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arXiv · 2608.15057

On branched coverings of the projective line over the integers

Ryoji Shimizu·Naganori Yamaguchi
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Abstract

We investigate the \'etale 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~$q$.

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Ryoji Shimizu, Naganori Yamaguchi. 2026-08-15. On branched coverings of the projective line over the integers. https://arxiv.org/abs/2608.15057

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