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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. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, particularly in Algebraic geometry, Complex analysis and Number theory, an Abelian variety is a projective algebraic variety In Mathematics, an algebraic number field (or simply number field) F is a finite (and hence algebraic) Field extension of the In Abstract algebra, an Abelian group ( G,+ is called finitely generated if there exist finitely many elements x 1 The case with A an elliptic curve E and K the rational number field Q is Mordell's theorem, answering a question apparently posed by Poincaré around 1908; it was proved by Louis Mordell in 1922. In Mathematics, a rational number is a number which can be expressed as a Ratio of two Integers Non-integer rational numbers (commonly called fractions Jules Henri Poincaré ( 29 April 1854 &ndash 17 July 1912) (ˈʒyl ɑ̃ˈʁi pwɛ̃kaˈʁe was a French Mathematician Louis Joel Mordell ( 28 January 1888 - 12 March 1972) was a British mathematician known for pioneering research in Number theory.

The tangent-chord process (one form of addition theorem on a cubic curve) had been known as far back as the seventeenth century. In Mathematics, an addition theorem is a formula such as that for the Exponential function e x + y = In Mathematics, a cubic plane curve is a Plane algebraic curve C defined by a cubic equation F ( x, y, As a means of recording the passage of Time, the 17th Century was that Century which lasted from 1601 - 1700 in the Gregorian calendar The process of infinite descent of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group E(Q)/2E(Q) which forms a major step in the proof. In Mathematics, a proof by infinite descent is a particular kind of proof by Mathematical induction. Pierre de Fermat pjɛːʁ dəfɛʁ'ma ( 17 August 1601 or 1607/8 &ndash 12 January 1665) was a French Lawyer at the In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G Certainly the finiteness of this group is a necessary condition for E(Q) to be finitely-generated; and it shows that the rank is finite. In Mathematics, the rank, or torsion-free rank, of an Abelian group measures how large a group is in terms of how large a Vector space over the This turns out to be the essential difficulty. It can be proved by direct analysis of the doubling of a point on E.

Some years later André Weil took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928. André Weil should not be confused with two other mathematicians with similar names Hermann Weyl (1885-1955 who made substantial contributions More abstract methods were required, to carry out a proof with the same basic structure. The second half of the proof needs some type of height function, in terms of which to bound the 'size' of points of A(K). 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 Some measure of the co-ordinates will do; heights are logarithmic, so that (roughly speaking) it is a question of how many digits are required to write down a set of homogeneous coordinates. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations For an abelian variety, there is no a priori preferred representation, though, as a projective variety. This article is about algebraic varieties For the term "a variety of algebras" and an explanation of the difference between a variety of algebras and an algebraic variety

Both halves of the proof have been improved significantly, by subsequent technical advances: in Galois cohomology as applied to descent, and in the study of the best height functions (which are quadratic forms). In Mathematics, Galois cohomology is the study of the Group cohomology of Galois modules that is the application of Homological algebra to In Mathematics, a quadratic form is a Homogeneous polynomial of degree two in a number of variables The theorem left unanswered a number of questions:

See also: arithmetic of abelian varieties. In Mathematics, the arithmetic of abelian varieties is the study of the Number theory of an Abelian variety, or family of those

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