Arithmetic geometry

Last updated
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

Elliptic curve Algebraic curve

In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. An elliptic curve is defined over a field K and describes points in K2, the Cartesian product of K with itself. If the field has characteristic different from 2 and 3 then the curve can be described as a plane algebraic curve which, after a linear change of variables, consists of solutions (x,y) to:

Faltingss theorem Curves of genus > 1 over the rationals have only finitely many rational points

In arithmetic geometry, the Mordell conjecture is the conjecture made by Louis Mordell that a curve of genus greater than 1 over the field Q of rational numbers has only finitely many rational points. In 1983 it was proved by Gerd Faltings, and is now known as Faltings's theorem. The conjecture was later generalized by replacing Q by any number field.

The modularity theorem states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles 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.

Jean-Pierre Serre French mathematician

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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.

Pierre Deligne Belgian mathematician

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

Yuri Manin Russian mathematician

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Tate conjecture

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.

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Modular elliptic curve

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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 rational number field Q 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.

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