Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves

Abstract

Let ${\cal M}_{g,[n]}$, for $2g-2+n>0$, be the D-M moduli stack of smooth curves of genus $g$ labeled by $n$ unordered distinct points. The main result of the paper is that a finite, connected \'etale cover ${\cal M}^\l$ of ${\cal M}_{g,[n]}$, defined over a sub-$p$-adic field $k$, is "almost" anabelian in the sense conjectured by Grothendieck for curves and their moduli spaces. The precise result is the following. Let $\pi_1({\cal M}^\l_{\ol{k}})$ be the geometric algebraic fundamental group of ${\cal M}^\l$ and let ${Out}^*(\pi_1({\cal M}^\l_{\ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${\cal M}^\l$ (this is the "$\ast$-condition" motivating the "almost" above). Let us denote by ${Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group $G_k$ of $k$. Let us assume, moreover, that the generic point of the D-M stack ${\cal M}^\l$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({\cal M}^\l)\cong{Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}})).$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.