Editing Diophantine geometry

{{Short description|Mathematics of varieties with integer coordinates}}
{{General geometry|branches}}
In [[mathematics]], '''Diophantine geometry''' is the study of [[Diophantine equation]]s 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.{{sfn|Hindry|Silverman|2000|loc=Preface|p=vii}} Diophantine geometry is part of the broader field of [[arithmetic geometry]].

Four [[theorem]]s in Diophantine geometry that are of fundamental importance include:{{sfn|Hindry|Silverman|2000|loc=Preface|p=viii}}

* [[Mordell–Weil theorem]]
* [[Roth's theorem]]
* [[Siegel's theorem on integral points|Siegel's theorem]]
* [[Faltings's theorem]]

==Background==
[[Serge Lang]] published a book ''Diophantine Geometry'' in the area in 1962, and by this book he coined the term "Diophantine geometry".{{sfn|Hindry|Silverman|2000|loc=Preface|p=vii}} The traditional arrangement of material on Diophantine equations was by [[degree of a polynomial|degree]] and number of [[variable (mathematics)|variables]], as in [[Louis Mordell|Mordell]]'s ''Diophantine Equations'' (1969). Mordell's book starts with a remark on [[Homogeneous polynomial|homogeneous equation]]s ''f'' = 0 over the [[rational field]], attributed to [[C. F. Gauss]], that non-zero solutions in [[integer]]s (even primitive lattice points) exist if non-zero rational solutions do, and notes a caveat of [[L. E. Dickson]], which is about parametric solutions.{{sfn|Mordell|1969|p=1}} The [[Hilbert]]–[[Adolf Hurwitz|Hurwitz]] result from 1890 reducing the Diophantine geometry of curves of genus 0 to degrees 1 and 2 ([[conic section]]s) 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 point]]s 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:

{{blockquote|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."<ref>{{cite web|url=http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183526083 |title=Mordell : Review: Serge Lang, Diophantine geometry |publisher=Projecteuclid.org |date=2007-07-04 |access-date=2015-10-07}}</ref>}}

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.&nbsp;263).

Despite a bad press initially, Lang's conception has been sufficiently widely accepted for a 2006 tribute to call the book "visionary".<ref>{{cite web|author=Marc Hindry|title=''La géométrie diophantienne, selon Serge Lang''|publisher=Gazette des mathématiciens|url=http://www.math.jussieu.fr/~hindry/Lang.pdf|format=PDF|access-date=2015-10-07|archive-date=2012-02-26|archive-url=https://web.archive.org/web/20120226091923/http://www.math.jussieu.fr/~hindry/Lang.pdf|url-status=dead}}</ref> A larger field sometimes called [[arithmetic of abelian varieties]] now includes Diophantine geometry along with [[class field theory]], [[complex multiplication]], [[local zeta-function]]s and [[L-function]]s.<ref>{{Springer|id=A/a011720|title=Algebraic varieties, arithmetic of}}</ref> [[Paul Vojta]] wrote:

:While others at the time shared this viewpoint (e.g., [[André Weil|Weil]], [[John Tate (mathematician)|Tate]], [[Jean-Pierre Serre|Serre]]), it is easy to forget that others did not, as Mordell's review of ''Diophantine Geometry'' attests.<ref>{{cite web|url=https://www.ams.org/notices/200704/fea-lang-web.pdf |format=PDF |title=The Mathematical Contributions of Serge Lang |author1=Jay Jorgenson |author2=Steven G. Krantz |publisher=Ams.org |access-date=2015-10-07}}</ref>

==Approaches==
A single equation defines a [[hypersurface]], and [[simultaneous equations|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 function]]s, 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 polynomial|homogeneous equation]]s 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 (i.e. [[lattice point]]s) 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]] (a conjecture of [[L. J. Mordell]]) stating that an [[algebraic curve]] ''C'' of [[genus (curve)|genus]] ''g'' > 1 over the rational numbers has only finitely many [[rational point]]s. 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==
{{reflist}}

==References==
{{refbegin}}
* {{cite book | first1=Alan | last1=Baker | author-link1=Alan Baker (mathematician)| first2=Gisbert | last2= Wüstholz | author-link2=Gisbert Wüstholz | title=Logarithmic Forms and Diophantine Geometry | series=New Mathematical Monographs | volume=9 | publisher=[[Cambridge University Press]] | year=2007 | isbn=978-0-521-88268-2 | zbl=1145.11004 }}
* {{cite book | first1=Enrico | last1=Bombieri | author-link1=Enrico Bombieri | first2=Walter | last2=Gubler | title=Heights in Diophantine Geometry | series=New Mathematical Monographs | volume=4 | publisher=[[Cambridge University Press]] | year=2006 | isbn=978-0-521-71229-3 | zbl=1115.11034 }}
* {{cite book | first1=Marc | last1=Hindry | first2=Joseph H. | last2=Silverman | author-link2=Joseph H. Silverman | title=Diophantine Geometry: An Introduction | series=[[Graduate Texts in Mathematics]] | volume=201 | year=2000 | isbn=0-387-98981-1 | zbl=0948.11023 }}
* {{cite book | first=Serge | last=Lang | author-link=Serge Lang | title=Survey of Diophantine Geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | zbl=0869.11051 }}
*{{cite book|title=Diophantine Equations|last=Mordell|first=Louis J.|author-link=Louis J. Mordell|year=1969|publisher=Academic Press|isbn=978-0125062503}}
*{{Springer|id=d/d032630|title=Diophantine geometry}}
{{refend}}

==External links==
*[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183532391 Lang's review of Mordell's ''Diophantine Equations'']
*[http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183526083 Mordell's review of Lang's ''Diophantine Geometry'']

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[[Category:Diophantine geometry| ]]