  We study the section conjecture of anabelian geometry and the sufficiency of
the finite descent obstruction to the Hasse principle for the moduli spaces of
principally polarized abelian varieties and of curves over number fields. For
the former we show that the section conjecture fails and the finite descent
obstruction holds for a general class of adelic points, assuming several
well-known conjectures. This is done by relating the problem to a local-global
principle for Galois representations. For the latter, we prove some partial
results that indicate that the finite descent obstruction suffices. We also
show how this sufficiency implies the same for all hyperbolic curves.
