The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces

Abstract

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S₃ onto a genus 2 Riemann surface S₂ by Hol(S₃, S₂). Call two mappings f and g in Hol(S₃, S₂) equivalent whenever there exist conformal automorphisms α and β of S₃ and S₂ respectively with f ◦ α = β ◦ g. It is known that Hol(S₃, S₂) always consists of at most two equivalence classes.

We obtain the following results: If Hol(S₃, S₂) consists of two equivalence classes then both S₃ and S₂ can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S₃, S₂) there exist anticonformal automorphisms α− and β− with f ◦ α− = β− ◦ g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S₃, S₂) such that Hol(S₃, S₂) consists of two equivalence classes.