Université Paris 6 (IMJ-PRG)

l-adic cohomology was built to provide an etale cohomology with coefficients in a field of characteristic 0. This, via the Grothendieck trace formula, gives  a cohomological interpretation of L-functions - a fundamental tool in Deligne's theory of weights developed in Weil II. Instead of l-adic coefficients one can consider coefficients in ultra products of finite fields. I will state the fundamental theorem of Weil II for curves in this setting and explain briefly what are the difficulties to overcome to adjust Deligne's proof. I will then discuss how this ultra product variant of Weil II allows to extend to arbitrary coefficients  previous results of Gabber and Hui, Tamagawa and myself for constant $\mathbb{Z}_\ell$-coefficients.  For instance,  it implies that, in an $E$-rational compatible system of smooth $\overline{\mathbb{Q}}_\ell$-sheaves all what is true for $\overline{\mathbb{Q}}_\ell$-coefficients (semi simplicity, irreducibility, invariant dimensions etc) is true for $\overline{\mathbb{F}}_\ell$-coefficients provided $\ell$ is large enough or that the $\overline{\mathbb{Z}}_\ell$-models are unique with torsion-free cohomology provided $\ell$ is large enough.