Grothendiecks dessins d'enfants, their deformations, and algebraic solutions of the sixth Painlev\'e and Gauss hypergeometric equations

A. V. Kitaev

Algebra i analiz, Tome 17 (2005) no. 1, pp. 224-275.

Résumé

Grothendieck's dessins d'enfants are applied to the theory of the sixth Painlevé and Gauss hypergeometric functions, two classical special functions of isomonodromy type. It is shown that higher order transformations and the Schwarz table for the Gauss hypergeometric function are closely related to some particular Belyĭ functions. Moreover, deformations of the dessins d'enfants are introduced, and it is shown that one-dimensional deformations are a useful tool for construction of algebraic sixth Painlevé functions.

Bibliographie

[1] Andreev F. V., Kitaev A. V., “Some examples of $RS_3^2(3)$-transformations of ranks 5 and 6 as the higher order transformations for the hypergeometric function”, Ramanujan, 2003, no. 4, 455–476 | DOI | MR | Zbl

[2] Andreev F. V., Kitaev A. V., “Transformations $RS_3^2(3)$ of the ranks $\le4$ and algebraic solutions of the sixth Painleve equation”, Comm. Math. Phys., 228 (2002), 151–176 | DOI | MR | Zbl

[3] Belyi G. V., “O rasshireniyakh Galua maksimalnogo krugovogo polya”, Izv. AN SSSR. Ser. mat., 43:2 (1979), 267–276 | MR | Zbl

[4] Boalch P., The Klein solution to Painlevé's sixth equation, e-preprint (http://xyz.lanl.gov) , 2003, 1–38 math.AG/0308221 | Zbl

[5] Doran Ch. F., “Algebraic and geometric isomonodromic deformations”, J. Differential Geom., 59 (2001), 33–85 | MR | Zbl

[6] Dubrovin B., Mazzocco M., “Monodromy of certain Painlevé-VI transcendents and reflection groups”, Invent. Math., 141 (2000), 55–147 | DOI | MR | Zbl

[7] ibid., 243–283 | MR | Zbl

[8] Granville A., Tucker T. J., “It's as easy as abc”, Notices Amer. Math. Soc., 49:10 (2002), 1224–1231 | MR | Zbl

[9] Jimbo M., Miwa T., “Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. II”, Phys. D, 2 (1981), 407–448 | DOI | MR

[10] Kitaev A. V., “Special functions of the isomonodromy type”, Acta Appl. Math., 64:1 (2000), 1–32 | DOI | MR | Zbl

[11] Kitaev A. V., “Spetsialnye funktsii izomonodromnogo tipa, ratsionalnye preobrazovaniya spektralnogo parametra i algebraicheskie resheniya shestogo uravneniya Penleve”, Algebra i analiz, 14:3 (2002), 121–139 ; e-preprint , 2000, 1–13 http://arxiv.org/abs/nlin/0102020 | MR | Zbl

[12] Kitaev A. V., “On similarity reductions of the three-wave resonant system to the Painlevé equations”, J. Phys. A, 23 (1990), 3543–3553 | DOI | MR | Zbl

[13] Kitaev A. V., “Quadratic transformations for the sixth Painlevé equation”, Lett. Math. Phys., 21 (1991), 105–111 | DOI | MR | Zbl

[14] Kitaev A. V., Korotkin D. A., “On solutions of the Schlesinger equations in terms of $\Theta$-functions”, Int. Math. Res. Not., , no. 17 (1998), 877–905 | DOI | MR

[15] Magot N., Zvonkin A., “Belyi functions for Archimedean solids”, Discrete Math., 217:1–3 (2000), 249–271 | DOI | MR | Zbl

[16] Manin Yu. I., “Sixth Painlevé equation, universal elliptic curve, and mirror of $\mathbf P^2$”, Geometry of Differential Equations, Amer. Math. Soc. Transl. Ser. 2, 186, Amer. Math. Soc., Providence, RI, 1998, 131–151 | MR | Zbl

[17] Noumi M., Takano K., Yamada Ya., “Bäcklund transformations and the manifolds of Painlevé systems”, Funkcial. Ekvac., 45:2 (2002), 237–258 | MR | Zbl

[18] Okamoto K., “Studies on the Painleve equations. I. Sixth Painlevé equation $P_{\rm VI}$”, Ann. Mat. Pura Appl. (4), 146 (1987), 337–381 | DOI | MR | Zbl

[19] Schwarz H. A., “Über diejenigen Fälle, in welchen die Gaussische hypergeometrische Reihe eine algebraische Function ihres vierten Elementes darstellt”, J. Reine Angew. Math., 75 (1873), 292–335

[20] Singerman D., “Riemann surfaces, Belyi functions and hypermaps”, Topics on Riemann Surfaces and Fuchsian Groups (Madrid, 1998), London Math. Soc. Lecture Note Ser., 287, Cambridge Univ. Press, Cambridge, 2001, 43–68 | MR | Zbl