The Lamé family of connections on the projective line

Frank Loray[1]; Marius van der Put[2]; Felix Ulmer[1]

  • [1] Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France
  • [2] Department of mathematics, University of Groningen, P.O.Box 800, 9700 AV, Groningen, The Netherlands

Annales de la faculté des sciences de Toulouse Mathématiques (2008)

  • Volume: 17, Issue: 2, page 371-409
  • ISSN: 0240-2963

Abstract

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This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and - 1 / 4 . This Lamé family of connections has been extensively studied in the literature. The differential Galois group of a Lamé connection is never maximal : it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections. The irreducible Lamé connections are derived from the rank 1 regular connections on the elliptic curve w 2 = z ( z - 1 ) ( z - t ) ; those connections having a finite Galois group are known to be related to points of finite order on the elliptic curve. In the paper, we provide a very efficient algorithm to compute the locus of those Lamé connections having a finite Galois group of a given order. We also give an efficient algorithm to compute the minimal polynomial for the corresponding field extension. We do this computation for low order and recover this way known algebraic solutions of the Painlevé VI equation and of the classical Lamé equation. In the final section we compare our moduli space with the classical one due to Okamoto.

How to cite

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Loray, Frank, van der Put, Marius, and Ulmer, Felix. "The Lamé family of connections on the projective line." Annales de la faculté des sciences de Toulouse Mathématiques 17.2 (2008): 371-409. <http://eudml.org/doc/10090>.

@article{Loray2008,
abstract = {This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and $-1/4$. This Lamé family of connections has been extensively studied in the literature. The differential Galois group of a Lamé connection is never maximal : it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections. The irreducible Lamé connections are derived from the rank $1$ regular connections on the elliptic curve $w^2=z(z-1)(z-t)$; those connections having a finite Galois group are known to be related to points of finite order on the elliptic curve. In the paper, we provide a very efficient algorithm to compute the locus of those Lamé connections having a finite Galois group of a given order. We also give an efficient algorithm to compute the minimal polynomial for the corresponding field extension. We do this computation for low order and recover this way known algebraic solutions of the Painlevé VI equation and of the classical Lamé equation. In the final section we compare our moduli space with the classical one due to Okamoto.},
affiliation = {Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France; Department of mathematics, University of Groningen, P.O.Box 800, 9700 AV, Groningen, The Netherlands; Université de Rennes 1, Campus de Beaulieu, 35042 Rennes Cedex, France},
author = {Loray, Frank, van der Put, Marius, Ulmer, Felix},
journal = {Annales de la faculté des sciences de Toulouse Mathématiques},
keywords = {Lamé connection; differential Galois group; Painlevé VI equation; Lamé equation},
language = {eng},
month = {6},
number = {2},
pages = {371-409},
publisher = {Université Paul Sabatier, Toulouse},
title = {The Lamé family of connections on the projective line},
url = {http://eudml.org/doc/10090},
volume = {17},
year = {2008},
}

TY - JOUR
AU - Loray, Frank
AU - van der Put, Marius
AU - Ulmer, Felix
TI - The Lamé family of connections on the projective line
JO - Annales de la faculté des sciences de Toulouse Mathématiques
DA - 2008/6//
PB - Université Paul Sabatier, Toulouse
VL - 17
IS - 2
SP - 371
EP - 409
AB - This paper deals with rank two connections on the projective line having four simple poles with prescribed local exponents 1/4 and $-1/4$. This Lamé family of connections has been extensively studied in the literature. The differential Galois group of a Lamé connection is never maximal : it is either dihedral (finite or infinite) or reducible. We provide an explicit moduli space of those connections having a free underlying vector bundle and compute the algebraic locus of those reducible connections. The irreducible Lamé connections are derived from the rank $1$ regular connections on the elliptic curve $w^2=z(z-1)(z-t)$; those connections having a finite Galois group are known to be related to points of finite order on the elliptic curve. In the paper, we provide a very efficient algorithm to compute the locus of those Lamé connections having a finite Galois group of a given order. We also give an efficient algorithm to compute the minimal polynomial for the corresponding field extension. We do this computation for low order and recover this way known algebraic solutions of the Painlevé VI equation and of the classical Lamé equation. In the final section we compare our moduli space with the classical one due to Okamoto.
LA - eng
KW - Lamé connection; differential Galois group; Painlevé VI equation; Lamé equation
UR - http://eudml.org/doc/10090
ER -

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