Lamé equations with algebraic solutions

Beukers, Frits; Waall, Alexa van der

doi:10.1016/j.jde.2003.10.017

Cited by 26 publications

(102 citation statements)

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“…Γ(N )) and now, broadly speaking, the equivalence classes of Lamé equations with index n = 1 and full monodromy group isomorphic to D N are in correspondence with the zeroes of the modular forms Z N (τ ; k 1 , k 2 ), with gcd(k 1 , k 2 , N) = 1. From the analysis given in the proof of [BW,Thm. 6.1] it follows that for N ≥ 3 we have Moreover, the proof tells us that for N ≥ 3 equality in (4.2) holds if and only if the modular forms Z N (τ ; k 1 , k 2 ), where k 1 , k 2 ∈ Z and gcd(k 1 , k 2 , N) = 1, have no multiple zeroes in the upper half plane.…”

Section: Applicationsmentioning

confidence: 99%

“…The counting method in [BW,Sec. 6] can also be used (as main ingredient) to obtain upper bounds for F L(n, N ) when n = 2, 3.…”

Section: Applicationsmentioning

confidence: 99%

“…It is known that if there exists a basis of algebraic solutions, then the (projective) monodromy group is dihedral; see for example [Bal,Thm. 1.5], [BW,Cor. 3.4] or [vdW,Thm.…”

Section: Introductionmentioning

confidence: 99%

“…This can be proven in different ways; see for example [Chi,Remark 0.2 and Sec. 1] [BW,Thm. 4.6], [vdW,Thms.…”

Section: Introductionmentioning

confidence: 99%

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Counting integral Lamé equations by means of dessins d’enfants

Dahmen

2006

Trans. Amer. Math. Soc.

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Section: Applicationsmentioning

confidence: 99%

“…The counting method in [BW,Sec. 6] can also be used (as main ingredient) to obtain upper bounds for F L(n, N ) when n = 2, 3.…”

Section: Applicationsmentioning

confidence: 99%

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Counting integral Lamé equations by means of dessins d’enfants

Dahmen

2006

Trans. Amer. Math. Soc.

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“…Another approach is the Klein pullback method: Klein ([18], also [1,5,2]) showed that if the projective differential Galois group is finite, then the equation is a pullback of an equation in a finite list of well-known standard hypergeometric equations. This means that the solutions are of the form e g H(f ) where f, g ∈ k and H is a standard hypergeometric function H(x) = 2 F1([a, b], [c], x) whose parameters a, b, c appear in a finite list.…”

Section: Introductionmentioning

confidence: 99%

Solving second order linear differential equations with Klein's theorem

Hoeij

Weil

2005

Proceedings of the 2005 International Symposium on Symbolic and Algebraic Computation

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Given a second order linear differential equations with coefficients in a field k = C(x), the Kovacic algorithm finds all Liouvillian solutions, that is, solutions that one can write in terms of exponentials, logarithms, integration symbols, algebraic extensions, and combinations thereof. A theorem of Klein states that, in the most interesting cases of the Kovacic algorithm (i.e when the projective differential Galois group is finite), the differential equation must be a pullback (a change of variable) of a standard hypergeometric equation. This provides a way to represent solutions of the differential equation in a more compact way than the format provided by the Kovacic algorithm. Formulas to make Klein's theorem effective were given in [4,2,3]. In this paper we will give a simple algorithm based on such formulas. To make the algorithm more easy to implement for various differential fields k, we will give a variation on the earlier formulas, namely we will base the formulas on invariants of the differential Galois group instead of semi-invariants.

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Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

Vidūnas

Kitaev

2016

J Math Sci

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Various Schlesinger transformations can be combined with a direct pull-back of a hypergeometric 2×2 system to obtain RS 2 4 -pullback transformations to isomonodromic 2 × 2 Fuchsian systems with 4 singularities. The corresponding Painlevé VI solutions are algebraic functions, possibly in different orbits under Okamoto transformations. This paper demonstrates a direct computation of Schlesinger transformations acting on several apparent singular points, and presents an algebraic procedure (via syzygies) of computing algebraic Painlevé VI solutions without deriving full RS-pullback transformations.An important observation is that the same rational covering R(x) can be used in several RS-pullback transformations. Application of different RS-transformations to respectively different matrix differential equations gives different algebraic Painlevé VI solutions. For example, [26] demonstrates usage of the same degree 10 covering to pullback three different hypergeometric equations (with the local monodromy differences 1/2, 1/3, k/7 with k = 1, 2 or 3) and obtain three algebraic Painlevé VI solutions unrelated by fractional-linear or Okamoto transformations.In our other concrete examples, we start with matrix hypergeometric equations with the icosahedral monodromy group. The pullbacked Fuchsian equations have the icosahedral monodromy group as well. Corresponding Painlevé VI solutions are called icosahedral [3]. There are 52 types of icosahedral Painlevé VI solutions in total [3], up to branching representation of the icosahedral monodromy group (or equivalently, up to Okamoto transformations). We recompute icosahedral solutions of Boalch types 26, 27, 31, 32. Second order ordinary Fuchsian equations (or 2 × 2 first order matrix Fuchsian equations) with a finite monodromy group are always pullbacks of a standard hypergeometric equation with the same monodromy group, as asserted by celebrated Klein's theorem [19]. In particular, existence of pull-back transformations for the four icosahedral examples follows from Klein's theorem. R. Fuchs [12] soon considered extension of Klein's theorem to algebraic solutions of Painlevé equations. Recently, Ohyama and Okumura [22] showed that algebraic solutions of Painlevé equations from the first to the fifth do arise from pullback transformations of confluent hypergeometric equations, affirming the formulation of R. Fuchs. The pullback method for computing algebraic Painlevé VI solutions was previously suggested in [18], [2], [17], [7]. The alternative representation-theoretic approach is due to Dubrovin-Mazzocco [9]. Recently, it was used [20] to complete classification of algebraic Painlevé VI solutions.The article is organized as follows. Section 2 presents two almost Belyi coverings we employ. The coverings have degree 8 and 12; they were previously used in [18]. Section 3 demonstrates two examples of full RS-pullback transformations, both with respect to the degree 8 covering. In Section 4 we formulate basic algebraic facts useful in computations of RS-pullback transformations....

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Lamé equations with algebraic solutions

Cited by 26 publications

References 9 publications

Counting integral Lamé equations by means of dessins d’enfants

Counting integral Lamé equations by means of dessins d’enfants

Solving second order linear differential equations with Klein's theorem

Computation of RS-Pullback Transformations for Algebraic Painlevé VI Solutions

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