|
ZilberâPink Conjecture
In mathematics, the ZilberâPink conjecture is a far-reaching generalisation of many famous Diophantine conjectures and statements, such as AndrĂ©âOort, ManinâMumford, and MordellâLang. For algebraic tori and semiabelian varieties it was proposed by Boris Zilber and independently by Enrico Bombieri, David Masser, Umberto Zannier in the early 2000's. For semiabelian varieties the conjecture implies the MordellâLang and ManinâMumford conjectures. Richard Pink proposed (again independently) a more general conjecture for Shimura varieties which also implies the AndrĂ©âOort conjecture. In the case of algebraic tori, Zilber called it the Conjecture on Intersection with Tori (CIT). The general version is now known as the ZilberâPink conjecture. It states roughly that atypical or unlikely intersections of an algebraic variety with certain special varieties are accounted for by finitely many special varieties. Statement Atypical and unlikely intersections The inters ... [...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] |
Diophantine Geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Four theorems in Diophantine geometry which are of fundamental importance include: * MordellâWeil Theorem * Roth's Theorem * Siegel's Theorem * Faltings's Theorem Background Serge Lang published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine Geometry". The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's ''Diophantine Equations'' (1969). Mordell's book starts with a remark on homogeneous equations ''f'' = 0 over the rational field, attributed to C. F. Gauss, that non-zero solutions in integers (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of L. E. D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AndrĂ©âOort Conjecture
In mathematics, the AndrĂ©âOort conjecture is a problem in Diophantine geometry, a branch of number theory, that can be seen as a non-abelian analogue of the ManinâMumford conjecture, which is now a theorem (and is actually proven in several genuinely different ways). The conjecture concerns itself with a characterization of the Zariski closure of sets of special points in Shimura varieties. A special case of the conjecture was stated by Yves AndrĂ© in 1989 and a more general statement (albeit with a restriction on the type of the Shimura variety) was conjectured by Frans Oort in 1995. The modern version is a natural generalization of these two conjectures. Statement The conjecture in its modern form is as follows. Each irreducible component of the Zariski closure of a set of special points in a Shimura variety is a special subvariety. AndrĂ©'s first version of the conjecture was just for one dimensional irreducible components, while Oort proposed that it should be true f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arithmetic Of Abelian Varieties
In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety ''A'' over a number field ''K''; or more generally (for global fields or more general finitely-generated rings or fields). Integer points on abelian varieties There is some tension here between concepts: ''integer point'' belongs in a sense to affine geometry, while ''abelian variety'' is inherently defined in projective geometry. The basic results, such as Siegel's theorem on integral points, come from the theory of diophantine approximation. Rational points on abelian varieties The basic result, the MordellâWeil theorem in Diophantine geometry, says that ''A''(''K''), the group of points on ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MordellâLang Conjecture
This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality. Diophantine geometry in general is the study of algebraic varieties ''V'' over fields ''K'' that are finitely generated over their prime fieldsâincluding as of special interest number fields and finite fieldsâand over local fields. Of those, only the complex numbers are algebraically closed; over any other ''K'' the existence of points of ''V'' with coordinates in ''K'' is something to be proved and studied as an extra topic, even knowing the geometry of ''V''. Arithmetic geometry can be more generally defined as the study of schemes of finite type over the spectrum of the ring of integers. Arithmetic geometry has also been defined as the application of the tec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Torus
In mathematics, an algebraic torus, where a one dimensional torus is typically denoted by \mathbf G_, \mathbb_m, or \mathbb, is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups \mathbf G_. These groups were named by analogy with the theory of ''tori'' in Lie group theory (see Cartan subgroup). For example, over the complex numbers \mathbb the algebraic torus \mathbf G_ is isomorphic to the group scheme \mathbb^* = \text(\mathbb ,t^, which is the scheme theoretic analogue of the Lie group U(1) \subset \mathbb. In fact, any \mathbf G_-action on a complex vector space can be pulled back to a U(1)-action from the inclusion U(1) \subset \mathbb^* as real manifolds. Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them such as symmetric spaces and buil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiabelian 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 from a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Boris Zilber
Boris Zilber (russian: ĐĐŸŃĐžŃ ĐĐŸŃĐžŃĐŸĐČĐžŃ ĐОлŃбДŃ, born 1949) is a Soviet-British mathematician who works in mathematical logic, specifically model theory. He is a professor of mathematical logic at the University of Oxford. He obtained his doctorate (Candidate of Sciences) from the Novosibirsk State University in 1975 under the supervision of Mikhail Taitslin and his habilitation (Doctor of Sciences) from the Saint Petersburg State University in 1986. He received the Senior Berwick Prize (2004) and the PĂłlya Prize (2015) from the London Mathematical Society. He also gave the Tarski Lectures The Alfred Tarski Lectures are an annual distinction in mathematical logic and series of lectures held at the University of California, Berkeley. Established in tribute to Alfred Tarski on the fifth anniversary of his death, the award has been give ... in 2002. References External links Prof. Zilber's homepage 20th-century British mathematicians 21st-century B ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enrico Bombieri
Enrico Bombieri (born 26 November 1940, Milan) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently Professor Emeritus in the School of Mathematics at the Institute for Advanced Study in Princeton, New Jersey. Bombieri won the Fields Medal in 1974 for his contributions to large sieve mathematics, conceptualized by Linnick 1941, and its application to the distribution of prime numbers. Career Bombieri published his first mathematical paper in 1957 when he was 16 years old. In 1963 at age 22 he earned his first degree (Laurea) in mathematics from the UniversitĂ degli Studi di Milano under the supervision of Giovanni Ricci and then studied at Trinity College, Cambridge with Harold Davenport. Bombieri was an assistant professor (1963â1965) and then a full professor (1965â1966) at the UniversitĂ di Cagliari, at the UniversitĂ di Pisa in 1966â1974, and then at the Scuola No ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
David Masser
David William Masser (born 8 November 1948) is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known for his work in transcendental number theory, Diophantine approximation, and Diophantine geometry. With Joseph Oesterlé in 1985, Masser formulated the abc conjecture, which has been called "the most important unsolved problem in Diophantine analysis".. Early life and education Masser was born on 8 November 1948 in London, England. He graduated from Trinity College, Cambridge with a B.A. (Hons) in 1970. In 1974, he obtained his M.A. and Ph.D. at the University of Cambridge, with a doctoral thesis under the supervision of Alan Baker titled ''Elliptic Functions and Transcendence''. Career Masser was a Lecturer at the University of Nottingham from 1973 to 1975, before spending the 1975-1976 year as a Research Fellow of Trinity College at the University of Cambridge. He returned to the University of Nottingham to serve as a L ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Umberto Zannier
Umberto Zannier (born 25 May 1957, in Spilimbergo, Italy) is an Italian mathematician, specializing in number theory and Diophantine geometry. Education Zannier earned a Laurea degree from University of Pisa and studied at the Scuola Normale Superiore di Pisa with Ph.D. supervised by Enrico Bombieri. Career Zannier was from 1983 to 1987 a researcher at the University of Padua, from 1987 to 1991 an associate professor at the University of Salerno, and from 1991 to 2003 a full professor at the Università IUAV di Venezia. From 2003 to the present he has been a Professor in Geometry at the Scuola Normale Superiore di Pisa. In 2010 he gave the Hermann Weyl Lectures at the Institute for Advanced Study. He was a visiting professor at several institutions, including the Institut Henri Poincaré in Paris, the ETH Zurich, and the Erwin Schrödinger Institute in Vienna. With Jonathan Pila he developed a method (now known as the Pila-Zannier method) of applying O-minimality to number-theo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shimura Variety
In number theory, a Shimura variety is a higher-dimensional analogue of a modular curve that arises as a quotient variety of a Hermitian symmetric space by a congruence subgroup of a reductive algebraic group defined over Q. Shimura varieties are not algebraic varieties but are families of algebraic varieties. Shimura curves are the one-dimensional Shimura varieties. Hilbert modular surfaces and Siegel modular varieties are among the best known classes of Shimura varieties. Special instances of Shimura varieties were originally introduced by Goro Shimura in the course of his generalization of the complex multiplication theory. Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit models defined over a number field, the reflex field of the Shimura variety. In the 1970s, Pierre Deligne created an axiomatic framework for the work of Shimura. In 1979, Robert Langlands remarked that Shimura varieties form a natural realm of ex ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
