# Arithmetic geometry

Geometry :
Projecting a sphere to a plane :
OutlineHistory (Timeline) :
Branches
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Non-Euclidean
Elliptic
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Non-Archimedean geometry
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Arithmetic
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Complex
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Discrete/Combinatorial
Digital
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Computational
Fractal
Incidence
Noncommutative geometry
Noncommutative algebraic geometry :
ConceptsFeaturesDimension
Straightedge and compass constructions
Angle
Curve
Diagonal
Orthogonality (Perpendicular)
Parallel
Vertex
Congruence
Similarity
Symmetry :
Zero-dimensional
Point :
One-dimensional
Line
segment
ray
Length :
Two-dimensional
Plane
Area
Polygon
Triangle
Altitude
Hypotenuse
Pythagorean theorem
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Square
Rectangle
Rhombus
Rhomboid
Quadrilateral
Trapezoid
Kite
Circle
Diameter
Circumference
Area :
Three-dimensional
Volume
Cube
cuboid
Cylinder
Dodecahedron
Icosahedron
Octahedron
Pyramid
Platonic Solid
Sphere
Tetrahedron :
Four- / other-dimensional
Tesseract
Hypersphere :
Geometers :
by name
Aida
Aryabhata
Ahmes
Alhazen
Apollonius
Archimedes
Atiyah
Baudhayana
Bolyai
Brahmagupta
Cartan
Coxeter
Descartes
Euclid
Euler
Gauss
Gromov
Hilbert
Huygens
Jyeṣṭhadeva
Kātyāyana
Khayyám
Klein
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Manava
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Minggatu
Pascal
Pythagoras
Parameshvara
Poincaré
Riemann
Sakabe
Sijzi
al-Tusi
Veblen
Virasena
Yang Hui
al-Yasamin
Zhang
List of geometers :
by period
BCE
Ahmes
Baudhayana
Manava
Pythagoras
Euclid
Archimedes
Apollonius
1–1400s
Zhang
Kātyāyana
Aryabhata
Brahmagupta
Virasena
Alhazen
Sijzi
Khayyám
al-Yasamin
al-Tusi
Yang Hui
Parameshvara
1400s–1700s
Jyeṣṭhadeva
Descartes
Pascal
Huygens
Minggatu
Euler
Sakabe
Aida
1700s–1900s
Gauss
Lobachevsky
Bolyai
Riemann
Klein
Poincaré
Hilbert
Minkowski
Cartan
Veblen
Coxeter
Present day
Atiyah
Gromov :
vte :
In mathematics, arithmetic geometry is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.
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.

## 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.
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. p-adic Hodge theory gives tools to examine when cohomological properties of varieties over the complex numbers extend to those over p-adic fields.

## 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.
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.

### 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.
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.
In 1949, André 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. Bernard Dwork proved one of the four Weil conjectures (rationality of the local zeta function) in 1960. Grothendieck developed étale cohomology theory to prove two of the Weil conjectures (together with Michael Artin and Jean-Louis Verdier) by 1965. The last of the Weil conjectures (an analogue of the Riemann hypothesis) would be finally proven in 1974 by Pierre Deligne.

### 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. 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.
In the 1960s, Goro Shimura introduced Shimura varieties as generalizations of modular curves. Since the 1979, Shimura varieties have played a crucial role in the Langlands program as a natural realm of examples for testing conjectures.
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. In 1996, the proof of the torsion conjecture was extended to all number fields by Loïc Merel.
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).
In 2001, the proof of the local Langlands conjectures for GL$_n$ was based on the geometry of certain Shimura varieties.
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.