$p$-adic motivic cohomology in arithmetic

Abstract

We will discuss recent progress by many people in the program of representing motivic cohomology with torsion coefficients of arithmetic schemes by various arithmetic p-adic cohomologies: étale, logarithmic de Rham–Witt, and syntomic. To illustrate possible applications in arithmetic geometry we will sketch proofs of the absolute purity conjecture in étale cohomology and comparison theorems of p-adic Hodge theory.