Abstract
We investigate smooth one-parameter families of complex tori over the Riemann sphere. The main problem is to describe such families in terms of projections of their branch-points. Earlier we investigated the problem for the case where, for every torus of the family, there is only one point lying over infinity. Here we consider the general case. We show that the uniformizing functions satisfy a partial differential equation and derive a system of differential equations for their critical points, poles, and moduli of tori. Based on the system we suggest an approximate method allowing to find an elliptic function uniformizing a given genus one ramified covering of the Riemann sphere.
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Notes
In Example 1, giving approximate values of the parameters, we restrict ourselves by \(8\) digits after the decimal point. In Example 2, we will give the values with accuracy \(10^{-10}\).
REFERENCES
D. Dautova, S. Nasyrov, and M. Vuorinen, ‘‘Conformal module of the exterior of two rectilinear slits,’’ Comput. Methods Funct. Theory (2020); arXiv:1908.02459. https://doi.org/10.1007/s40315-020-00315-y
H. M. Farkas and I. Kra, Riemann Surfaces (Springer, New York, 1980).
A. Hurwitz, ‘‘Über Riemannsche Flächen mit gegeben Verzweigungpunkten,’’ Math. Ann. 39, 1–61 (1891).
A. Hurwitz, ‘‘Über die Anzahl der Riemannsche Flac̈hen mit gegeben Verzweigungpunkten,’’ Math. Ann. 55, 53–66 (1902).
S. K. Lando, ‘‘Ramified coverings of the two-dimensional sphere and the intersection theory in spaces of meromorphic functions on algebraic curves,’’ Russ. Math. Surv. 57, 463–533 (2002). https://doi.org/10.1070/RM2002v057n03ABEH000511
S. K. Lando and A. K. Zvonkin, Graphs on Surfaces and Their Applications, Vol. 141 of Encyclopaedia Math. Sci. (Springer, Berlin, Heidelberg, New York, 2004). https://doi.org/10.1007/978-3-540-38361-1
E. Lloyd, ‘‘Riemann surface transformation groups,’’ J. Combin. Theory, Ser. A 13, 17–27 (1972).
A. D. Mednykh, ‘‘Nonequivalent coverings of Riemann surfaces with a prescribed ramification type,’’ Sib. Mat. Zh. 25 (4), 120–142 (1984).
A. D. Mednykh, ‘‘A new method for counting coverings over manifold with finitely generated fundamental group,’’ Dokl. Math. 74, 498–502 (2006). https://doi.org/10.1134/S1064562406040089
S. R. Nasyrov, Geometric Problems of the Theory of Ramified Coverings of Riemann Surfaces (Magarif, Kazan, 2008) [in Russian].
S. R. Nasyrov, ‘‘Determination of the polynomial uniformizing a given compact Riemann surface,’’ Math. Notes 91, 558–567 (2012). https://doi.org/10.1134/S0001434612030303
S. R. Nasyrov, ‘‘Uniformization of simply-connected ramified coverings of the sphere by rational functions,’’ Lobachevskii J. Math. 39 (2), 252–258 (2018). https://doi.org/10.1134/S1995080218020208
S. R. Nasyrov, ‘‘Uniformization of one-parametric families of complex tori,’’ Russ. Math. (Iz. VUZ) 61 (8), 36–45 (2017). https://doi.org/10.3103/S1066369X17080047
S. R. Nasyrov, ‘‘Families of elliptic functions and uniformization of complex tori with a unique point over infinity,’’ Probl. Anal. Issues Anal. 7 (25) (2), 98–111 (2018). https://doi.org/10.15393/j3.art.2018.5290
G. Springer, Inroduction to Riemann Surfaces (Addison-Wesley, Reading, MA, 1958).
H. Weyl, ‘‘Über das Hurwitzsche Problem der Bestimmung der Anzahl Riemannscher Flächen von gegebener Verzweigungsart,’’ Comm. Math. Helv. 3, 103–113 (1931).
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This work was supported by the Russian Foundation for Basic Research and the Government of the Republic of Tatarstan, grant no. 18-41-160003.
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(Submitted by F. G. Avkhadiev)
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Nasyrov, S. Families of Elliptic Functions, Realizing Coverings of the Sphere, with Branch-Points and Poles of Arbitrary Multiplicities. Lobachevskii J Math 41, 2223–2230 (2020). https://doi.org/10.1134/S1995080220110153
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DOI: https://doi.org/10.1134/S1995080220110153