Weak Bombieri–Lang conjecture: If \(X\) is a surface of general type defined over a number field \(k\), then the \(k\)-rational points of \(X\) do not form a dense set in the Zariski topology on \(X\). [1]
General Bombieri–Lang conjecture: If \(X\) is an algebraic variety of general type defined over a number field \(k\), then the \(k\)-rational points of \(X\) do not form a dense set in the Zariski topology on \(X\). [2][3][4]
Refined Bombieri–Lang conjecture: If \(X\) is an algebraic variety of general type defined over a number field \(k\), then there is a dense open subset \(U\) of \(X\) such that for all number field extensions \(k'\) over \(k\), the set of \(k'\)-rational points in \(U\) is finite. [4]
| Author | Date Published | |
| Enrico Bombieri and Serge Lang (independently) | 1980 and 1972 Respectively |
Q103002Erdős–Ulam problem
Q809001 Uniform Boundedness Conjecture for Rational Points
Q809002 Mazur's Conjecture B
[1] 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, 654, American Mathematical Society, pp. 53–73, arXiv:1407.7750
[2] Poonen, Bjorn (2012), Uniform boundedness of rational points and preperiodic points, arXiv:1206.7104
[3] Conceição, 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
[4] Hindry, Marc; Silverman, Joseph H. (2000), "F.5.2. The Bombieri–Lang Conjecture", Diophantine Geometry: An Introduction, Graduate Texts in Mathematics, 201, Springer-Verlag, New York, pp. 479–482, doi:10.1007/978-1-4612-1210-2, ISBN 0-387-98975-7, MR 1745599
Bombieri–Lang Conjecture; general type; Zariski topology