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Siegel Theta Series
In mathematics, a Siegel theta series is a Siegel modular form associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice. Definition Suppose that ''L'' is a positive definite lattice. The Siegel theta series of degree ''g'' is defined by :\Theta_L^g(T) = \sum_\exp(\pi i Tr(\lambda T \lambda^t)) where ''T'' is an element of the Siegel upper half plane of degree ''g''. This is a Siegel modular form of degree ''d'' and weight dim(''L'')/2 for some subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ... of the Siegel modular group. If the lattice ''L'' is even and unimodular then this is a Siegel modular form for the full Siegel modular group. When the degree is 1 this is just the usual theta function of a lattice. References *{{citati ... [...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 variety, Siegel modular varieties, which are basic models for what a moduli space for abelian varieties (with some extra level structure (algebraic geometry), 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 matrix, 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 for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lattice (discrete Subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure. In the special case of subgroups of R''n'', this amounts to the usual geometric notion of a lattice as a periodic subset of points, and both the algebraic structure of lattices and the geometry of the space of all lattices are relatively well understood. The theory is particularly rich for lattices in semisimple Lie groups or more generally in semisimple algebraic groups over local fields. In particular there is a wealth of rigidity results in this setting, and a celebrated theorem of Grigory Margulis states that in most cases all lattices are obtained as arithmetic groups. Lattices are also well-studied in some other classes of groups, in particular groups associated to Kac–Moody algebras and automorphisms groups of regular trees (the latter are known as ''tree lattices''). Lattices a ... [...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 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''. See also * Siegel theta series In mathematics, a Siegel theta series is a Siegel modular form associated to a positive definite lattice, generalizing the 1-variable theta function of a lattice. Definition Suppose that ''L'' is a positive definite lattice. The Siegel theta ser ... * Theta constant References ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ∗, a subset of is called a subgroup of if also forms a group under the operation ∗. More precisely, is a subgroup of if the Restriction (mathematics), restriction of ∗ to is a group operation on . This is often denoted , read as " is a subgroup of ". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group is a subgroup which is a subset, proper subset of (that is, ). This is often represented notationally by , read as " is a proper subgroup of ". Some authors also exclude the trivial group from being proper (that is, ). If is a subgroup of , then is sometimes called an overgroup of . The same definitions apply more generally when is an arbitrary se ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodular Lattice
In geometry and mathematical group theory, a unimodular lattice is an integral Lattice (group), lattice of Lattice (group)#Dividing space according to a lattice, determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E8 lattice, ''E''8 lattice and the Leech lattice are two famous examples. Definitions * A lattice is a free abelian group of finite free abelian group, rank with a symmetric bilinear form (·, ·). * The lattice is integral if (·,·) takes integer values. * The dimension of a lattice is the same as its free module, rank (as a Z-module (mathematics), module). * The norm of a lattice element ''a'' is (''a'', ''a''). * A lattice is positive definite if the norm of all nonzero elements is positive. * The determinant of a lattice is the determinant of the Gram matrix, a matrix (mathematics), matrix with entries (''ai'', ''aj'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
