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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 parallelogram square rectangle rhombus rhomboid quadrilateral trapezoid kite circle diameter circumference area three dimensional volume cube cuboid cylinder pyramid sphere 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 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 minggatu euler sakabe aida 1700s 1900s gauss lobachevsky bolyai riemann klein poincaré hilbert minkowski cartan veblen coxeter present day atiyah gromov v t e 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 contents 1 overview 2 history 2 1 19th century early arithmetic geometry 2 2 early to mid 20th century algebraic developments and the weil conjectures 2 3 mid to late 20th century developments in modularity p adic methods and beyond 3 see also 4 references overview edit 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 edit 19th century early arithmetic geometry edit 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 edit 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 edit 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 gl n 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 edit 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 frobenioid references edit 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 american journal of mathematics vol 82 no 3 82 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 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 with appendix by dorian goldfeld rational isogenies of prime degree inventiones mathematicae 44 2 129 162 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 v t e number theory fields algebraic number theory class field theory non abelian class field theory iwasawa theory iwasawa tate theory kummer theory analytic number theory analytic theory of l functions probabilistic number theory sieve theory geometric number theory computational number theory transcendental number theory diophantine geometry arakelov theory hodge arakelov theory arithmetic combinatorics additive number theory arithmetic geometry anabelian geometry p adic hodge theory arithmetic topology arithmetic dynamics key concepts numbers natural numbers prime numbers rational numbers irrational numbers algebraic numbers transcendental numbers p adic numbers p adic analysis arithmetic modular arithmetic chinese remainder 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