Branch of algebraic geometry focused on problems in number theory
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 .
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]
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
References
^ Sutherland, Andrew V. (September 5, 2013).
"Introduction to Arithmetic Geometry" (PDF) . Retrieved 22 March 2019 .
^ 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
n Lab
^
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 . 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 (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 (1): 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 . Archived from
the original (PDF) on 2011-05-10. Retrieved 2019-03-22 .
^ 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 . Vol. 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). "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 .
^
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.
Bibcode :
1983InMat..73..349F .
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. Vol. 151.
Princeton University Press .
ISBN
978-0-691-09090-0 .
MR
1876802 .
^
"Fields Medals 2018" .
International Mathematical Union . Retrieved 2 August 2018 .
^ Scholze, Peter.
"Perfectoid spaces: A survey" (PDF) . University of Bonn . Retrieved 4 November 2018 .
Fields Key concepts Advanced concepts