Real forms of a Riemann surface of even genus

Abstract

Natanzon proved that a Riemann surface X X of genus g ≥ 2 g \ge 2 has at most 2 ( g + 1 ) 2(\sqrt g+1) conjugacy classes of symmetries, and this bound is attained for infinitely many genera g g . The aim of this note is to prove that a Riemann surface of even genus g g has at most four conjugacy classes of symmetries and this bound is attained for an arbitrary even g g as well. An equivalent formulation in terms of algebraic curves is that a complex curve of an even genus g g has at most four real forms which are not birationally equivalent.