Title: Congruences for the number of rational points of varieties
defined over finite fields.
Speaker: Helene Esnault
Abstract: The classical theorem of Chevalley-Warning asserts that if
$X\subset \P^n$ is a hypersurface of degree $d\le n$ defined over a finite field
$\F_q$, then $X$ carries a rational point. Ax and Katz, in a celebrated theorem,
showed later on that the sharp formulation of this theorem says that if $X$ is
defined by equations of degrees $d_1\ge \ldots \ge d_r$, and $\kappa$ is a
natural number fulfilling $d_1\kappa +d_2+\ldots +d_r \le n$, then the number of
rational points of $X$ is the same as the number of rational points of $\P^n$
modulo $q^\kappa$, that is $1+q+\ldots +q^{\kappa -1}$.
Smooth hypersurfaces of degree $d\le n$ are precisely Fano hypersurfaces. We
will show how Bloch-Beilinson motivic philosophy, transposed to varieties over
finite fields, allows to prove that all smooth Fano varieties carry a rational
point over a finite field (Lang-Manin conjecture), thereby generalizing
Chevalley-Warning-Ax-Katz' theorem. However, lacking a good motivic
understanding in the singular case, we are not able to handle the case of
singular Fano varieties. Yet we can nevertheless explain cohomologically
Ax-Katz' theorem showing that the eigenvalues of Frobenius acting on \'etale
cohomology have the required divisibility property (joint with N. Katz).