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Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre 2
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 39-54.

On étudie la possibilité de construire des solutions algébriques partielles des équations d’isomonodromie pour les connexions holomorphes de rang 2 sur les courbes de genre 2 en adaptant la méthode de Doran-Andreev-Kitaev par les familles de Hurwitz. Nous classifions tous les cas où la connexion est à monodromie Zariski dense.

We study the possiblility to construct partial algebraic solutions of the isomonodromy equations for holomorphic connexions of rank 2 on curves of genus 2 by adapting the Doran-Andreev-Kitaev method of Hurwitz families. We classify all cases where the connexion is Zariski dense monodromy.

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     author = {Karamoko Diarra},
     title = {Solutions alg\'ebriques partielles des \'equations isomonodromiques sur les courbes de genre $2$},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {39--54},
     publisher = {Universit\'e Paul Sabatier, Institut de Math\'ematiques},
     address = {Toulouse},
     volume = {6e s{\'e}rie, 24},
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     year = {2015},
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Karamoko Diarra. Solutions algébriques partielles des équations isomonodromiques sur les courbes de genre $2$. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 24 (2015) no. 1, pp. 39-54. doi : 10.5802/afst.1441. https://afst.centre-mersenne.org/articles/10.5802/afst.1441/

[1] Abu Osman (M. T.) et Rosenberger (G.).— Embedding property of surface groups. Bull. Malaysian Math. Soc. 3, p. 21-27 (1980). | MR | Zbl

[2] Andreev (F. V.) et Kitaev (A. V.).— Transformations RS 4 2 (3) of the ranks ≤4 and algebraic solutions of the sixth Painlevé equation. Comm. Math. Phys. 228, p. 151-176 (2002). | MR | Zbl

[3] Boalch (P.).— From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. London Math. Soc. 90, p. 167-208 (2005). | MR | Zbl

[4] Boalch (P.).— From Klein to Painlevé via Fourier, Laplace and Jimbo. Proc. Londo, The fifty-two icosahedral solutions to Painlevé VI. J. Reine Angew. Math. 596, p. 183-214 (2006). | MR | Zbl

[5] Boalch (P.).— Six results on Painlevé VI. Théories asymptotiques et équations de Painlevé, 1-20, Sémin. Congr., 14, Soc. Math. France, Paris, 2006. | MR | Zbl

[6] Boalch (P.).— Higher genus icosahedral Painlevé curves. Funkcial. Ekvac. 50, no. 1, p. 19-32 (2007). | MR | Zbl

[7] Boalch (P.).— Some explicit solutions to the Riemann-Hilbert problem. Differential equations and quantum groups, 85-112, IRMA Lect. Math. Theor. Phys., 9, Eur. Math. Soc., Zürich, 2007. | MR

[8] Boalch (P.).— Towards a non-linear Schwarz’s list. The many facets of geometry, 210-236, Oxford Univ. Press, Oxford, 2010. | MR | Zbl

[9] Chiarellotto (B.).— On Lamé Operators which are Pullbacks of Hypergeometric Ones. Trans. Amer. Math. Soc. 347, p. 2735-2780 (1995). | MR | Zbl

[10] Corlette (K.) et Simpson (C.).— On the classification of rank-two representations of quasiprojective fundamental groups. Compos. Math. 144, p. 1271-1331 (2008). | MR | Zbl

[11] Cousin (G.).— Un exemple de feuilletage modulaire déduit d’une solution algébrique de l’équation de Painlevé VI. arXiv :1201.2755

[12] Diarra (K.).— Construction de déformations isomonodromiques par revêtements, thèse de doctorat, Université de Rennes 1 (2011). http ://tel.archives-ouvertes.fr/

[13] Diarra (K.).— Construction et classification de certaines solutions algébriques des systèmes de Garnier. Bull. Braz. Math. Soc., New Series 44, p. 1-26 (2013). | MR | Zbl

[14] Doran (C. F.).— Algebraic and Geometric Isomonodromic Deformations. J. Differential Geometry 59, p. 33-85 (2001). | MR | Zbl

[15] Dubrovin (B.) et Mazzocco (M.).— Monodromy of certain Painlevé-VI transcendents and reflection groups. Invent. Math. 141, p. 55-147 (2000). | MR | Zbl

[16] Heu (V.) et Loray (F.).— Flat rank 2 vector bundles on genus 2 curves. http ://hal.archives-ouvertes.fr/hal-00927061/fr/

[17] Hitchin (N.).— Poncelet polygons and the Painlevé equations. Geometry and analysis (Bombay, 1992), 151-185, Tata Inst. Fund. Res., Bombay, 1995. | MR | Zbl

[18] Hitchin (N.).— A lecture on the octahedron. Bull. London Math. Soc. 35, p. 577-600 (2003). | MR | Zbl

[19] Hulpke (A.), Kuusalo (T.), Näätänen (M. ) et Rosenberger (G.).— On orbifold coverings by genus 2 surfaces. Sci. Ser. A Math. Sci. (N.S.) 11, p. 45-55 (2005). | MR | Zbl

[20] Iwasaki (K.), Kimura (H.), Shimomura (S.) et Yoshida (M.).— From Gauss to Painlevé. A modern theory of special functions. Aspects of Mathematics, E16. Friedr. Vieweg and Sohn, Braunschweig, 1991. | MR | Zbl

[21] Kitaev (A. V.).— Special functions of isomonodromy type, rational transformations of the spectral parameter, and algebraic solutions of the sixth Painlevé equation. Algebra i Analiz 14, p. 121-139 (2002). | MR | Zbl

[22] Kitaev (A. V.).— Grothendieck’s dessins d’enfants, their deformations, and algebraic solutions of the sixth Painlevé and Gauss hypergeometric equations. Algebra i Analiz 17, p. 224-275 (2005). | MR | Zbl

[23] Klein (F.).— Vorlesungen über das Ikosaedar, B. G. Teubner, Leipzig (1884). | JFM

[24] Krichever (I.).— Isomonodromy equations on algebraic curves, canonical transformations and Whitham equations. Mosc. Math. J. 2, p. 717-752 (2002). | MR | Zbl

[25] Lisovyy (O.) et Tykhyy (Y.).— Algebraic solutions of the sixth Painlevé equation. arXiv :0809.4873 [math.CA] | MR

[26] Mazzocco (M.).— Rational solutions of the Painlevé VI equation. Kowalevski Workshop on Mathematical Methods of Regular Dynamics (Leeds, 2000). J. Phys. A 34, p. 2281-2294 (2001). | MR | Zbl

[27] Mazzocco (M.).— Picard and Chazy solutions to the Painlevé VI equation. Math. Ann. 321, p. 157-195 (2001). | MR | Zbl

[28] Mednykh (A.).— Counting conjugacy classes of subgroups in a finitely generated group. J. Algebra 320, p. 2209-2217 (2008). | MR | Zbl

[29] Pascali (M. A.) et Petronio (C.).— Branched covers of the sphere and the prime-degree conjecture. Ann. Mat. Pura Appl. 191, p. 563-594 (2012). | MR | Zbl

[30] Pervova (E.) et Petronio (C.).— On the existence of branched coverings between surfaces with prescribed branch data. I. Algebr. Geom. Topol. 6, p. 1957-1985 (2006). | MR | Zbl

[31] Pervova (E.) et Petronio (C.).— On the existence of branched coverings between surfaces with prescribed branch data. II. J. Knot Theory Ramifications 17, p. 787-816 (2008). | MR | Zbl

[32] Simpson (C.).— Moduli of representations of the fundamental group of a smooth projective variety. II. Inst. Hautes Études Sci. Publ. Math. 80, p. 5-79 (1994). | EuDML | Numdam | MR | Zbl

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