P-Groups acting on Riemann surfaces

Abstract

Let p be a prime integer and let r≥3 be an integer so that p≥5r−7. We show that a closed Riemann surface S of genus g≥2 has at most one p-group H of conformal automorphisms so that S/H has genus zero and exactly r cone points. This, in particular, asserts that, for r=3 and p≥11, the minimal field of definition of S coincides with that of (S,H). Another application of this fact, for the case that S is pseudo-real, is that Aut(S)/H must be either trivial or a cyclic group and that r is necessarily even. This generalizes a result due to Bujalance–Costa for the case of pseudo-real cyclic p-gonal Riemann surfaces.