ON REAL FORMS OF A BELYI ACTION OF THE ALTERNATING GROUPS

Abstract

In virtue of the Belyi Theorem a complex algebraic curve can be defined over the algebraic numbers if and only if the corresponding Riemann surface can be uniformized by a subgroup of a Fuchsian triangle group. Such surfaces are known as Belyi surfaces. Here we study certain natural actions of the alternating groups ${\mathrm{A}}_n$ on them. We show that they are symmetric and calculate the number of connected components, called ovals, of the corresponding real forms. We show that all symmetries with ovals are conjugate and we calculate the number of purely imaginary real forms both in case of ${\mathrm{A}}_n$ considered here and ${\mathrm{S}}_n$ considered in an earlier paper [2].