arithmetic geometry

{{short description|Branch of algebraic geometry focused on problems in number theory}}

File:Example of a hyperelliptic curve.svg defined by $y^2=x(x+1)(x-3)(x+2)(x-2)$ has only finitely many rational points (such as the points $(-2, 0)$ and $(-1, 0)$ ) by Faltings's theorem.]]

In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory.{{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}} Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.{{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}}{{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}}

In more abstract terms, arithmetic geometry can be defined as the study of schemes of finite type over the spectrum of the ring of integers.{{nlab|id=arithmetic+geometry|title=Arithmetic geometry}}

Overview

The classical objects of interest in arithmetic geometry are rational points: sets of solutions of a system of polynomial equations over number fields, finite fields, p-adic fields, or function fields, i.e. fields that are not algebraically closed excluding the real numbers. Rational points can be directly characterized by height functions which measure their arithmetic complexity.{{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 }}

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 invariants associated to algebraic varieties. p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.{{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}}

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 coefficients exist if non-zero rational solutions exist.{{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}}

In the 1850s, Leopold Kronecker formulated the Kronecker–Weber theorem, introduced the theory of divisors, and made numerous other connections between number theory and algebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as his twelfth problem, which outlines a goal to have number theory operate only with rings that are quotients of polynomial rings over the integers.{{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 }}

=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.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}}.

Modern foundations of algebraic geometry were developed based on contemporary commutative algebra, including valuation theory and the theory of ideals by Oscar Zariski and others in the 1930s and 1940s.{{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}}

In 1949, André Weil posed the landmark Weil conjectures about the local zeta-functions of algebraic varieties over finite fields.{{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}} 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.{{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 }} Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960.{{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}} Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with Michael Artin and Jean-Louis Verdier) by 1965.{{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/}}{{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}} }} The last of the Weil conjectures (an analogue of the Riemann hypothesis) would be finally proven in 1974 by Pierre Deligne.{{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 }}

=Mid-to-late 20th century: developments in modularity, p-adic methods, and beyond=

Between 1956 and 1957, Yutaka Taniyama and Goro Shimura posed the Taniyama–Shimura conjecture (now known as the modularity theorem) relating elliptic curves to modular forms.{{cite journal|last=Taniyama|first=Yutaka |journal=Sugaku|volume=7|page=269|year=1956|title=Problem 12|language=ja}}{{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 }} This connection would ultimately lead to the first proof of Fermat's Last Theorem in number theory through algebraic geometry techniques of modularity lifting developed by Andrew Wiles in 1995.{{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}}

In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves.{{cite book|last=Shimura|first=Goro|title=The Collected Works of Goro Shimura|publisher=Springer Nature|isbn=978-0387954158|year=2003}} Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.{{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}}

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 curves.{{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/}}{{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}} In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel.{{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 }}

In 1983, Gerd Faltings proved the Mordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstrates finite generation of the set of rational points as opposed to finiteness).{{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}}{{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 }}

In 2001, the proof of the local Langlands conjectures for GL_n was based on the geometry of certain Shimura varieties.{{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}}

In the 2010s, Peter Scholze developed perfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application to Galois representations and certain cases of the weight-monodromy conjecture.{{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}}{{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}}