Real forms of a Riemann surface of even genus

Natanzon proved that a Riemann surface of genus has at most conjugacy classes of symmetries, and this bound is attained for infinitely many genera . The aim of this note is to prove that a Riemann surface of even genus has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus has at most four real forms which are not birationally equivalent.