Anabelian geometry and descent obstructions on moduli spaces

TL;DR

This paper investigates the section conjecture and descent obstructions on moduli spaces, revealing failures and partial successes in their applicability to rational points over number fields.

Contribution

It demonstrates the failure of the section conjecture for certain moduli spaces and provides partial results supporting the sufficiency of descent obstructions for hyperbolic curves.

Findings

Section conjecture fails for moduli spaces of abelian varieties.

Finite descent obstruction holds for a broad class of adelic points under certain conjectures.

Partial results suggest descent obstruction sufficiency for hyperbolic curves.

Abstract

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.