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Theta Constant
In mathematics, a theta constant or Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction ''θ''''m''(''τ'') = θ''m''(''τ'',''0'') of a theta function ''θ''''m''(τ,''z'') with rational characteristic ''m'' to ''z'' = 0. The variable ''τ'' may be a complex number in the upper half-plane in which case the theta constants are modular forms, or more generally may be an element of a Siegel upper half plane in which case the theta constants are Siegel modular forms. The theta function of a lattice is essentially a special case of a theta constant. Definition The theta function ''θ''''m''(''τ'',''z'') = ''θ''''a'',''b''(''τ'',''z'')is defined by : \theta_(\tau,z) = \sum_ \exp\left pi (\xi+a)\tau(\xi+a)^t + 2\pi i(\xi+a)(z+b)^t\right/math> where * ''n'' is a positive integer, called the genus or rank. * ''m'' = (''a'',''b'') is called the characteristic * ''a'',''b'' are in R''n'' * ''τ'' is a complex ''n'' by ''n'' matri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions". Throughout this article, (e^)^ should b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Siegel Modular Form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular forms are Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure) should be and are constructed as quotients of the Siegel upper half-space rather than the upper half-plane by discrete groups. Siegel modular forms are holomorphic functions on the set of symmetric ''n'' × ''n'' matrices with positive definite imaginary part; the forms must satisfy an automorphy condition. Siegel modular forms can be thought of as multivariable modular forms, i.e. as special functions of several complex variables. Siegel modular forms were first investigated by for the purpose of studying quadratic forms analytically. These primarily arise in various branches of number theory, su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theta Function Of A Lattice
In mathematics, the theta function of a lattice is a function whose coefficients give the number of vectors of a given norm. Definition One can associate to any (positive-definite) lattice Λ a theta function given by :\Theta_\Lambda(\tau) = \sum_e^\qquad\mathrm\,\tau > 0. The theta function of a lattice is then a holomorphic function on the upper half-plane. Furthermore, the theta function of an even unimodular lattice of rank ''n'' is actually a modular form In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ... of weight ''n''/2. The theta function of an integral lattice is often written as a power series in q = e^ so that the coefficient of ''q''''n'' gives the number of lattice vectors of norm 2''n''. References * Theta functions {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Theta Functions
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field theory. The most common form of theta function is that occurring in the theory of elliptic functions. With respect to one of the complex variables (conventionally called ), a theta function has a property expressing its behavior with respect to the addition of a period of the associated elliptic functions, making it a quasiperiodic function. In the abstract theory this quasiperiodicity comes from the cohomology class of a line bundle on a complex torus, a condition of descent. One interpretation of theta functions when dealing with the heat equation is that "a theta function is a special function that describes the evolution of temperature on a segment domain subject to certain boundary conditions". Throughout this article, (e^)^ sho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Forms
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ''G''(A''F''), for an algebraic group ''G'' and an algebraic number field ''F'', is a complex-valued function on ''G''(A''F'') that is left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
