Mordell–Weil theorem
Encyclopedia
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 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.
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(Q)/2E(Q) which forms a major step in the proof. 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. 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. 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). 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
. For an abelian variety, there is no a priori preferred representation, though, as a projective 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 form
s). The theorem left unanswered a number of questions:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Mordell–Weil theorem states that for an abelian variety
Abelian variety
In mathematics, particularly in algebraic geometry, complex analysis and 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...
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
Elliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
E and K the rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
field Q is Mordell's theorem, answering a question apparently posed by Poincaré
Henri Poincaré
Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and a philosopher of science...
around 1908; it was proved by Louis Mordell
Louis Mordell
Louis Joel Mordell was a British mathematician, known for pioneering research in number theory. He was born in Philadelphia, USA, in a Jewish family of Lithuanian extraction...
in 1922.
The tangent-chord process (one form of addition theorem
Addition theorem
In mathematics, an addition theorem is a formula such as that for the exponential functionthat expresses, for a particular function f, f in terms of f and f...
on a cubic curve) had been known as far back as the seventeenth century. The process of infinite descent
Infinite descent
In mathematics, a proof by infinite descent is a particular kind of proof by contradiction which relies on the fact that the natural numbers are well ordered. One typical application is to show that a given equation has no solutions. Assuming a solution exists, one shows that another exists, that...
of Fermat was well known, but Mordell succeeded in establishing the finiteness of the quotient group
Quotient group
In mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
E(Q)/2E(Q) which forms a major step in the proof. Certainly the finiteness of this group is a necessary condition for E(Q) to be finitely-generated; and it shows that the rank
Rank of an abelian group
In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
is finite. 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
André Weil
André Weil was an influential mathematician of the 20th century, renowned for the breadth and quality of his research output, its influence on future work, and the elegance of his exposition. He is especially known for his foundational work in number theory and algebraic geometry...
took up the subject, producing the generalisation to Jacobians of higher genus curves over arbitrary number fields in his doctoral dissertation published in 1928. 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). 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
Homogeneous coordinates
In mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcül, are a system of coordinates used in projective geometry much as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points,...
. For an abelian variety, there is no a priori preferred representation, though, as a projective variety.
Both halves of the proof have been improved significantly, by subsequent technical advances: in Galois cohomology
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...
as applied to descent, and in the study of the best height functions (which are quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
s). The theorem left unanswered a number of questions:
- Calculation of the rank (still a demanding computational problem, and not always effective, as far as it is currently known).
- Meaning of the rank: see Birch and Swinnerton-Dyer conjectureBirch and Swinnerton-Dyer conjectureIn mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Its status as one of the most challenging mathematical questions has become widely recognized; the conjecture was chosen as one of the seven Millennium Prize Problems listed by the Clay...
. - For a curveAlgebraic curveIn algebraic geometry, an algebraic curve is an algebraic variety of dimension one. The theory of these curves in general was quite fully developed in the nineteenth century, after many particular examples had been considered, starting with circles and other conic sections.- Plane algebraic curves...
C in its Jacobian varietyJacobian varietyIn mathematics, the Jacobian variety J of a non-singular algebraic curve C of genus g is the moduli space of degree 0 line bundles...
as A, can the intersection of C with A(K) be infinite? (Not unless C = A, according to Mordell's conjecture, proved by Faltings.) - In the same context, can C contain infinitely many torsion points of A? (No, according to the Manin-Mumford conjecture proved by Raynaud, other than in the elliptic curve case.)