(in terms of their degree), things can be more complicated. For example, the Hasse principle fails for the smooth cubic surface in over by Ian Cassels and Richard Guy. Jean-Louis Colliot-Thélène has conjectured that the Brauer–Manin obstruction is the only obstruction to the Hasse principle for cubic surfaces. More generally, that should hold for every rationally connected variety over a number field. In some cases, it is known that has "many" rational points whenever it has one. For example, extending work of Beniamino Segre and Yuri Manin, János Kollár showed: for a cubic hypersurface of dimension at least 2 over a perfect field with not a cone, is unirational over if it has a -rational point. (In particular, for infinite, unirationality implies that the set of -rational points is Zariski dense in .) The Manin conjecture is a more precise statement that would describe the asymptotics of the number of rational points of bounded height on a Fano variety. Counting points over finite fields A variety over a finite field has only finitely many -rational points. The Weil conjectures, proved by André Weil in dimension 1 and by Pierre Deligne in any dimension, give strong estimates for the number of -points in terms of the Betti numbers of . For example, if is a smooth projective curve of genus over a field of order (a prime power), then For a smooth hypersurface of degree in over a field of order , Deligne's theorem gives the bound: There are also significant results about when a projective variety over a finite field has at least one -rational point. For example, the Chevalley–Warning theorem implies that any hypersurface of degree in over a finite field has a -rational point if . For smooth , this also follows from Hélène Esnault's theorem that every smooth projective rationally chain connected variety, for example every Fano variety, over a finite field has a -rational point. See also Arithmetic dynamics Birational geometry Functor represented by a scheme Notes References External links Diophantine geometry