# Arithmetic geometry

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{{Short description|Branch of algebraic geometry}}
{{General geometry|branches}}
[[File:Example of a hyperelliptic curve.svg|thumb|The [hyperelliptic curve](/source/hyperelliptic_curve) defined by {{math|''y''<sup>2</sup> {{=}} ''x''(''x'' + 1)(''x'' − 3)(''x'' + 2)(''x'' − 2)}} has only finitely many [rational point](/source/rational_point)s (such as the points {{math|(−2, 0)}} and {{math|(−1, 0)}}) by [Faltings' theorem](/source/Faltings'_theorem).]]

In mathematics, '''arithmetic geometry''' is roughly the application of techniques from [algebraic geometry](/source/algebraic_geometry) to problems in [number theory](/source/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](/source/Diophantine_geometry), the study of [rational point](/source/rational_point)s of [algebraic varieties](/source/algebraic_variety).<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 [schemes](/source/scheme_(mathematics)) of [finite type](/source/Finite_morphism) over the [spectrum](/source/spectrum_of_a_ring) of the [ring of integers](/source/ring_of_integers).<ref>{{nlab|id=arithmetic+geometry|title=Arithmetic geometry}}</ref>

==Overview==
The classical objects of interest in arithmetic geometry are rational points: [sets of solutions](/source/solution_set) of a [system of polynomial equations](/source/system_of_polynomial_equations) over [number field](/source/number_field)s, [finite field](/source/finite_field)s, [{{mvar|p}}-adic fields](/source/p-adic_field), or [function field](/source/Algebraic_function_field)s, i.e. [field](/source/field_(mathematics))s that are not [algebraically closed](/source/algebraically_closed) excluding the [real number](/source/real_number)s. Rational points can be directly characterized by [height function](/source/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](/source/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](/source/%C3%A9tale_cohomology) provides [topological invariant](/source/Topological_property)s associated to algebraic varieties.<ref name="grothendieck-cohomology"/> [{{mvar|p}}-adic Hodge theory](/source/p-adic_Hodge_theory) gives tools to examine when cohomological properties of varieties over the [complex number](/source/complex_number)s extend to those over [{{mvar|p}}-adic fields](/source/p-adic_field).<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](/source/Carl_Friedrich_Gauss) observed that non-zero [integer](/source/integer) solutions to [homogeneous polynomial](/source/homogeneous_polynomial) equations with [rational](/source/rational_number) 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](/source/Leopold_Kronecker) formulated the [Kronecker–Weber theorem](/source/Kronecker%E2%80%93Weber_theorem), introduced the theory of [divisor](/source/Divisor_(algebraic_geometry))s, and made numerous other connections between number theory and [algebra](/source/algebra). He then conjectured his "[liebster Jugendtraum](/source/Kronecker's_Jugendtraum)" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his [twelfth problem](/source/Hilbert's_problems), which outlines a goal to have number theory operate only with rings that are quotients of [polynomial ring](/source/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](/source/Andr%C3%A9_Weil) demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the [Mordell–Weil theorem](/source/Mordell%E2%80%93Weil_theorem) which demonstrates that the set of rational points of an [abelian variety](/source/abelian_variety) is a [finitely generated abelian group](/source/finitely_generated_abelian_group).<ref>{{cite journal|first=A. |last=Weil |author-link=André Weil |title=L'arithmétique sur les courbes algébriques |journal=Acta Mathematica |volume=52 |date=1929 |pages=281–315}}. Reprinted in Volume 1 of his collected papers, {{isbn|0-387-90330-5}}.</ref>

Modern foundations of algebraic geometry were developed based on contemporary [commutative algebra](/source/commutative_algebra), including [valuation theory](/source/valuation_theory) and the theory of [ideals](/source/ideal_(ring_theory)) by [Oscar Zariski](/source/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](/source/Springer-Verlag) | location=Berlin, New York | edition=2nd suppl. | series=Classics in Mathematics | isbn=978-3-540-58658-6 | year=2004 | mr=0469915}}</ref>

In 1949, [André Weil](/source/Andr%C3%A9_Weil) posed the landmark [Weil conjectures](/source/Weil_conjectures) about the [local zeta-function](/source/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](/source/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](/source/Alexander_Grothendieck) to recast the foundations making use of [sheaf theory](/source/sheaf_theory) (together with [Jean-Pierre Serre](/source/Jean-Pierre_Serre)), and later scheme theory, in the 1950s and 1960s.<ref>{{cite journal | last1 = Serre | first1 = Jean-Pierre | year = 1955 | title = Faisceaux Algébriques Cohérents | journal = The Annals of Mathematics | volume = 61 | issue = 2| pages = 197–278 | doi=10.2307/1969915| jstor = 1969915 }}</ref> [Bernard Dwork](/source/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](/source/American_Journal_of_Mathematics) | issn=0002-9327 | volume=82 | issue=3 | pages=631–648 | doi=10.2307/2372974 }}</ref> Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with [Michael Artin](/source/Michael_Artin) and [Jean-Louis Verdier](/source/Jean-Louis_Verdier)) by 1965.<ref name="grothendieck-cohomology">{{cite book | last1=Grothendieck | first1=Alexander | author1-link=Alexander Grothendieck | title=Proceedings of the International Congress of Mathematicians (Edinburgh, 1958) | publisher=[Cambridge University Press](/source/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](/source/Soci%C3%A9t%C3%A9_Math%C3%A9matique_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](/source/Riemann_hypothesis)) would be finally proven in 1974 by [Pierre Deligne](/source/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](/source/Publications_Math%C3%A9matiques_de_l'IH%C3%89S) | volume=43 | issn=1618-1913 | issue=1 | pages=273–307| doi=10.1007/BF02684373 | url-access=subscription }}</ref>

===Mid-to-late 20th century: developments in modularity, ''{{mvar|p}}''-adic methods, and beyond===
Between 1956 and 1957, [Yutaka Taniyama](/source/Yutaka_Taniyama) and [Goro Shimura](/source/Goro_Shimura) posed the [Taniyama–Shimura conjecture](/source/Modularity_theorem) (now known as the modularity theorem) relating [elliptic curves](/source/elliptic_curves) to [modular forms](/source/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 [the first proof](/source/Wiles's_proof_of_Fermat's_Last_Theorem) of [Fermat's Last Theorem](/source/Fermat's_Last_Theorem) in number theory through algebraic geometry techniques of [modularity lifting](/source/Lift_(mathematics)) developed by [Andrew Wiles](/source/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 varieties](/source/Shimura_variety) as generalizations of [modular curve](/source/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](/source/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=33 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](/source/Barry_Mazur) proved the [torsion conjecture](/source/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](/source/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](/source/Publications_Math%C3%A9matiques_de_l'IH%C3%89S)|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](/source/Inventiones_Mathematicae)|others=with appendix by [Dorian Goldfeld](/source/Dorian_Goldfeld)|bibcode=1978InMat..44..129M}}</ref> In 1996, the proof of the torsion conjecture was extended to all number fields by [Loïc Merel](/source/Lo%C3%AFc_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](/source/Inventiones_Mathematicae) | volume=124 | issue=1 | pages=437–449 | bibcode=1996InMat.124..437M }}</ref>

In 1983, [Gerd Faltings](/source/Gerd_Faltings) proved the [Mordell conjecture](/source/Faltings'_theorem), demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates [finite generation](/source/finitely_generated_abelian_group) 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](/source/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](/source/Inventiones_Mathematicae) |volume=75 |issue=2 |pages=381 |doi=10.1007/BF01388572 | mr=0732554 | language=de |doi-access=free }}</ref>

=== Early 21st century ===
In 2001, the proof of the [local Langlands conjectures for {{math|GL<sub>''n''</sub>}}](/source/Local_Langlands_conjectures) 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](/source/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](/source/Peter_Scholze) developed [perfectoid space](/source/perfectoid_space)s and new cohomology theories in arithmetic geometry over {{mvar|p}}-adic fields with application to [Galois representations](/source/Galois_representations) and certain cases of the [weight-monodromy conjecture](/source/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](/source/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|publisher=University of Bonn|access-date=4 November 2018}}</ref>

==See also==
* [Anabelian geometry](/source/Anabelian_geometry)
* [Arithmetic dynamics](/source/Arithmetic_dynamics)
* [Arithmetic of abelian varieties](/source/Arithmetic_of_abelian_varieties)
* [Birch and Swinnerton-Dyer conjecture](/source/Birch_and_Swinnerton-Dyer_conjecture)
* [Category theory](/source/Category_theory)
* [Frobenioid](/source/Frobenioid)
* [Moduli of algebraic curves](/source/Moduli_of_algebraic_curves)
* [Siegel modular variety](/source/Siegel_modular_variety)
* [Siegel's theorem on integral points](/source/Siegel's_theorem_on_integral_points)

==References==
{{Reflist}}

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{{DEFAULTSORT:Arithmetic Geometry}}
Category:Arithmetic geometry

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Adapted from the Wikipedia article [Arithmetic geometry](https://en.wikipedia.org/wiki/Arithmetic_geometry) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Arithmetic_geometry?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.