(This is a reading seminar presented by two graduate students.) In this reading seminar, we will present an introduction to étale cohomology and the Weil conjectures based on Milne's lecture notes. Beginning with the étale topology and the theory of sheaves on étale sites, we will develop the basic constructions and properties of étale cohomology. We will then explain several fundamental results of the theory including the purity theorem, the base change theorems, and the comparison theorem. Finally, we will discuss how these results culminate in the proof of the Weil conjectures.

Zeta and L-functions have been classically associated, first to schemes of finite type over Z, and then to the l-adic cohomology of smooth projective varieties over a global field. In the latter definition, independence from l and actual existence remain partly conjectural in characteristic 0. In these lectures, I will explain how to associate unconditionally an L-function to objects in triangulated categories of motives. For the motive of a smooth projective variety this definition differs from the one above in general, but only up to a finite number of Euler factors. This might be useful to tackle the Beilinson conjectures.