Bombieri–Lang conjecture
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. Statement. The weak Bombieri–Lang conjecture for surfaces states that if formula_1 is a smooth surface of general type defined over a number field formula_2, then the formula_2-rational points of formula_1 do not form a dense set in the Zariski topology on formula_1. The general form of the Bombieri–Lang conjecture states that if formula_1 is a positive-dimensional algebraic variety  of general type defined over a number field formula_2, then the formula_2-rational points of formula_1 do not form a dense set in the Zariski topology. The refined form of the Bombieri–Lang conjecture states that if formula_1 is an algebraic variety of general type defined over a number field formula_2, then there is a dense open subset formula_12 of formula_1 such that for all number field extensions formula_14 over formula_2, the set of formula_14-rational points in formula_12 is finite. 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. 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, formulated in the "refined form" of the Bombieri–Lang conjecture. 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. 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. 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 formula_18 depending only on formula_19 and formula_20 such that the number of rational points of any genus formula_19 curve formula_1 over any degree formula_20 number field is at most formula_18.