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

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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 fα = βg. 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 fα− = 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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Correspondence to A. D. Mednykh.

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Original Russian Text Copyright © 2016 Mednykh A.D. and Mednykh I.A.

The authors were supported by the Russian Foundation for Basic Research (Grants 15–01–07906; 16–31–00138) and the Government of the Russian Federation for the State Maintenance Program for the Leading Scientific Schools at Siberian Federal University (Grant 14.Y26.31.0006).

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Mednykh, A.D., Mednykh, I.A. The equivalence classes of holomorphic mappings of genus 3 Riemann surfaces onto genus 2 Riemann surfaces. Sib Math J 57, 1055–1065 (2016). https://doi.org/10.1134/S0037446616060124

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  • DOI: https://doi.org/10.1134/S0037446616060124

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