论文标题
带有交流和对称单片的投影线的驯服盖
Tamely Ramified Covers of the Projective Line with Alternating and Symmetric Monodromy
论文作者
论文摘要
让$ k $是特征$ p $的代数封闭的字段，让$ x $ the juptive line the投影线上$ k $，删除了三分。我们调查了哪些有限的$ g $可能会出现，这是因为$ x $的有限étale封面组在三个删除点上被驯服的$ x $。这提供了有关投影线的驯服基本组的新信息。特别是，我们表明，对于每个Prime $ p \ ge 5 $，都有一个卑鄙的封面，带有sonodromy the Symmetric Group $ s_n $或交替的组$ a_n $，用于无限的许多$ n $。这些封面来自带有$ psl_2（\ mathbb {f} _ \ ell）$的椭圆形曲线的模量空间，并且分析使用了Bourgain，Gamburd和Sarnak的工作，并适应了Meiri和Puder的工作，大约是Mark Friples Modulo $ \ Ell $。
Let $k$ be an algebraically closed field of characteristic $p$ and let $X$ the projective line over $k$ with three points removed. We investigate which finite groups $G$ can arise as the monodromy group of finite étale covers of $X$ that are tamely ramified over the three removed points. This provides new information about the tame fundamental group of the projective line. In particular, we show that for each prime $p\ge 5$, there are families of tamely ramified covers with monodromy the symmetric group $S_n$ or alternating group $A_n$ for infinitely many $n$. These covers come from the moduli spaces of elliptic curves with $PSL_2(\mathbb{F}_\ell)$-structure, and the analysis uses work of Bourgain, Gamburd, and Sarnak, and adapts work of Meiri and Puder, about Markoff triples modulo $\ell$.
