Syntomic cohomology and Beilinson's Tate conjecture for $K_2$

Details Syntomic cohomology and Beilinson's Tate conjecture for $K_2$ Syntomic cohomology and Beilinson's Tate conjecture for $K_2$

2009-04-23

arxiv.org/abs/0904.3672v3

Mathematics

Algebraic Geometry

67 pages. The earlier version was entitled `Beilinson's Tate conjecture for $K_2$ and finiteness of torsion zero-cycles on ell

Scientific paper

In this paper, we study an analogue of the Tate conjecture for $K_2$ of U, the complement of split multiplicative fibers in an elliptic surface. A main result is to give an upper bound of the rank of the Galois fixed part of the etale cohomology $H^2(\bar{U},Q_p(2))$. As an application, we give an elliptic K3 surface $X$ over a p-adic field for which the torsion part of the Chow group $CH_0(X)$ of 0-cycles is finite. This would be the first example of a surface $X$ over a p-adic field whose geometric genus is non-zero and for which the torsion part of $CH_0(X)$ is finite.

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