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Conductor Of An Abelian Variety
In mathematics, in Diophantine geometry, the conductor of an abelian variety defined over a local or global field ''F'' is a measure of how "bad" the bad reduction at some prime is. It is connected to the ramification in the field generated by the torsion points. Definition For an abelian variety ''A'' defined over a field ''F'' as above, with ring of integers ''R'', consider the Néron model of ''A'', which is a 'best possible' model of ''A'' defined over ''R''. This model may be represented as a scheme over :Spec(''R'') (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism :Spec(''F'') → Spec(''R'') gives back ''A''. Let ''A''0 denote the open subgroup scheme of the Néron model whose fibres are the connected components. For a maximal ideal ''P'' of ''R'' with residue field ''k'', ''A''0''k'' is a group variety over ''k'', hence an extension of an abelian variety by a linear group. This linear group is an extension of a torus ... [...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] |
Spectrum Of A Ring
In commutative algebra, the prime spectrum (or simply the spectrum) of a ring ''R'' is the set of all prime ideals of ''R'', and is usually denoted by \operatorname; in algebraic geometry it is simultaneously a topological space equipped with the sheaf of rings \mathcal. Zariski topology For any ideal ''I'' of ''R'', define V_I to be the set of prime ideals containing ''I''. We can put a topology on \operatorname(R) by defining the collection of closed sets to be :\. This topology is called the Zariski topology. A basis for the Zariski topology can be constructed as follows. For ''f'' ∈ ''R'', define ''D''''f'' to be the set of prime ideals of ''R'' not containing ''f''. Then each ''D''''f'' is an open subset of \operatorname(R), and \ is a basis for the Zariski topology. \operatorname(R) is a compact space, but almost never Hausdorff: in fact, the maximal ideals in ''R'' are precisely the closed points in this topology. By the same reasoning, it is not, in general, a ... [...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] |
Semistable Abelian Variety
In algebraic geometry, a semistable abelian variety is an abelian variety defined over a global or local field, which is characterized by how it reduces at the primes of the field. For an abelian variety A defined over a field F with ring of integers R, consider the Néron model of A, which is a 'best possible' model of A defined over R. This model may be represented as a scheme over \mathrm(R) (cf. spectrum of a ring) for which the generic fibre constructed by means of the morphism \mathrm(F) \to \mathrm(R) gives back A. The Néron model is a smooth group scheme, so we can consider A^0, the connected component of the Néron model which contains the identity for the group law. This is an open subgroup scheme of the Néron model. For a residue field k, A^0_k is a group variety over k, hence an extension of an abelian variety by a linear group. If this linear group is an algebraic torus, so that A^0_k is a semiabelian variety, then A has ''semistable reduction'' at the prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Good Reduction
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 fields—including as of special interest number fields and finite fields—and 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 tech ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unipotent Group
In mathematics, a unipotent element ''r'' of a ring ''R'' is one such that ''r'' − 1 is a nilpotent element; in other words, (''r'' − 1)''n'' is zero for some ''n''. In particular, a square matrix ''M'' is a unipotent matrix if and only if its characteristic polynomial ''P''(''t'') is a power of ''t'' − 1. Thus all the eigenvalues of a unipotent matrix are 1. The term quasi-unipotent means that some power is unipotent, for example for a diagonalizable matrix with eigenvalues that are all roots of unity. In the theory of algebraic groups, a group element is unipotent if it acts unipotently in a certain natural group representation. A unipotent affine algebraic group is then a group with all elements unipotent. Definition Definition with matrices Consider the group \mathbb_n of upper-triangular matrices with 1's along the diagonal, so they are the group of matrices :\mathbb_n = \left\. Then, a unipotent group can be defined ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Residue Field
In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a local ring and ''m'' is then its unique maximal ideal. This construction is applied in algebraic geometry, where to every point ''x'' of a scheme ''X'' one associates its residue field ''k''(''x''). One can say a little loosely that the residue field of a point of an abstract algebraic variety is the 'natural domain' for the coordinates of the point. Definition Suppose that ''R'' is a commutative local ring, with maximal ideal ''m''. Then the residue field is the quotient ring ''R''/''m''. Now suppose that ''X'' is a scheme and ''x'' is a point of ''X''. By the definition of scheme, we may find an affine neighbourhood ''U'' = Spec(''A''), with ''A'' some commutative ring. Considered in the neighbourhood ''U'', the point ''x'' correspond ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Fibre
In algebraic geometry, a generic point ''P'' of an algebraic variety ''X'' is, roughly speaking, a point at which all generic properties are true, a generic property being a property which is true for almost every point. In classical algebraic geometry, a generic point of an affine or projective algebraic variety of dimension ''d'' is a point such that the field generated by its coordinates has transcendence degree ''d'' over the field generated by the coefficients of the equations of the variety. In scheme theory, the spectrum of an integral domain has a unique generic point, which is the zero ideal. As the closure of this point for the Zariski topology is the whole spectrum, the definition has been extended to general topology, where a generic point of a topological space ''X'' is a point whose closure is ''X''. Definition and motivation A generic point of the topological space ''X'' is a point ''P'' whose closure is all of ''X'', that is, a point that is dense in ''X' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scheme (mathematics)
In mathematics, a scheme is a mathematical structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations ''x'' = 0 and ''x''2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers). Scheme theory was introduced by Alexander Grothendieck in 1960 in his treatise "Éléments de géométrie algébrique"; one of its aims was developing the formalism needed to solve deep problems of algebraic geometry, such as the Weil conjectures (the last of which was proved by Pierre Deligne). Strongly based on commutative algebra, scheme theory allows a systematic use of methods of topology and homological algebra. Scheme theory also unifies algebraic geometry with much of number theory, which eventually led to Wiles's proof of Fermat's Last Theorem. Formally, a scheme is a topological space together with ... [...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: * Mordell–Weil 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] |
Néron Model
In algebraic geometry, the Néron model (or Néron minimal model, or minimal model) for an abelian variety ''AK'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''AK'' from Spec(''K'') to Spec(''R''), in other words the "best possible" group scheme ''AR'' defined over ''R'' corresponding to ''AK''. They were introduced by for abelian varieties over the quotient field of a Dedekind domain ''R'' with perfect residue fields, and extended this construction to semiabelian varieties over all Dedekind domains. Definition Suppose that ''R'' is a Dedekind domain with field of fractions ''K'', and suppose that ''AK'' is a smooth separated scheme over ''K'' (such as an abelian variety). Then a Néron model of ''AK'' is defined to be a smooth separated scheme ''AR'' over ''R'' with fiber ''AK'' that is universal in the following sense. :If ''X'' is a smooth separated scheme over ''R'' then any ''K''-morphism from ''X''''K'' to ''AK'' ca ... [...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] |
