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

Abstract

Let ${\mathcal{M}}_{g,[n]}$, 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 étale cover ${\mathcal{M}}^{ł}$ of ${\mathcal{M}}_{g,[n]}$, 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 $π_1({\cal M}^ł_{\ol{k}})$ be the geometric algebraic fundamental group of ${\mathcal{M}}^{ł}$ and let ${Out}^*(π_1({\cal M}^ł_{\ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of ${\mathcal{M}}^{ł}$ (this is the "$\ast$-condition" motivating the "almost" above). Let us denote by ${Out}^*_{G_k}(π_1({\cal M}^ł_{\ol{k}}))$ 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 ${\mathcal{M}}^{ł}$ has a trivial automorphisms group. Then, there is a natural isomorphism: $${Aut}_k({\cal M}^ł)\cong{Out}^*_{G_k}(π_1({\cal M}^ł_{\ol{k}})).$$ This partially extends to moduli spaces of curves the anabelian properties proved by Mochizuki for hyperbolic curves over sub-$p$-adic fields.

Superseded by arXiv:2004.04135 and hal-02992317

References in corpus (2)

• Geometry of the Complex of Curves I: Hyperbolicity
• Foundations of Topological Stacks I