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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 the ... [...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] |
Cyclotomic Field
In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , 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 ) – 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 , let ; this is a primitive th root of unity. Then the th cyclotomic field is the extension of generated by . Properties * The th cyclotomic polynomial : \Phi_n(x) = \!\!\!\prod_\stackrel\!\!\! \left(x-e^\right) = \!\!\!\prod_\stackrel\!\!\! (x-^k) :is irreducible, so it is the minimal polynomial of over . * The conjugates of in are therefore the other primiti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 managing editors are Camillo De Lellis (Institute for Advanced Study, Princeton) and Jean-Benoît Bost (University of Paris-Sud Paris-Sud University (French: ''Université Paris-Sud''), also known as University of Paris — XI (or as Université d'Orsay before 1971), was a French research university distributed among several campuses in the southern suburbs of Paris, in ...). Abstracting and indexing The journal is abstracted and indexed in: References External links *{{Official website, https://www.springer.com/journal/222 Mathematics journals Publications established in 1966 English-language journals Springer Science+Business Media academic journals Monthly journals ... [...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] |
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 bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 bo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teichmüller Character
In number theory, the Teichmüller character ω (at a prime ''p'') is a character of (Z/''q''Z)×, 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, ω 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 of the natural surjection . The image of an element under this map is called its Teichmüller representative. The restriction of ω to ''k''× 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 multiplica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kummer Congruence
In mathematics, Kummer's congruences are some congruences involving Bernoulli numbers, found by . used Kummer's congruences to define the p-adic zeta function. Statement The simplest form of Kummer's congruence states that : \frac\equiv \frac \pmod p \text h\equiv k \pmod where ''p'' is a prime, ''h'' and ''k'' are positive even integers not divisible by ''p''−1 and the numbers ''B''''h'' are Bernoulli numbers. More generally if ''h'' and ''k'' are positive even integers not divisible by ''p'' − 1, then : (1-p^)\frac\equiv (1-p^)\frac \pmod whenever : h\equiv k\pmod where φ(''p''''a''+1) is the Euler totient function, evaluated at ''p''''a''+1 and ''a'' is a non negative integer. At ''a'' = 0, the expression takes the simpler form, as seen above. The two sides of the Kummer congruence are essentially values of the p-adic zeta function, and the Kummer congruences imply that the ''p''-adic zeta function for negative integers is continuous, so can be extended b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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, specializing in number theory. He is best known for proving Fermat's Last Theorem, for which he was awarded the 2016 Abel Prize and the 2017 Copley Medal by the Royal Society. He was appointed Knight Commander of the Order of the British Empire in 2000, and in 2018, was appointed the first Regius Professor of Mathematics at Oxford. Wiles is also a MacArthur Fellows Program, 1997 MacArthur Fellow. Education and early life Wiles was born on 11 April 1953 in Cambridge, England, Cambridge, England, the son of Maurice Wiles, Maurice Frank Wiles (1923–2005) and Patricia Wiles (née Mowll). From 1952-1955, his father worked as the chaplain at Ridley Hall, Cambridge, and later became the Regius Professor of Divinity at the University of Oxford. Wiles attended King's College School, Cambridge, and The Leys School, Cambridge. Wiles states that h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 MIT, although he did not graduate from the latter on account of failing a then-present ROTC requirement. He was nonetheless accepted for graduate studies at Princeton University, from where he received his PhD in mathematics in 1959 after completing a doctoral dissertation titled "On embeddings of spheres." He then became a Junior Fellow at Harvard University from 1961 to 1964. He is the Gerhard Gade University Professor and a Senior Fellow at Harvard. He is the brother of Joseph Mazur and the father of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
