Inmathematics,Diophantine geometry is the study ofDiophantine equations by means of powerful methods inalgebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations.^[1] Diophantine geometry is part of the broader field ofarithmetic geometry.

Fourtheorems in Diophantine geometry that are of fundamental importance include:^[2]

• Mordell–Weil theorem
• Roth's theorem
• Siegel's theorem
• Faltings's theorem

Serge Lang published a bookDiophantine Geometry in the area in 1962, and by this book he coined the term "Diophantine geometry".^[1] The traditional arrangement of material on Diophantine equations was bydegree and number ofvariables, as inMordell'sDiophantine Equations (1969). Mordell's book starts with a remark onhomogeneous equationsf = 0 over therational field, attributed toC. F. Gauss, that non-zero solutions inintegers (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat ofL. E. Dickson, which is about parametric solutions.^[3] TheHilbert–Hurwitz result from 1890 reducing the Diophantine geometry of curves of genus 0 to degrees 1 and 2 (conic sections) occurs in Chapter 17, as doesMordell's conjecture.Siegel's theorem on integral points occurs in Chapter 28.Mordell's theorem on the finite generation of the group ofrational points on anelliptic curve is in Chapter 16, and integer points on theMordell curve in Chapter 26.

In a hostile review of Lang's book, Mordell wrote:

In recent times, powerful new geometric ideas and methods have been developed by means of which important new arithmetical theorems and related results have been found and proved and some of these are not easily proved otherwise. Further, there has been a tendency to clothe the old results, their extensions, and proofs in the new geometrical language. Sometimes, however, the full implications of results are best described in a geometrical setting. Lang has these aspects very much in mind in this book, and seems to miss no opportunity for geometric presentation. This accounts for his title "Diophantine Geometry."^[4]

He notes that the content of the book is largely versions of theMordell–Weil theorem,Thue–Siegel–Roth theorem, Siegel's theorem, with a treatment ofHilbert'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 usedabelian 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".^[5] A larger field sometimes calledarithmetic of abelian varieties now includes Diophantine geometry along withclass field theory,complex multiplication,local zeta-functions andL-functions.^[6]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 ofDiophantine Geometry attests.^[7]

A single equation defines ahypersurface, andsimultaneous Diophantine equations give rise to a generalalgebraic varietyV overK; the typical question is about the nature of the setV(K) of points onV with co-ordinates inK, and by means ofheight 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 ofhomogeneous equations andhomogeneous co-ordinates is fundamental, for the same reasons thatprojective geometry is the dominant approach in algebraic geometry. Rational number solutions therefore are the primary consideration; but integral solutions (i.e.,integer lattice points) can be treated in the same way as anaffine variety may be considered inside a projective variety that has extrapoints at infinity.

The general approach of Diophantine geometry is illustrated byFaltings's theorem (a conjecture ofL. J. Mordell) stating that analgebraic curveC ofgenusg > 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 caseg = 0. The theory consists both of theorems and many conjectures and open questions.

• Glossary of arithmetic and Diophantine geometry
• Arakelov geometry

• Lang's review of Mordell'sDiophantine Equations
• Mordell's review of Lang'sDiophantine Geometry

• Diophantine geometry

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