Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

Sigma-function solution to the general Somos-6 recurrence via hyperelliptic Prym varieties

Yuri N. Fedorov and Andrew N. W. Hone Department of Mathematics I, Politechnic university of Catalonia, Barcelona, Spain.   E-mail: Yuri.Fedorov@upc.eduSchool of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury CT2 7NF, U.K.   E-mail: A.N.W.Hone@kent.ac.uk
Abstract

We construct the explicit solution of the initial value problem for sequences generated by the general Somos-6 recurrence relation, in terms of the Kleinian sigma-function of genus two. For each sequence there is an associated genus two curve , such that iteration of the recurrence corresponds to translation by a fixed vector in the Jacobian of . The construction is based on a Lax pair with a spectral curve of genus four admitting an involution with two fixed points, and the Jacobian of arises as the Prym variety Prym.

1 Introduction

Somos sequences are integer sequences generated by quadratic recurrence relations, which can be regarded as nonlinear analogues of the Fibonacci numbers. They are also known as Gale-Robinson sequences, and as well as arising from reductions of bilinear partial difference equations in the theory of discrete integrable systems, they appear in number theory, statistical mechanics, string theory and algebraic combinatorics [4, 10, 12, 15].

This article is concerned with the general form of the sixth-order recurrence

(1.1)

with three arbitrary coefficients . It was an empirical observation of Somos [29] that in the case the initial values generate a sequence of integers (A006722 in Sloane’s Online Encyclopedia of Integer Sequences), which begins

(1.2)

Consequently, the relation (1.1) with generic coefficients is referred to as the Somos-6 recurrence, and the corresponding sequence as a Somos-6 sequence.

The first proof that the original Somos-6 sequence (1.2) consists entirely of integers was an unpublished result of Hickerson (see [16]); it relied on showing that the Somos-6 recurrence has the Laurent property, meaning that the iterates are Laurent polynomials in the initial data with integer coefficients. To be precise, in the general case the iterates satisfy

which was proved by Fomin and Zelevinsky as an offshoot of their development of cluster algebras [15]. The latter proof made essential use of the fact that (1.1) is a reduction of the cube recurrence, a partial difference equation which is better known in the theory of integrable systems as Miwa’s equation, or the bilinear form of the discrete BKP equation (see [9], for instance). In the general case , (1.1) does not arise from mutations in a cluster algebra, although it does appear in the broader framework of Laurent phenomenon algebras [23].

As was found independently by several people (see e.g. [18, 19, 27, 28] and references), the sequences generated by general bilinear recurrences of order 4 or 5 are associated with sequences of points on elliptic curves, and can be written in terms of the corresponding Weierstrass sigma-function. It was shown in [20] that sequences produced by (1.1) are the first ones which go beyond genus one: in general, they are parametrized by a sigma-function in two variables. To be precise, given a genus 2 algebraic curve defined by the affine model

(1.3)

in the plane, let denote the associated Kleinian sigma-function with , as described in [2] (see also [6, 7]). It gives rise to the Kleinian hyperelliptic functions , which are meromorphic on the Jacobian variety and generalize the Weierstrass elliptic function.

Theorem 1 ([20]).

For arbitrary , the sequence with th term

(1.4)

satisfies the recurrence (1.1) with coefficients

(1.5)

where

(1.6)

provided that satisfies the constraint

(1.7)

The preceding statement differs slightly from that of Theorem 1.1 in [20], in that we have used an alternative (but equivalent) expression for in (1.6), and have included an additional parameter which is needed in what follows. Now while the above result means that the expression (1.4) is a solution of (1.1) with suitable coefficients, it does not guarantee that it is the general solution, in the sense that the sequence can always be written in this way, for a generic choice of initial data and coefficients. The ultimate purpose of this paper is to show that this is indeed the case. Our main result is the solution of the initial value problem by explicit reconstruction of the parameters appearing in (1.4), which yields the following.

Theorem 2.

For a sequence of complex numbers generated by the recurrence (1.1) with generic values of the initial data and coefficients , there exists a genus 2 curve with affine model (1.3) and period lattice , points with satisfying (1.7), and constants such that the terms and coefficients are parametrized by the corresponding Kleinian functions according to (1.4) and (1.5), respectively.

In order to solve the reconstruction problem, it will be convenient to work with a reduced version of the Somos-6 recurrence. The parameters in (1.4) correspond to the group of scaling symmetries , which maps solutions to solutions, and considering invariance under this symmetry leads to certain quantities , as described in the next paragraph. The parameter corresponds to covariance under the further scaling , which maps solutions of (1.1) to solutions of the same recurrence with rescaled coefficients; in due course we will consider quantities that are also invariant with respect to this additional symmetry.

The reduced Somos-6 map.

Sequences generated by iteration of the Somos-6 recurrence (1.1) are equivalent to the orbits of the birational map

As was observed in [20], this map is Poisson with respect to the log-canonical bracket , which has four independent Casimir functions

(1.8)

these quantities are also invariant under the scaling transformation . The map induces a recurrence of order 4 for a corresponding sequence , that is

(1.9)

which is equivalent to iteration of a birational map in with coordinates . We will refer to as the reduced Somos 6 map.

The map defined by (1.9) preserves the meromorphic volume form

for arbitrary values of , and has two independent rational first integrals, here denoted , which are presented explicitly in section 2 below. According to [20], the map is also integrable in the Liouville–Arnold sense [25], at least in the case . In this paper are concerned with the general case , where a symplectic structure for the map is not known.

On the other hand, a genus 2 curve (1.3) and the corresponding sigma-function solution (1.4), (1.5) of the Somos-6 map imply that the solution of (1.9) is

(1.10)

In view of the addition formula for the genus 2 sigma-function [2], the right hand side of (1.10) can be written in terms of Kleinian functions as

(1.11)

where . Note that, when is considered as a function on the Jacobian, is singular if and only if , the theta divisor in (using the notation in [6]). Then, upon setting , we have the map

(1.12)

which is a meromorphic embedding , where denotes the theta divisor together with its translates by and . Once Theorem 2 is proved (see section 6), we are able to recover , and from the coefficients and initial data of the map, so that we arrive at

Theorem 3.

Generic complex invariant manifolds of the map are isomorphic to open subsets of .

For the purposes of our discussion, it will be more convenient to describe the reduced Somos-6 map in an alternative set of coordinates. We introduce the quantities

(1.13)

so that , and are birationally related to . Thus, after conjugating by a birational change of variables, we can rewrite it in the form , where

(1.14)

with the coefficients

(1.15)

Observe that, from the analytic formulae (1.5) and (1.10), the quantities and the coefficients are independent of the parameter .

Outline of the paper.

In the next section, we describe the first of our main tools, namely the Lax pair for the map , which (as announced in [20]) is obtained from the associated Lax representation for the discrete BKP equation. The corresponding spectral curve yields the first integrals . However, is not the required genus 2 curve , but rather it is trigonal of genus 4, having an involution with two fixed points. Then it turns out that the 2-dimensional Jacobian of , which is the complex invariant manifold of the map according to Theorem 3, can be identified with the Prym subvariety of . (An analogous situation was described recently for an integrable Hénon-Heiles system [11].)

To obtain an explicit algebraic description of and, therefore, of the curve , we make use of recent work by Levin [24] on the general case of double covers of hyperelliptic curves with two branch points. All relevant details are given in section 3.

In section 4 it is shown how the discrete Lax pair allows a description of the map as a translation on by a certain vector. This translation is subsequently identified with a specific degree zero divisor on representing the required vector , and in section 5 we also explicitly find degree zero divisors on representing the vectors . This enables us to rewrite the determinantal constraint (1.7) in terms of the above three divisors, and then observe that it is trivially satisfied.

In section 6, all of the required ingredients are ready to present the reconstruction of the sigma-function solution (1.4) from the initial data and coefficients, which proves Theorem 2. We also provide a couple of explicit examples, including the original Somos-6 sequence (1.2). The paper ends with some conclusions, followed by an appendix which includes the derivation of the Lax pair and another technical result.

2 The Lax pair, its spectral curve, and related Jacobian varieties

The key to the solution of the initial value problem for the Somos-6 recurrence is the Lax representation of the map .

Theorem 4.

The mapping is equivalent to the discrete Lax equation

(2.1)

with

(2.2)
(2.3)

where as in (1.14), and

  • The equation (2.1) can be checked directly with computer algebra. For the rather more straightforward origin of this complicated-looking Lax pair, see the appendix. ∎

The characteristic equation defines the spectral curve , which, after elimination of the common factor , is given by

(2.4)

where are independent first integrals, namely

(2.5)
(2.6)
Remark 5.

Replacing the variables by the expressions (1.13) yields the first integrals of the reduced map in the original variables . These are seen to be rescaled versions of the quantities derived in [20] from higher order bilinear relations, according to

(2.7)

One can also verify that, for generic values of , the complex invariant manifold is irreducible.

The curve is trigonal of genus 4 and has an interesting involution with two fixed points, namely and .

We compactify by embedding it in with homogeneous coordinates , where . The compact curve has a singularity at . After regularization, this point gives two points at infinity: the first one is with the Laurent expansion

with respect to a local parameter near ; and the second is , with the Laurent expansion

The third point at infinity comes from and has the expansion

Under the action of , these points are in involution with the following three finite points:

The above three pairs of points on will play an important role, so we depict them on the diagram above, with arrows denoting the involution .

The curve can be viewed as a 3-fold cover of with affine coordinate . As follows from the above description, the points are ordinary branch points of the covering, and there is no branching at the points . It follows that the divisors of zeros and poles of the coordinates on are

(2.8)

Observe that a generic complex 2-dimensional invariant manifold of the reduced Somos-6 map cannot be the Jacobian of , as the latter has genus 4. The curve is a 2-fold covering of a curve whose genus is 2, by the Riemann–Hurwitz formula. The involution extends to which then contains two Abelian subvarieties: the Jacobian of , which is invariant under , and the 2-dimensional Prym variety, denoted , which is anti-invariant with respect to . It will play a key role in the description of the complex invariant manifolds of the map . For this purpose it is convenient to recall some properties of Prym varieties corresponding to our case.

3 Hyperelliptic Prym varieties

Generic double cover of a hyperelliptic curve with two branch points.

Consider a genus hyperelliptic curve : , where is a polynomial of degree with simple roots. As was shown in [24], any double cover of ramified at two finite points (which are not related to each other by the hyperelliptic involution on , i.e., ) can be written as a space curve of the form

(3.1)

where is a polynomial of degree such that

with being a polynomial of degree . (Here or may or may not coincide with roots of .) Thus admits the involution , with fixed points . Then the genus of is , and it was shown by Mumford [26] and Dalaljan [8] that

  • contains two -dimensional Abelian subvarieties: Jac and the Prym subvariety , with the former invariant under the extension of to , and the latter anti-invariant;

  • Prym is principally polarized and is the Jacobian of a hyperelliptic curve .

It was further shown recently by Levin [24] that the second curve can be written explicitly as

(3.2)

which is equivalent to the plane curve . The latter can be transformed explicitly to a hyperelliptic form by an algorithm given in [24].

In order to apply the above results to obtain an explicit description of the Prym variety in our case, we will need

Proposition 6.

1) The quotient of by the involution is the genus 2 curve given by the equation

(3.3)

where , ,

The double cover is described by the relations

(3.4)

and the images of the branch points on are .

2) The curve is equivalent to the following curve in hyperelliptic form:

(3.5)

The birational transformation between and is described by the relations

(3.6)
(3.7)

The branch points on are, respectively, with

(3.8)
(3.9)
  • Applying the substitution (3.4) to the polynomial we get the product , where

    and this product is zero due to the equation (2.4). Hence satisfy . The proof of the other items is a direct calculation (which we made with Maple). ∎

is isomorphic to the Jacobian of a second genus 2 curve