Syntomic cohomology and $p$-adic motivic cohomology

TL;DR

This paper proves a mixed characteristic analog of the Beilinson-Lichtenbaum Conjecture, linking p-adic motivic cohomology with differential forms, and extends previous results to all Tate twists using syntomic and p-adic cycle comparisons.

Contribution

It establishes a broad generalization of the Beilinson-Lichtenbaum Conjecture for p-adic motivic cohomology in mixed characteristic, utilizing syntomic complexes and recent comparison theorems.

Findings

Description of p-adic motivic cohomology via differential forms

Extension of Geisser's results to all Tate twists

Use of syntomic and p-adic cycle comparison theorems

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 and uses as a critical new ingredient the comparison theorem between syntomic complexes and p-adic nearby cycles proved recently in Colmez-Niziol.