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Atkin–Lehner Theory
In mathematics, Atkin–Lehner theory is part of the theory of modular forms describing when they arise at a given integer ''level'' ''N'' in such a way that the theory of Hecke operators can be extended to higher levels. Atkin–Lehner theory is based on the concept of a newform, which is a cusp form 'new' at a given ''level'' ''N'', where the levels are the nested congruence subgroups: :\Gamma_0(N) = \left\ of the modular group, with ''N'' ordered by divisibility. That is, if ''M'' divides ''N'', Γ0(''N'') is a subgroup of Γ0(''M''). The oldforms for Γ0(''N'') are those modular forms ''f(τ)'' of level ''N'' of the form ''g''(''d τ'') for modular forms ''g'' of level ''M'' with ''M'' a proper divisor of ''N'', where ''d'' divides ''N/M''. The newforms are defined as a vector subspace of the modular forms of level ''N'', complementary to the space spanned by the oldforms, i.e. the orthogonal space with respect to the Petersson inner product. The Hecke operators, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections with number theory. Modular forms appear in other areas, such as algebraic topology, sphere packing, and string theory. A modular function is a function that is invariant with respect to the modular group, but without the condition that be Holomorphic function, holomorphic in the upper half-plane (among other requirements). Instead, modular functions are Meromorphic function, meromorphic (that is, they are holomorphic on the complement of a set of isolated points, which are poles of the function). Modular form theory is a special case of the more general theory of automorphic form ... [...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] |
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] |
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] |
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] |
Divisibility
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgroup of ''G'' if the restriction of ∗ to is a group operation on ''H''. This is often denoted , read as "''H'' is a subgroup of ''G''". The trivial subgroup of any group is the subgroup consisting of just the identity element. A proper subgroup of a group ''G'' is a subgroup ''H'' which is a proper subset of ''G'' (that is, ). This is often represented notationally by , read as "''H'' is a proper subgroup of ''G''". Some authors also exclude the trivial group from being proper (that is, ). If ''H'' is a subgroup of ''G'', then ''G'' is sometimes called an overgroup of ''H''. The same definitions apply more generally when ''G'' is an arbitrary semigroup, but this article will only deal with subgroups of groups. Subgroup tests Suppose th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Petersson Inner Product
In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms. It was introduced by the German mathematician Hans Petersson. Definition Let \mathbb_k be the space of entire modular forms of weight k and \mathbb_k the space of cusp forms. The mapping \langle \cdot , \cdot \rangle : \mathbb_k \times \mathbb_k \rightarrow \mathbb, :\langle f , g \rangle := \int_\mathrm f(\tau) \overline (\operatorname\tau)^k d\nu (\tau) is called Petersson inner product, where :\mathrm = \left\ is a fundamental region of the modular group \Gamma and for \tau = x + iy :d\nu(\tau) = y^dxdy is the hyperbolic volume form. Properties The integral is absolutely convergent and the Petersson inner product is a positive definite Hermitian form. For the Hecke operators T_n, and for forms f,g of level \Gamma_0, we have: :\langle T_n f , g \rangle = \langle f , T_n g \rangle This can be used to show that the space of cusp forms of level \Gamma_0 has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Self-adjoint Operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itself) that is its own adjoint. If ''V'' is finite-dimensional with a given orthonormal basis, this is equivalent to the condition that the matrix of ''A'' is a Hermitian matrix, i.e., equal to its conjugate transpose ''A''. By the finite-dimensional spectral theorem, ''V'' has an orthonormal basis such that the matrix of ''A'' relative to this basis is a diagonal matrix with entries in the real numbers. In this article, we consider generalizations of this concept to operators on Hilbert spaces of arbitrary dimension. Self-adjoint operators are used in functional analysis and quantum mechanics. In quantum mechanics their importance lies in the Dirac–von Neumann formulation of quantum mechanics, in which physical observables such as positi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
C*-algebra
In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous linear operators on a complex Hilbert space with two additional properties: * ''A'' is a topologically closed set in the norm topology of operators. * ''A'' is closed under the operation of taking adjoints of operators. Another important class of non-Hilbert C*-algebras includes the algebra C_0(X) of complex-valued continuous functions on ''X'' that vanish at infinity, where ''X'' is a locally compact Hausdorff space. C*-algebras were first considered primarily for their use in quantum mechanics to model algebras of physical observables. This line of research began with Werner Heisenberg's matrix mechanics and in a more mathematically developed form with Pascual Jordan around 1933. Subsequently, John von Neumann attempted to establi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modeled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective. Examples of operators to which the spectral theorem appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators. Properties The algebra is a commutative ring. In the classical elliptic modular form theory, the Hecke operators ''T''''n'' with ''n'' coprime to the level acting on the space of cusp forms of a given weight are self-adjoint with respect to the Petersson inner product. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators. Each of these basic forms possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime ''p'' is the reciprocal of the Hecke polynomial, a quadratic polynomial in ''p''−''s''. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of ''τ''(''n''). See al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
