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{{short description|Branch of algebraic geometry focused on problems in number theory}}
{{General geometry|branches}}
[[File:Example of a hyperelliptic curve.svg|thumb|The [[hyperelliptic curve]] defined by <math>y^2=x(x+1)(x-3)(x+2)(x-2)</math> has only finitely many [[rational point]]s (such as the points <math>(-2, 0)</math> and <math>(-1, 0)</math>) by [[Faltings's theorem]].]]
In mathematics, '''arithmetic geometry''' is roughly the application of techniques from [[algebraic geometry]] to problems in [[number theory]].<ref>{{cite web|title=Introduction to Arithmetic Geometry|last=Sutherland|first=Andrew V.|url=https://ocw.mit.edu/courses/mathematics/18-782-introduction-to-arithmetic-geometry-fall-2013/lecture-notes/MIT18_782F13_lec1.pdf|date=September 5, 2013|access-date=22 March 2019}}</ref> Arithmetic geometry is centered around [[Diophantine geometry]], the study of [[rational point]]s of [[algebraic variety|algebraic varieties]].<ref name="Quanta">{{cite web|url=https://www.quantamagazine.org/peter-scholze-and-the-future-of-arithmetic-geometry-20160628/|title=Peter Scholze and the Future of Arithmetic Geometry|last=Klarreich|first=Erica|date=June 28, 2016|access-date=March 22, 2019}}</ref><ref name="poonen-notes">{{cite web|title=Introduction to Arithmetic Geometry|last=Poonen|first=Bjorn|author-link=Bjorn Poonen|url=http://math.mit.edu/~poonen/782/782notes.pdf|year=2009|access-date=March 22, 2019}}</ref>

In more abstract terms, arithmetic geometry can be defined as the study of [[scheme (mathematics)|schemes]] of [[Finite morphism#Morphisms of finite type|finite type]] over the [[spectrum of a ring|spectrum]] of the [[ring of integers]].<ref>{{nlab|id=arithmetic+geometry|title=Arithmetic geometry}}</ref>

==Overview==
The classical objects of interest in arithmetic geometry are rational points: [[solution set|sets of solutions]] of a [[system of polynomial equations]] over [[number field]]s, [[finite field]]s, [[p-adic field]]s, or [[Algebraic function field|function field]]s, i.e. [[field (mathematics)|field]]s that are not [[algebraically closed]] excluding the [[real number]]s. Rational points can be directly characterized by [[height function]]s which measure their arithmetic complexity.<ref>{{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 | pages=43–67 }}</ref>

The structure of algebraic varieties defined over non-algebraically closed fields has become a central area of interest that arose with the modern abstract development of algebraic geometry. Over finite fields, [[étale cohomology]] provides [[Topological property|topological invariant]]s associated to algebraic varieties.<ref name="grothendieck-cohomology"/> [[p-adic Hodge theory]] gives tools to examine when cohomological properties of varieties over the [[complex number]]s extend to those over [[p-adic field]]s.<ref>{{cite journal | last=Serre | first=Jean-Pierre | author-link=Jean-Pierre Serre | title=Résumé des cours, 1965–66 | journal=Annuaire du Collège de France | location=Paris | year=1967 | pages=49–58}}</ref>

==History==
===19th century: early arithmetic geometry===
In the early 19th century, [[Carl Friedrich Gauss]] observed that non-zero [[integer]] solutions to [[homogeneous polynomial]] equations with [[rational number|rational]] coefficients exist if non-zero rational solutions exist.<ref>{{cite book|title=Diophantine Equations|last=Mordell|first=Louis J.|author-link=Louis J. Mordell|year=1969|publisher=Academic Press|isbn=978-0125062503|page=1}}</ref>

In the 1850s, [[Leopold Kronecker]] formulated the [[Kronecker–Weber theorem]], introduced the theory of [[Divisor (algebraic geometry)|divisor]]s, and made numerous other connections between number theory and [[algebra]]. He then conjectured his "[[Kronecker's Jugendtraum|liebster Jugendtraum]]" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his [[Hilbert's problems|twelfth problem]], which outlines a goal to have number theory operate only with rings that are quotients of [[polynomial ring]]s over the integers.<ref name="Princeton">{{cite book| last1 = Gowers| first1 = Timothy| last2 = Barrow-Green| first2 = June| last3 = Leader| first3 = Imre| title = The Princeton companion to mathematics| url = https://archive.org/details/princetoncompanio00gowe| year = 2008| publisher = Princeton University Press| isbn = 978-0-691-11880-2| pages = 773–774 }}</ref>

===Early-to-mid 20th century: algebraic developments and the Weil conjectures===
In the late 1920s, [[André Weil]] demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the [[Mordell–Weil theorem]] which demonstrates that the set of rational points of an [[abelian variety]] is a [[finitely generated abelian group]].<ref>A. Weil, ''L'arithmétique sur les courbes algébriques'', Acta Math 52, (1929) p. 281-315, reprinted in vol 1 of his collected papers {{isbn|0-387-90330-5}}.</ref>

Modern foundations of algebraic geometry were developed based on contemporary [[commutative algebra]], including [[valuation theory]] and the theory of [[ideal (ring theory)|ideals]] by [[Oscar Zariski]] and others in the 1930s and 1940s.<ref>{{cite book | last1=Zariski | first1=Oscar | author1-link=Oscar Zariski | editor1-last=Abhyankar | editor1-first=Shreeram S. | editor1-link=Shreeram Shankar Abhyankar| editor2-last=Lipman | editor2-first=Joseph | editor2-link=Joseph Lipman| editor3-last=Mumford | editor3-first=David | editor3-link=David Mumford | title=Algebraic surfaces | orig-year=1935 | url=https://books.google.com/books?id=d6Zzhm9eCmgC | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=second supplemented | series=Classics in mathematics | isbn=978-3-540-58658-6 | year=2004 | mr=0469915}}</ref>

In 1949, [[André Weil]] posed the landmark [[Weil conjectures]] about the [[local zeta-function]]s of algebraic varieties over finite fields.<ref>{{cite journal | last1=Weil | first1=André | author1-link=André Weil | title=Numbers of solutions of equations in finite fields | doi=10.1090/S0002-9904-1949-09219-4 | mr=0029393 | year=1949 | journal=[[Bulletin of the American Mathematical Society]] | issn=0002-9904 | volume=55 | pages=497–508 | issue=5| doi-access=free }} Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil {{isbn|0-387-90330-5}}</ref> These conjectures offered a framework between algebraic geometry and number theory that propelled [[Alexander Grothendieck]] to recast the foundations making use of [[sheaf theory]] (together with [[Jean-Pierre Serre]]), and later scheme theory, in the 1950s and 1960s.<ref>{{cite journal | last1 = Serre | first1 = Jean-Pierre | year = 1955 | title = Faisceaux Algebriques Coherents | journal = The Annals of Mathematics | volume = 61 | issue = 2| pages = 197–278 | doi=10.2307/1969915| jstor = 1969915 }}</ref> [[Bernard Dwork]] proved one of the four Weil conjectures (rationality of the local zeta function) in 1960.<ref>{{cite journal | last1=Dwork | first1=Bernard | author1-link=Bernard Dwork | title=On the rationality of the zeta function of an algebraic variety | jstor=2372974 | mr=0140494 | year=1960 | journal=[[American Journal of Mathematics]] | issn=0002-9327 | volume=82 | pages=631–648 | doi=10.2307/2372974 | issue=3 | publisher=American Journal of Mathematics, Vol. 82, No. 3}}</ref> Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with [[Michael Artin]] and [[Jean-Louis Verdier]]) by 1965.<ref name="grothendieck-cohomology">{{cite book | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Proc. Internat. Congress Math. (Edinburgh, 1958) | publisher=[[Cambridge University Press]] | mr=0130879 | year=1960 | chapter=The cohomology theory of abstract algebraic varieties | pages=103–118|chapter-url=http://grothendieckcircle.org/}}</ref><ref>{{cite book | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Séminaire Bourbaki | chapter-url=http://www.numdam.org/item?id=SB_1964-1966__9__41_0 | publisher=[[Société Mathématique de France]] | location=Paris | mr=1608788 | year=1995 | volume=9 | chapter=Formule de Lefschetz et rationalité des fonctions L | pages=41–55|orig-year=1965 |ref= {{harvid|Grothendieck|1965}} }}</ref> The last of the Weil conjectures (an analogue of the [[Riemann hypothesis]]) would be finally proven in 1974 by [[Pierre Deligne]].<ref>{{cite journal | last1=Deligne | first1=Pierre | author1-link=Pierre Deligne | title=La conjecture de Weil. I | url=http://www.numdam.org/item?id=PMIHES_1974__43__273_0 | mr=0340258 | year=1974 | journal=[[Publications Mathématiques de l'IHÉS]] | volume=43 | issn=1618-1913 | issue=1 | pages=273–307| doi=10.1007/BF02684373 }}</ref>

===Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond===
Between 1956 and 1957, [[Yutaka Taniyama]] and [[Goro Shimura]] posed the [[Modularity theorem|Taniyama–Shimura conjecture]] (now known as the modularity theorem) relating [[elliptic curves]] to [[modular forms]].<ref>{{cite journal|last=Taniyama|first=Yutaka |journal=Sugaku|volume=7|page=269|year=1956|title=Problem 12|language=ja}}</ref><ref>{{cite journal | last1=Shimura | first1=Goro | title=Yutaka Taniyama and his time. Very personal recollections | doi=10.1112/blms/21.2.186 | mr=976064 | year=1989 | journal=The Bulletin of the London Mathematical Society | issn=0024-6093 | volume=21 | issue=2 | pages=186–196| doi-access=free }}</ref> This connection would ultimately lead to [[Wiles's proof of Fermat's Last Theorem|the first proof]] of [[Fermat's Last Theorem]] in number theory through algebraic geometry techniques of [[Lift (mathematics)|modularity lifting]] developed by [[Andrew Wiles]] in 1995.<ref name="wiles1995">{{cite journal|last=Wiles|first=Andrew|author-link=Andrew Wiles|year=1995|title=Modular elliptic curves and Fermat's Last Theorem|url=http://math.stanford.edu/~lekheng/flt/wiles.pdf|journal=Annals of Mathematics|volume=141|issue=3|pages=443–551|oclc=37032255|doi=10.2307/2118559|jstor=2118559|citeseerx=10.1.1.169.9076|access-date=2019-03-22|archive-date=2011-05-10|archive-url=https://web.archive.org/web/20110510062158/http://math.stanford.edu/%7Elekheng/flt/wiles.pdf|url-status=dead}}</ref>

In the 1960s, Goro Shimura introduced [[Shimura variety|Shimura varieties]] as generalizations of [[modular curve]]s.<ref>{{cite book|last=Shimura|first=Goro|title=The Collected Works of Goro Shimura|publisher=Springer Nature|isbn=978-0387954158|year=2003}}</ref> Since the 1979, Shimura varieties have played a crucial role in the [[Langlands program]] as a natural realm of examples for testing conjectures.<ref>{{cite book|title=Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics|publisher=Chelsea Publishing Company|editor-last1=Borel|editor-first1=Armand|editor-link1=Armand Borel|editor-last2=Casselman|editor-first2=William|editor-link2=Bill Casselman (mathematician)|year=1979|volume=XXXIII Part 1|last=Langlands|first=Robert|author-link=Robert Langlands|chapter-url=http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/autoreps-ps.pdf|chapter=Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen|pages=205–246}}</ref>

In papers in 1977 and 1978, [[Barry Mazur]] proved the [[torsion conjecture]] giving a complete list of the possible torsion subgroups of elliptic curves over the rational numbers. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain [[modular curve]]s.<ref>{{cite journal|last=Mazur|first=Barry|author-link=Barry Mazur|title=Modular curves and the Eisenstein ideal|volume=47|issue=1|pages=33–186|year=1977|doi=10.1007/BF02684339|mr=0488287|journal=[[Publications Mathématiques de l'IHÉS]]|url=http://www.numdam.org/item/PMIHES_1977__47__33_0/}}</ref><ref>{{cite journal|last=Mazur|first=Barry|title=Rational isogenies of prime degree|volume=44|issue=2|pages=129–162|year=1978|doi=10.1007/BF01390348|mr=0482230|journal=[[Inventiones Mathematicae]]|others=with appendix by [[Dorian Goldfeld]]|bibcode=1978InMat..44..129M}}</ref> In 1996, the proof of the torsion conjecture was extended to all number fields by [[Loïc Merel]].<ref>{{cite journal | last1=Merel | first1=Loïc | author1-link=Loïc Merel | title=Bornes pour la torsion des courbes elliptiques sur les corps de nombres | trans-title=Bounds for the torsion of elliptic curves over number fields | language=fr | doi=10.1007/s002220050059 |mr=1369424 | year=1996 | journal=[[Inventiones Mathematicae]] | volume=124 | issue=1 | pages=437–449 | bibcode=1996InMat.124..437M }}</ref>

In 1983, [[Gerd Faltings]] proved the [[Faltings's theorem|Mordell conjecture]], demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates [[finitely generated abelian group|finite generation]] of the set of rational points as opposed to finiteness).<ref>{{cite journal |author-link=Gerd Faltings| last=Faltings |first=Gerd |year=1983 |title=Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal=[[Inventiones Mathematicae]] |volume=73 |issue=3 |pages=349–366 |doi=10.1007/BF01388432 | mr=0718935 | trans-title=Finiteness theorems for abelian varieties over number fields | language=de | bibcode=1983InMat..73..349F}}</ref><ref>{{cite journal |last=Faltings |first=Gerd |year=1984 |title=Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern |journal=[[Inventiones Mathematicae]] |volume=75 |issue=2 |pages=381 |doi=10.1007/BF01388572 | mr=0732554 | language=de |doi-access=free }}</ref>

In 2001, the proof of the [[Local Langlands conjectures#Local Langlands conjectures for GLn|local Langlands conjectures for GL<sub>n</sub>]] was based on the geometry of certain Shimura varieties.<ref>{{cite book | author1-link=Michael Harris (mathematician)| last1=Harris | first1=Michael | author2-link=Richard Taylor (mathematician)| last2=Taylor | first2=Richard | title=The geometry and cohomology of some simple Shimura varieties | url=https://books.google.com/books?id=sigBbO69hvMC | publisher=[[Princeton University Press]] | series=Annals of Mathematics Studies | isbn=978-0-691-09090-0 | mr=1876802 | year=2001 | volume=151}}</ref>

In the 2010s, [[Peter Scholze]] developed [[perfectoid space]]s and new cohomology theories in arithmetic geometry over p-adic fields with application to [[Galois representations]] and certain cases of the [[weight-monodromy conjecture]].<ref>{{cite web |title=Fields Medals 2018 |url=https://www.mathunion.org/imu-awards/fields-medal/fields-medals-2018 |publisher=[[International Mathematical Union]] |access-date=2 August 2018}}</ref><ref>{{cite web|last=Scholze|first=Peter|url=http://www.math.uni-bonn.de/people/scholze/CDM.pdf|title=Perfectoid spaces: A survey|website=University of Bonn|access-date=4 November 2018}}</ref>

==See also==
*[[Arithmetic dynamics]]
*[[Arithmetic of abelian varieties]]
*[[Birch and Swinnerton-Dyer conjecture]]
*[[Moduli of algebraic curves]]
*[[Siegel modular variety]]
*[[Siegel's theorem on integral points]]
*[[Category theory]]
*[[Frobenioid]]

==References==
{{reflist}}

{{Number theory |expanded}}
{{Areas of mathematics | state=collapsed}}

{{Authority control}}

{{DEFAULTSORT:Arithmetic Geometry}}
[[Category:Arithmetic geometry| ]]

Arithmetic geometry - Revision history

imported>Chrisdmiddleton: removed Category:Fields of mathematics using HotCat