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摘要翻译：
设${\cal M}_{g,[n]}$,对于$2g-2+n>0$,是$g$属光滑曲线的D-M模叠加，其标记为$n$无序不同点。本文的主要结果是：定义在子P$-adic域$K$上的有限连通etale覆盖${Cal M}_{g,[n]}$在Grothendieck对曲线及其模空间的推测意义上是“几乎”anabelian的。精确的结果如下。设$\pi_1({\cal M}^\l_{\ol{k}}）$为${\cal M}^\l$的几何代数基本群，设${Out}^*(\pi_1({\cal M}^\l_{\ol{k}}）)$为其外部自同构的群，这些自同构保持与围绕${\cal M}^\l$的Deligne-Mumford边界的简单环对应的元素的共轭类（这是引发上述“几乎”的“$\ast$-条件”）。让我们用${Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}}））表示由元素组成的子群，这些元素与$k$的绝对伽罗瓦群$G_k$的自然作用交换。此外，让我们假设D-M堆栈的泛型点${\cal M}^\l$具有一个平凡的自同构群。然后，存在一个自然同构:$${Aut}_k({\cal M}^\l)\cong{Out}^*_{G_k}(\pi_1({\cal M}^\l_{\ol{k}}））。$$这将Mochizuki对次$p$-adic域上的双曲曲线证明的anabelian性质部分地推广到曲线的模空间。
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英文标题：
《Profinite complexes of curves, their automorphisms and anabelian
properties of moduli stacks of curves》
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作者：
M. Boggi, P. Lochak
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最新提交年份：
2011
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分类信息：
一级分类：Mathematics 数学
二级分类：Algebraic Geometry 代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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一级分类：Mathematics 数学
二级分类：Number Theory 数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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英文摘要：
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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PDF链接：
https://arxiv.org/pdf/0706.0859
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