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40 relations: Adele ring, Adolf Hurwitz, Analytic function, Automorphic form, Braid group, Commutative ring, Correspondence (mathematics), Cusp form, Determinant, Dirichlet series, Don Zagier, Double coset, Eichler–Shimura congruence relation, Eigenform, Eigenfunction, Euler product, Group algebra, Harmonic analysis, Hecke algebra acting on modular forms, Homogeneous function, Homothetic transformation, Jean-Pierre Serre, Lattice (group), Mathematical Proceedings of the Cambridge Philosophical Society, Mathematics, Mathematische Annalen, Mellin transform, Modular curve, Modular form, Modular group, Multiplicative function, Petersson inner product, Ramanujan tau function, Self-adjoint operator, Spectral theorem, Springer Science+Business Media, Srinivasa Ramanujan, Subgroup, Upper half-plane, Vector space.
Adele ring
In mathematics, the adele ring (also adelic ring or ring of adeles) is defined in class field theory, a branch of algebraic number theory.
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Adolf Hurwitz
Adolf Hurwitz (26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory.
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Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series.
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Automorphic form
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.
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Braid group
In mathematics, the braid group on strands (denoted), also known as the Artin braid group, is the group whose elements are equivalence classes of n-braids (e.g. under ambient isotopy), and whose group operation is composition of braids (see). Example applications of braid groups include knot theory, where any knot may be represented as the closure of certain braids (a result known as Alexander's theorem); in mathematical physics where Artin's canonical presentation of the braid group corresponds to the Yang–Baxter equation (see); and in monodromy invariants of algebraic geometry.
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Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative.
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Correspondence (mathematics)
In mathematics and mathematical economics, correspondence is a term with several related but distinct meanings.
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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.
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Determinant
In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.
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Dirichlet series
In mathematics, a Dirichlet series is any series of the form where s is complex, and a_n is a complex sequence.
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Don Zagier
Don Bernard Zagier (born 29 June 1951) is an American mathematician whose main area of work is number theory.
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Double coset
In group theory, a field of mathematics, a double coset is a collection of group elements which are equivalent under the symmetries coming from two subgroups.
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Eichler–Shimura congruence relation
In number theory, the Eichler–Shimura congruence relation expresses the local ''L''-function of a modular curve at a prime p in terms of the eigenvalues of Hecke operators.
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Eigenform
An eigenform (meaning simultaneous Hecke eigenform with modular group SL(2,Z)) is a modular form which is an eigenvector for all Hecke operators Tm, m.
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Eigenfunction
In mathematics, an eigenfunction of a linear operator D defined on some function space is any non-zero function f in that space that, when acted upon by D, is only multiplied by some scaling factor called an eigenvalue.
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Euler product
In number theory, an Euler product is an expansion of a Dirichlet series into an infinite product indexed by prime numbers.
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Group algebra
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 algebra), such that representations of the algebra are related to representations of the group.
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Harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an extended form of Fourier analysis).
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Hecke algebra acting on modular forms
In number theory in mathematics, the Hecke algebra is the algebra generated by Hecke operators.
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Homogeneous function
In mathematics, a homogeneous function is one with multiplicative scaling behaviour: if all its arguments are multiplied by a factor, then its value is multiplied by some power of this factor.
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Homothetic transformation
In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point S called its center and a nonzero number λ called its ratio, which sends in other words it fixes S, and sends any M to another point N such that the segment SN is on the same line as SM, but scaled by a factor λ. In Euclidean geometry homotheties are the similarities that fix a point and either preserve (if) or reverse (if) the direction of all vectors.
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Jean-Pierre Serre
Jean-Pierre Serre (born 15 September 1926) is a French mathematician who has made contributions to algebraic topology, algebraic geometry, and algebraic number theory.
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Lattice (group)
In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.
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Mathematical Proceedings of the Cambridge Philosophical Society
Mathematical Proceedings of the Cambridge Philosophical Society is a mathematical journal published by Cambridge University Press for the Cambridge Philosophical Society.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
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Mellin transform
In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform.
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Modular curve
In number theory and algebraic geometry, a modular curve Y(Γ) is a Riemann surface, or the corresponding algebraic curve, constructed as a quotient of the complex upper half-plane H by the action of a congruence subgroup Γ of the modular group of integral 2×2 matrices SL(2, Z).
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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 of the modular group, and also satisfying a growth condition.
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Modular group
In mathematics, the modular group is the projective special linear group PSL(2,Z) of 2 x 2 matrices with integer coefficients and unit determinant.
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Multiplicative function
In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1).
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Petersson inner product
In mathematics the Petersson inner product is an inner product defined on the space of entire modular forms.
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Ramanujan tau function
The Ramanujan tau function, studied by, is the function \tau:\mathbb\to\mathbb defined by the following identity: where q.
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Self-adjoint operator
In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.
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Spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Srinivasa Ramanujan
Srinivasa Ramanujan (22 December 188726 April 1920) was an Indian mathematician who lived during the British Rule in India. Though he had almost no formal training in pure mathematics, he made substantial contributions to mathematical analysis, number theory, infinite series, and continued fractions, including solutions to mathematical problems considered to be unsolvable.
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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 ∗.
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Upper half-plane
In mathematics, the upper half-plane H is the set of complex numbers with positive imaginary part: The term arises from a common visualization of the complex number x + iy as the point (x,y) in the plane endowed with Cartesian coordinates.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Hecke correspondence, Hecke operators, Modular eigenform.
References
[1] https://en.wikipedia.org/wiki/Hecke_operator
