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.
References
De Franchis M., “Un teorema sulle involuzioni irrazionali,” Rend. Circ. Mat. Palermo, 36, 368 (1913).
Howard A. and Sommese A. J., “On the theorem of de Franchis,” Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 10, 429–436 (1983).
Alzati A. and Pirola G. P., “Some remarks on the de Franchis theorem,” Ann. Univ. Ferrara Sez. VII (N. S.), 36, 45–52 (1990).
Tanabe M., “A bound for the theorem of de Franchis,” Proc. Amer. Math. Soc., 127, 2289–2295 (1999).
Tanabe M., “Holomorphic maps of Riemann surfaces and Weierstrass points,” Kodai Math. J., 28, No. 2, 423–429 (2005).
Ito M. and Yamamoto H., “Holomorphic mappings between compact Riemann surfaces,” Proc. Edinburgh Math. Soc., 52, 109–126 (2009).
Mednykh I. A., “Classification up to equivalence of the holomorphic mappings of Riemann surfaces of low genus,” Sib. Math. J., 51, No. 6, 1091–1103 (2010).
Mednykh I. A., “On the sharp upper bound for the number of holomorphic mappings of Riemann surfaces of low genus,” Sib. Math. J., 53, No. 2, 259–273 (2012).
Farkas H. M. and Kra I., Riemann Surfaces, Springer-Verlag, New York (1980) (Graduate Texts Math.; 71).
Farkas H. M., “Unramified double coverings of hyperelliptic surfaces,” J. Anal. Math., 30, 150–155 (1976).
Accola R. D. M., “On lifting the hyperelliptic involution,” Proc. Amer. Math. Soc., 122, No. 2, 341–347 (1994).
Scott P., The Geometries of 3-Manifolds [Russian translation], Mir, Moscow (1986).
Thurston W. P., The Geometry and Topology of Three-Manifolds, Princeton Univ. Math. Dept., Princeton (1978).
Martens H., “A remark on Abel’s Theorem and the mapping of linear series,” Comment. Math. Helv., 52, 557–559 (1977).
Greenleaf N. and May C. L., “Bordered Klein surfaces with maximal symmetry,” Trans. Amer. Math. Soc., 274, No. 1, 265–283 (1982).
Natanzon S. M., “Klein surfaces,” Russian Math. Surveys, 45, No. 6, 53–108 (1990).
Bujalance E., Cirre F. J., Gamboa J. M., and Gromadzki G., Symmetries of Compact Riemann Surfaces, Springer-Verlag, Berlin and Heidelberg (2010).
Shaska T., “Determining the automorphism group of a hyperelliptic curve,” in: Proc. 2003 Intern. Symp. Symbolic and Algebraic Computation (Philadelphia, PA), ACM, New York, 2003, pp. 248–254.
Bolza O., “On binary sextics with linear transformations into themselves,” Amer. J. Math., 10, 47–60 (1888).
Igusa J., “Arithmetic variety of moduli for genus two,” Ann. Math., 72, No. 3, 612–649 (1960).
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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