Bombieri–Lang Conjecture

The Bombieri–Lang conjecture predicts that Diophantine behaviour is controlled by geometry: if $X$ is a smooth projective variety of general type over a number field $K$, then $X(K)$ is not Zariski dense, so the rational points lie in a proper closed subvariety. Lang formulated the general-type principle in the early 1970s [Lang1971RationalPoints]; the conjecture then developed through his exchanges with Bombieri, with no single canonical proposal date, and Lang later placed it in his hyperbolic-Diophantine framework [Lang1986Hyperbolic].

In dimension one the conjecture is the Mordell conjecture, proved by Faltings: a curve of genus at least two over a number field has only finitely many rational points. In higher dimensions important special families are known, and the geometric (function-field) analogue has seen notable recent progress [Xie2026BombieriLang].

The uniform and Lang–Vojta strengthenings are stronger still, but even the weak form stated here is unproved: no argument covers all higher-dimensional varieties of general type over number fields, and the conjecture remains open.