Appears in collection : Conference on Arithmetic Geometry in honor of Luc Illusie

The absolute prismatic site of a p-adic formal scheme carries and organizes interesting arithmetic and geometric information attached to the formal scheme. In this talk, after recalling the definition of this site, I will discuss a stacky approach to absolute prismatic cohomology and its concomitant structures (joint with Lurie, and due independently to Drinfeld). As a geometric application, I'll explain Drinfeld's refinement of the Deligne-Illusie theorem on Hodge-to-de Rham degeneration. On the arithmetic side, I'll describe a new classification of crystalline Galois representations of the Galois group of a local field in terms of F-crystals on the site (joint with Scholze).