A Class of Pseudo-Real Riemann Surfaces with Diagonal Automorphism Group

A Riemann surface S having field of moduli R, but not a field of definition, is called pseudo-real. This means that S has anticonformal automorphisms, but none of them is an involution. A Riemann surface is said to be plane if it can be described by a smooth plane model of some degree d >= 4 in P-C(2). We characterize pseudo-real-plane Riemann surfaces S, whose conformal automorphism group Aut(+) (S) is PGL(3 )(C)-conjugate to a finite non-trivial group that leaves invariant infinitely many points of P-C(2). In particular, we show that such pseudo-real-plane Riemann surfaces exist only if Aut(+) (S) is cyclic of even order n dividing the degree d. Explicit families of pseudo-real-plane Riemann surfaces are given for any degree d = 2pm with m > 1 odd, p prime and n = d/p.