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P-adic Zeta Function
In mathematics, a ''p''-adic zeta function, or more generally a ''p''-adic ''L''-function, is a function analogous to the Riemann zeta function, or more general ''L''-functions, but whose domain and target are ''p-adic'' (where ''p'' is a prime number). For example, the domain could be the ''p''-adic integers Z''p'', a profinite ''p''-group, or a ''p''-adic family of Galois representations, and the image could be the ''p''-adic numbers Q''p'' or its algebraic closure. The source of a ''p''-adic ''L''-function tends to be one of two types. The first source—from which Tomio Kubota and Heinrich-Wolfgang Leopoldt gave the first construction of a ''p''-adic ''L''-function —is via the ''p''-adic interpolation of special values of ''L''-functions. For example, Kubota–Leopoldt used Kummer's congruences for Bernoulli numbers to construct a ''p''-adic ''L''-function, the ''p''-adic Riemann zeta function ζ''p''(''s''), whose values at negative odd integers are those of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Cyclotomic Field
In algebraic number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to \Q, the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's Last Theorem. It was in the process of his deep investigations of the arithmetic of these fields (for prime n)—and more precisely, because of the failure of unique factorization in their rings of integers—that Ernst Kummer first introduced the concept of an ideal number and proved his celebrated congruences. Definition For n \geq 1, let :\zeta_n=e^\in\C. This is a primitive nth root of unity. Then the nth cyclotomic field is the field extension \mathbb(\zeta_n) of \mathbb generated by \zeta_n. Properties * The nth cyclotomic polynomial :: \Phi_n(x) = \prod_\stackrel\!\!\! \left(x-e^\right) = \prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of \zeta_n o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Inventiones Mathematicae
''Inventiones Mathematicae'' is a mathematical journal published monthly by Springer Science+Business Media. It was established in 1966 and is regarded as one of the most prestigious mathematics journals in the world. The current (2023) managing editors are Jean-Benoît Bost (University of Paris-Sud) and Wilhelm Schlag (Yale University Yale University is a Private university, private Ivy League research university in New Haven, Connecticut, United States. Founded in 1701, Yale is the List of Colonial Colleges, third-oldest institution of higher education in the United Stat ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Academic journals established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
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 '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P-adic Distribution
In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of ''p''-adic numbers. Definition If ''X'' is a topological space, a distribution on ''X'' with values in an abelian group ''G'' is a finitely additive function from the compact open subsets of ''X'' to ''G''. Equivalently, if we define the space of test functions to be the locally constant and compactly supported integer-valued functions, then a distribution is an additive map from test functions to ''G''. This is formally similar to the usual definition of distributions, which are continuous linear maps from a space of test functions on a manifold to the real numbers. ''p''-adic measures A ''p''-adic measure is a special case of a ''p''-adic distribution, analogous to a measure on a measurable space. A ''p''-adic distribution taking values in a normed space is called a ''p''-adic measure if the values on compact open subsets are bou ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
P-adic Measure
In mathematics, a p-adic distribution is an analogue of ordinary distributions (i.e. generalized functions) that takes values in a ring of ''p''-adic numbers. Definition If ''X'' is a topological space, a distribution on ''X'' with values in an abelian group ''G'' is a finitely additive function from the compact open subsets of ''X'' to ''G''. Equivalently, if we define the space of test functions to be the locally constant and compactly supported integer-valued functions, then a distribution is an additive map from test functions to ''G''. This is formally similar to the usual definition of distributions, which are continuous linear maps from a space of test functions on a manifold to the real numbers. ''p''-adic measures A ''p''-adic measure is a special case of a ''p''-adic distribution, analogous to a measure on a measurable space. A ''p''-adic distribution taking values in a normed space is called a ''p''-adic measure if the values on compact open subsets are b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Teichmüller Character
In number theory, the Teichmüller character \omega (at a prime p) is a character of (\Z/q\Z)^\times, where q = p if p is odd and q = 4 if p = 2, taking values in the roots of unity of the ''p''-adic integers. It was introduced by Oswald Teichmüller. Identifying the roots of unity in the p-adic integers with the corresponding ones in the complex numbers, \omega can be considered as a usual Dirichlet character of conductor q. More generally, given a complete discrete valuation ring O whose residue field k is perfect of characteristic p, there is a unique multiplicative section \omega:k\to O of the natural surjection O\to k. The image of an element under this map is called its Teichmüller representative. The restriction of \omega to k^\times is called the Teichmüller character. Definition If x is a p-adic integer, then \omega(x) is the unique solution of \omega(x)^p = \omega(x) that is congruent to x mod p. It can also be defined by :\omega(x)=\lim_ x^ The multiplicative gr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Kummer Congruence
Kummer is a German surname. Notable people with the surname include: *Bernhard Kummer (1897–1962), German Germanist *Clare Kummer (1873–1958), American composer, lyricist and playwright *Clarence Kummer (1899–1930), American jockey * Christopher Kummer (born 1975), German economist *Corby Kummer (born 1957), American journalist * Dirk Kummer (born 1966), German actor, director, and screenwriter *Eberhard Kummer (1940–2019), Austrian concert singer, lawyer, and medieval music expert * Eduard Kummer, also known as the following Ernst Kummer *Eloise Kummer (1916–2008), American actress *Ernst Kummer (1810–1893), German mathematician **Kummer configuration, a mathematical structure discovered by Ernst Kummer **Kummer surface, a related geometrical structure discovered by Ernst Kummer * Ferdinand von Kummer (1816–1900), German general *Frederic Arnold Kummer (1873–1943), American author, playwright, and screenwriter *Friedrich August Kummer (1797–1879), German cellist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Generalized Bernoulli Number
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and indepe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Andrew Wiles
Sir Andrew John Wiles (born 11 April 1953) is an English mathematician and a Royal Society Research Professor at the University of Oxford, specialising in number theory. He is best known for Wiles's proof of Fermat's Last Theorem, proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal and for which he was appointed a Order of the British Empire, Knight Commander of the Order of the British Empire in 2000. In 2018, Wiles was appointed the first Regius Professor of Mathematics at Oxford. Wiles is also a MacArthur Fellows Program, 1997 MacArthur Fellow. Wiles was born in Cambridge to theologian Maurice Frank Wiles and Patricia Wiles. While spending much of his childhood in Nigeria, Wiles developed an interest in mathematics and in Fermat's Last Theorem in particular. After moving to Oxford and graduating from there in 1974, he worked on unifying Galois representations, elliptic curves and modular forms, starting with Barry Mazur's gene ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Barry Mazur
Barry Charles Mazur (; born December 19, 1937) is an American mathematician and the Gerhard Gade University Professor at Harvard University. His contributions to mathematics include his contributions to Wiles's proof of Fermat's Last Theorem in number theory, Mazur's torsion theorem in arithmetic geometry, the Mazur swindle in geometric topology, and the Mazur manifold in differential topology. Life Born in New York City, Mazur attended the Bronx High School of Science, and left after his junior year to attend MIT; he did not graduate from the university on account of failing a then-present ROTC requirement. He was nonetheless accepted for graduate studies at Princeton University, where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled ''On embeddings of spheres''. Thus, his only academic degree is a PhD. He then became a Junior Fellow at Harvard, Harvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Seni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
