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Manin Obstruction
In mathematics, in the field of arithmetic algebraic geometry, the Manin obstruction (named after Yuri Manin) is attached to a variety ''X'' over a global field, which measures the failure of the Hasse principle for ''X''. If the value of the obstruction is non-trivial, then ''X'' may have points over all local fields but not over the global field. The Manin obstruction is sometimes called the Brauer–Manin obstruction, as Manin used the Brauer group of X to define it. For abelian varieties the Manin obstruction is just the Tate–Shafarevich group and fully accounts for the failure of the local-to-global principle (under the assumption that the Tate–Shafarevich group is finite). There are however examples, due to Alexei Skorobogatov Alexei Nikolaievich Skorobogatov (russian: Алексе́й Никола́евич Скоробога́тов) is a British-Russian mathematician and Professor in Pure Mathematics at Imperial College London specialising in algebraic geometry. Hi ... [...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] |
Arithmetic Geometry
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic variety, algebraic varieties. In more abstract terms, arithmetic geometry can be defined as the study of scheme (mathematics), schemes of Finite morphism#Morphisms of finite type, finite type over the spectrum of a ring, spectrum of the ring of integers. Overview The classical objects of interest in arithmetic geometry are rational points: solution set, sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or Algebraic function field, function fields, i.e. field (mathematics), fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined over ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuri Manin
Yuri Ivanovich Manin (russian: Ю́рий Ива́нович Ма́нин; born 16 February 1937) is a Russian mathematician, known for work in algebraic geometry and diophantine geometry, and many expository works ranging from mathematical logic to theoretical physics. Moreover, Manin was one of the first to propose the idea of a quantum computer in 1980 with his book ''Computable and Uncomputable''. Life and career Manin gained a doctorate in 1960 at the Steklov Mathematics Institute as a student of Igor Shafarevich. He is now a Professor at the Max-Planck-Institut für Mathematik in Bonn, and a professor emeritus at Northwestern University. Manin's early work included papers on the arithmetic and formal groups of abelian varieties, the Mordell conjecture in the function field case, and algebraic differential equations. The Gauss–Manin connection is a basic ingredient of the study of cohomology in families of algebraic varieties. He wrote a book on cubic surfaces and cubic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Global Field
In mathematics, a global field is one of two type of fields (the other one is local field) which are characterized using valuations. There are two kinds of global fields: * Algebraic number field: A finite extension of \mathbb *Global function field: The function field of an algebraic curve over a finite field, equivalently, a finite extension of \mathbb_q(T), the field of rational functions in one variable over the finite field with q=p^n elements. An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s. Formal definitions A ''global field'' is one of the following: ;An algebraic number field An algebraic number field ''F'' is a finite (and hence algebraic) field extension of the field of rational numbers Q. Thus ''F'' is a field that contains Q and has finite dimension when considered as a vector space over Q. ;The function field of an algebraic curve over a finite field A function field of a variety is t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hasse Principle
In mathematics, Helmut Hasse's local–global principle, also known as the Hasse principle, is the idea that one can find an integer solution to an equation by using the Chinese remainder theorem to piece together solutions modulo powers of each different prime number. This is handled by examining the equation in the completions of the rational numbers: the real numbers and the ''p''-adic numbers. A more formal version of the Hasse principle states that certain types of equations have a rational solution if and only if they have a solution in the real numbers ''and'' in the ''p''-adic numbers for each prime ''p''. Intuition Given a polynomial equation with rational coefficients, if it has a rational solution, then this also yields a real solution and a ''p''-adic solution, as the rationals embed in the reals and ''p''-adics: a global solution yields local solutions at each prime. The Hasse principle asks when the reverse can be done, or rather, asks what the obstruction is: wh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Local Field
In mathematics, a field ''K'' is called a (non-Archimedean) local field if it is complete with respect to a topology induced by a discrete valuation ''v'' and if its residue field ''k'' is finite. Equivalently, a local field is a locally compact topological field with respect to a non-discrete topology. Sometimes, real numbers R, and the complex numbers C (with their standard topologies) are also defined to be local fields; this is the convention we will adopt below. Given a local field, the valuation defined on it can be of either of two types, each one corresponds to one of the two basic types of local fields: those in which the valuation is Archimedean and those in which it is not. In the first case, one calls the local field an Archimedean local field, in the second case, one calls it a non-Archimedean local field. Local fields arise naturally in number theory as completions of global fields. While Archimedean local fields have been quite well known in mathematics for at lea ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brauer Group
Brauer or Bräuer is a surname of German origin, meaning "brewer". Notable people with the name include:- * Alfred Brauer (1894–1985), German-American mathematician, brother of Richard * Andreas Brauer (born 1973), German film producer * Arik Brauer (1929–2021), Austrian painter, poet, and actor, father of Timna Brauer * August Brauer (1863-1917), German zoologist * Friedrich Moritz Brauer (1832–1904), Austrian entomologist and museum director * Georg Brauer (1908–2001), German chemist * Ingrid Arndt-Brauer (born 1961), German politician; member of the Bundestag * Jono Brauer (born 1981), Australian Olympic skier * Max Brauer (1887–1973), German politician; First Mayor of Hamburg * Michael Brauer (contemporary), American audio engineer * Rich Brauer (born 1954), American politician from Illinois; state legislator since 2003 * Richard Brauer (1901–1977), German-American mathematician * Richard H. W. Brauer (contemporary), American art museum director; eponym of the Bra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abelian Variety
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory. An abelian variety can be defined by equations having coefficients in any field; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of complex numbers. Such abelian varieties turn out to be exactly those complex tori that can be embedded into a complex projective space. Abelian varieties defined over algebraic number fields are a special case, which is important also from the viewpoint of number theory. Localization techniques lead naturally fr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tate–Shafarevich Group
In arithmetic geometry, the Tate–Shafarevich group of an abelian variety (or more generally a group scheme) defined over a number field consists of the elements of the Weil–Châtelet group that become trivial in all of the completions of (i.e. the -adic fields obtained from , as well as its real and complex completions). Thus, in terms of Galois cohomology, it can be written as :\bigcap_v\mathrm\left(H^1\left(G_K,A\right)\rightarrow H^1\left(G_,A_v\right)\right). This group was introduced by Serge Lang and John Tate and Igor Shafarevich. Cassels introduced the notation , where is the Cyrillic letter " Sha", for Shafarevich, replacing the older notation or . Elements of the Tate–Shafarevich group Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of that have -rational points for every place of , but no -rational point. Thus, the group measures the extent to which the Hasse principle fails to ho ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Alexei Skorobogatov
Alexei Nikolaievich Skorobogatov (russian: Алексе́й Никола́евич Скоробога́тов) is a British-Russian mathematician and Professor in Pure Mathematics at Imperial College London specialising in algebraic geometry. His work has focused on rational points, the Hasse principle, the Manin obstruction, exponential sums, and error-correcting codes. Education He completed his dissertation under the supervision of Yuri Manin, for which he was awarded a Ph.D. degree. Awards In 2001 he was awarded a Whitehead Prize by the London Mathematical Society. He was elected as a Fellow of the 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, ... in the 2020 Class, for "contributions to the Diophantine geometry of surfaces and higher dimen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in Berlin, it expanded internationally in the 1960s, and through mergers in the 1990s and a sale to venture capitalists it fused with Wolters Kluwer and eventually became part of Springer Nature in 2015. Springer has major offices in Berlin, Heidelberg, Dordrecht, and New York City. History Julius Springer founded Springer-Verlag in Berlin in 1842 and his son Ferdinand Springer grew it from a small firm of 4 employees into Germany's then second largest academic publisher with 65 staff in 1872.Chronology ". Springer Science+Business Media. In 1964, Springer expanded its business internationally, o ... [...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] |
