On Singerman symmetries of a class of Belyi Riemann surfaces

By virtue of the Belyi Theorem an 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 and an important class of them consists of Riemann surfaces having the so-called large group of automorphisms. Necessary and sufficient algebraic conditions for these surfaces to be symmetric were found by Singerman in the middle of the seventies and, by a recent result of Köck and Singerman, the algebraic numbers above can be chosen to be real if and only if the respective surface is symmetric. The aim of this paper is to give, in similar terms, the formulas for the number of ovals of the corresponding symmetries, which we refer to as the Singerman symmetries.