In mathematics, diophantine geometry is one approach to the theory of
Diophantine equations, 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 numbers 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.
1 Background 2 See also 3 References 4 Notes 5 Further reading 6 External links
“ 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, 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-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.
Glossary of arithmetic and Diophantine geometry Category:Diophantine geometry
Hazewinkel, Michiel, ed. (2001) , "Diophantine geometry", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
^ "Mordell : Review: Serge Lang, Diophantine geometry". Projecteuclid.org. 2007-07-04. Retrieved 2015-10-07. ^ Marc Hindry. "La géométrie diophantienne, selon Serge Lang" (PDF). Gazette des mathématiciens. Retrieved 2015-10-07. ^ Hazewinkel, Michiel, ed. (2001) , "Algebraic varieties, arithmetic of", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4 ^ Jay Jorgenson; Steven G. Krantz. "The Mathematical Contributions of Serge Lang" (PDF). Ams.org. Retrieved 2015-10-07.
Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. 9. Cambridge University Press. ISBN 978-0-521-88268-2. Zbl 1145.11004. Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. 4. Cambridge University Press. ISBN 978-0-521-71229-3. Zbl 1115.11034. Hindry, Marc; Silverman, Joseph H. (2000). Diophantine Geometry: An Introduction. Graduate Texts in Mathematics. 201. ISBN 0-387-98981-1. Zbl 0948.11023. Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051.
Lang's review of Mordell's Diophantine Equations Mordell's review of Lang's Di