Bombieri–Lang conjecture

In arithmetic geometry, the Bombieri–Lang conjecture is an unproved conjecture about the rational points on an algebraic surface, named after Enrico Bombieri and Serge Lang. It states that, if $X$ is a smooth surface of general type defined over the rational numbers, then the rational points of $X$ do not form a dense set in the Zariski topology on $X$ .

It 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. If true, the Bombieri–Lang conjecture would also 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.

The conjecture can also be extended to algebraic varieties of general type and higher dimension, and to number fields other than the rational numbers.

References

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
Tao, Terence (December 20, 2014), "The Erdos-Ulam problem, varieties of general type, and the Bombieri-Lang conjecture", What's new

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