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).