Counting real Galois covers of the projective line
Counting real Galois covers of the projective line
复制标题
计算投影线的真实伽罗瓦覆盖
DOI：
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发表时间：
2005
期刊：
影响因子：
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通讯作者：
A. Cadoret
中科院分区：
文献类型：
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作者：
A. Cadoret
对于给定分支类型的 P1 的伽罗瓦覆盖？本质上，假设给定的单性群 G 和分支轨迹是在 R 上定义的？我们问：有多少个覆盖是在 R 上定义的，有多少不是？ J.-P。 Serre 表明，具有给定分枝类型的所有 Galois 覆盖的数量可以从 G 的特征表中计算出来。我们将 Serre 的计算方法应用于 R 上定义的 Galois 覆盖的更精细的情况，其中存在 P. D?s 和 M. Fried 的群论表征。我们获得了问题的明确答案。作为一种应用，我们展示了未在其模域上定义的新覆盖族，其单向群可以选择任意大。我们还给出了在具有 Q-有理分支轨迹的全实代数数域 Qtr 上定义的伽罗瓦覆盖的例子。
For Galois covers of P1 of a given rami?cation type ? essentially, a given monodromy group G and branch locus, assumed to be de?ned over R ?we ask: How many covers are de?ned over R and how many are not? J.-P. Serre showed that the number of all Galois covers with given rami?cation type can be computed from the character table of G. We adapt Serre?s method of calculation to the more re?ned situation of Galois covers de?ned over R, for which there is a group-theoretic characterization due to P. D?s and M. Fried. We obtain explicit answers to our problem. As an application, we exhibit new families of covers not de?ned over their ?eld of moduli, the monodromy group of which can be chosen arbitrarily large. We also give examples of Galois covers de?ned over the ?eld Qtr of totally real algebraic numbers with Q-rational branch locus.