The weak Bombieri–Lang conjecture for surfaces states that if is a smooth surface of general type defined over a number field , then the -rational points of do not form a dense set in the Zariski topology on .[1]
The general form of the Bombieri–Lang conjecture states that if is a positive-dimensional algebraic variety of general type defined over a number field , then the -rational points of do not form a dense set in the Zariski topology.[2][3][4]
The refined form of the Bombieri–Lang conjecture states that if is an algebraic variety of general type defined over a number field , then there is a dense open subset of such that for all number field extensions over , the set of -rational points in is finite.[4]
History
The Bombieri–Lang conjecture was independently posed by Enrico Bombieri and Serge Lang. In a 1980 lecture at the University of Chicago, Enrico Bombieri posed a problem about the degeneracy of rational points for surfaces of general type.[1] Independently in a series of papers starting in 1971, Serge Lang conjectured a more general relation between the distribution of rational points and algebraic hyperbolicity,[1][5][6][7] formulated in the "refined form" of the Bombieri–Lang conjecture.[4]
Generalizations and implications
The Bombieri–Lang conjecture is an analogue for surfaces of Faltings's theorem, which states that algebraic curves of genus greater than one only have finitely many rational points.[8]
If true, the Bombieri–Lang conjecture would resolve the Erdős–Ulam problem, as it would imply that there do not exist dense subsets of the Euclidean plane all of whose pairwise distances are rational.[8][9]
^ abcDas, Pranabesh; Turchet, Amos (2015), "Invitation to integral and rational points on curves and surfaces", in Gasbarri, Carlo; Lu, Steven; Roth, Mike; Tschinkel, Yuri (eds.), Rational Points, Rational Curves, and Entire Holomorphic Curves on Projective Varieties, Contemporary Mathematics, vol. 654, American Mathematical Society, pp. 53–73, arXiv:1407.7750
^ abPoonen, Bjorn (2012), Uniform boundedness of rational points and preperiodic points, arXiv:1206.7104
^ abConceição, Ricardo; Ulmer, Douglas; Voloch, José Felipe (2012), "Unboundedness of the number of rational points on curves over function fields", New York Journal of Mathematics, 18: 291–293
^ abcHindry, Marc; Silverman, Joseph H. (2000), "F.5.2. The Bombieri–Lang Conjecture", Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, vol. 201, Springer-Verlag, New York, pp. 479–482, doi:10.1007/978-1-4612-1210-2, ISBN0-387-98975-7, MR 1745599
^Lang, Serge (1971), "Transcendental numbers and diophantine approximations", Bulletin of the American Mathematical Society, vol. 77, no. 5, pp. 635–678, doi:10.1090/S0002-9904-1971-12761-1, ISSN 0002-9904
^Lang, Serge (1974), "Higher dimensional diophantine problems", Bulletin of the American Mathematical Society, vol. 80, no. 5, pp. 779–788, doi:10.1090/S0002-9904-1974-13516-0, ISSN 0002-9904
^ abTao, Terence (December 20, 2014), "The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture", What's new
^Shaffaf, Jafar (May 2018), "A solution of the Erdős–Ulam problem on rational distance sets assuming the Bombieri–Lang conjecture", Discrete & Computational Geometry, 60 (8), arXiv:1501.00159, doi:10.1007/s00454-018-0003-3
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bombieri, lang, conjecture, arithmetic, geometry, unsolved, problem, conjectured, enrico, bombieri, serge, lang, about, zariski, density, rational, points, algebraic, variety, general, type, contents, statement, history, generalizations, implications, referenc. In arithmetic geometry the Bombieri Lang conjecture is an unsolved problem conjectured by Enrico Bombieri and Serge Lang about the Zariski density of the set of rational points of an algebraic variety of general type Contents 1 Statement 2 History 3 Generalizations and implications 4 ReferencesStatement EditThe weak Bombieri Lang conjecture for surfaces states that if X displaystyle X is a smooth surface of general type defined over a number field k displaystyle k then the k displaystyle k rational points of X displaystyle X do not form a dense set in the Zariski topology on X displaystyle X 1 The general form of the Bombieri Lang conjecture states that if X displaystyle X is a positive dimensional algebraic variety of general type defined over a number field k displaystyle k then the k displaystyle k rational points of X displaystyle X do not form a dense set in the Zariski topology 2 3 4 The refined form of the Bombieri Lang conjecture states that if X displaystyle X is an algebraic variety of general type defined over a number field k displaystyle k then there is a dense open subset U displaystyle U of X displaystyle X such that for all number field extensions k displaystyle k over k displaystyle k the set of k displaystyle k rational points in U displaystyle U is finite 4 History EditThe Bombieri Lang conjecture was independently posed by Enrico Bombieri and Serge Lang In a 1980 lecture at the University of Chicago Enrico Bombieri posed a problem about the degeneracy of rational points for surfaces of general type 1 Independently in a series of papers starting in 1971 Serge Lang conjectured a more general relation between the distribution of rational points and algebraic hyperbolicity 1 5 6 7 formulated in the refined form of the Bombieri Lang conjecture 4 Generalizations and implications EditThe Bombieri Lang conjecture is an analogue for surfaces of Faltings s theorem which states that algebraic curves of genus greater than one only have finitely many rational points 8 If true the Bombieri Lang conjecture would resolve the Erdos Ulam problem as it would imply that there do not exist dense subsets of the Euclidean plane all of whose pairwise distances are rational 8 9 In 1997 Lucia Caporaso Barry Mazur Joe Harris and Patricia Pacelli showed that the Bombieri Lang conjecture implies a uniform boundedness conjecture for rational points there is a constant B g d displaystyle B g d depending only on g displaystyle g and d displaystyle d such that the number of rational points of any genus g displaystyle g curve X displaystyle X over any degree d displaystyle d number field is at most B g d displaystyle B g d 2 3 References Edit a b c Das Pranabesh Turchet Amos 2015 Invitation to integral and rational points on curves and surfaces in Gasbarri Carlo Lu Steven Roth Mike Tschinkel Yuri eds Rational Points Rational Curves and Entire Holomorphic Curves on Projective Varieties Contemporary Mathematics vol 654 American Mathematical Society pp 53 73 arXiv 1407 7750 a b Poonen Bjorn 2012 Uniform boundedness of rational points and preperiodic points arXiv 1206 7104 a b Conceicao Ricardo Ulmer Douglas Voloch Jose Felipe 2012 Unboundedness of the number of rational points on curves over function fields New York Journal of Mathematics 18 291 293 a b c Hindry Marc Silverman Joseph H 2000 F 5 2 The Bombieri Lang Conjecture Diophantine Geometry An Introduction Graduate Texts in Mathematics vol 201 Springer Verlag New York pp 479 482 doi 10 1007 978 1 4612 1210 2 ISBN 0 387 98975 7 MR 1745599 Lang Serge 1971 Transcendental numbers and diophantine approximations Bulletin of the American Mathematical Society vol 77 no 5 pp 635 678 doi 10 1090 S0002 9904 1971 12761 1 ISSN 0002 9904 Lang Serge 1974 Higher dimensional diophantine problems Bulletin of the American Mathematical Society vol 80 no 5 pp 779 788 doi 10 1090 S0002 9904 1974 13516 0 ISSN 0002 9904 Lang Serge 1983 Fundamentals of Diophantine geometry New York Springer Verlag p 224 ISBN 0 387 90837 4 a b Tao Terence December 20 2014 The Erdos Ulam problem varieties of general type and the Bombieri Lang conjecture What s new Shaffaf Jafar May 2018 A solution of the Erdos Ulam problem on rational distance sets assuming the Bombieri Lang conjecture Discrete amp Computational Geometry 60 8 arXiv 1501 00159 doi 10 1007 s00454 018 0003 3 This algebraic geometry related article is a stub You can help Wikipedia by expanding it vte Retrieved from https en wikipedia org w index php title Bombieri Lang conjecture amp oldid 1103672890, wikipedia, wiki, book, books, library,