Rational Singularities and Rational Points

الأصل · EN

If X is a projective, geometrically irreducible variety defined over a finite field, such that it is smooth and its Chow group of 0-cycles fulfills base change, i.e. CH₀(X×(X))=, then the second author's theorem asserts that its number of rational points satisfies |X()| ≡ 1 modulo q. If X is not smooth, this is no longer true. Indeed J. Kollár constructed an example of a rationally connected surface over without any rational points. Based on the work by Berthelot-Bloch and the second author computing the slope <1 piece of rigid cohomology, we define a notion of Witt-rational singularities in characteristic p>0. The theorem is then that if X/ is a projective, geometrically irreducible variety, such that it has Witt-rational singularities and its Chow group of 0-cycles fulfills base change, then |X()| ≡ 1 modulo q.