A pp-adic Regulator Map and Finiteness Results for Arithmetic Schemes

  • S. Saito

  • K. Sato


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