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Mordell–Weil Group
In arithmetic geometry, the Mordell–Weil group is an abelian group associated to any abelian variety A defined over a number field K, it is an arithmetic invariant of the Abelian variety. It is simply the group of K-points of A, so A(K) is the Mordell–Weil grouppg 207. The main structure theorem about this group is the Mordell–Weil theorem which shows this group is in fact a finitely-generated abelian group. Moreover, there are many conjectures related to this group, such as the Birch and Swinnerton-Dyer conjecture which relates the rank of A(K) to the zero of the associated L-function at a special point. Examples Constructing explicit examples of the Mordell–Weil group of an abelian variety is a non-trivial process which is not always guaranteed to be successful, so we instead specialize to the case of a specific elliptic curve E/\mathbb. Let E be defined by the Weierstrass equationy^2 = x(x-6)(x+6)over the rational numbers. It has discriminant \Delta_E = 2^\cdot 3^6 (and ... [...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] |
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] |
Mordell–Weil Theorem
In mathematics, the Mordell–Weil theorem states that for an abelian variety A over a number field K, the group A(K) of ''K''-rational points of A is a finitely-generated abelian group, called the Mordell–Weil group. The case with A an elliptic curve E and K the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties. History The ''tangent-chord process'' (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(\mathbb)/2E(\mathbb) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(\mathbb) to be finitely generated; and it shows that the rank is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Birch And Swinnerton-Dyer Conjecture
In mathematics, the Birch and Swinnerton-Dyer conjecture (often called the Birch–Swinnerton-Dyer conjecture) describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. , only special cases of the conjecture have been proven. The modern formulation of the conjecture relates arithmetic data associated with an elliptic curve ''E'' over a number field ''K'' to the behaviour of the Hasse–Weil ''L''-function ''L''(''E'', ''s'') of ''E'' at ''s'' = 1. More specifically, it is conjectured that the rank of the abelian group ''E''(''K'') of points of ''E'' is the order of the zero of ''L''(''E'', ''s'') at ''s'' = 1, and the first non-zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may give rise to an ''L''-function via analytic continuation. The Riemann zeta function is an example of an ''L''-function, and one important conjecture involving ''L''-functions is the Riemann hypothesis and its generalization. The theory of ''L''-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the ''L''-series for a Dirichlet character are constructed, and their general properties, in most cases still out of reach of proof, are set out in a systematic way. Because of the Euler product formula there is a deep connection between ''L''-functions and the theory of prime numbers. The mathematical field that studies L-func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weierstrass Equation
In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which consists of solutions for: :y^2 = x^3 + ax + b for some coefficients and in . The curve is required to be non-singular, which means that the curve has no cusps or self-intersections. (This is equivalent to the condition , that is, being square-free in .) It is always understood that the curve is really sitting in the projective plane, with the point being the unique point at infinity. Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curve ... [...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] |
Kummer Theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of ''n''th roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. The main statements do not depend on the nature of the field – apart from its characteristic, which should not divide the integer ''n'' – and therefore belong to abstract algebra. The theory of cyclic extensions of the field ''K'' when the characteristic of ''K'' does divide ''n'' is called Artin–Schreier theory. Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots. The main burden in class field theory is to dispense with extra roots of unity ('descending' back to smaller fields); which is somet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperelliptic Curve
In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' distinct roots, and ''h''(''x'') is a polynomial of degree 3. Therefore, in giving such an equation to specify a non-singular curve, it is almost always assumed that a non-singular model (also called a smooth completion), equivalent in the sense of birational geometry, is meant. To be more precise, the equation defines a quadratic extension of C(''x''), and it is that function field that is meant. The singular point at infinity can be removed (since this is a curve) by the normalization ( integral closure) process. It turns out that after doing this, there is an open cover of the curve by two affine charts: the one already given by y^2 = f(x) and another one given by w^2 = v^f(1/v) . The glueing maps between the two charts are given by (x, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galois Cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natural way on some abelian groups, for example those constructed directly from ''L'', but also through other Galois representations that may be derived by more abstract means. Galois cohomology accounts for the way in which taking Galois-invariant elements fails to be an exact functor. History The current theory of Galois cohomology came together around 1950, when it was realised that the Galois cohomology of ideal class groups in algebraic number theory was one way to formulate class field theory, at the time it was in the process of ridding itself of connections to L-functions. Galois cohomology makes no assumption that Galois groups are abelian groups, so this was a non-abelian theory. It was formulated abstractly as a theory of class ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Twisted Abelian Variety
Twisted may refer to: Film and television * ''Twisted'' (1986 film), a horror film by Adam Holender starring Christian Slater * ''Twisted'' (1996 film), a modern retelling of ''Oliver Twist'' * ''Twisted'', a 2011 Singapore Chinese film directed by Chai Yee Wei * ''Twisted'' (2004 film), a thriller starring Ashley Judd and Andy Garcia * ''Twisted'', a parody musical by StarKid Productions * ''Twisted'' (TV series), 2013 * "Twisted" (''Star Trek: Voyager''), a television episode * ''Twisted'' (web series), an Indian erotic thriller web series Software and games * '' Twisted: The Game Show'', a 1994 3DO game * Twisted (software), an event-driven networking framework * '' WarioWare: Twisted!'', a 2005 game for the Game Boy Advance Books * ''Twisted'' (book), a short story collection by crime writer Jeffery Deaver ** ''More Twisted'', a second short story collection by Deaver * '' Twisted'', a novel by Laurie Halse Anderson * ''Twisted'', a ''Pretty Little Liars'' novel by Sar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1002
Year 1002 (MII) was a common year starting on Thursday (link will display the full calendar) of the Julian calendar. Events By place Europe * January 23 – Emperor Otto III dies, at the age of 22, of smallpox at Castle of Paterno (near Rome) after a 19-year reign. He leaves no son, nor a surviving brother who can succeed by hereditary right to the throne.Reuter, Timothy (1992). ''The New Cambridge Medieval History, Volume III'', p. 259. . Otto is buried in Aachen Cathedral alongside the body of Charlemagne. * February 15 – At an assembly at Pavia of Lombard nobles and ''secondi milites'' (the minor nobles), Arduin of Ivrea (grandson of former King Berengar II) is restored to his domains and crowned as King of Italy in the Basilica of San Michele Maggiore. Arduin is supported by Arnulf II, archbishop of Milan. * June 7 – Henry II, a cousin of Otto III, is elected and crowned as King of Germany by Archbishop Willigis at Mainz. Henry does not recogn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
