Diophantine geometry
Encyclopedia
In mathematics, diophantine geometry is one approach to the theory of Diophantine equation
s, formulating questions about such equations in terms of algebraic geometry
over a ground field
K that is not algebraically closed, such as the field of rational number
s or a finite field
, or more general commutative ring
such as the integers. A single equation defines a hypersurface
, and simultaneous Diophantine equations give rise to a general algebraic variety
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' theorem
(a conjecture of L. J. Mordell) starting that an algebraic curve
C of genus 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.
published a book Diophantine Geometry in the area, in 1962. 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 integers (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
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
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 algebraic varieties" now includes diophantine geometry with class field theory
, complex multiplication
, local zeta-function
s and L-function
s. Paul Vojta
wrote:
Diophantine equation
In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...
s, formulating questions about such equations in terms of 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...
over a ground field
Ground field
In mathematics, a ground field is a field K fixed at the beginning of the discussion. It is used in various areas of algebra: for example in linear algebra where the concept of a vector space may be developed over any field; and in algebraic geometry, where in the foundational developments of André...
K that is not algebraically closed, such as the field of 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...
s or a finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
, or more general commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
such as the integers. A single equation defines a hypersurface
Hypersurface
In geometry, a hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimensions; then any submanifold of M of n − 1 dimensions is a hypersurface...
, and simultaneous Diophantine equations give rise to a general algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
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
Projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant under projective transformations. This means that, compared to elementary geometry, projective geometry has a different setting, projective space, and a selective set of basic geometric concepts...
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' theorem
Faltings' theorem
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...
(a conjecture of L. J. Mordell) starting that an algebraic curve
Algebraic curve
In 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 of genus 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.
Background
Serge LangSerge Lang
Serge Lang was a French-born American mathematician. He was known for his work in number theory and for his mathematics textbooks, including the influential Algebra...
published a book Diophantine Geometry in the area, in 1962. 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 integers (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
Conic section
In 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...
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
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...
is in Chapter 16, and integer points on the Mordell curve
Mordell curve
In algebra, a Mordell curve is an elliptic curve of the form y2 = x3 + n, where n is an integer and where n ≠ 0. In Mordell curves, if is a solution, it therefore follows that is as well....
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
Hilbert's irreducibility theorem
In number theory, Hilbert's irreducibility theorem, conceived by David Hilbert, states that every finite number of irreducible polynomials in a finite number of variables and having rational number coefficients admit a common specialization of a proper subset of the variables to rational numbers...
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 algebraic varieties" now includes diophantine geometry with class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
, complex multiplication
Complex multiplication
In mathematics, complex multiplication is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense In mathematics, complex multiplication is the...
, local zeta-function
Local zeta-function
In number theory, a local zeta-functionis a function whose logarithmic derivative is a generating functionfor the number of solutions of a set of equations defined over a finite field F, in extension fields Fk of F.-Formulation:...
s and L-function
L-function
The theory of L-functions has become a very substantial, and still largely conjectural, part of contemporary analytic number theory. In it, broad generalisations of the Riemann zeta function and the L-series for a Dirichlet character are constructed, and their general properties, in most cases...
s. Paul Vojta
Paul Vojta
Paul Alan Vojta is an American mathematician, known for his work in number theory on diophantine geometry and diophantine approximation....
wrote:
- While others at the time shared this viewpoint (e.g., WeilAndré WeilAndré 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...
, TateJohn TateJohn Torrence Tate Jr. is an American mathematician, distinguished for many fundamental contributions in algebraic number theory, arithmetic geometry and related areas in algebraic geometry.-Biography:...
, SerreJean-Pierre SerreJean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
), it is easy to forget that others did not, as Mordell's review of Diophantine Geometry attests.
See also
- Glossary of arithmetic and Diophantine geometryGlossary of arithmetic and Diophantine geometryThis is a glossary of arithmetic and Diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry...
- :Category:Diophantine geometry