Faltings' theorem
Encyclopedia
In number theory
, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing Q by a finite extension. It was proved by , and is now known as Faltings' theorem.
g over Q. Then the set of rational points on C may be determined as follows:
, and a number of tools from algebraic geometry
, including the theory of Néron model
s. A very different proof, based on diophantine approximation, was found by Paul Vojta
. A more elementary variant of Vojta's proof was given by Enrico Bombieri
.
The reduction of the Mordell conjecture to the Shafarevich conjecture was due to . A sample application of Faltings' theorem is to a weak form of Fermat's Last Theorem
: for any fixed n > 4 there are at most finitely many primitive solutions to an + bn = cn. (The curve
xn + yn = 1
has genus greater than 1.)
Another higher-dimensional generalization of Faltings' theorem is the Bombieri-Lang conjecture that if X is a pseudo-canonical variety (i.e., variety of general type) over a number field k, then X(k) is not Zariski dense in X. Even more general conjectures have been put forth by Paul Vojta
.
The Mordell conjecture for function fields was proved by and by . found and fixed a gap in Manin's proof.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the Mordell conjecture is the conjecture made by that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. The conjecture was later generalized by replacing Q by a finite extension. It was proved by , and is now known as Faltings' theorem.
Background
Let C be a non-singular algebraic curve of genusGenus (mathematics)
In mathematics, genus has a few different, but closely related, meanings:-Orientable surface:The genus of a connected, orientable surface is an integer representing the maximum number of cuttings along non-intersecting closed simple curves without rendering the resultant manifold disconnected. It...
g over Q. Then the set of rational points on C may be determined as follows:
- Case g = 0: no points or infinitely many; C is handled as a conic sectionConic sectionIn mathematics, a conic section is a curve obtained by intersecting a cone with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2...
. - Case g = 1: no points, or C is an elliptic curveElliptic curveIn 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...
and its rational points form a finitely generated abelian groupFinitely generated abelian groupIn abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...
(Mordell's Theorem, later generalized to the Mordell-Weil theorem). Moreover Mazur's torsion theoremMazur's torsion theoremIn algebraic geometry Mazur's torsion theorem, due to Barry Mazur, classifies the possible torsion subgroups of the group of rational points on an elliptic curve defined over the rational numbers....
restricts the structure of the torsion subgroup. - Case g > 1: according to the Mordell conjecture, now Faltings' Theorem, C has only a finite number of rational points.
Proofs
Faltings' original proof used the known reduction to a case of the Tate conjectureTate conjecture
In mathematics, the Tate conjecture is a 1963 conjecture of John Tate linking algebraic geometry, and more specifically the identification of algebraic cycles, with Galois modules coming from étale cohomology...
, and a number of tools from algebraic geometry
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, including the theory of Néron model
Néron model
In algebraic geometry, a Néron model for an abelian variety AK defined over a local field K is the "best possible" group scheme AO defined over the ring of integers R of the local field K that becomes isomorphic to AK after base change from R to K.They were introduced by André Néron...
s. A very different proof, based on diophantine approximation, was found by Paul Vojta
Paul Vojta
Paul Alan Vojta is an American mathematician, known for his work in number theory on diophantine geometry and diophantine approximation....
. A more elementary variant of Vojta's proof was given by Enrico Bombieri
Enrico Bombieri
Enrico Bombieri is a mathematician who has been working at the Institute for Advanced Study in Princeton, New Jersey. Bombieri's research in number theory, algebraic geometry, and mathematical analysis have earned him many international prizes --- a Fields Medal in 1974 and the Balzan Prize in 1980...
.
Consequences
Faltings' 1983 paper had as consequences a number of statements which had previously been conjectured:- The Mordell conjecture that a curve of genus greater than 1 over a number field has only finitely many rational points;
- The Shafarevich conjecture that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a given finite set of places; and
- The Isogeny theorem that abelian varieties with isomorphic Tate moduleTate moduleIn mathematics, a Tate module of an abelian group, named for John Tate, is a module constructed from an abelian group A. Often, this construction is made in the following situation: G is a commutative group scheme over a field K, Ks is the separable closure of K, and A = G...
s (as Ql-modules with Galois action) are isogenousIsogenyIn mathematics, an isogeny is a morphism of varieties between two abelian varieties that is surjective and has a finite kernel....
.
The reduction of the Mordell conjecture to the Shafarevich conjecture was due to . A sample application of Faltings' theorem is to a weak form of Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
: for any fixed n > 4 there are at most finitely many primitive solutions to an + bn = cn. (The curve
xn + yn = 1
has genus greater than 1.)
Generalizations
Because of the Mordell-Weil theorem, Faltings' theorem can be reformulated as a statement about the intersection of a curve C with a finitely generated subgroup Γ of an abelian variety A. Generalizing by replacing C by an arbitrary subvariety of A and Γ by an arbitrary finite-rank subgroup of A leads to the Mordell-Lang conjecture, which has been proved.Another higher-dimensional generalization of Faltings' theorem is the Bombieri-Lang conjecture that if X is a pseudo-canonical variety (i.e., variety of general type) over a number field k, then X(k) is not Zariski dense in X. Even more general conjectures have been put forth by Paul Vojta
Paul Vojta
Paul Alan Vojta is an American mathematician, known for his work in number theory on diophantine geometry and diophantine approximation....
.
The Mordell conjecture for function fields was proved by and by . found and fixed a gap in Manin's proof.