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Profinite complexes of curves, their aut...
Boggi, M....

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Profinite complexes of curves, their automorphisms and anabelian properties of moduli stacks of curves by Boggi, M. ( Author )
Lemonjar Open Access
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06-06-2007
Let Mg,[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 Mł of Mg,[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 $\pi_1({\cal M}^ł_{\ol{k}})$ be the geometric algebraic fundamental group of Mł and let ${Out}^*(\pi_1({\cal M}^ł_{\ol{k}}))$ be the group of its exterior automorphisms which preserve the conjugacy classes of elements corresponding to simple loops around the Deligne-Mumford boundary of Mł (this is the "∗-condition" motivating the "almost" above). Let us denote by ${Out}^*_{G_k}(\pi_1({\cal M}^ł_{\ol{k}}))$ the subgroup consisting of elements which commute with the natural action of the absolute Galois group Gk of k. Let us assume, moreover, that the generic point of the D-M stack Mł has a trivial automorphisms group. Then, there is a natural isomorphism: {Aut}_k({\cal M}^ł)\cong{Out}^*_{G_k}(\pi_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.
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English
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MYR 0.00
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http://arxiv.org/abs/0706.0859
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