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

Denote the set of all holomorphic mappings of a genus 3 Riemann surface S (3) onto a genus 2 Riemann surface S (2) by Hol(S (3), S (2)). Call two mappings f and g in Hol(S (3), S (2)) equivalent whenever there exist conformal automorphisms alpha and beta of S (3) and S (2) respectively with f au broken vertical bar alpha = beta au broken vertical bar g. It is known that Hol(S (3), S (2)) always consists of at most two equivalence classes. We obtain the following results: If Hol(S (3), S (2)) consists of two equivalence classes then both S (3) and S (2) can be defined by real algebraic equations; furthermore, for every pair of inequivalent mappings f and g in Hol(S (3), S (2)) there exist anticonformal automorphisms alpha- and beta- with f au broken vertical bar alpha- = beta- au broken vertical bar g. Up to conformal equivalence, there exist exactly three pairs of Riemann surfaces (S (3), S (2)) such that Hol(S (3), S (2)) consists of two equivalence classes.