The Monodromy Group of pq-Covers

Abstract

In this work, we study the monodromy group of covers φ ∘ ψ of curves $\mathcal {Y}\xrightarrow {\quad {\psi }}$ $\mathcal {X} \xrightarrow {\quad \varphi } \mathbb {P}^{1}$, where ψ is a q-fold cyclic étale cover and φ is a totally ramified p-fold cover, with p and q different prime numbers with p odd. We show that the Galois group $\mathcal {G}$ of the Galois closure $\mathcal {Z}$ of φ ∘ ψ is of the form $\mathcal {G} = \mathbb {Z}_{q}^{s} \rtimes \mathcal {U}$, where 0 ≤ s ≤ p − 1 and $\mathcal {U}$ is a simple transitive permutation group of degree p. Since the simple transitive permutation group of prime degree p are known, and we construct examples of such covers with these Galois groups, the result is very different from the previously known case when the cover φ was assumed to be cyclic, in which case the Galois group is of the form $\mathcal {G} = \mathbb {Z}_{q}^{s} \rtimes \mathbb {Z}_{p}$. Furthermore, we are able to characterize the subgroups ${\mathscr{H}}$ and $\mathcal {N}$ of $\mathcal {G}$ such that $\mathcal {Y} = \mathcal {Z}/\mathcal {N}$ and $X = \mathcal {Z}/{\mathscr{H}}$.