Arithmetic Geometry

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## Overview

## History

### 19th century: early arithmetic geometry

### Early-to-mid 20th century: algebraic developments and the Weil conjectures

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

## See also

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Arithmetic Geometry

In mathematics, **arithmetic geometry** is roughly the application of techniques from algebraic geometry to problems in number theory.^{[1]} Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.^{[2]}^{[3]}

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.^{[4]}

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.^{[5]}

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.^{[6]}p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.^{[7]}

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.^{[8]}

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.^{[9]}

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.^{[10]}

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.^{[11]}

In 1949, André Weil posed the landmark Weil conjectures about the local zeta-functions of algebraic varieties over finite fields.^{[12]} 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.^{[13]}Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960.^{[14]} Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with Michael Artin and Jean-Louis Verdier) by 1965.^{[6]}^{[15]} The last of the Weil conjectures (an analogue of the Riemann hypothesis) would be finally proven in 1974 by Pierre Deligne.^{[16]}

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.^{[17]}^{[18]} 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.^{[19]}

In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves.^{[20]} Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.^{[21]}

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.^{[22]}^{[23]} In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel.^{[24]}

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).^{[25]}^{[26]}

In 2001, the proof of the local Langlands conjectures for GL_{n} was based on the geometry of certain Shimura varieties.^{[27]}

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.^{[28]}^{[29]}

- Arithmetic dynamics
- Arithmetic of abelian varieties
- Birch and Swinnerton-Dyer conjecture
- Moduli of algebraic curves
- Siegel modular variety
- Siegel's theorem on integral points

**^**Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF). Retrieved 2019.**^**Klarreich, Erica (June 28, 2016). "Peter Scholze and the Future of Arithmetic Geometry". Retrieved 2019.**^**Poonen, Bjorn (2009). "Introduction to Arithmetic Geometry" (PDF). Retrieved 2019.**^**Arithmetic geometry in*nLab***^**Lang, Serge (1997).*Survey of Diophantine Geometry*. Springer-Verlag. pp. 43-67. ISBN 3-540-61223-8. Zbl 0869.11051.- ^
^{a}^{b}Grothendieck, Alexander (1960). "The cohomology theory of abstract algebraic varieties".*Proc. Internat. Congress Math. (Edinburgh, 1958)*. Cambridge University Press. pp. 103-118. MR 0130879. **^**Serre, Jean-Pierre (1967). "Résumé des cours, 1965-66".*Annuaire du Collège de France*. Paris: 49-58.**^**Mordell, Louis J. (1969).*Diophantine Equations*. Academic Press. p. 1. ISBN 978-0125062503.**^**Gowers, Timothy; Barrow-Green, June; Leader, Imre (2008).*The Princeton companion to mathematics*. Princeton University Press. pp. 773-774. ISBN 978-0-691-11880-2.**^**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.**^**Zariski, Oscar (2004) [1935]. Abhyankar, Shreeram S.; Lipman, Joseph; Mumford, David (eds.).*Algebraic surfaces*. Classics in mathematics (second supplemented ed.). Berlin, New York: Springer-Verlag. ISBN 978-3-540-58658-6. MR 0469915.**^**Weil, André (1949). "Numbers of solutions of equations in finite fields".*Bulletin of the American Mathematical Society*.**55**(5): 497-508. doi:10.1090/S0002-9904-1949-09219-4. ISSN 0002-9904. MR 0029393. Reprinted in Oeuvres Scientifiques/Collected Papers by André Weil ISBN 0-387-90330-5**^**Serre, Jean-Pierre (1955). "Faisceaux Algebriques Coherents".*The Annals of Mathematics*.**61**(2): 197-278. doi:10.2307/1969915. JSTOR 1969915.**^**Dwork, Bernard (1960). "On the rationality of the zeta function of an algebraic variety".*American Journal of Mathematics*. American Journal of Mathematics, Vol. 82, No. 3.**82**(3): 631-648. doi:10.2307/2372974. ISSN 0002-9327. JSTOR 2372974. MR 0140494.**^**Grothendieck, Alexander (1995) [1965]. "Formule de Lefschetz et rationalité des fonctions L".*Séminaire Bourbaki*.**9**. Paris: Société Mathématique de France. pp. 41-55. MR 1608788.**^**Deligne, Pierre (1974). "La conjecture de Weil. I".*Publications Mathématiques de l'IHÉS*.**43**(43): 273-307. doi:10.1007/BF02684373. ISSN 1618-1913. MR 0340258.**^**Taniyama, Yutaka (1956). "Problem 12".*Sugaku*(in Japanese).**7**: 269.**^**Shimura, Goro (1989). "Yutaka Taniyama and his time. Very personal recollections".*The Bulletin of the London Mathematical Society*.**21**(2): 186-196. doi:10.1112/blms/21.2.186. ISSN 0024-6093. MR 0976064.**^**Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF).*Annals of Mathematics*.**141**(3): 443-551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255.**^**Shimura, Goro (2003).*The Collected Works of Goro Shimura*. Springer Nature. ISBN 978-0387954158.**^**Langlands, Robert (1979). "Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen" (PDF). In Borel, Armand; Casselman, William (eds.).*Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics*. XXXIII Part 1. Chelsea Publishing Company. pp. 205-246.**^**Mazur, Barry (1977). "Modular curves and the Eisenstein ideal".*Publications Mathématiques de l'IHÉS*.**47**(1): 33-186. doi:10.1007/BF02684339. MR 0488287.**^**Mazur, Barry (1978). with appendix by Dorian Goldfeld. "Rational isogenies of prime degree".*Inventiones Mathematicae*.**44**(2): 129-162. Bibcode:1978InMat..44..129M. doi:10.1007/BF01390348. MR 0482230.**^**Merel, Loïc (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres" [Bounds for the torsion of elliptic curves over number fields].*Inventiones Mathematicae*(in French).**124**(1): 437-449. Bibcode:1996InMat.124..437M. doi:10.1007/s002220050059. MR 1369424.**^**Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields].*Inventiones Mathematicae*(in German).**73**(3): 349-366. doi:10.1007/BF01388432. MR 0718935.**^**Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern".*Inventiones Mathematicae*(in German).**75**(2): 381. doi:10.1007/BF01388572. MR 0732554.**^**Harris, Michael; Taylor, Richard (2001).*The geometry and cohomology of some simple Shimura varieties*. Annals of Mathematics Studies.**151**. Princeton University Press. ISBN 978-0-691-09090-0. MR 1876802.**^**"Fields Medals 2018". International Mathematical Union. Retrieved 2018.**^**Scholze, Peter. "Perfectoid spaces: A survey" (PDF).*University of Bonn*. Retrieved 2018.

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