A $p$-adic Regulator Map and Finiteness Results for Arithmetic Schemes

Abstract

A main theme of the paper is a conjecture of Bloch-Kato on the image of $p$-adic regulator maps for a proper smooth variety $X$ over an algebraic number field $k$. The conjecture for a regulator map of particular degree and weight is related to finiteness of two arithmetic objects: One is the $p$-primary torsion part of the Chow group in codimension 2 of $X$. Another is an unramified cohomology group of $X$. As an application, for a regular model $\mathcal{X}$ of $X$ over the integer ring of $k$, we prove an injectivity result on the torsion cycle class map of codimension 2 with values in a new $p$-adic cohomology of $\mathcal{X}$ introduced by the second author, which is a candidate of the conjectural étale motivic cohomology with finite coefficients of Beilinson-Lichtenbaum.