Rigidity and Real Residue Class Fields

Introduction and acknowledgements: Consider a cover φ : X → Px of the Riemann sphere (uniformized by x) by a projective nonsingular curve X with r > 2 branch points. Assume that both the curves and the map are defined over Q. Generalizing Serre [Se] we consider not necessarily Galois covers with any number r of branch points (not necessarily in R). We show how to compute the action of complex conjugation on the fiber in X over a real value of x0 ∈ Px. It is an “exceptional cover” for which all of the residue class fields above x0 are real. The group of the Galois closure of such an exceptional cover must be a quotient of a universal group generated by elements of order 2.