Syntomic cohomology and Beilinson’s Tate conjecture for 𝐾₂

Abstract

We study Beilinson’s Tate conjecture for K 2 K_2 using the theory of syntomic cohomology. As an application, we construct integral indecomposable elements of K 1 K_1 of elliptic surfaces. Moreover, we give the first example of a surface X X with p g ≠ 0 p_g\ne 0 over a p p -adic field such that the torsion of C H 0 ( X ) \mathrm {CH}_0(X) is finite.