Bombieri–Lang conjecture

The weak Bombieri–Lang conjecture for surfaces states that if ${\displaystyle X}$ is a smooth surface of general type defined over a number field ${\displaystyle k}$, then the ${\displaystyle k}$-rational points of ${\displaystyle X}$ do not form a dense set in the Zariski topology on ${\displaystyle X}$.[1]

The general form of the Bombieri–Lang conjecture states that if ${\displaystyle X}$ is a positive-dimensional algebraic variety of general type defined over a number field ${\displaystyle k}$, then the ${\displaystyle k}$-rational points of ${\displaystyle X}$ do not form a dense set in the Zariski topology.[2][3][4]

The refined form of the Bombieri–Lang conjecture states that if ${\displaystyle X}$ is an algebraic variety of general type defined over a number field ${\displaystyle k}$, then there is a dense open subset ${\displaystyle U}$ of ${\displaystyle X}$ such that for all number field extensions ${\displaystyle k'}$ over ${\displaystyle k}$, the set of ${\displaystyle k'}$-rational points in ${\displaystyle U}$ is finite.[4]

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]

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]

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 ${\displaystyle B_{g,d}}$ depending only on ${\displaystyle g}$ and ${\displaystyle d}$ such that the number of rational points of any genus ${\displaystyle g}$ curve ${\displaystyle X}$ over any degree ${\displaystyle d}$ number field is at most ${\displaystyle B_{g,d}}$.[2][3]