least 2. Lang also made finer conjectures relating finiteness of rational points to Kobayashi hyperbolicity. For example, the Bombieri–Lang conjecture predicts that a smooth hypersurface of degree in projective space over a number field does not have Zariski dense rational points if . Not much is known about that case. The strongest known result on the Bombieri–Lang conjecture is Faltings's theorem on subvarieties of abelian varieties (generalizing the case of curves). Namely, if is a subvariety of an abelian variety over a number field , then all -rational points of are contained in a finite union of translates of abelian subvarieties contained in . (So if contains no translated abelian subvarieties of positive dimension, then is finite.) Varieties with many rational points In the opposite direction, a variety over a number field is said to have potentially dense rational points if there is a finite extension field of such that the -rational points of are Zariski dense in . Frédéric Campana conjectured that a variety is potentially dense if and only if it has no rational fibration over a positive-dimensional orbifold of general type. A known case is that every cubic surface in over a number field has potentially dense rational points, because (more strongly) it becomes rational over some finite extension of (unless it is the cone over a plane cubic curve). Campana's conjecture would also imply that a K3 surface (such as a smooth quartic surface in ) over a number field has potentially dense rational points. That is known only in special cases, for example if has an elliptic fibration. One may ask when a variety has a rational point without extending the base field. In the case of a hypersurface of degree in over a number field, there are good results when is much smaller than , often based on the Hardy–Littlewood circle method. For example, the Hasse–Minkowski theorem says that the Hasse principle holds for quadric hypersurfaces over a number field (the case ). Christopher Hooley proved the Hasse principle for smooth cubic hypersurfaces in over when . In higher dimensions, even more is true: every smooth cubic in over has a rational point when , by Roger Heath-Brown. More generally, Birch's theorem says that for any odd positive integer , there is an integer such that for all , every hypersurface of degree in over has a rational point. For hypersurfaces of smaller dimension