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Computation of highly ramified coverings. (English) Zbl 1226.14043

This paper is divided into two sections. In the first section, certain explicit families of “almost Belyi” coverings of the projective line (over the algebraic closure \(\overline{\mathbb{Q}}\) of \(\mathbb{Q}\)) are constructed. In this case, “almost Belyi” means that the covering is branched at \(0\), \(1\), and \(\infty\), as well as a fourth point, above which the ramification is simple (meaning that there is only one ramification point above the branch point, and its ramification index is 2). The coverings constructed are of degrees 12, 11, and 20, and they all have genus zero, thus are defined by a single rational function. The strategy is to fix the ramification type, and then give an ansatz for what form the rational function should take. One can then take the logarithmic derivative of the ansatz form directly, and also realize (independently) that the logarithmic derivative must have certain poles and zeros. This leads to a set of equations for the coefficients of the rational form that can be solved by computer. In the case of the degree 12 covering, this is straightforward. For the other two coverings, further tricks (including solving the undetermined coefficient equations in various finite fields) are required to make the computation feasible.
In the second section, these almost Belyi coverings are applied to give solutions to Painlevé VI equations. This is possible because pulling back via these coverings can transform hypergeometric differential equations into \(2 \times 2\) matrix Fuchsian systems with 4 regular singular points. The paper computes an example for each of the coverings from the first section.
Reviewer: Andrew Obus (Bonn)

MSC:

14H57 Dessins d’enfants theory
57M12 Low-dimensional topology of special (e.g., branched) coverings
34M55 Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies
33E17 Painlevé-type functions

Software:

BRAID; SINGULAR
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] F. V. Andreev and A. V. Kitaev, Some examples of \?\?²\(_{3}\)(3)-transformations of ranks 5 and 6 as the higher order transformations for the hypergeometric function, Ramanujan J. 7 (2003), no. 4, 455 – 476. · Zbl 1042.33004 · doi:10.1023/B:RAMA.0000012428.77217.bc
[2] F. V. Andreev and A. V. Kitaev, Transformations \?\?²\(_{4}\)(3) of the ranks \le 4 and algebraic solutions of the sixth Painlevé equation, Comm. Math. Phys. 228 (2002), no. 1, 151 – 176. · Zbl 1019.34086 · doi:10.1007/s002200200653
[3] Elizabeth A. Arnold, Modular algorithms for computing Gröbner bases, J. Symbolic Comput. 35 (2003), no. 4, 403 – 419. · Zbl 1046.13018 · doi:10.1016/S0747-7171(02)00140-2
[4] G. V. Belyĭ, Galois extensions of a maximal cyclotomic field, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 2, 267 – 276, 479 (Russian). · Zbl 0409.12012
[5] Philip Boalch, The fifty-two icosahedral solutions to Painlevé VI, J. Reine Angew. Math. 596 (2006), 183 – 214. · Zbl 1112.34072 · doi:10.1515/CRELLE.2006.059
[6] Philip Boalch, Some explicit solutions to the Riemann-Hilbert problem, Differential equations and quantum groups, IRMA Lect. Math. Theor. Phys., vol. 9, Eur. Math. Soc., Zürich, 2007, pp. 85 – 112. · Zbl 1356.34092
[7] Philip Boalch, Higher genus icosahedral Painlevé curves, Funkcial. Ekvac. 50 (2007), no. 1, 19 – 32. · Zbl 1159.34060 · doi:10.1619/fesi.50.19
[8] Jean-Marc Couveignes, Tools for the computation of families of coverings, Aspects of Galois theory (Gainesville, FL, 1996) London Math. Soc. Lecture Note Ser., vol. 256, Cambridge Univ. Press, Cambridge, 1999, pp. 38 – 65. · Zbl 1016.14012
[9] Steven Diaz, Ron Donagi, and David Harbater, Every curve is a Hurwitz space, Duke Math. J. 59 (1989), no. 3, 737 – 746. · Zbl 0712.14013 · doi:10.1215/S0012-7094-89-05933-4
[10] Charles F. Doran, Algebraic and geometric isomonodromic deformations, J. Differential Geom. 59 (2001), no. 1, 33 – 85. · Zbl 1043.34098
[11] B. Dubrovin and M. Mazzocco, Monodromy of certain Painlevé-VI transcendents and reflection groups, Invent. Math. 141 (2000), no. 1, 55 – 147. · Zbl 0960.34075 · doi:10.1007/PL00005790
[12] Alexandre Grothendieck, Esquisse d’un programme, Geometric Galois actions, 1, London Math. Soc. Lecture Note Ser., vol. 242, Cambridge Univ. Press, Cambridge, 1997, pp. 5 – 48 (French, with French summary). With an English translation on pp. 243 – 283. · Zbl 0901.14001
[13] G.-M. Greuel, G. Pfister, H. Schönemann, SINGULAR 2.0.3. A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, University of Kaiserslautern (2005). http://www.singular.uni-kl.de.
[14] Robin Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. · Zbl 0367.14001
[15] Joachim A. Hempel, Existence conditions for a class of modular subgroups of genus zero, Bull. Austral. Math. Soc. 66 (2002), no. 3, 517 – 525. · Zbl 1023.20024 · doi:10.1017/S0004972700040351
[16] Michio Jimbo, Tetsuji Miwa, and Kimio Ueno, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and \?-function, Phys. D 2 (1981), no. 2, 306 – 352. , https://doi.org/10.1016/0167-2789(81)90013-0 Michio Jimbo and Tetsuji Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II, Phys. D 2 (1981), no. 3, 407 – 448. , https://doi.org/10.1016/0167-2789(81)90021-X Michio Jimbo and Tetsuji Miwa, Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. III, Phys. D 4 (1981/82), no. 1, 26 – 46. · Zbl 1194.34169 · doi:10.1016/0167-2789(81)90003-8
[17] A. V. Kitaev, Special functions of isomonodromy type, rational transformations of the spectral parameter, and algebraic solutions of the sixth Painlevé equation, Algebra i Analiz 14 (2002), no. 3, 121 – 139 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 14 (2003), no. 3, 453 – 465. · Zbl 1044.34052
[18] A. V. Kitaev, Grothendieck’s dessins d’enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations, Algebra i Analiz 17 (2005), no. 1, 224 – 275; English transl., St. Petersburg Math. J. 17 (2006), no. 1, 169 – 206.
[19] Alexander V. Kitaev, Remarks towards a classification of \?\?\(_{4}\)²(3)-transformations and algebraic solutions of the sixth Painlevé equation, Théories asymptotiques et équations de Painlevé, Sémin. Congr., vol. 14, Soc. Math. France, Paris, 2006, pp. 199 – 227 (English, with English and French summaries). · Zbl 1142.34060
[20] E. M. Kreĭnes, Families of geometric parasitic solutions of systems of equations for the Belyĭ function of genus zero, Fundam. Prikl. Mat. 9 (2003), no. 1, 103 – 111 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 128 (2005), no. 6, 3396 – 3401. · doi:10.1007/s10958-005-0278-9
[21] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982), no. 4, 515 – 534. · Zbl 0488.12001 · doi:10.1007/BF01457454
[22] Kay Magaard, Sergey Shpectorov, and Helmut Völklein, A GAP package for braid orbit computation and applications, Experiment. Math. 12 (2003), no. 4, 385 – 393. · Zbl 1068.12002
[23] Mohamed El Marraki, Nicolas Hanusse, Jörg Zipperer, and Alexander Zvonkin, Cacti, braids and complex polynomials, Sém. Lothar. Combin. 37 (1996), Art. B37b, 36 (English, with English and French summaries). · Zbl 0976.57004
[24] G. Shabat, On a class of families of Belyi functions, Formal power series and algebraic combinatorics (Moscow, 2000) Springer, Berlin, 2000, pp. 575 – 580. · Zbl 0962.14024
[25] Leila Schneps, Dessins d’enfants on the Riemann sphere, The Grothendieck theory of dessins d’enfants (Luminy, 1993) London Math. Soc. Lecture Note Ser., vol. 200, Cambridge Univ. Press, Cambridge, 1994, pp. 47 – 77. · Zbl 0823.14017
[26] Raimundas Vidunas and Alexander V. Kitaev, Quadratic transformations of the sixth Painlevé equation with application to algebraic solutions, Math. Nachr. 280 (2007), no. 16, 1834 – 1855. · Zbl 1137.34041 · doi:10.1002/mana.200510582
[27] R. Vidūnas and A. V. Kitaev, Computation of \( RS\)-pullback transformations for algebraic Painlevé VI solutions. Available at http://arxiv.org/abs/0705.2963.
[28] R. Vidūnas and A. V. Kitaev, Schlesinger transformations for algebraic Painlevé VI solutions. Available at http://arxiv.org/abs/0810.2766.
[29] R. Vidūnas, Algebraic Transformations of Gauss Hypergeometric Functions, Accepted by Funk. Ekvac. Available at http://www.arxiv.org/math.CA/0408269 (2004).
[30] Raimundas Vidūnas, Transformations of some Gauss hypergeometric functions, J. Comput. Appl. Math. 178 (2005), no. 1-2, 473 – 487. · Zbl 1076.33002 · doi:10.1016/j.cam.2004.09.053
[31] P. S. Wang, M. J. T. Guy and J. H. Davenport, \( P\)-adic reconstruction of rational numbers, SIGSAM Bulletin, Vol. 16, ACM, 1982, pp. 2-3. · Zbl 0489.68032
[32] L. Zapponi, Galois action on diameter four trees, preprint http://www.arxiv.org/math.AG/ 0108031 (2001).
[33] Alexander Zvonkin, Megamaps: construction and examples, Discrete models: combinatorics, computation, and geometry (Paris, 2001) Discrete Math. Theor. Comput. Sci. Proc., AA, Maison Inform. Math. Discrèt. (MIMD), Paris, 2001, pp. 329 – 339. · Zbl 1007.05099
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