The Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one

We study the Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one. A consequence of the main results is that the isomorphism class of a certain moduli space of hyperbolic curves of genus one over a sub-$p$-adic field is completely determined by the isomorphism class of the etale fundamental group of the moduli space over the absolute Galois group of the sub-$p$-adic field. We also prove related results in absolute anabelian geometry.