The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces |
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| Authors: | A D Mednykh I A Mednykh |
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| Institution: | 1.Sobolev Institute of Mathematics, Novosibirsk State University, Novosibirsk Siberian Federal University,Krasnoyarsk,Russia |
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| Abstract: | 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 α and β of S 3 and S 2 respectively with f ? α = β ? 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 α? and β? with f ? α? = β? ? 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. |
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