Triangular Schlesinger systems and superelliptic curves

https://doi.org/10.1016/j.physd.2021.132947Get rights and content

Highlights

  • Schlesinger systems for triangular matrices with arithmetic constraints are studied.

  • Solutions are constructed via periods of differentials on superelliptic curves.

  • Rational solutions of Painlevé VI equations are derived.

  • Algebraic solutions of Garnier systems are provided.

Abstract

We study the Schlesinger system of partial differential equations in the case when the unknown matrices of arbitrary size (p×p) are triangular and the eigenvalues of each matrix form an arithmetic progression with a rational difference q, the same for all matrices. We show that such a system possesses a family of solutions expressed via periods of meromorphic differentials on the Riemann surfaces of superelliptic curves. We determine the values of the difference q, for which our solutions lead to explicit polynomial or rational solutions of the Schlesinger system. As an application of the (2×2)-case, we obtain explicit sequences of rational solutions and of one-parameter families of rational solutions of Painlevé VI equations. Using similar methods, we provide algebraic solutions of particular Garnier systems.

Introduction

We consider the Schlesinger system [1] dB(i)=j=1,jiN[B(i),B(j)]aiajd(aiaj),i=1,,N,for (p×p)-matrices B(1),,B(N) depending on the variable a=(a1,,aN) which belongs to some disc D of the space Nij{ai=aj}. Written in a PDEs form, this becomes B(i)aj=[B(i),B(j)]aiaj(ij),B(i)ai=j=1,jiN[B(i),B(j)]aiaj.These equations govern an isomonodromic family of Fuchsian linear differential systems dydz=(i=1NB(i)(a)zai)y,y(z)p,with varying singular points a1,,aN. As follows from the isomonodromic nature of the Schlesinger system, the eigenvalues βik of the matrices B(i) that solve this system are constant (see proof of Theorem 3 from [2]). These eigenvalues are called the exponents of the Schlesinger system and of the related isomonodromic family (3) of Fuchsian systems, at their varying singular points z=ai.

As known, due to B. Malgrange [3], the Schlesinger system is completely integrable in D, that is, for any initial data B0(1),,B0(N)Mat(p,) and any a0D, it has the unique solution B(1)(a),,B(N)(a) such that B(i)(a0)=B0(i), i=1,,N. Moreover, (the pull-backs of) the matrix functions B(i) are continued meromorphically to the universal cover Z of the space Nij{ai=aj} and their polar locus ΘZ, called the Malgrange divisor, is described as a zero set of a function τ, holomorphic on the whole space Z. Being locally descended to D, this global τ-function, up to a holomorphic non-vanishing in D factor, coincides with the local one satisfying Miwa’s formula [4] dlnτ(a)=12i=1Nj=1,jiNtr(B(i)(a)B(j)(a))aiajd(aiaj).

In the present paper we are going to focus on upper triangular matrix solutions B(i)=(bikl)1k,lp, that is on those with bikl=0 for k>l, with specific arithmetic restrictions on the exponents. Triangular solutions of the Schlesinger system are those and only those with triangular initial data, since any set of N triangular matrices evolving with respect to this system remains triangular, due to the form of the system. Note that the exponents in this case coincide with the diagonal entries: βik=bikk.

Motivation for the problem we are going to consider comes from the basic p=2,N=3 case and classical algebraic geometry. It is well known, see [5], that in this case when the matrices are traceless triangular, with a1=0,a2=1,a3=x, the off-diagonal matrix element b112=b112(x) of the matrix B(1) satisfies a hypergeometric equation: x(1x)b1xx12+[c(a+b+1)x]b1x12abb112=0,where a=2j=13βj1,b=2β31, and c=12(β11+β31). In the special case (β11,β21,β31)=(14,14,14), one recognizes the classical Picard–Fuchs equation: x(1x)b1xx12+(12x)b1x1214b112=0,whose solutions are given by linear combinations of the periods of the differential duv on the elliptic curve v2=u(u1)(ux)(see for example [6] and [7], formula (2.25), p. 61). Let us note that in this case βi1βi2=±12.

The last observation motivates us to consider the following particular case: each tuple {βi1,,βip} forms an arithmetic progression with the same rational difference q=nm, where n0 and m>0 are coprime. Generalizing the relationship with the Picard–Fuchs equations, we prove that the corresponding triangular system (2) possesses a family of solutions having algebro-geometric nature, namely they are expressed via periods of meromorphic differentials on the Riemann surfaces Xa of a (varying) algebraic plane curve of superelliptic type Γˆa={(z,w)2wm=(za1)(zaN)}.These expressions for the matrix entries bikl(a) are presented in Theorem 1 from Section 2.1.

Superelliptic curves are of much interest nowadays as well as some other related classes of curves, like Zm curves or (m,N)-curves, see [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and references therein. There are some differences and ambiguity across the literature in definitions of these classes. For us, (following J. Sander, Yu. Zarhin and others) superelliptic curves are those which can be represented by an equation of the form: wm=PN(z),where PN is any polynomial of degree N. Due to the nature of the matter considered in the present paper, the zeros of PN are additionally assumed to be simple, thus the superelliptic curves considered here are smooth in the affine part. It is possible to extend the study to a more general case of superelliptic curves with singularities in the affine part, which we are going to address in a separate publication.

Note that triangular and, more generally, reducible, Schlesinger systems of arbitrary size p were already studied by B. Dubrovin and M. Mazzocco in [20], where the main question was the following: when are solutions of one Schlesinger system for N (p×p)-matrices expressed via solutions of some other “simpler” Schlesinger systems of smaller matrix size or involving less than N matrices? (See also some investigations of triangular Schlesinger systems in this context in the case of small dimensions p=2, p=3 in [21], [22].) However, there was no restriction imposed on the exponents, and thus there was no discussion of the integration of such systems in an explicit, in particular algebro-geometric, form. Nevertheless, it was mentioned that triangular solutions are expressed via solutions of Lauricella differential systems. For the latter, there are already known representations by integrals of multivalued functions over several kinds of chains in (see, for example, [23] where the question of the linear independence of such integrals is also solved). We propose an alternative analysis based on the algebro-geometric approach which also helps to obtain some elementary expressions such as polynomial or rational ones, as we clarify below. On the other hand, in the previous papers which provide particular algebro-geometric solutions to the Schlesinger system [24], [25], [26] for p=2, and [10], [27] for an arbitrary p in the case of quasi-permutation monodromy matrices of the family (3) the specific character of the triangular case has not been taken into consideration. The first article on triangular algebro-geometric solutions of Schlesinger systems (in the case p=2) is the recent [28], where the hyperelliptic case m=2 is studied, and our present work is an improvement and extension of the latter.

Concluding our introduction, let us be more specific and describe in general some features of the proposed algebro-geometric approach. The first main result of this paper is Theorem 1 in Section 2.1 which provides families of algebro-geometric solutions of the system (2). Theorem 2 from Section 2.3 answers a delicate question about the dimension of the families of the solutions obtained in Theorem 1.

In the case of n>0 and when m and N are coprime, the mentioned meromorphic differentials have only one pole, therefore are all of the second kind, i. e. have no residues. Thus their integration over elements of the homology group H1(Xa,Z) is well defined. As observed before Theorem 1 in the case when n is positive and the greatest common divisor of m and N is bigger than 1, denoted (m,N)>1, or when n is negative, the involved meromorphic differentials have several poles P1,,Ps and are of the third kind in general, i. e. have non-zero residues, therefore one should use elements of H1(Xa{P1,,Ps},Z) to integrate them correctly. We observe another effect in this case: taking small loops encircling the poles of the differentials, one expresses the matrix entries bikl(a) via the residues of the differentials, which turn out to be polynomials or rational functions in the variables a1,,aN. These are the results of Theorem 3 in Section 3.1 for n positive and of Theorem 4 from Section 3.2 for n negative.

As a consequence of Theorem 3, we calculate explicitly a rational solution of the Painlevé VI equation with the parameters α=(n+1)22,β=n218,γ=n218,δ=9n218,for each positive integer n not divisible by 3, additionally observing quite a regular asymptotic behaviour of its zeros and poles with respect to n tending to infinity, see Section 4.1, Theorem 5 and Proposition 1. In the same fashion, Theorem 6 from Section 4.2 gives a one-parameter family of rational solutions of the Painlevé VI equation with the parameters α=(3n+1)22,β=n22,γ=n22,δ=1n22,for each negative integer n. The last Section 5 is devoted to the applications to Garnier systems. Some algebraic solutions of particular Garnier systems are computed explicitly in Section 5.1, Theorem 7, and Section 5.2, Theorem 8.

Section snippets

An upper triangular Schlesinger system

Let us note that the generally non-linear system (1) in the case of triangular (p×p)-matrices B(i) splits into a set of p(p1)2 inhomogeneous linear systems, each system has N unknowns b1kl(a),,bNkl(a) with k,l fixed. Indeed, first for each fixed k=1,,p1 one considers a homogeneous linear system dbik,k+1(a)=j=1,jiN(βik,k+1bjk,k+1(a)βjk,k+1bik,k+1(a))d(aiaj)aiaj,withβik,k+1=βikβik+1,whereβik=bikk, with respect to the unknowns b1k,k+1(a),,bNk,k+1(a). Written in a vector form for the

Polynomial and rational solutions of the Schlesinger system

Our differentials Ωi(j)(a) defined on the compact Riemann surface Xa have poles at points at infinity or at finite ramification points, depending on the sign of n. In general, the residues at these poles of Ωi(j)(a) are non-zero and, according to Theorem 1, give rise to solutions of the Schlesinger system (2). In this section we show that such solutions are polynomial in a1,,aN in the case of n>0 and rational in a1,,aN in the case of n<0. This will lead us, in subsequent sections, to rational

Application to Painlevé VI equations

As is well known, in the case p=2, N=3 (assuming (a1,a2,a3)=(0,1,x), x{0,1}) the Schlesinger system for traceless (2×2)-matrices B(1)(x), B(2)(x), B(3)(x), dB(1)dx=[B(3),B(1)]x,dB(2)dx=[B(3),B(2)]x1,B(1)+B(2)+B(3)=β00β(if β=0, the last matrix sum is a Jordan cell), corresponds to the sixth Painlevé equation PVI(α,β,γ,δ) d2ydx2=121y+1y1+1yxdydx21x+1x1+1yxdydx +y(y1)(yx)x2(x1)2α+βxy2+γx1(y1)2+δx(x1)(yx)2.

The parameters (α,β,γ,δ) of PVI are computed from the eigenvalues ±βi

Application to Garnier systems

Here we consider Garnier systems GM(θ) (a multidimensional generalization of Painlevé VI equations) depending on M+3 complex parameters θ1,,θM+2,θ. These are completely integrable PDEs systems of second order [42], [43]. They can be written in a Hamiltonian form obtained by K. Okamoto [44], uiaj=Hjvi,viaj=Hjui,i,j=1,,M,for the unknown functions (u,v)=(u1,,uM,v1,,vM) of the variable a=(a1,,aM), where the Hamiltonians Hj=Hj(a,u,v,θ) are rational functions of their arguments (see

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

We thank Vladimir Leksin who had drawn attention of the second author to the paper [28], which has led to the present work, as well as Irina Goryuchkina for helping us to verify by Maple the solutions of Example 4 (they indeed satisfy the corresponding Garnier systems!).

We thank the anonymous referees for useful suggestions which improved this paper.

V. S. is grateful to the Natural Sciences and Engineering Research Council of Canada for the financial support through a Discovery grant and to the

References (57)

  • BennettM.A. et al.

    Rational points on Erdös–Selfridge superelliptic curves

    Compos. Math.

    (2016)
  • BuchstaberV.M. et al.

    σ-Function of (n,s)-curves

    Russian Math. Surveys

    (1999)
  • EnolskiV. et al.

    Singular ZN curves, Riemann–Hilbert problem and modular solutions of the Schlesinger equation

    Int. Math. Res. Not. IMRN

    (2004)
  • EnolskiV. et al.

    Thomae type formulae for singular ZN curves

    Lett. Math. Phys.

    (2006)
  • HidalgoR. et al.

    On the field of moduli of superelliptic curves

  • LegrandF.

    Twists of superelliptic curves without rational points

    Int. Math. Res. Not. IMRN

    (2018)
  • MatsutaniS. et al.

    The al function of a cyclic trigonal curve of genus three

    Collect. Math.

    (2015)
  • OcchipintiT. et al.

    Low-dimensional factors of superelliptic Jacobians

    Eur. J. Math.

    (2015)
  • ProkopchukA.V. et al.

    Rational points of superelliptic curves, and polynomial mappings of fields (Russian)

    Vesci Nac. Akad. Navuk Belarusi. Ser. Fiz.-Mat. Navuk

    (2006)
  • J.W.Sander

    Rational points on a class of superelliptic curves

    J. Lond. Math. Soc. (2)

    (1999)
  • XueJ. et al.

    Centers of hodge groups of superelliptic Jacobians

    Transform. Groups

    (2010)
  • Yu.G.Zarhin

    Superelliptic Jacobians

  • DubrovinB. et al.

    On the reductions and classical solutions of the schlesinger equations

  • GontsovR.

    On movable singularities of garnier systems

    Math. Notes

    (2010)
  • GontsovR. et al.

    On the reducibility of schlesinger isomonodromic families

  • MimachiK. et al.

    Irreducibility and reducibility of lauricella’s system of differential equations ED and the Jordan–pochhammer differential equation EJP

    Kyushu J. Math.

    (2012)
  • DeiftP. et al.

    On the algebro-geometric integration of the schlesinger equations

    Comm. Math. Phys.

    (1999)
  • KitaevA. et al.

    On solutions of the schlesinger equations in terms of Θ-functions

    Int. Math. Res. Not. IMRN

    (1998)
  • View full text