Faltings's theorem is a result in
arithmetic geometry, according to which a curve of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
greater than 1 over the field
of
rational number
In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example,
The set of all ...
s has only finitely many
rational point
In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the fiel ...
s. This was conjectured in 1922 by
Louis Mordell, and known as the Mordell conjecture until its 1983 proof by
Gerd Faltings. The conjecture was later generalized by replacing
by any
number field.
Background
Let
be a
non-singular algebraic curve of
genus
Genus (; : genera ) is a taxonomic rank above species and below family (taxonomy), family as used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In bino ...
over
. Then the set of rational points on
may be determined as follows:
* When
, there are either no points or infinitely many. In such cases,
may be handled as a
conic section
A conic section, conic or a quadratic curve is a curve obtained from a cone's surface intersecting a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a special case of the ellipse, tho ...
.
* When
, if there are any points, then
is an
elliptic curve and its rational points form a
finitely generated abelian group. (This is ''Mordell's Theorem'', later generalized to the
Mordell–Weil theorem.) Moreover,
Mazur's torsion theorem restricts the structure of the torsion subgroup.
* When
, according to Faltings's theorem,
has only a finite number of rational points.
Proofs
Igor Shafarevich conjectured 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 fixed finite set of
places.
Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.
Gerd Faltings proved Shafarevich's finiteness conjecture using a known reduction to a case of the
Tate conjecture
In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The ...
, together with tools from
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
, including the theory of
Néron models. The main idea of Faltings's proof is the comparison of
Faltings heights and
naive heights via
Siegel modular varieties.
Later proofs
*
Paul Vojta gave a proof based on
Diophantine approximation
In number theory, the study of Diophantine approximation deals with the approximation of real numbers by rational numbers. It is named after Diophantus of Alexandria.
The first problem was to know how well a real number can be approximated ...
.
Enrico Bombieri
Enrico Bombieri (born 26 November 1940) is an Italian mathematician, known for his work in analytic number theory, Diophantine geometry, complex analysis, and group theory. Bombieri is currently professor emeritus in the School of Mathematics ...
found a more elementary variant of Vojta's proof.
*Brian Lawrence and
Akshay Venkatesh gave a proof based on
-adic Hodge theory, borrowing also some of the easier ingredients of Faltings's original proof.
Consequences
Faltings's 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 ''Isogeny theorem'' that abelian varieties with isomorphic
Tate modules (as
-modules with Galois action) are
isogenous.
A sample application of Faltings's theorem is to a weak form of
Fermat's Last Theorem
In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive number, positive integers , , and satisfy the equation for any integer value of greater than . The cases ...
: for any fixed
there are at most finitely many primitive integer solutions (pairwise
coprime
In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...
solutions) to
, since for such
the
Fermat curve has genus greater than 1.
Generalizations
Because of the
Mordell–Weil theorem, Faltings's theorem can be reformulated as a statement about the intersection of a curve
with a finitely generated subgroup
of an abelian variety
. Generalizing by replacing
by a
semiabelian variety,
by an arbitrary subvariety of
, and
by an arbitrary finite-rank subgroup of
leads to the
Mordell–Lang conjecture, which was proved in 1995 by
McQuillan following work of Laurent,
Raynaud, Hindry,
Vojta, and
Faltings.
Another higher-dimensional generalization of Faltings's theorem is the
Bombieri–Lang conjecture that if
is a
pseudo-canonical variety (i.e., a variety of general type) over a number field
, then
is not
Zariski dense in
. Even more general conjectures have been put forth by
Paul Vojta.
The Mordell conjecture for function fields was proved by
Yuri Ivanovich Manin and by
Hans Grauert. In 1990,
Robert F. Coleman found and fixed a gap in Manin's proof.
Notes
Citations
References
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* → Contains an English translation of
*
*
*
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*
* → Gives Vojta's proof of Faltings's Theorem.
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*
* (Translation: )
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{{Authority control
Diophantine geometry
Theorems in number theory
Theorems in algebraic geometry