Syntomic cohomology and -adic motivic cohomology

Abstract

We prove a mixed characteristic analog of the Beilinson-Lichtenbaum conjecture for p-adic motivic cohomology. It gives a description, in the stable range, of p-adic motivic cohomology (defined using algebraic cycles) in terms of differential forms. This generalizes a result of Geisser from small Tate twists to all twists. We use as a critical new ingredient the comparison theorem between syntomic complexes and p-adic nearby cycles proved recently by Colmez and Niziol.