In
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 varieties.
...
, the Mordell conjecture is the conjecture made by
Louis Mordell
Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Educatio ...
that a curve of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nomencla ...
greater than 1 over the field Q 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 (e.g. ). The set of all ratio ...
s has only finitely many
rational points
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 ...
. In 1983 it was proved by
Gerd Faltings
Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry.
Education
From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathem ...
, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any
number field
In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension).
Thus K is a f ...
.
Background
Let ''C'' be a
non-singular
In the mathematical field of algebraic geometry, a singular point of an algebraic variety is a point that is 'special' (so, singular), in the geometric sense that at this point the tangent space at the variety may not be regularly defined. In c ...
algebraic curve of
genus
Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nomencla ...
''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 section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a speci ...
.
* Case ''g'' = 1: no points, or ''C'' is an
elliptic curve
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 ...
and its rational points form a
finitely generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
(''Mordell's Theorem'', later generalized to the
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 ellip ...
). Moreover,
Mazur's torsion theorem
In algebraic geometry and number theory, the torsion conjecture or uniform boundedness conjecture for torsion points for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in term ...
restricts the structure of the torsion subgroup.
* Case ''g'' > 1: according to the Mordell conjecture, now Faltings's theorem, ''C'' has only a finite number of rational points.
Proofs
Igor Shafarevich
Igor Rostislavovich Shafarevich (russian: И́горь Ростисла́вович Шафаре́вич; 3 June 1923 – 19 February 2017) was a Soviet and Russian mathematician who contributed to algebraic number theory and algebraic geometry. ...
conjectured that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed
polarization
Polarization or polarisation may refer to:
Mathematics
*Polarization of an Abelian variety, in the mathematics of complex manifolds
*Polarization of an algebraic form, a technique for expressing a homogeneous polynomial in a simpler fashion by ...
degree over a fixed number field with
good reduction outside a fixed finite set of
place
Place may refer to:
Geography
* Place (United States Census Bureau), defined as any concentration of population
** Census-designated place, a populated area lacking its own municipal government
* "Place", a type of street or road name
** Often ...
s.
Aleksei Parshin showed that Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.
Gerd Faltings
Gerd Faltings (; born 28 July 1954) is a German mathematician known for his work in arithmetic geometry.
Education
From 1972 to 1978, Faltings studied mathematics and physics at the University of Münster. In 1978 he received his PhD in mathem ...
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 con ...
, together with tools from
algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrica ...
, including the theory of
Néron model In algebraic geometry, the Néron model (or Néron minimal model, or minimal model)
for an abelian variety ''AK'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''AK'' from Spec(''K'') to Spec(''R''), ...
s. The main idea of Faltings's proof is the comparison of
Faltings heights and
naive heights via
Siegel modular varieties.
Later proofs
*
Paul Vojta
Paul Alan Vojta (born September 30, 1957) is an American mathematician, known for his work in number theory on Diophantine geometry and Diophantine approximation.
Contributions
In formulating Vojta's conjecture, he pointed out the possible exist ...
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 by ...
.
Enrico Bombieri
Enrico Bombieri (born 26 November 1940, Milan) 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 Mathema ...
found a more elementary variant of Vojta's proof.
*Brian Lawrence and
Akshay Venkatesh
Akshay Venkatesh (born 21 November 1981) is an Australian mathematician and a professor (since 15 August 2018) at the School of Mathematics at the Institute for Advanced Study. His research interests are in the fields of counting, equidistrib ...
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
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 functio ...
with isomorphic
Tate module
In 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'', ...
s (as Q
''ℓ''-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 integers , , and satisfy the equation for any integer value of greater than 2. The cases and have been k ...
: for any fixed ''n'' ≥ 4 there are at most finitely many primitive integer solutions (pairwise
coprime
In mathematics, 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 equival ...
solutions) to ''a''
''n'' + ''b''
''n'' = ''c''
''n'', since for such ''n'' the
Fermat curve
In mathematics, the Fermat curve is the algebraic curve in the complex projective plane defined in homogeneous coordinates (''X'':''Y'':''Z'') by the Fermat equation
:X^n + Y^n = Z^n.\
Therefore, in terms of the affine plane its equation is
:x^ ...
''x''
''n'' + ''y''
''n'' = 1 has genus greater than 1.
Generalizations
Because of the
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 ellip ...
, Faltings's 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 ''A'' by a
semiabelian variety, ''C'' by an arbitrary subvariety of ''A'', and Γ by an arbitrary finite-rank subgroup of ''A'' leads to the
Mordell–Lang conjecture, which was proved in 1995 by
McQuillan
McQuillan and MacQuillan are surnames of Irish origin. There are several unrelated origins of the surnames McQuillan and MacQuillan.
The Ulster variant of the surname was claimed to be an anglicisation of the Gaelic ''Mac Uighilín'' (''son of ...
following work of Laurent,
Raynaud, Hindry,
Vojta, and
Faltings.
Another higher-dimensional generalization of Faltings's theorem is the
Bombieri–Lang conjecture that if ''X'' is a
pseudo-canonical variety (i.e., a variety of general type) over a number field ''k'', then ''X''(''k'') is not
Zariski dense
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
in ''X''. Even more general conjectures have been put forth by
Paul Vojta
Paul Alan Vojta (born September 30, 1957) is an American mathematician, known for his work in number theory on Diophantine geometry and Diophantine approximation.
Contributions
In formulating Vojta's conjecture, he pointed out the possible exist ...
.
The Mordell conjecture for function fields was proved by
Yuri Ivanovich Manin and by
Hans Grauert
Hans Grauert (8 February 1930 in Haren, Emsland, Germany – 4 September 2011) was a German mathematician. He is known for major works on several complex variables, complex manifolds and the application of sheaf theory in this area, which i ...
. In 1990,
Robert F. Coleman found and fixed a gap in Manin's proof.
Notes
Citations
References
*
*
* → Contains an English translation of
*
*
*
*
*
* → Gives Vojta's proof of Faltings's Theorem.
*
*
* (Translation: )
*
*
*
*
*
*
*
{{Authority control
Diophantine geometry
Theorems in number theory
Theorems in algebraic geometry