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

Anabelian geometry and descent obstructions on moduli spaces