A p-adic regulator map and finiteness results for arithmetic schemes

Details A p-adic regulator map and finiteness results for arithmetic schemes A p-adic regulator map and finiteness results for arithmetic schemes

2006-12-04

arxiv.org/abs/math/0612081v5

Mathematics

Algebraic Geometry

53 pages. The title and the notation has been changed, and Appendix B has been added

Scientific paper

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 ${\mathscr X}$ of $X$ over the integer ring of $k$, we show an injectivity result on torsion of a cycle class map from the Chow group in codimension 2 of ${\mathscr X}$ to a new $p$-adic cohomology of ${\mathscr X}$ introduced by the second author, which is a candidate of the conjectural \'etale motivic cohomology with finite coefficients of Beilinson-Lichtenbaum.

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