Abstract

In this work, we study the monodromy group of covers ? ? ? of curves Y??X???1, where ? is a q-fold cyclic étale cover and ? is a totally ramified p-fold cover, with p and q different prime numbers with p odd. We show that the Galois group G of the Galois closure Z of ? ? ? is of the form G=?qs?U, where 0 ? s ? p ? 1 and 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 ? was assumed to be cyclic, in which case the Galois group is of the form G=?qs??p. Furthermore, we are able to characterize the subgroups H and N of G such that Y=Z/N and X=Z/H. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2022.