|
Trace Formula (other)
Trace formula may refer to: *Arthur–Selberg trace formula, also known as invariant trace formula, Jacquet's relative trace formula, simple trace formula, stable trace formula *Grothendieck trace formula, an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology, used to express the Hasse–Weil zeta function. * Gutzwiller trace formula: See Quantum chaos * Kuznetsov trace formula, an extension of the Petersson trace formula. *Local trace formula, an analog of Arthur–Selberg trace formula *Petersson trace formula *Selberg trace formula *Behrend's trace formula, or Behrend's fixed point formula a generalization of the Grothendieck–Lefschetz trace formula, that may be interpreted as a Selberg trace formula. See also * List of zeta functions * List of fixed point theorems In mathematics, a fixed-point theorem is a result saying that a function ''F'' will have at least one fixed point (a point ''x'' for which ''F''(''x'') = ''x''), under some ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arthur–Selberg Trace Formula
In mathematics, the Arthur–Selberg trace formula is a generalization of the Selberg trace formula from the group SL2 to arbitrary reductive groups over global fields, developed by James Arthur in a long series of papers from 1974 to 2003. It describes the character of the representation of on the discrete part of in terms of geometric data, where is a reductive algebraic group defined over a global field and is the ring of adeles of ''F''. There are several different versions of the trace formula. The first version was the unrefined trace formula, whose terms depend on truncation operators and have the disadvantage that they are not invariant. Arthur later found the invariant trace formula and the stable trace formula which are more suitable for applications. The simple trace formula is less general but easier to prove. The local trace formula is an analogue over local fields. Jacquet's relative trace formula is a generalization where one integrates the kernel function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grothendieck Trace Formula
In algebraic geometry, the Grothendieck trace formula expresses the number of points of a variety over a finite field in terms of the trace of the Frobenius endomorphism on its cohomology groups. There are several generalizations: the Frobenius endomorphism can be replaced by a more general endomorphism, in which case the points over a finite field are replaced by its fixed points, and there is also a more general version for a sheaf over the variety, where the cohomology groups are replaced by cohomology with coefficients in the sheaf. The Grothendieck trace formula is an analogue in algebraic geometry of the Lefschetz fixed-point theorem in algebraic topology. One application of the Grothendieck trace formula is to express the zeta function of a variety over a finite field, or more generally the L-series of a sheaf, as a sum over traces of Frobenius on cohomology groups. This is one of the steps used in the proof of the Weil conjectures. Behrend's trace formula generalizes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lefschetz Fixed-point Theorem
In mathematics, the Lefschetz fixed-point theorem is a formula that counts the fixed points of a continuous mapping from a compact topological space X to itself by means of traces of the induced mappings on the homology groups of X. It is named after Solomon Lefschetz, who first stated it in 1926. The counting is subject to an imputed multiplicity at a fixed point called the fixed-point index. A weak version of the theorem is enough to show that a mapping without ''any'' fixed point must have rather special topological properties (like a rotation of a circle). Formal statement For a formal statement of the theorem, let :f\colon X \rightarrow X\, be a continuous map from a compact triangulable space X to itself. Define the Lefschetz number \Lambda_f of f by :\Lambda_f:=\sum_(-1)^k\mathrm(f_*, H_k(X,\Q)), the alternating (finite) sum of the matrix traces of the linear maps induced by f on H_k(X,\Q), the singular homology groups of X with rational coefficients. A simple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Topology
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence. Although algebraic topology primarily uses algebra to study topological problems, using topology to solve algebraic problems is sometimes also possible. Algebraic topology, for example, allows for a convenient proof that any subgroup of a free group is again a free group. Main branches of algebraic topology Below are some of the main areas studied in algebraic topology: Homotopy groups In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, which records information about loops in a space. Intuitively, homotopy groups record information about the basic shape, or holes, of a topological space. Homolog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse–Weil Zeta Function
In mathematics, the Hasse–Weil zeta function attached to an algebraic variety ''V'' defined over an algebraic number field ''K'' is a meromorphic function on the complex plane defined in terms of the number of points on the variety after reducing modulo each prime number ''p''. It is a global ''L''-function defined as an Euler product of local zeta functions. Hasse–Weil ''L''-functions form one of the two major classes of global ''L''-functions, alongside the ''L''-functions associated to automorphic representations. Conjecturally, these two types of global ''L''-functions are actually two descriptions of the same type of global ''L''-function; this would be a vast generalisation of the Taniyama-Weil conjecture, itself an important result in number theory. For an elliptic curve over a number field ''K'', the Hasse–Weil zeta function is conjecturally related to the group of rational points of the elliptic curve over ''K'' by the Birch and Swinnerton-Dyer conjecture. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quantum Chaos
Quantum chaos is a branch of physics which studies how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics and classical chaos?" The correspondence principle states that classical mechanics is the classical limit of quantum mechanics, specifically in the limit as the ratio of Planck's constant to the action of the system tends to zero. If this is true, then there must be quantum mechanisms underlying classical chaos (although this may not be a fruitful way of examining classical chaos). If quantum mechanics does not demonstrate an exponential sensitivity to initial conditions, how can exponential sensitivity to initial conditions arise in classical chaos, which must be the correspondence principle limit of quantum mechanics?''Quantum Signatures of Chaos'', Fritz Haake, Edition: 2, Springer, 2001, , . Michael Berry, "Quantum Chaology", p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kuznetsov Trace Formula
In analytic number theory, the Kuznetsov trace formula is an extension of the Petersson trace formula. The Kuznetsov or ''relative trace'' formula connects Kloosterman sums at a deep level with the spectral theory of automorphic forms. Originally this could have been stated as follows. Let : g: \mathbb\rightarrow \mathbb be a sufficiently "well behaved" function. Then one calls identities of the following type ''Kuznetsov trace formula'': : \sum_ c^ K(m,n,c) g\left(\frac\right) = \text\ +\ \text. The integral transform part is some integral transform of ''g'' and the spectral part is a sum of Fourier coefficients, taken over spaces of holomorphic and non-holomorphic modular forms twisted with some integral transform of ''g''. The Kuznetsov trace formula was found by Kuznetsov while studying the growth of weight zero automorphic functions. Using estimates on Kloosterman sums he was able to derive estimates for Fourier coefficients of modular forms in cases where Pierre Delign ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Trace Formula
In mathematics, the local trace formula is a local analogue of the Arthur–Selberg trace formula that describes the character of the representation of ''G''(F) on the discrete part of ''L''2(''G''(F)), for ''G'' a reductive algebraic group over a local field In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compa ... F. References * {{numtheory-stub Automorphic forms Theorems in number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersson Trace Formula
In analytic number theory, the Petersson trace formula is a kind of orthogonality relation between coefficients of a holomorphic modular form. It is a specialization of the more general Kuznetsov trace formula. In its simplest form the Petersson trace formula is as follows. Let \mathcal be an orthonormal basis of S_k(\Gamma(1)), the space of cusp forms of weight k>2 on SL_2(\mathbb). Then for any positive integers m,n we have : \frac \sum_ \bar(m) \hat(n) = \delta_ + 2\pi i^ \sum_\frac J_\left(\frac\right), where \delta is the Kronecker delta function, S is the Kloosterman sum and J is the Bessel function of the first kind. References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 spec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Behrend's Trace Formula
In algebraic geometry, Behrend's trace formula is a generalization of the Grothendieck–Lefschetz 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 Harder–Narasimhan 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 defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
