# Symmetry Types of Hyperelliptic Riemann Surfaces

The aim of the publication is to present classification of all real structures of hyper-elliptic Riemann surfaces, together with their full group of analytic and antianalytic automorphisms, their topological invariants, their description in terms of polynomials equations and explicit formulae for the corresponding real structures. A real structure τ on a Riemann surface X is an antianalytic involution on X. A real form on X is a class of equivalence of τ modulo conjugation by elements of the full group of analytic and antianalytic authomorphisms. For example, if the curve X is defined by a set of polynomials with real coefficients, then the complex conjugation gives real structure on X. Most complex curves have no real forms but some have more than one. A number of connected components of the fixed point set of the involution τ and connectivity of its complement in X together form an invariant, which characterizes the conjugacy class of τ. The symmetry type of X is the (finite) set of these invariants for all real forms of X. In the first chapter, the authors introduce combinatorial methods used in the classification. The second chapter is devoted to a description of the full group of automorphisms and computation of the number of real forms. The third chapter contains a computation of symmetry types of hyper-elliptic Riemann surfaces, divided into ten different subcases.

**Submitted by Anonymous |

**8 / Sep / 2011