tailieunhanh - Đề tài " Deligne’s integrality theorem in unequal characteristic and rational points over finite fields "

A Pierre Deligne, a l ’occasion de son 60-i`me anniversaire, ` e en t´moignage de profonde admiration e Abstract If V is a smooth projective variety defined over a local field K with fi¯ nite residue field, so that its ´tale cohomology over the algebraic closure K is e supported in codimension 1, then the mod p reduction of a projective regular model carries a rational point. As a consequence, if the Chow group of 0-cycles of V over a large algebraically closed field is trivial, then the mod p reduction of a projective regular model carries a rational. | Annals of Mathematics Deligne s integrality theorem in unequal characteristic and rational points over finite fields By H el ene Esnault Annals of Mathematics 164 2006 715 730 Deligne s integrality theorem in unequal characteristic and rational points over finite fields By Helene Esnault A Pierre Deligne a l occasion de son 60-ieme anniversaire en témoignage de profonde admiration Abstract If V is a smooth projective variety defined over a local field K with finite residue field so that its etale cohomology over the algebraic closure K is supported in codimension 1 then the mod p reduction of a projective regular model carries a rational point. As a consequence if the Chow group of 0-cycles of V over a large algebraically closed field is trivial then the mod p reduction of a projective regular model carries a rational point. 1. Introduction If Y is a smooth geometrically irreducible projective variety over a finite field k we singled out in 10 a motivic condition forcing the existence of a rational point. Indeed if the Chow group of 0-cycles of Y fulfills base change CH0 Y Xk k Y Z Q Q then the number of rational points of Y is congruent to 1 modulo k . In general it is hard to control the Chow group of 0-cycles but if Y is rationally connected for example if Y is a Fano variety then the base change condition is fulfilled and thus rationally connected varieties over a finite field have a rational point. Recall the Leitfaden of the proof. By S. Bloch s decomposition of the diagonal acting on cohomology as a correspondence 2 Appendix to Lecture 1 the base change condition implies that etale cohomology Hm Y Q is supported in codimension 1 for all m 1 that is that etale cohomology for m 1 lives in coniveau 1. Here t is a prime number not dividing k . On the other hand by Deligne s integrality theorem 6 Cor. the coniveau condition implies that the eigenvalues of the geometric Frobenius acting on Hm Y O are divisible by k as algebraic Partially supported by the .