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Izvestiya: Mathematics, 2016, Volume 80, Issue 1, Pages 113–166
DOI: https://doi.org/10.1070/IM8310
(Mi im8310)
 

This article is cited in 7 scientific papers (total in 7 papers)

Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symmetry

F. Loray

Institute of Mathematical Research of Rennes, France
References:
Abstract: A Lamé connection is a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection $(E,\nabla)$ over an elliptic curve $X\colon \{y^2=x(x-1)(x-t)\}$, $t\neq 0,1$, having a single pole at infinity. When this connection is irreducible, we show that it is invariant under the standard involution and can be pushed down to a logarithmic $\mathrm{sl}(2,\mathbb C)$-connection on $\mathbb P^1$ with poles at $0$, $1$, $t$ and $\infty$. Therefore the isomonodromic deformation $(E_t,\nabla_t)$ of an irreducible Lamé connection, when the elliptic curve $X_t$ varies in the Legendre family, is parametrized by a solution $q(t)$ of the Painlevé VI differential equation $\mathrm{P}_{\mathrm{VI}}$. The variation of the underlying vector bundle $E_t$ along the deformation is computed in terms of the Tu moduli map: it is given by another solution $\tilde q(t)$ of $\mathrm{P}_{\mathrm{VI}}$, which is related to $q(t)$ by the Okamoto symmetry $s_2 s_1 s_2$ (Noumi–Yamada notation). Motivated by the Riemann–Hilbert problem for the classical Lamé equation, we raise the question whether the Painlevé transcendents do have poles. Some of these results were announced in [6].
Keywords: complex ordinary differential equations, isomonodromic deformations, Lamé differential equation, Painlevé equation.
Received: 18.10.2014
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2016, Volume 80, Issue 1, Pages 119–176
DOI: https://doi.org/10.4213/im8310
Bibliographic databases:
Document Type: Article
UDC: 514.763.8
Language: English
Original paper language: Russian
Citation: F. Loray, “Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symmetry”, Izv. RAN. Ser. Mat., 80:1 (2016), 119–176; Izv. Math., 80:1 (2016), 113–166
Citation in format AMSBIB
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\paper Isomonodromic deformation of Lam\'e connections, Painlev\'e~VI equation and Okamoto symmetry
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  • https://doi.org/10.1070/IM8310
  • https://www.mathnet.ru/eng/im/v80/i1/p119
  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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    Abstract page:474
    Russian version PDF:209
    English version PDF:20
    References:98
    First page:59
     
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