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

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

2007-06-06

arxiv.org/abs/0706.0859v2

Mathematics

Algebraic Geometry

This paper has been withdrawn because of a flaw in the paper "Profinite Teichm\"uller theory" of the first author, on which th

Scientific paper

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.

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