Arithmetic geometry

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

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]

Contents

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]

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. [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]

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. [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]

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. [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]

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. [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 GLn 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]

See also

Related Research Articles

<span class="mw-page-title-main">Faltings's theorem</span> Curves of genus > 1 over the rationals have only finitely many rational points

Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and known as the Mordell conjecture until its 1983 proof by Gerd Faltings. The conjecture was later generalized by replacing by any number field.

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms in a particular way. Andrew Wiles and Richard Taylor proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.

Gorō Shimura was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multiplication of abelian varieties and Shimura varieties, as well as posing the Taniyama–Shimura conjecture which ultimately led to the proof of Fermat's Last Theorem.

<span class="mw-page-title-main">Jean-Pierre Serre</span> French mathematician

Jean-Pierre Serre is a French mathematician who has made contributions to algebraic topology, algebraic geometry and algebraic number theory. He was awarded the Fields Medal in 1954, the Wolf Prize in 2000 and the inaugural Abel Prize in 2003.

In mathematics, the Weil conjectures were highly influential proposals by André Weil. They led to a successful multi-decade program to prove them, in which many leading researchers developed the framework of modern algebraic geometry and number theory.

In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally.

<span class="mw-page-title-main">Pierre Deligne</span> Belgian mathematician

Pierre René, Viscount Deligne is a Belgian mathematician. He is best known for work on the Weil conjectures, leading to a complete proof in 1973. He is the winner of the 2013 Abel Prize, 2008 Wolf Prize, 1988 Crafoord Prize, and 1978 Fields Medal.

In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectures. Étale cohomology theory can be used to construct ℓ-adic cohomology, which is an example of a Weil cohomology theory in algebraic geometry. This has many applications, such as the proof of the Weil conjectures and the construction of representations of finite groups of Lie type.

In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. It is named after mathematicians Bryan John Birch and Peter Swinnerton-Dyer, who developed the conjecture during the first half of the 1960s with the help of machine computation. As of 2024, only special cases of the conjecture have been proven.

In mathematics, a Galois module is a G-module, with G being the Galois group of some extension of fields. The term Galois representation is frequently used when the G-module is a vector space over a field or a free module over a ring in representation theory, but can also be used as a synonym for G-module. The study of Galois modules for extensions of local or global fields and their group cohomology is an important tool in number theory.

In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study these equations. Diophantine geometry is part of the broader field of arithmetic geometry.

<span class="mw-page-title-main">Yutaka Taniyama</span> Japanese mathematician

Yutaka Taniyama was a Japanese mathematician known for the Taniyama–Shimura conjecture.

Hilbert's twelfth problem is the extension of the Kronecker–Weber theorem on abelian extensions of the rational numbers, to any base number field. It is one of the 23 mathematical Hilbert problems and asks for analogues of the roots of unity that generate a whole family of further number fields, analogously to the cyclotomic fields and their subfields. Leopold Kronecker described the complex multiplication issue as his liebster Jugendtraum, or "dearest dream of his youth", so the problem is also known as Kronecker's Jugendtraum.

<span class="mw-page-title-main">Tate conjecture</span>

In number theory and algebraic geometry, the Tate conjecture is a 1963 conjecture of John Tate that would describe the algebraic cycles on a variety in terms of a more computable invariant, the Galois representation on étale cohomology. The conjecture is a central problem in the theory of algebraic cycles. It can be considered an arithmetic analog of the Hodge conjecture.

This is a glossary of arithmetic and diophantine geometry in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

<span class="mw-page-title-main">Modular elliptic curve</span>

A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve. The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.

In mathematics, the Mordell–Weil theorem states that for an abelian variety over a number field , the group of K-rational points of is a finitely-generated abelian group, called the Mordell–Weil group. The case with an elliptic curve and the field of rational numbers is Mordell's theorem, answering a question apparently posed by Henri Poincaré around 1901; it was proved by Louis Mordell in 1922. It is a foundational theorem of Diophantine geometry and the arithmetic of abelian varieties.

In arithmetic geometry, the Tate–Shafarevich groupШ(A/K) of an abelian variety A (or more generally a group scheme) defined over a number field K consists of the elements of the Weil–Châtelet group , where is the absolute Galois group of K, that become trivial in all of the completions of K (i.e., the real and complex completions as well as the p-adic fields obtained from K by completing with respect to all its Archimedean and non Archimedean valuations v). Thus, in terms of Galois cohomology, Ш(A/K) can be defined as

<span class="mw-page-title-main">Siegel modular variety</span> Algebraic variety that is a moduli space for principally polarized abelian varieties

In mathematics, a Siegel modular variety or Siegel moduli space is an algebraic variety that parametrizes certain types of abelian varieties of a fixed dimension. More precisely, Siegel modular varieties are the moduli spaces of principally polarized abelian varieties of a fixed dimension. They are named after Carl Ludwig Siegel, the 20th-century German number theorist who introduced the varieties in 1943.

References

  1. ↑ Sutherland, Andrew V. (September 5, 2013). "Introduction to Arithmetic Geometry" (PDF). Retrieved 22 March 2019.
  2. ↑ Klarreich, Erica (June 28, 2016). "Peter Scholze and the Future of Arithmetic Geometry" . Retrieved March 22, 2019.
  3. ↑ Poonen, Bjorn (2009). "Introduction to Arithmetic Geometry" (PDF). Retrieved March 22, 2019.
  4. ↑ Arithmetic geometry at the nLab
  5. ↑ Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. pp. 43–67. ISBN   3-540-61223-8. Zbl   0869.11051.
  6. 1 2 Grothendieck, Alexander (1960). "The cohomology theory of abstract algebraic varieties". Proc. Internat. Congress Math. (Edinburgh, 1958). Cambridge University Press. pp. 103–118. MR   0130879.
  7. ↑ Serre, Jean-Pierre (1967). "Résumé des cours, 1965–66". Annuaire du Collège de France. Paris: 49–58.
  8. ↑ Mordell, Louis J. (1969). Diophantine Equations. Academic Press. p. 1. ISBN   978-0125062503.
  9. ↑ 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.
  10. ↑ 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.
  11. ↑ 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.
  12. ↑ 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
  13. ↑ Serre, Jean-Pierre (1955). "Faisceaux Algebriques Coherents". The Annals of Mathematics. 61 (2): 197–278. doi:10.2307/1969915. JSTOR   1969915.
  14. ↑ 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.
  15. ↑ 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.
  16. ↑ Deligne, Pierre (1974). "La conjecture de Weil. I". Publications Mathématiques de l'IHÉS . 43 (1): 273–307. doi:10.1007/BF02684373. ISSN   1618-1913. MR   0340258.
  17. ↑ Taniyama, Yutaka (1956). "Problem 12". Sugaku (in Japanese). 7: 269.
  18. ↑ 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.
  19. ↑ 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. Archived from the original (PDF) on 2011-05-10. Retrieved 2019-03-22.
  20. ↑ Shimura, Goro (2003). The Collected Works of Goro Shimura. Springer Nature. ISBN   978-0387954158.
  21. ↑ 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. Vol. XXXIII Part 1. Chelsea Publishing Company. pp. 205–246.
  22. ↑ Mazur, Barry (1977). "Modular curves and the Eisenstein ideal". Publications Mathématiques de l'IHÉS . 47 (1): 33–186. doi:10.1007/BF02684339. MR   0488287.
  23. ↑ Mazur, Barry (1978). "Rational isogenies of prime degree". Inventiones Mathematicae . 44 (2). with appendix by Dorian Goldfeld: 129–162. Bibcode:1978InMat..44..129M. doi:10.1007/BF01390348. MR   0482230.
  24. ↑ 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.
  25. ↑ 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. Bibcode:1983InMat..73..349F. doi:10.1007/BF01388432. MR   0718935.
  26. ↑ Faltings, Gerd (1984). "Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern". Inventiones Mathematicae (in German). 75 (2): 381. doi: 10.1007/BF01388572 . MR   0732554.
  27. ↑ Harris, Michael; Taylor, Richard (2001). The geometry and cohomology of some simple Shimura varieties. Annals of Mathematics Studies. Vol. 151. Princeton University Press. ISBN   978-0-691-09090-0. MR   1876802.
  28. ↑ "Fields Medals 2018". International Mathematical Union . Retrieved 2 August 2018.
  29. ↑ Scholze, Peter. "Perfectoid spaces: A survey" (PDF). University of Bonn. Retrieved 4 November 2018.
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