2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/63157We prove a Kodaira-type vanishing theorem for the Witt vector sheaf on a Fano variety over a perfect field of characteristic p. As a corollary, we deduce that the number of rational points on a Fano variety over a finite field with q=p^n elements is congruent to 1 mod q. This refines a result of Esnault which say that the number is congruent to 1 mod p.A correct proof of the original vanishing theorem (mod torsion) has been given in the meanwhile by Esnault. This note strengthens that result slightlyAlgebraic GeometryNumber Theory14G05; 11G25text