Regular dessins with moduli fields of the form Q(ζp, \sqrt[p]{q})

Abstract

Gareth Jones asked during the 2014 SIGMAP conference for examples of regular dessins with nonabelian fields of moduli. In this paper, we first construct dessins whose moduli fields are nonabelian Galois extensions of the form Q(ζp, \sqrt[p]{q}), where p is an odd prime and ζp is a pth root of unity and q ∈ Q is not a pth power, and we then show that their regular closures have the same moduli fields. Finally, in the special case p = q = 3 we give another example of a regular dessin with moduli field Q(ζ3, ∛(3)) of degree 219 · 34 and genus 14155777.