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Selberg Trace Formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given by the trace of certain functions on . The simplest case is when is cocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of the Frobenius formula for the character of an induced representation of finite groups. When is the cocompact subgroup of the real numbers , the Selberg trace formula is essentially the Poisson summation formula. The case when is not compact is harder, because there is a continuous spectrum, described using Eisenstein series. Selberg worked out the non-compact case when is the group ; the extension to higher rank groups is the Arthur–Selberg trace formula. When is the fundamental group of a Riemann surface, the Selberg trace formula describes the spectru ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compact Riemann Surface
In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed versions of the complex plane: locally near every point they look like patches of the complex plane, but the global topology can be quite different. For example, they can look like a sphere or a torus or several sheets glued together. The main interest in Riemann surfaces is that holomorphic functions may be defined between them. Riemann surfaces are nowadays considered the natural setting for studying the global behavior of these functions, especially multi-valued functions such as the square root and other algebraic functions, or the logarithm. Every Riemann surface is a two-dimensional real analytic manifold (i.e., a surface), but it contains more structure (specifically a complex structure) which is needed for the unambiguous definition of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eichler–Shimura Congruence Relation
In number theory, the Eichler–Shimura congruence relation expresses the local ''L''-function of a modular curve at a prime ''p'' in terms of the eigenvalues of Hecke operators. It was introduced by and generalized by . Roughly speaking, it says that the correspondence on the modular curve inducing the Hecke operator ''T''''p'' is congruent mod ''p'' to the sum of the Frobenius map ''Frob'' and its transpose ''Ver''. In other words, :''T''''p'' = ''Frob'' + ''Ver'' as endomorphisms of the Jacobian ''J''0(''N'')F''p'' of the modular curve ''X''0''N'' over the finite field F''p''. The Eichler–Shimura congruence relation and its generalizations to Shimura varieties play a pivotal role in the Langlands program, by identifying a part of the Hasse–Weil zeta function of a modular curve or a more general modular variety, with the product of Mellin transforms of weight 2 modular forms or a product of analogous automorphic ''L''-functions. References * * * * Goro Shimura ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of scheme (mathematics), schemes of Finite morphism#Morphisms of finite type, finite type over the spectrum of a ring, spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: solution set, sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or Algebraic function field, function fields, i.e. field (mathematics), fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann–Roch Theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeros and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus ''g'', in a way that can be carried over into purely algebraic settings. Initially proved as Riemann's inequality by , the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student . It was later generalized to algebraic curves, to higher-dimensional varieties and beyond. Preliminary notions A Riemann surface X is a topological space that is locally homeomorphic to an open subset of \Complex, the set of complex numbers. In addition, the transition maps between these open subsets are required to be holomorphic. The latter condition allows one to transfer the notions and methods of complex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Group
In mathematics, the modular group is the projective special linear group of matrices with integer coefficients and determinant 1. The matrices and are identified. The modular group acts on the upper-half of the complex plane by fractional linear transformations, and the name "modular group" comes from the relation to moduli spaces and not from modular arithmetic. Definition The modular group is the group of linear fractional transformations of the upper half of the complex plane, which have the form :z\mapsto\frac, where , , , are integers, and . The group operation is function composition. This group of transformations is isomorphic to the projective special linear group , which is the quotient of the 2-dimensional special linear group over the integers by its center . In other words, consists of all matrices :\begin a & b \\ c & d \end where , , , are integers, , and pairs of matrices and are considered to be identical. The group operation is the usual mult ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Congruence Subgroup
In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible matrix, invertible 2 × 2 integer matrices of determinant 1, in which the off-diagonal entries are ''even''. More generally, the notion of congruence subgroup can be defined for arithmetic subgroups of algebraic groups; that is, those for which we have a notion of 'integral structure' and can define reduction maps modulo an integer. The existence of congruence subgroups in an arithmetic group provides it with a wealth of subgroups, in particular it shows that the group is residually finite. An important question regarding the algebraic structure of arithmetic groups is the congruence subgroup problem, which asks whether all subgroups of finite Index of a subgroup, index are essentially congruence subgroups. Congruence subgroups of 2×2 matrices are fundamental objects in the classical the ... [...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 cusp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Martin Eichler
Martin Maximilian Emil Eichler (29 March 1912 – 7 October 1992) was a German number theorist. Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936. Eichler and Goro Shimura developed a method to construct elliptic curves from certain modular forms. The converse notion that every elliptic curve has a corresponding modular form would later be the key to the proof of Fermat's Last Theorem. Selected publications * ''Quadratische Formen und orthogonale Gruppen'', Springer 1952, 1974 * * ''Einführung in die Theorie der algebraischen Zahlen und Funktionen'', Birkhäuser 1963; Eng. trans. 1966''Introduction to the theory of algebraic numbers and functions'' in which a section on modular forms is added; pbk 2014 reprint of 1963 German original * ''Projective varieties and modular forms'' 1971 (Riemann–Roch theorem); * with Don Zagier: ''The Theory of Jacobi forms'', Birkhäuser 1985; ''Über die Einheiten der Divisionsalgebren'', Mathem. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Operator
In mathematics, in particular in the theory of modular forms, a Hecke operator, studied by , is a certain kind of "averaging" operator that plays a significant role in the structure of vector spaces of modular forms and more general automorphic representations. History used Hecke operators on modular forms in a paper on the special cusp form of Ramanujan, ahead of the general theory given by . Mordell proved that the Ramanujan tau function, expressing the coefficients of the Ramanujan form, : \Delta(z)=q\left(\prod_^(1-q^n)\right)^= \sum_^ \tau(n)q^n, \quad q=e^, is a multiplicative function: : \tau(mn)=\tau(m)\tau(n) \quad \text (m,n)=1. The idea goes back to earlier work of Adolf Hurwitz, who treated algebraic correspondences between modular curves which realise some individual Hecke operators. Mathematical description Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer some functio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Geodesic
In differential geometry and dynamical systems, a closed geodesic on a Riemannian manifold is a geodesic that returns to its starting point with the same tangent direction. It may be formalized as the projection of a closed orbit of the geodesic, geodesic flow on the tangent space of the manifold. Definition In a Riemannian manifold (''M'',''g''), a closed geodesic is a curve \gamma:\mathbb R\rightarrow M that is a geodesic for the metric ''g'' and is periodic. Closed geodesics can be characterized by means of a variational principle. Denoting by \Lambda M the space of smooth 1-periodic curves on ''M'', closed geodesics of period 1 are precisely the critical point (mathematics), critical points of the energy function E:\Lambda M\rightarrow\mathbb R, defined by : E(\gamma)=\int_0^1 g_(\dot\gamma(t),\dot\gamma(t))\,\mathrmt. If \gamma is a closed geodesic of period ''p'', the reparametrized curve t\mapsto\gamma(pt) is a closed geodesic of period 1, and therefore it is a critical poi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
