In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, Diophantine geometry is the study of
Diophantine equations by means of powerful methods in
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 geometrical ...
. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations.
Four
theorems in Diophantine geometry which are of fundamental importance include:
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Mordell–Weil Theorem
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Roth's Theorem
In mathematics, Roth's theorem is a fundamental result in diophantine approximation to algebraic numbers. It is of a qualitative type, stating that algebraic numbers cannot have many rational number approximations that are 'very good'. Over half ...
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Siegel's Theorem
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Faltings's Theorem
Background
Serge Lang published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine Geometry". The traditional arrangement of material on Diophantine equations was by
degree and number of
variables, as in Mordell's ''Diophantine Equations'' (1969). Mordell's book starts with a remark on homogeneous equations ''f'' = 0 over the
rational field, attributed to
C. F. Gauss, that non-zero solutions in
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of
L. E. Dickson, which is about parametric solutions. The Hilbert–Hurwitz result from 1890 reducing the Diophantine geometry of curves of genus 0 to degrees 1 and 2 (
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 ...
s) occurs in Chapter 17, as does Mordell's conjecture.
Siegel's theorem on integral points occurs in Chapter 28.
Mordell's theorem on the finite generation of the group of
rational points on 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. I ...
is in Chapter 16, and integer points on the
Mordell curve in Chapter 26.
In a hostile review of Lang's book, Mordell wrote:
He notes that the content of the book is largely versions of the
Mordell–Weil theorem,
Thue–Siegel–Roth theorem, Siegel's theorem, with a treatment of
Hilbert's irreducibility theorem and applications (in the style of Siegel). Leaving aside issues of generality, and a completely different style, the major mathematical difference between the two books is that Lang used
abelian varieties and offered a proof of Siegel's theorem, while Mordell noted that the proof "is of a very advanced character" (p. 263).
Despite a bad press initially, Lang's conception has been sufficiently widely accepted for a 2006 tribute to call the book "visionary". A larger field sometimes called
arithmetic of abelian varieties now includes Diophantine geometry along with
class field theory
In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is cre ...
,
complex multiplication,
local zeta-function In number theory, the local zeta function (sometimes called the congruent zeta function or the Hasse–Weil zeta function) is defined as
:Z(V, s) = \exp\left(\sum_^\infty \frac (q^)^m\right)
where is a non-singular -dimensional projective al ...
s and
L-function
In mathematics, an ''L''-function is a meromorphic function on the complex plane, associated to one out of several categories of mathematical objects. An ''L''-series is a Dirichlet series, usually convergent on a half-plane, that may giv ...
s.
Paul Vojta wrote:
:While others at the time shared this viewpoint (e.g.,
Weil,
Tate,
Serre), it is easy to forget that others did not, as Mordell's review of ''Diophantine Geometry'' attests.
Approaches
A single equation defines a
hypersurface, and
simultaneous Diophantine equations give rise to a general
algebraic variety
Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. ...
''V'' over ''K''; the typical question is about the nature of the set ''V''(''K'') of points on ''V'' with co-ordinates in ''K'', and by means of
height functions quantitative questions about the "size" of these solutions may be posed, as well as the qualitative issues of whether any points exist, and if so whether there are an infinite number. Given the geometric approach, the consideration of
homogeneous equations and
homogeneous co-ordinates is fundamental, for the same reasons that
projective geometry is the dominant approach in algebraic geometry. Rational number solutions therefore are the primary consideration; but integral solutions (i.e.
lattice points) can be treated in the same way as an
affine variety may be considered inside a projective variety that has extra
points at infinity.
The general approach of Diophantine geometry is illustrated by
Faltings's theorem (a conjecture of
L. J. Mordell) stating that an
algebraic curve ''C'' 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 nom ...
''g'' > 1 over the rational numbers has only finitely many
rational points. The first result of this kind may have been the theorem of Hilbert and Hurwitz dealing with the case ''g'' = 0. The theory consists both of theorems and many conjectures and open questions.
See also
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Glossary of arithmetic and Diophantine geometry
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Arakelov geometry In mathematics, Arakelov theory (or Arakelov geometry) is an approach to Diophantine geometry, named for Suren Arakelov. It is used to study Diophantine equations in higher dimensions.
Background
The main motivation behind Arakelov geometry is ...
Citations
References
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External links
Lang's review of Mordell's ''Diophantine Equations''Mordell's review of Lang's ''Diophantine Geometry''
{{Areas of mathematics , state=collapsed
Fields of mathematics