The weak BombieriāLang 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 BombieriāLang 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 BombieriāLang 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 BombieriāLang 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 BombieriāLang conjecture.[4]
Generalizations and implications
The BombieriāLang 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 BombieriāLang conjecture would resolve the
ErdÅsāUlam 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]
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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. 53ā73,
arXiv:1407.7750