Geometry
Projecting asphere to aplane
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Concepts Features Dimension Straightedge and compass constructions Angle Curve Diagonal Orthogonality (Perpendicular) Parallel Vertex Congruence Similarity Symmetry
Zero-dimensional Point
One-dimensional Line Line segment Ray Curve Length
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Three-dimensional Surface area Volume Polyhedron Platonic Solid Tetrahedron cuboid Cube Octahedron Dodecahedron Icosahedron Pyramid Solid of revolution Sphere Great circle Cylinder Cone
Four-/other-dimensional 4-polytope Simplex 5-cell Hypercube Tesseract n-sphere Hypersphere
Geometers
by name Aida Aryabhata Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva Kātyāyana Khayyám Klein Lobachevsky Manava Minkowski 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 Chern Present day Atiyah Gromov
v t e

In mathematics,arithmetic geometry is roughly the application of techniques fromalgebraic geometry to problems innumber theory.^[1] Arithmetic geometry is centered aroundDiophantine geometry, the study ofrational points ofalgebraic varieties.^[2][3]

In more abstract terms, arithmetic geometry can be defined as the study ofschemes offinite type over thespectrum of thering of integers.^[4]

The classical objects of interest in arithmetic geometry are rational points:sets of solutions of asystem of polynomial equations overnumber fields,finite fields,p-adic fields, orfunction fields, i.e.fields that are notalgebraically closed excluding thereal numbers. Rational points can be directly characterized byheight 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 providestopological invariants associated to algebraic varieties.^[6]p-adic Hodge theory gives tools to examine when cohomological properties of varieties over thecomplex numbers extend to those overp-adic fields.^[7]

In the early 19th century,Carl Friedrich Gauss observed that non-zerointeger solutions tohomogeneous polynomial equations withrational coefficients exist if non-zero rational solutions exist.^[8]

In the 1850s,Leopold Kronecker formulated theKronecker–Weber theorem, introduced the theory ofdivisors, and made numerous other connections between number theory andalgebra. He then conjectured his "liebster Jugendtraum" ("dearest dream of youth"), a generalization that was later put forward by Hilbert in a modified form as histwelfth problem, which outlines a goal to have number theory operate only with rings that are quotients ofpolynomial rings over the integers.^[9]

In the late 1920s,André Weil demonstrated profound connections between algebraic geometry and number theory with his doctoral work leading to theMordell–Weil theorem which demonstrates that the set of rational points of anabelian variety is afinitely generated abelian group.^[10]

Modern foundations of algebraic geometry were developed based on contemporarycommutative algebra, includingvaluation theory and the theory ofideals byOscar Zariski and others in the 1930s and 1940s.^[11]

In 1949,André Weil posed the landmarkWeil conjectures about thelocal zeta-functions of algebraic varieties over finite fields.^[12] These conjectures offered a framework between algebraic geometry and number theory that propelledAlexander Grothendieck to recast the foundations making use ofsheaf theory (together withJean-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 withMichael Artin andJean-Louis Verdier) by 1965.^[6][15] The last of the Weil conjectures (an analogue of theRiemann hypothesis) would be finally proven in 1974 byPierre Deligne.^[16]

Between 1956 and 1957,Yutaka Taniyama andGoro Shimura posed theTaniyama–Shimura conjecture (now known as the modularity theorem) relatingelliptic curves tomodular forms.^[17][18] This connection would ultimately lead tothe first proof ofFermat's Last Theorem in number theory through algebraic geometry techniques ofmodularity lifting developed byAndrew Wiles in 1995.^[19]

In the 1960s, Goro Shimura introducedShimura varieties as generalizations ofmodular curves.^[20] Since the 1979, Shimura varieties have played a crucial role in theLanglands program as a natural realm of examples for testing conjectures.^[21]

In papers in 1977 and 1978,Barry Mazur proved thetorsion 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 certainmodular curves.^[22][23] In 1996, the proof of the torsion conjecture was extended to all number fields byLoïc Merel.^[24]

In 1983,Gerd Faltings proved theMordell conjecture, demonstrating that a curve of genus greater than 1 has only finitely many rational points (where the Mordell–Weil theorem only demonstratesfinite generation of the set of rational points as opposed to finiteness).^[25][26]

In 2001, the proof of thelocal Langlands conjectures for GL_n was based on the geometry of certain Shimura varieties.^[27]

In the 2010s,Peter Scholze developedperfectoid spaces and new cohomology theories in arithmetic geometry over p-adic fields with application toGalois representations and certain cases of theweight-monodromy conjecture.^[28][29]

• Anabelian geometry
• Frobenioid
• Arithmetic dynamics
• Arithmetic of abelian varieties
• Birch and Swinnerton-Dyer conjecture
• Moduli of algebraic curves
• Siegel modular variety
• Siegel's theorem on integral points
• Category theory

• Arithmetic geometry

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