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Mosc. Math. J., 2008, Volume 8, Number 2, Pages 233–271 (Mi mmj312)  

This article is cited in 8 scientific papers (total in 8 papers)

On uniformization of algebraic curves

Yu. V. Brezhnev

Kaliningrad State University

Abstract: Based on Burnside's parametrization of the algebraic curve $y^2=x^5-x$ we obtain the remaining attributes of its uniformization: associated Fuchsian equations and their solutions, accessory parameters, monodromies, conformal maps, fundamental polygons, etc. As a generalization, we propose a way of uniformization of arbitrary curves by zero genus groups. In the hyperelliptic case all the objects of the theory are explicitly described. We consider a large number of examples and, briefly, applications: Abelian integrals, metrics of Poincaré, differential equations of the Jacobi–Chazy and Picard–Fuchs type, and others.

Key words and phrases: Uniformization of algebraic curves, Riemann surfaces, Fuchsian equations/groups, monodromy groups, accessory parameters, modular equations, conformal maps, curvelinear polygons, $\theta$-functions, Abelian integrals, metrics of Poincaré, moduli spaces.

DOI: https://doi.org/10.17323/1609-4514-2008-8-2-233-271

Full text: http://www.ams.org/.../abst8-2-2008.html
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Bibliographic databases:

MSC: 30F10, 30F35
Received: January 30, 2006
Language:

Citation: Yu. V. Brezhnev, “On uniformization of algebraic curves”, Mosc. Math. J., 8:2 (2008), 233–271

Citation in format AMSBIB
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\by Yu.~V.~Brezhnev
\paper On uniformization of algebraic curves
\jour Mosc. Math.~J.
\yr 2008
\vol 8
\issue 2
\pages 233--271
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\crossref{https://doi.org/10.17323/1609-4514-2008-8-2-233-271}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2462436}
\zmath{https://zbmath.org/?q=an:1167.30018}
\isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000261829700002}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. Yu. V. Brezhnev, “A $\tau$-function solution of the sixth Painlevé transcendent”, Theoret. and Math. Phys., 161:3 (2009), 1616–1633  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    2. Brezhnev Yu.V., “On uniformization of Burnside's curve $y^2=x^5-x$”, J. Math. Phys., 50:10 (2009), 103519, 23 pp.  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    3. D. V. Artamonov, “The Schlesinger system and isomonodromic deformations of bundles with connections on Riemann surfaces”, Theoret. and Math. Phys., 171:3 (2012), 739–753  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib  elib
    4. Brezhnev Yu.V., “the Sixth Painlevé Transcendent and Uniformization of Algebraic Curves”, J. Differ. Equ., 260:3 (2016), 2507–2556  crossref  mathscinet  zmath  isi  elib  scopus
    5. A. D. Bryuno, “O reshenii algebraicheskogo uravneniya”, Preprinty IPM im. M. V. Keldysha, 2016, 070, 20 pp.  mathnet  crossref
    6. A. D. Bryuno, “Reshenie algebraicheskogo uravneniya algoritmami stepennoi geometrii”, Preprinty IPM im. M. V. Keldysha, 2017, 034, 28 pp.  mathnet  crossref
    7. Bruno A.D., “Algorithms For Solving An Algebraic Equation”, Program. Comput. Softw., 44:6 (2018), 533–545  crossref  mathscinet  isi  scopus
    8. A. D. Bruno, “On the Parametrization of an Algebraic Curve”, Math. Notes, 106:6 (2019), 885–893  mathnet  crossref  crossref  isi
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