Diophantine geometry

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In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic 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 of arithmetic geometry.

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

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

Background

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 the

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".

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.^[7]

Approaches

A single equation defines a

.

The general approach of Diophantine geometry is illustrated by

. 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

• Glossary of arithmetic and Diophantine geometry
• Arakelov geometry

Citations

References

External links

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