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

TL;DR

This paper demonstrates that certain moduli stacks of curves over sub-p-adic fields exhibit anabelian properties, linking their automorphisms to Galois actions on their fundamental groups, extending Mochizuki's results.

Contribution

It establishes an almost anabelian correspondence for finite covers of moduli stacks of curves, generalizing Mochizuki's hyperbolic curve results to moduli spaces.

Findings

Isomorphism between automorphisms of the moduli stack and Galois-equivariant outer automorphisms of the fundamental group

Extension of Mochizuki's anabelian properties to moduli spaces of curves

Conditions under which the automorphism group of the stack is determined by fundamental group automorphisms

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.