%0 Journal Article
%T Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces
%A Spencer Bloch
%A H¨¦l¨¨ne Esnault
%A Marc Levine
%J Mathematics
%D 2003
%I arXiv
%X Let $X\subset \P^n$ be a possibly singular hypersurface of degree $d\le n$, defined over a finite field $k$. We show that the diagonal, suitably interpreted, is decomposable. This gives a proof that the eigenvalues of the Frobenius action on its $\ell$-adic cohomology $H^i(\bar{X}, \Q_\ell)$, for $\ell \neq {\rm char}(k)$, are divisible by $q$, without using the result on the existence of rational points by Ax and Katz.
%U http://arxiv.org/abs/math/0302109v1