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Fontaine's Period Rings
In mathematics, Fontaine's period rings are a collection of commutative rings first defined by Jean-Marc Fontaine that are used to classify ''p''-adic Galois representations. The ring BdR The ring \mathbf_ is defined as follows. Let \mathbf_p denote the completion of \overline. Let :\tilde^+ = \varprojlim_ \mathcal_/(p) So an element of \tilde^+ is a sequence (x_1,x_2,\ldots) of elements x_i\in \mathcal_/(p) such that x_^p \equiv x_i \pmod p. There is a natural projection map f:\tilde^+ \to \mathcal_/(p) given by f(x_1,x_2,\dotsc) = x_1. There is also a multiplicative (but not additive) map t:\tilde^+\to \mathcal_ defined by t(x_,x_2,\dotsc) = \lim_ \tilde x_i^, where the \tilde x_i are arbitrary lifts of the x_i to \mathcal_. The composite of t with the projection \mathcal_\to \mathcal_/(p) is just f. The general theory of Witt vectors In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set ... [...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] |
Commutative Ring
In mathematics, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra. Complementarily, noncommutative algebra is the study of ring properties that are not specific to commutative rings. This distinction results from the high number of fundamental properties of commutative rings that do not extend to noncommutative rings. Definition and first examples Definition A ''ring'' is a set R equipped with two binary operations, i.e. operations combining any two elements of the ring to a third. They are called ''addition'' and ''multiplication'' and commonly denoted by "+" and "\cdot"; e.g. a+b and a \cdot b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition; i.e., a \cdot \left(b + c\right) = \left(a \cdot b\right) + \left(a \cdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Marc Fontaine
Jean-Marc Fontaine (13 March 1944 – 29 January 2019) was a French mathematician. He was one of the founders of p-adic Hodge theory. He was a professor at Paris-Sud 11 University from 1988 to his death. Life In 1962 Fontaine entered the École Polytechnique, from 1965 to 1971 was a researcher at CNRS and received his doctorate in 1972. From 1971 to 72 he was at the University of Paris VI and from 1972 to 1988 was at the University of Grenoble (only Maître de Conferences, but later a professor). From 1989 he was professor at the University of Paris-Sud XI in Orsay. Among his first works was the classification of ''p''-divisible groups (= Barsotti–Tate group) over the ring of integers of a local field and the field of ''p''-adic periods, a ''p''-adic analogue of the field of complex numbers. Fontaine is one of the founders of p-adic Hodge theory. He proved that there are no non-trivial abelian varieties over the rational numbers with good reduction everywhere (''Il n'y a p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Representation
In mathematics, a Galois module is a ''G''-module, with ''G'' being the Galois group of some extension of fields. The term Galois representation is frequently used when the ''G''-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for ''G''-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory. Examples *Given a field ''K'', the multiplicative group (''Ks'')× of a separable closure of ''K'' is a Galois module for the absolute Galois group. Its second cohomology group is isomorphic to the Brauer group of ''K'' (by Hilbert's theorem 90, its first cohomology group is zero). *If ''X'' is a smooth proper scheme over a field ''K'' then the ℓ-adic cohomology groups of its geometric fibre are Galois modules for the absolute Galois group of ''K''. Ramification theory Let ''K'' be a valued field (with valuation denoted ''v'') ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Witt Vectors
In mathematics, a Witt vector is an infinite sequence of elements of a commutative ring. Ernst Witt showed how to put a ring structure on the set of Witt vectors, in such a way that the ring of Witt vectors W(\mathbb_p) over the finite field of order p is the ring of p-adic integers. They have a highly non-intuitive structure upon first glance because their additive and multiplicative structure depends on an infinite set of recursive formulas which do not behave like addition and multiplication formulas for standard p-adic integers. The main idea behind Witt vectors is instead of using the standard p-adic expansiona = a_0+a_1p+a_2p^2 + \cdotsto represent an element in \mathbb_p, we can instead consider an expansion using the Teichmüller character\omega: \mathbb_p^* \to \mathbb_p^*which sends each element in the solution set of x^-1 in \mathbb_p to an element in the solution set of x^-1 in \mathbb_p. That is, we expand out elements in \mathbb_p in terms of roots of unity instead of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Teichmüller Representative
Teichmüller is a German surname (German for ''pond miller'') and may refer to: * Anna Teichmüller (1861–1940), German composer * :de:Frank Teichmüller (19?? – now), former German IG Metall district manager "coast" * Gustav Teichmüller (1832–1888), German philosopher * :de:Marlies Teichmüller (1914–2000), German geologist * (Paul Julius) Oswald Teichmüller (1913–1943), German mathematician * Robert Teichmüller (1863–1939), pianist and professor * :de:Rolf Teichmüller (1904–1983), German geologist See also * Teichmüller space * Inter-universal Teichmüller theory Inter-universal Teichmüller theory (abbreviated as IUT or IUTT) is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic geometry. According to Mochizuki, it is "an arith ... {{DEFAULTSORT:Teichmueller German-language surnames ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Number Theory
Algebraic number theory is a branch of number theory that uses the techniques of abstract algebra to study the integers, rational numbers, and their generalizations. Number-theoretic questions are expressed in terms of properties of algebraic objects such as algebraic number fields and their rings of integers, finite fields, and Algebraic function field, function fields. These properties, such as whether a ring (mathematics), ring admits unique factorization, the behavior of ideal (ring theory), ideals, and the Galois groups of field (mathematics), fields, can resolve questions of primary importance in number theory, like the existence of solutions to Diophantine equations. History of algebraic number theory Diophantus The beginnings of algebraic number theory can be traced to Diophantine equations, named after the 3rd-century Alexandrian mathematician, Diophantus, who studied them and developed methods for the solution of some kinds of Diophantine equations. A typical Diophantin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to group theory, which makes them simpler and easier to understand. Galois introduced the subject for studying roots of polynomials. This allowed him to characterize the polynomial equations that are solvable by radicals in terms of properties of the permutation group of their roots—an equation is ''solvable by radicals'' if its roots may be expressed by a formula involving only integers, th roots, and the four basic arithmetic operations. This widely generalizes the Abel–Ruffini theorem, which asserts that a general polynomial of degree at least five cannot be solved by radicals. Galois theory has been used to solve classic problems including showing that two problems of antiquity cannot be solved as they were stated (doubling the cub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory Of Groups
Representation may refer to: Law and politics *Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a group of people * Representation, in contract law a pre-contractual statement that may (if untrue) result in liability for misrepresentation * Labor representation, or worker representation, the work of a union representative who represents and defends the interests of fellow labor union members * Legal representation, provided by a barrister, lawyer, or other advocate * Lobbying or interest representation, attempts to influence the actions, policies, or decisions of officials * "No taxation without representation", a 1700s slogan that summarized one of the American colonists' 27 colonial grievances in the Thirteen Colonies, which was one of the major causes of the American Revolution * Permanent representation, a type of diplomatic mission ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
