Note on linear relations in Galois cohomology and etale K-theory of curves

In this paper, we investigate a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. In [G. Banaszak and P. Krason, On a local to global principle in etale K-groups of curves, J. K-Theory Appl. Algebra Geom. Topol. 12 (2013) 183-201], G. Banaszak and the author obtained the sufficient condition for the validity of the local to global principle for etale K-theory of a curve. This condition in fact has been established by means of an analysis of the corresponding problem in the Galois cohomology. We show that in some cases, this result is the best possible i.e. if this condition does not hold we obtain counterexamples. We also give some examples of curves and their Jacobians. Finally, we prove the dynamical version of the local to global principle for etale K-theory of a curve. The dynamical local to global principle for the groups of Mordell-Weil type has recently been considered by S. Baranczuk in [S. Baranczuk, On a dynamical local-global principle in Mordell-Weil type groups, Expo. Math. 35(2) (2017) 206-211]. We show that all our results remain valid for Quillen K-theory of X if the Bass and Quillen-Lichtenbaum conjectures hold true for X.

Automated summary

This paper investigates a local to global principle for Galois cohomology of number fields with coefficients in the Tate module of an abelian variety. It discusses sufficient conditions, counterexamples in some cases, examples of curves and their Jacobians, and proves the dynamical version of the local to global principle for etale K-theory of a curve. Furthermore, it shows that all results remain valid for Quillen K-theory of X if certain conjectures hold true.