关于整数射影直线的分支覆盖
On branched coverings of the projective line over the integers
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中文总结 AI 辅助
该研究探讨整数射影直线水平除子补集的平展基本群，证明其无非平凡有限可解商群的充要条件，还给出特定条件下不存在同构于$\boldsymbol{\text{PSL}}_{2}(q)$商群的结论。
中文摘要 AI 辅助
我们研究了$\boldsymbol{\text{Spec}\thinspace \boldsymbol{\text{Z}}}$上射影直线$\boldsymbol{\text{P}}^{1}_{\boldsymbol{\text{Z}}}$的水平除子补集的平展基本群。我们证明：当且仅当该除子在素数2处具有正常交叉时，该群无非平凡有限可解商群。此外，若该除子在素数2处具有正常交叉，且要么有三个不可约分量，要么在素数3处具有正常交叉，则对于某些素数幂$q$，不存在同构于$\boldsymbol{\text{PSL}}_{2}(q)$的商群。
英文摘要
We investigate the étale fundamental group of the complement of a horizontal divisor on $\mathbb{P}^{1}_{\mathbb{Z}}$. We prove that this group has no nontrivial finite solvable quotient if and only if the divisor is normal crossings at the prime~$2$. Moreover, if the divisor is normal crossings at the prime~$2$ and either has three irreducible components or is normal crossings at the prime~$3$, we show that no quotient isomorphic to $\mathrm{PSL}_{2}(q)$ can occur for certain prime powers~$q$.
