Diophantine geometry

Four theorems of fundamental importance in Diophantine geometry are:

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

Another major theorem is Mazur's torsion theorem. More modern examples include the André–Oort conjecture, the Bogomolov conjecture and also the uniform Mordell conjecture.

Background

Serge Lang published a book Diophantine Geometry in 1962, which introduced the term "Diophantine geometry" and led to disagreement over the most promising framework for the solution of Diophantine equations.

The traditional arrangement of material on Diophantine equations was by degree and number of variables, as in Mordell's Diophantine Equations (1969). The Hilbert–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 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 negative 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."

He notes that the content of the book is largely versions of the Mordell–Weil theorem, Thue–Siegel–Roth theorem, and 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 widely accepted, and a 2006 tribute called 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:

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.