Arithmetic geometry
| Geometry |
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| Geometers |
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 points of algebraic varieties with coordinates in the integers, rational numbers, and their generalizations.[2][3] These generalizations typically are fields that are not algebraically closed, such as number fields, finite fields, function fields, and p-adic fields (but not the real numbers which are used in real algebraic geometry).
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]
History
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.[5]
In the 1920s, André Weil demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to the Mordell–Weil theorem in Diophantine geometry (1928, and shortly applied in Siegel's theorem on integral points).[6]
Following the development of the foundations of algebraic geometry based on contemporary commutative algebra, including valuation theory and the theory of ideals by Oscar Zariski[7], André Weil, and many others, Weil posed the landmark Weil conjectures about the local zeta-functions of algebraic varieties over finite fields. 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.[8] Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960.[9] Grothendieck developed étale cohomology theory and, together with Michael Artin and Jean-Louis Verdier, proved two of the Weil conjectures by 1965.[10][11] The last of the Weil conjectures (an analogue of the Riemann hypothesis) would be finally proven in 1974 by Pierre Deligne.[12]
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.[13][14] 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.[15]
In 1977, Barry Mazur proved Mazur's torsion theorem, which gives 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.[16]
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.[17][18]
See also
References
- ^ Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF). Retrieved 22 March 2019.
{{cite web}}: Cite has empty unknown parameter:|1=(help) - ^ Klarreich, Erica (June 28, 2016). "Peter Scholze and the Future of Arithmetic Geometry". Retrieved March 22, 2019.
- ^ Poonen, Bjorn (2009). "Introduction to Arithmetic Geometry" (PDF). Retrieved March 22, 2019.
- ^ Arithmetic geometry at the nLab
- ^ 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
- ^ Serre, Jean-Pierre (1955). "Faisceaux Algebriques Coherents". The Annals of Mathematics. 61 (2): 197–278. doi:10.2307/1969915.
- ^ Dwork, Bernard (1960), "On the rationality of the zeta function of an algebraic variety", American Journal of Mathematics, 82 (3), American Journal of Mathematics, Vol. 82, No. 3: 631–648, doi:10.2307/2372974, ISSN 0002-9327, JSTOR 2372974, MR 0140494
- ^ Grothendieck, Alexander (1960), "The cohomology theory of abstract algebraic varieties", Proc. Internat. Congress Math. (Edinburgh, 1958), Cambridge University Press, pp. 103–118, MR 0130879
- ^ Grothendieck, Alexander (1995) [1965], "Formule de Lefschetz et rationalité des fonctions L", Séminaire Bourbaki, vol. 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): 273–307, 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.
- ^ Mazur, Barry (1977). "Modular curves and the Eisenstein ideal". Publications Mathématiques de l'IHÉS. 47 (1): 33–186. doi:10.1007/BF02684339. MR 0488287.
{{cite journal}}: Invalid|ref=harv(help) - ^ "Fields Medals 2018". International Mathematical Union. Retrieved 2 August 2018.
- ^ Scholze, Peter. "Perfectoid spaces: A survey" (PDF). University of Bonn. Retrieved 4 November 2018.
