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Shimura's Reciprocity Law
In mathematics, Shimura's reciprocity law, introduced by , describes the action of ideles of imaginary quadratic fields on the values of modular functions at singular moduli. It forms a part of the Kronecker Jugendtraum, explicit class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ... for such fields. There are also higher-dimensional generalizations. References * {{numtheory-stub Theorems in number theory ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idele
In abstract algebra, an adelic algebraic group is a semitopological group defined by an algebraic group ''G'' over a number field ''K'', and the adele ring ''A'' = ''A''(''K'') of ''K''. It consists of the points of ''G'' having values in ''A''; the definition of the appropriate topology is straightforward only in case ''G'' is a linear algebraic group. In the case of ''G'' being an abelian variety, it presents a technical obstacle, though it is known that the concept is potentially useful in connection with Tamagawa numbers. Adelic algebraic groups are widely used in number theory, particularly for the theory of automorphic representations, and the arithmetic of quadratic forms. In case ''G'' is a linear algebraic group, it is an affine algebraic variety in affine ''N''-space. The topology on the adelic algebraic group G(A) is taken to be the subspace topology in ''A''''N'', the Cartesian product of ''N'' copies of the adele ring. In this case, G(A) is a topological group. History ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Quadratic Field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers. Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 and 1. If d>0, the corresponding quadratic field is called a real quadratic field, and, if d<0, it is called an imaginary quadratic field or a complex quadratic field, corresponding to whether or not it is a subfield of the field of the s. Quadratic fields have been studied in great depth, initially as part of the theory of s. There remain some unsolved prob ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Modular Function
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. 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 in the upper half-plane (among other requirements). Instead, modular functions are 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 forms which are functions defined on Lie groups which transform nicely with ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Moduli
Singular may refer to: * Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms * Singular homology * SINGULAR, an open source Computer Algebra System (CAS) * Singular or sounder, a group of boar, see List of animal names * Singular matrix, a matrix that is not invertible * Singular measure, a measure or probability distribution whose support has zero Lebesgue (or other) measure * Singular cardinal, an infinite cardinal number that is not a regular cardinal * The property of a ''singularity'' or ''singular point'' in various meanings; see Singularity (other) * Singular (band), a Thai jazz pop duo *'' Singular: Act I'', a 2018 studio album by Sabrina Carpenter *'' Singular: Act II'', a 2019 studio album by Sabrina Carpenter See also * Singulair, Merck trademark for the drug Montelukast * Cingular Wireless AT&T Mobility LLC, also known as AT&T Wireless and marketed as simply AT&T, is an American telecommunications company ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kronecker Jugendtraum
Kronecker's Jugendtraum or Hilbert's twelfth problem, of the 23 mathematical Hilbert problems, is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. That is, it asks for analogues of the roots of unity, as complex numbers that are particular values of the exponential function; the requirement is that such numbers should generate a whole family of further number fields that are analogues of the cyclotomic fields and their subfields. The classical theory of complex multiplication, now often known as the ''Kronecker Jugendtraum'', does this for the case of any imaginary quadratic field, by using modular functions and elliptic functions chosen with a particular period lattice related to the field in question. Goro Shimura extended this to CM fields. In the special case of totally real fields, a solution was given by Dasgupta and Kakde. This provides an effective method to construct the maximal abelian extension of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Class Field Theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credited as one of pioneers of the notion of a class field. However, this notion was already familiar to Kronecker and it was actually Weber who coined the term before Hilbert's fundamental papers came out. The relevant ideas were developed in the period of several decades, giving rise to a set of conjectures by Hilbert that were subsequently proved by Takagi and Artin (with the help of Chebotarev's theorem). One of the major results is: given a number field ''F'', and writing ''K'' for the maximal abelian unramified extension of ''F'', the Galois group of ''K'' over ''F'' is canonically isomorphic to the ideal class group of ''F''. This statement was generalized to the so called Artin reciprocity law; in the idelic language, writing ''CF' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
