In mathematics, Diophantine geometry is the study of points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations. These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and p-adic fields. It is a sub-branch of arithmetic geometry and is one approach to the theory ofDiophantine equations, formulating questions about such equations in terms of algebraic geometry. A single equation defines a hypersurface, and simultaneous Diophantine equations give rise to a general algebraic varietyV over K; the typical question is about the nature of the set V 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 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 stating that an algebraic curveC of genusg > 1 over the rational numbers has only finitely manyrational 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
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. 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 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 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. 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". 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, complex multiplication, local zeta-functions and L-functions. Paul Vojta wrote:
