Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Bloch, Spencer; Esnault, Hélène; Levine, Marc

Decomposition of the diagonal and eigenvalues of Frobenius for Fano hypersurfaces

Authors

Bloch, Spencer

Esnault, Hélène

Levine, Marc

Description

Let $X\subset ¶^n$ 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.
16 pages