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Modular Symbol
In mathematics, modular symbols, introduced independently by Bryan John Birch and by , span a vector space closely related to a space of modular forms, on which the action of the Hecke algebra can be described explicitly. This makes them useful for computing with spaces of modular forms. Definition The abelian group of (universal weight 2) modular symbols is spanned by symbols for α, β in the rational projective line Q ∪ subject to the relations * + = Informally, represents a homotopy class of paths from α to β in the upper half-plane. The group ''GL''2(Q) acts on the rational projective line In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the ..., and this induces an action on the modular symbols. There is a pairing between cusp forms ''f'' of weight 2 and modular symb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bryan John Birch
Bryan John Birch FRS (born 25 September 1931) is a British mathematician. His name has been given to the Birch and Swinnerton-Dyer conjecture. Biography Bryan John Birch was born in Burton-on-Trent, the son of Arthur Jack and Mary Edith Birch. He was educated at Shrewsbury School and Trinity College, Cambridge. He married Gina Margaret Christ in 1961. They have three children. As a doctoral student at the University of Cambridge, he was officially working under J. W. S. Cassels. More influenced by Harold Davenport, he proved Birch's theorem, one of the results to come out of the Hardy–Littlewood circle method. He then worked with Peter Swinnerton-Dyer on computations relating to the Hasse–Weil L-functions of elliptic curves. Their subsequently formulated conjecture relating the rank of an elliptic curve to the order of zero of an L-function has been an influence on the development of number theory from the mid-1960s onwards. He introduced modular symbols in about 19 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Form
In mathematics, a modular form is a holomorphic function on the complex upper half-plane, \mathcal, that roughly satisfies a functional equation with respect to the group action of the modular group and a growth condition. The theory of modular forms has origins in complex analysis, with important connections with number theory. Modular forms also appear in other areas, such as algebraic topology, sphere packing, and string theory. Modular form theory is a special case of the more general theory of automorphic forms, which are functions defined on Lie groups that transform nicely with respect to the action of certain discrete subgroups, generalizing the example of the modular group \mathrm_2(\mathbb Z) \subset \mathrm_2(\mathbb R). Every modular form is attached to a Galois representation. The term "modular form", as a systematic description, is usually attributed to Erich Hecke. The importance of modular forms across multiple field of mathematics has been humorously re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Group Action (mathematics)
In mathematics, a group action of a group G on a set (mathematics), set S is a group homomorphism from G to some group (under function composition) of functions from S to itself. It is said that G acts on S. Many sets of transformation (function), transformations form a group (mathematics), group under function composition; for example, the rotation (mathematics), rotations around a point in the plane. It is often useful to consider the group as an abstract group, and to say that one has a group action of the abstract group that consists of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally, a group of transformations of a mathematical structure, structure acts also on various related structures; for example, the above rotation group also acts on triangles by transforming triangles into triangles. If a group acts on a structure, it will usually also act on objects built from that st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hecke Algebra
In mathematics, the Hecke algebra is the algebra generated by Hecke operators, which are named after Erich Hecke. 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 multi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after the Norwegian mathematician Niels Henrik Abel. The concept of an abelian group underlies many fundamental algebraic structures, such as fields, rings, vector spaces, and algebras. The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood and fully classified. Definition An abelian group is a set A, together with an operation ･ , that combines any two elements a and b of A to form another element of A, denoted a \cdot b. The sym ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Upper Half-plane
In mathematics, the upper half-plane, is the set of points in the Cartesian plane with The lower half-plane is the set of points with instead. Arbitrary oriented half-planes can be obtained via a planar rotation. Half-planes are an example of two-dimensional half-space. A half-plane can be split in two quadrants. Affine geometry The affine transformations of the upper half-plane include # shifts (x,y)\mapsto (x+c,y), c\in\mathbb, and # dilations (x,y)\mapsto (\lambda x,\lambda y), \lambda > 0. Proposition: Let and be semicircles in the upper half-plane with centers on the boundary. Then there is an affine mapping that takes A to B. :Proof: First shift the center of to Then take \lambda=(\text\ B)/(\text\ A) and dilate. Then shift to the center of Inversive geometry Definition: \mathcal := \left\ . can be recognized as the circle of radius centered at and as the polar plot of \rho(\theta) = \cos \theta. Proposition: in and are collinear points. In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Line
In projective geometry and mathematics more generally, a projective line is, roughly speaking, the extension of a usual line by a point called a '' point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case). There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field ''K'', commonly denoted P1(''K''), as the set of one-dimensional subspaces of a two-dimensional ''K''-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals is a manifold; see '' Real projective line'' for details. Homogeneous coordinates An arbitrary point in the projective line P1(''K'') may be represented by an equivalence class of '' homogeneous coordi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
