In mathematics, in particular in the theory of modular forms, a Hecke operator 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). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, the general notion of automorphic form is the extension to Analytic functions perhaps of Several complex variables, of the theory of
These operators can be realised in a number of contexts; the simplest meaning is combinatorial, namely as taking for a given integer n some function f(Λ) defined on lattices to
with the sum taken over all the Λ′ that are subgroups of Λ of index n. In Mathematics, especially in Geometry and Group theory, a lattice in R n is a Discrete subgroup of In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of For example, with n=2 and two dimensions, there are three such Λ′. Modular forms are particular kinds of functions of a lattice, subject to conditions making them analytic functions and homogeneous with respect to enlargement of a lattice; these conditions are preserved by the summation and so Hecke operators take modular forms to modular forms. This article is about both real and complex analytic functions In Mathematics, a homogeneous function is a function with multiplicative scaling behaviour if the argument is multiplied by a factor then the result is multiplied by some power
Algebras of Hecke operators are called Hecke algebras, and the most significant basic fact of the theory is that these are commutative rings. Hecke algebra is the common name of several related types of Associative rings in algebra and Representation theory. In Ring theory, a branch of Abstract algebra, a commutative ring is a ring in which the multiplication operation has the commutative property Other mathematical rings are called Hecke algebras, without the obvious link to Hecke operators. These include certain quotients of the group algebra of a braid group. In Mathematics, the group algebra is any of various constructions to assign to a Locally compact group an Operator algebra (or more generally a Banach In Mathematics, the braid group on n strands, denoted by B n, is a certain group which has an intuitive geometrical representation
The theory of Hecke operators on modular forms is often said to have been founded by Mordell in a paper on the special cusp form of Ramanujan, ahead of the general theory given by Erich Hecke. Louis Joel Mordell ( 28 January 1888 - 12 March 1972) was a British mathematician known for pioneering research in Number theory. In Number theory, a branch of Mathematics, a cusp form is a particular kind of Modular form, distinguished in the case of modular forms for the Modular Erich Hecke ( September 20, 1887 &ndash February 13, 1947) was a German Mathematician. The idea may be considered to go back to earlier work of Hurwitz, who treated correspondences between modular curves which realise some individual Hecke operator. Hurwitz is a surname and may refer to Aaron Hurwitz, musician see Live on Breeze Hill Adolf Hurwitz (1859-1919 German mathematician In Number theory and Algebraic geometry, a modular curve is a Riemann surface, or the corresponding Algebraic curve, constructed as a quotient In fact the algebraic theory of correspondences (relations closed for the Zariski topology) is another and natural way to express the formal sum involved. This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations In Mathematics, namely Algebraic geometry, the Zariski topology is a particular Topology chosen for algebraic varieties that reflects the algebraic In Abstract algebra, a free abelian group is an Abelian group that has a "basis" in the sense that every element of the group can be written in
A third way to express Hecke operators is as double cosets in the modular group. In Mathematics, an ( H, K) double coset in G, where G is a group and H and K are Subgroups In Mathematics, the modular group Γ is a fundamental object of study in Number theory, Geometry, algebra, and many other areas of advanced In the contemporary adelic approach, this translates to double cosets with respect to some compact subgroups. In Number theory, the adele ring is a Topological ring which is built on the field of Rational numbers (or more generally any Algebraic In any case, the presence of this commutative operator algebra plays a significant role in the harmonic analysis of modular forms and generalisations. Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes
In the classical elliptic modular form theory, it is shown that the Hecke operators are a C*-algebra with respect to the Petersson inner product; and that therefore the spectral theory implies that there is a basis of modular forms that are eigenfunctions for all Hecke operators. In Mathematics, a modular form is a (complex Analytic function on the Upper half-plane satisfying a certain kind of Functional equation and C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. 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 In Mathematics, spectral theory is an inclusive term for theories extending the Eigenvector and Eigenvalue theory of a single Square matrix. In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in These basic forms all have Euler products (more precisely, their Mellin transforms are Dirichlet series that have Euler products with the factor for each prime p being of degree 2). In Number theory, an Euler product is an Infinite product expansion indexed by Prime numbers p, of a Dirichlet series. In Mathematics, the Mellin transform is an Integral transform that may be regarded as the multiplicative version of the Two-sided Laplace transform In Mathematics, a Dirichlet series is any series of the form \sum_{n=1}^{\infty} \frac{a_n}{n^s} where s and In the case treated by Mordell, there is a one-dimensional space of cusp forms of weight 12; so that the Euler product must apply to Ramanujan's form.
References
- Apostol, Tom M. (1990), Modular functions and Dirichlet series in number theory (2nd ed. Tom Mike Apostol (born 1923 is an American analytic number theorist and professor at the California Institute of Technology. ), Berlin, New York: Springer-Verlag, ISBN 978-0-387-97127-8 (See chapter 8. Springer Science+Business Media or Springer (ˈʃpʁɪŋɐ is a worldwide Publishing company based in Germany, which publishes textbooks academic )
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