On branched coverings of the projective line over the integers

Abstract

We investigate the étale fundamental group of the complement of a horizontal divisor on ${\mathbb{P}}_{\mathbb{Z}}^1$. 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~$q$.