivic ''L''-function can be analytically continued to a meromorphic function on the entire complex plane and satisfies a functional equation relating the ''L''-function ''L''(''s'', ... [...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] |
Functional Equation
In mathematics, a functional equation is, in the broadest meaning, an equation in which one or several functions appear as unknowns. So, differential equations and integral equations are functional equations. However, a more restricted meaning is often used, where a ''functional equation'' is an equation that relates several values of the same function. For example, the logarithm functions are essentially characterized by the ''logarithmic functional equation'' \log(xy)=\log(x) + \log(y). If the domain of the unknown function is supposed to be the natural numbers, the function is generally viewed as a sequence, and, in this case, a functional equation (in the narrower meaning) is called a recurrence relation. Thus the term ''functional equation'' is used mainly for real functions and complex functions. Moreover a smoothness condition is often assumed for the solutions, since without such a condition, most functional equations have very irregular solutions. For example, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bloch–Kato Conjecture (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 (regar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deligne's Conjecture (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 pi, 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 L-function, 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 problem, 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 L-function, -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 repl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Zeta Function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for and its analytic continuation elsewhere. The Riemann zeta function plays a pivotal role in analytic number theory and has applications in physics, probability theory, and applied statistics. Leonhard Euler first introduced and studied the function over the reals in the first half of the eighteenth century. Bernhard Riemann's 1859 article "On the Number of Primes Less Than a Given Magnitude" extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation, and established a relation between its zeros and the distribution of prime numbers. This paper also contained the Riemann hypothesis, a conjecture about the distribution of complex zeros of the Riemann zeta function that many mathematicians consider th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Selberg Class
In mathematics, the Selberg class is an axiomatic definition of a class of L-function, ''L''-functions. The members of the class are Dirichlet series which obey four axioms that seem to capture the essential properties satisfied by most functions that are commonly called ''L''-functions or zeta functions. Although the exact nature of the class is conjectural, the hope is that the definition of the class will lead to a classification of its contents and an elucidation of its properties, including insight into their relationship to automorphic forms and the Riemann hypothesis. The class was defined by Atle Selberg in , who preferred not to use the word "axiom" that later authors have employed. Definition The formal definition of the class ''S'' is the set of all Dirichlet series :F(s)=\sum_^\infty \frac absolutely convergent for Re(''s'') > 1 that satisfy four axioms (or assumptions as Selberg calls them): Comments on definition The condition that the real part o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic L-function
In mathematics, an automorphic ''L''-function is a function ''L''(''s'',π,''r'') of a complex variable ''s'', associated to an automorphic representation π of a reductive group ''G'' over a global field and a finite-dimensional complex representation ''r'' of the Langlands dual group ''L''''G'' of ''G'', generalizing the Dirichlet L-series of a Dirichlet character and the Mellin transform of a modular form. They were introduced by . and gave surveys of automorphic L-functions. Properties Automorphic L-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases). The L-function L(s, \pi, r) should be a product over the places v of F of local L functions. L(s, \pi, r) = \prod_v L(s, \pi_v, r_v) Here the automorphic representation \pi = \otimes\pi_v is a tensor product of the representations \pi_v of local groups. The L-function is expected to have an analytic continuation as a meromorphic function of all com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cusp Form
In number theory, a branch of mathematics, a cusp form is a particular kind of modular form with a zero constant coefficient in the Fourier series expansion. Introduction A cusp form is distinguished in the case of modular forms for the modular group by the vanishing of the constant coefficient ''a''0 in the Fourier series expansion (see ''q''-expansion) :\sum a_n q^n. This Fourier expansion exists as a consequence of the presence in the modular group's action on the upper half-plane via the transformation :z\mapsto z+1. For other groups, there may be some translation through several units, in which case the Fourier expansion is in terms of a different parameter. In all cases, though, the limit as ''q'' → 0 is the limit in the upper half-plane as the imaginary part of ''z'' → ∞. Taking the quotient by the modular group, this limit corresponds to a cusp of a modular curve (in the sense of a point added for compactification). So, the definition amounts to saying that a cu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Artin L-function
In mathematics, an Artin ''L''-function is a type of Dirichlet series associated to a linear representation ρ of a Galois group ''G''. These functions were introduced in 1923 by Emil Artin, in connection with his research into class field theory. Their fundamental properties, in particular the Artin conjecture described below, have turned out to be resistant to easy proof. One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin ''L''-functions into a larger framework, such as is provided by automorphic forms and the Langlands program. So far, only a small part of such a theory has been put on a firm basis. Definition Given \rho , a representation of G on a finite-dimensional complex vector space V, where G is the Galois group of the finite extension L/K of number fields, the Artin L-function L(\rho,s) is defined by an Euler product. For each prime ideal \mathfrak p in K's ring of integers, there is an Euler fac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
