A review of elliptic difference Painlevé equations
aa r X i v : . [ n li n . S I] F e b A REVIEW OF ELLIPTIC DIFFERENCE PAINLEV ´E EQUATIONS
NALINI JOSHI AND NOBUTAKA NAKAZONOA bstract . Discrete Painlev´e equations are nonlinear, nonautonomous di ff erence equations of second-order. They have coe ffi cients that are explicit functions of the independent variable n and there arethree di ff erent types of equations according to whether the coe ffi cient functions are linear, exponentialor elliptic functions of n . In this paper, we focus on the elliptic type and give a review of the construc-tion of such equations on the E lattice. The first such construction was given by Sakai [38]. We focuson recent developments giving rise to more examples of elliptic discrete Painlev´e equations.
1. I ntroduction
Discrete Painlev´e equations are nonlinear integrable ordinary di ff erence equations of second or-der. They have a long history (see § ffi cients are iterated on elliptic curves. Sakai’s equation is an elliptic di ff erence Painlev´eequation.Sakai’s unification is based on a deep geometric theory shared by all the discrete Painlev´e equa-tions, first described by Okamoto [33] for the classical Painlev´e equations (see § ffi ne root systems.This led Sakai to describe discrete Painlev´e equations as the result of translations on latticesdefined by a ffi ne Weyl groups [19]. In particular, Sakai’s elliptic di ff erence equation [38] is iteratedon the lattice generated by the a ffi ne exceptional Lie group E (1)8 (see also [30, 32]). More recently,other elliptic di ff erence equations of Painlev´e type have been discovered [4, 6, 21, 36] through otherapproaches. This review concentrates on describing the construction of such elliptic di ff erencePainlev´e equations by using Sakai’s geometric way.To describe the construction that underlies and explains all of these examples, we rely on thefollowing mathematical description. Fix a point in the E (1)8 lattice [7]. Then there are 240 nearestneighbors of this point in the lattice, lying at a distance whose squared length is equal to 2. Werefer to the 120 vectors between the initial fixed point and its possible nearest neighbors as nearest-neighbor-connecting vectors (NVs). Similarly, there are 2160 next-nearest neighbors, lying at adistance whose squared length is 4. The 1080 vectors between the fixed point and such next-nearestneighbors will be referred to as next-nearest-neighbor-connecting vectors (NNVs). Sakai’s ellipticdi ff erence equation is constructed in terms of translations expressed in terms of NVs. However,more recently deduced examples are obtained from NNVs. The example that led us to this key observation is the following elliptic di ff erence Painlev´e equa-tion, originally found by Ramani, Carstea and Grammaticos [36]: y n + (cid:16) k (cg − cz n )cz n dz n x n y n − (1 − k sz n )cg e dg e x n + (1 − k sg sz n )cz n dz n y n (cid:17) = (1 − k sz n )cg e dg e x n y n − (cg − cz n )cz n dz n − (1 − k sg sz n )cz n dz n x n , x n + (cid:16) k (cg − b cz n ) b cz n b dz n y n + x n − (1 − k b sz n )cg o dg o y n + + (1 − k sg b sz n ) b cz n b dz n x n (cid:17) = (1 − k b sz n )cg o dg o y n + x n − (cg − b cz n ) b cz n b dz n − (1 − k sg b sz n ) b cz n b dz n y n + , (1.1)where sz n = sn ( z n ) , b sz n = sn ( z n + γ e + γ o ) , sg e = sn ( γ e ) , (1.2a)sg o = sn ( γ o ) , cz n = cn ( z n ) , b cz n = cn ( z n + γ e + γ o ) , (1.2b)cg e = cn ( γ e ) , cg o = cn ( γ o ) , dz n = dn ( z n ) , (1.2c) b dz n = dn ( z n + γ e + γ o ) , dg e = dn ( γ e ) , dg o = dn ( γ o ) , (1.2d)and z n = z + γ e + γ o ) n . We will refer to this equation as the RCG equation. Here, sn, cn and dnare Jacobi elliptic functions and k is the modulus of the elliptic sine. For more information aboutJacobian elliptic functions and notations, see [9, Chapter 22] and [40].It turns out that the RCG equation (1.1) is a projective reduction of an NNV, i.e., the iterativestep is not a translation on the E (1)8 lattice, but its square is a translation corresponding to a NNV.In general, we can derive various discrete Painlev´e equations from elements of infinite order in thea ffi ne Weyl group that are not necessarily translations by taking a projection on a certain subspaceof the parameters. Kajiwara et al studied such procedures [23, 24] to obtain non-translation typediscrete Painlev´e equations and named these “projective reductions”. The RCG equation (1.1) is thefirst known case of an elliptic di ff erence Painlev´e equation obtained from such a reduction.We constructed a discrete Painlev´e equation, which has the RCG equation as a projective reduc-tion, by using NNVs on the E (1)8 lattice in [21]. Most of discrete Painlev´e equations admit the specialsolutions expressible in terms of solutions of linear equations when some of the parameters takespecial values (see, for example, [25] and references therein). It is known that projectively-reducedequations have di ff erent type of such solutions from translation-type discrete Painlev´e equations onthe same lattice [23, 24]. In this paper, we will also show the special solutions of Equation (1.1),which is quite di ff erent from those of a translation-type elliptic di ff erence Painlev´e equation reportedin [22, 25].1.1. Background.
Shohat studied polynomials Φ n ( x ) indexed by degree n ∈ N , defined in an inter-val ( −∞ , ∞ ), with a weight function p ( x ) = exp( − x /
4) such that Z ∞−∞ p ( x ) Φ m ( x ) Φ n ( x ) dx = , ( m , n , m , n ∈ N ) . (1.3)Shohat obtained the 3-term recurrence relation (see Equation (39) of [39]) Φ n ( x ) − ( x − c n ) Φ n − ( x ) + λ n Φ n − ( x ) = , n ≥ , (1.4)where Φ ( x ) ≡ Φ ( x ) = x − c , where c is independent of x , and deduced the following di ff erenceequation for λ n : λ n + (cid:0) λ n + + λ n + + λ n + (cid:1) = n + . (1.5)We now know that this equation is intimately related to one of the six classical Painlev´e equations,universal classes of second-order ordinary di ff erential equations (ODEs) studied by Painlev´e [34], REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 3
Fuchs [13] and Gambier [14]. Fokas et al [12] showed that the solutions of Equation (1.5) aresolutions of the fourth Painlev´e equation:P IV : w ′′ = w ′ w + w + tw + t − α ) w + β w . Actually, solutions of P IV : w = w n , n ∈ Z , satisfy a more general version of Shohat’s equation givenby w n (cid:0) w n + + w n + w n − (cid:1) = a n + b + c ( − n + d w n , (1.6)where a , b , c , d are constants (see [11]). This equation is an integrable equation in its own right,with fundamental properties such as a Lax pair [8].Equation (1.6) is one of many equations now known as discrete Painlev´e equations. In general,there exist three types of discrete Painlev´e equations. They are distinguished by the types of function t n appearing in the coe ffi cient of each equation.(i) If there exists k ∈ Z > such that t n + k − t n is a constant, then the equation is said to be of additive-type .(ii) If there exists k ∈ Z > such that t n + k / t n is a constant, denoted q ( , , multiplicative-type or q-di ff erence-type .(iii) If t n can be expressed by the elliptic function of n , then the equation is said to be of elliptic-type .We list a few discrete Painlev´e equations here.d-P II [35] : X n + + X n − = t n X n + a − X n , (1.7) q -P III [37] : X n + X n − = ( X − at n )( X − a − t n )( X − b )( X − b − ) , (1.8)d-P IV [37] : ( X n + + X n )( X n + X n − ) = ( X n − a )( X n − b )( X n − t n ) − c , (1.9) q -P V [37] : ( X n + X n − X n X n − − = t n ( X n − a )( X n − a − )( X n − b )( X n − b − )( X n − ct n )( X n − c − t n ) , (1.10)where a , b and c are constants. Here, t n + − t n is a constant for Equations (1.7) and (1.9) and t n + / t n is a constant for Equations (1.8) and (1.10), that is, Equations (1.7) and (1.9) are additive-type,while Equations (1.8) and (1.10) are multiplicative-type. Note that the notation for each equationoriginates from their discrete types and continuum limits. Moreover, an example of elliptic-type isgiven by Equation (1.1).Okamoto [33] described a geometric framework for studying the Painlev´e equations. He showedthat the initial-value (or phase) space of the Painlev´e equations, which is a foliated vector bundle[28], can be compactified and regularised by a minimum of eight blow-ups on a Hirzebruch surface(or nine in P ).This geometric theory also leads to a description of their symmetry groups, described in terms ofa ffi ne Weyl groups [31]. Such symmetries lead to transformations of the Painlev´e equations called B¨acklund transformations .Sakai’s geometric description of discrete Painlev´e equations, based on types of space of initialvalues, is well known [38]. This picture relies on compactifying and regularizing space of initialvalues. The spaces of initial values are constructed by blow up of P × P at base points (see §
3) andare classified into 22 types according to the configuration of the base points as follows:
NALINI JOSHI AND NOBUTAKA NAKAZONO
Discrete type Type of surfaceElliptic A (1)0 Multiplicative A (1) ∗ , A (1)1 , A (1)2 , A (1)3 , . . . , A (1)8 , A (1) ′ Additive A (1) ∗∗ , A (1) ∗ , A (1) ∗ , D (1)4 , . . . , D (1)8 , E (1)6 , E (1)7 , E (1)8 In each case, the root system characterizing the surface forms a subgroup of the 10-dimensionalPicard lattice. The symmetry group of each equation, formed by Cremona isometries, arises fromthe orthogonal complement of this root system inside the Picard lattice.1.2.
Periodic reduction of the Q4-equation.
In this section, we recall how to obtain Equation(1.1) from the lattice Krichever-Novikov system [2,17] (or, Q4 in the terminology of Adler-Bobenko-Suris [3]): sn ( α l ) ( u l , m u l + , m + u l , m + u l + , m + ) − sn ( β m ) ( u l , m u l , m + + u l + , m u l + , m + ) − sn ( α l − β m ) (cid:0) u l + , m u l , m + + u l , m u l + , m + (cid:1) + sn ( α l ) sn ( β m ) sn ( α l − β m ) (cid:0) + k u l , m u l + , m u l , m + u l + , m + (cid:1) = , (1.11)where α l and β m are parameters, u l , m is the dependent variable, l and m are independent variables(often taken to be integer) and k is the modulus of the elliptic function sn. Taking a periodic reduc-tion u l + , m − = u l , m of Equation (1.11), and identifying n = l + m , leads to an autonomous secondorder ordinary di ff erence equation [20]: (cid:0) sn ( α ) − sn ( β ) (cid:1) u n ( u n + + u n − ) − sn ( α − β ) ( u n + u n − + u n ) + sn ( α ) sn ( β ) sn ( α − β ) (1 + k u n u n + u n − ) = . (1.12)Ramani et al [36] deautonomised Equation (1.12) by the method of singularity confinement after achange of variables α → γ + z , β → γ − z . The resulting equation then becomes Equation (1.1)with x n = u n and y n = u n − .1.3. Outline of the paper. In §
2, we recall the basic definitions of reflection group theory anddefine translations, for the interested reader. The initial-value space of the RCG equation (1.1) isconstructed in §
3, where we also introduce related algebro-geometric concepts. In §
4, we constructCremona isometries on this initial value space, which roughly speaking, are mappings that preserveits geometric structure. In §
5, we give the resulting birational actions on the coordinates and param-eters of the initial value space. We show that by using these birational actions, we arived at the RCGequation. Some explicit special solutions of the RCG equation are described in §
6. In AppendixA we illustrate the geometric ideas for the case of A (1)4 , and in Appendix B we describe the genericknown examples of elliptic di ff erence Painlev´e equations.2. E (1)8 - lattice To understand how to construct discrete Painlev´e equations from symmetry groups, we need thetheory of finite and a ffi ne reflection groups. In this section, we recall the basic definitions of reflec-tion group theory before defining the transformation group W ( E (1)8 ) and describing its translationsoperators.Consider two n -dimensional real vector spaces V and V ∗ , spanned by the basis sets ∆ = (cid:8) α , . . . , α n (cid:9) and ∆ ∨ = (cid:8) α ∨ , . . . , α ∨ n (cid:9) respectively. The elements of ∆ are called simple roots , while those of ∆ ∨ are simple coroots . To define reflections, we use a bilinear pairing given by the entries of an n × n Cartan matrix A = ( A i j ) (see the definition of Cartan matrices in [5]): h α ∨ i , α j i = A i j , (2.1)for all i , j ∈ { , . . . , n } . If V and V ∗ are Euclidean spaces, this bilinear pairing is the usual innerproduct. REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 5
It is also important to define the fundamental weights h i , i = , . . . , n , which are given by h α ∨ i , h j i = δ i j , (1 ≤ i , j ≤ n ) . (2.2)Correspondingly, the integer linear combinations (or Z -modules) Q = n X k = Z α k , Q ∨ = n X k = Z α ∨ k , P = n X k = Z h k , (2.3)are called the root lattice , coroot lattice and weight lattice respectively. We are now in a position todefine reflections. For each i ∈ { , . . . , n } , the linear operator defined by s α i ( α j ) = α j − A ji α i , s α i ( α ∨ j ) = α ∨ j − A ji α ∨ i (2.4)for every j ∈ { , . . . , n } is a reflection operator . That is, it has the following properties:(i) s α i ( α i ) = − α i , s α i ( α ∨ i ) = − α ∨ i .(ii) s α i = s α i . h α ∨ j , α k i = h s α i ( α ∨ j ) , s α i ( α k ) i = h α ∨ j , α k i .The group W generated by s α , . . . , s α n is called a Weyl group. The root system of W is definedto be the subset Φ of Q given by Φ = W ( ∆ ). A root system is said to be irreducible if it is not acombination of mutually orthogonal root systems. Each irreducible root system Φ contains a uniqueroot given by e α = X i C i α i , (2.5)whose height (i.e., the sum of coe ffi cients in the expansion) is maximal, which is called the highestroot .For our case, n =
8, and the starting point is the Cartan matrix of type E A = − − − − − −
10 0 − − − − − − − − . (2.6)The astute reader might notice that this is not the standard one given in Bourbaki [5], but it is equiv-alent to it (under conjugate transforms of the bases). We make this choice because it correspondsto the identification of simple roots made in Section 4. Moreover, the highest root and coroot aregiven respectively by e α = α + α + α + α + α + α + α + α , (2.7a) e α ∨ = α ∨ + α ∨ + α ∨ + α ∨ + α ∨ + α ∨ + α ∨ + α ∨ . (2.7b)We now expand the root system by defining α by α + e α =
0. The corresponding extension ofthe coroot system is defined by α ∨ and δ ∨ (called the null coroot ): α ∨ + e α ∨ = δ ∨ . To construct thecorresponding a ffi ne Weyl group, we now extend the Cartan matrix A by adding a row and columngiven respectively by A j = h− e α ∨ , α j i , A j = h α ∨ j , − e α i , (2.8)along with A =
2. The extended Cartan matrix and corresponding root systems and groups arenow denoted with a superscript containing (1) to denote this extension.
NALINI JOSHI AND NOBUTAKA NAKAZONO
The extended Cartan matrix of type E (1)8 is given by A (1) = ( A i j ) i , j = = − − − − − − −
10 0 0 − − − − − − − − − . (2.9)The reflections have the form s i ( λ ) = λ − h α ∨ i , λ i α i , i = , . . . , , λ ∈ P . (2.10)In particular, this gives s i ( h i ) = h i − α i , i = , . . . ,
8. On the other hand, each root α i can be expressedin terms of weights h i by using the relationship α i = X j = A i j h j . Therefore, we obtain the reflections on the fundamental weights as follows: s ( h ) = − h + h , s ( h ) = − h + h , s ( h ) = h − h + h , s ( h ) = h − h + h + h , s ( h ) = h − h + h , s ( h ) = h − h + h , s ( h ) = h − h + h , s ( h ) = h + h − h , s ( h ) = h − h . (2.11)It is useful to express the actions of the reflections on the roots, and we obtain s : ( α , α ) ( − α , α + α ) , s : ( α , α ) ( − α , α + α ) , s : ( α , α , α ) ( α + α , − α , α + α ) , s : ( α , α , α , α ) ( α + α , − α , α + α , α + α ) , s : ( α , α , α ) ( α + α , − α , α + α ) , s : ( α , α , α ) ( α + α , − α , α + α ) , s : ( α , α , α ) ( α + α , − α , α + α ) , s : ( α , α ) ( α + α , − α ) , s : ( α , α ) ( α + α , − α ) . (2.12)Note that fundamental weights and simple roots which are not explicitly shown in Equation (2.11)and (2.12) remain unchanged under the action of the corresponding reflections.Under the linear actions on the weight lattice (2.10), W ( E (1)8 ) = h s , . . . , s i forms an a ffi ne Weylgroup of type E (1)8 . Indeed, the following fundamental relations hold:( s i s j ) l ij = , where l i j = , i = j , i = j − , j = , . . . , , or ( i , j ) = (3 , , (7 , , otherwise . (2.13)Note that representing the simple reflections s i by nodes and connecting i -th and j -th nodes by( l i j −
2) lines, we obtain the Dynkin diagram of type E (1)8 shown in Figure 1. Remark 2.1.
We can also define the action of W ( E (1)8 ) on the coroot lattice Q ∨ by replacing α i with α ∨ i in the actions (2.12) . Note that δ ∨ is invariant under the action of W ( E (1)8 ) . Then, the REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 7
Figure 1. Dynkin diagram of type E (1)8 . transformations in W ( E (1)8 ) preserve the form h· , ·i , that is, the following holds:w . h γ, λ i = h w ( γ ) , w ( λ ) i = h γ, λ i , (2.14) for arbitrary w ∈ W ( E (1)8 ) , γ ∈ Q ∨ and λ ∈ P. For each root α ∈ Q , we define the Kac translation T α on the weight lattice P by T α ( λ ) = λ − h δ ∨ , λ i α, λ ∈ P , (2.15)and on the coroot lattice Q by T α ( λ ∨ ) = λ ∨ + h λ ∨ , α i δ ∨ , λ ∨ ∈ Q . (2.16)We can easily verify that under the linear actions above the translation T α have the following prop-erties.(i) For any α, β ∈ Q , T α ◦ T β = T α + β .(ii) For any w ∈ W ( E (1)8 ) and α ∈ Q , w ◦ T α = T w ( α ) ◦ w . (2.17)(iii) For any α, β ∈ Q , T α ( β ) = β .For any α = P i = c i α i ∈ Q , we define the squared length of T α by | T α | : = h α ∨ , α i , (2.18)where α ∨ = P i = c i α ∨ i . Then, from (2.14) and (2.17), we obtain the following lemma. Lemma 2.2.
For any α, β ∈ Q, if T α and T β are conjugate to each other in W ( E (1)8 ) , then the squaredlengths of T α and T β are equal.Proof. Assume that T α and T β , where α, β ∈ Q , are conjugate to each other in W ( E (1)8 ). Let w ◦ T α ◦ w − = T β , where w ∈ W ( E (1)8 ). Since of (2.17), we obtain w ◦ T α ◦ w − = T w ( α ) , which gives T w ( α ) = T β . (2.19)Since of (2.14) and (2.18), the statement follows from | T α | = | T w ( α ) | = | T β | . (2.20) (cid:3) In [30] Murata et al consider the following translation as the time evolution of the Sakai’s ellipticdi ff erence equation: T ( M ) = s , (2.21)whose action on the coroot lattice Q ∨ is given by T ( M ) ( α ∨ ) = α ∨ + δ ∨ , T ( M ) ( α ∨ ) = α ∨ − δ ∨ , (2.22)while in [4, 21] Joshi et al showed that the squared time evolution of the RCG equation correspondsto the following translation: T ( JN ) = s , (2.23) NALINI JOSHI AND NOBUTAKA NAKAZONO whose action on the coroot lattice Q ∨ is given by T ( JN ) ( α ∨ ) = α ∨ − δ ∨ , T ( JN ) ( α ∨ ) = α ∨ + δ ∨ . (2.24)Note that for convenience we use the following notations for the composition of the reflections s i : s i ··· i m = s i ◦ · · · ◦ s i m , s i ··· i m = s i ··· i m ◦ s i ··· i m , (2.25)where i · · · i m ∈ { , . . . , } . Comparing (2.16) and (2.22) and comparing (2.16) and (2.24), we canrespectively express the translations T ( M ) and T ( JN ) by the Kac translations as the following: T ( M ) = T α , T ( JN ) = T α + α + α + α + α + α + α + α , (2.26)where | T α | = , | T α + α + α + α + α + α + α + α | = . (2.27)Therefore, the translations T ( M ) and T ( JN ) are not conjugate to each other in W ( E (1)8 ) and respectivelycorrespond to a NV and a NNV. Remark 2.3.
As another example, we will show the lattice and transformation group of type A (1)4 in § A.
3. T he initial value space of the
RCG equation
Consider the RCG equation (1.1) as a discrete dynamical system. It is reversible and so thesystem maps ( x n , y n ) to ( x n + , y n + ) at each forward time step or to ( x n − , y n − ) at each backwardstep. Because the coe ffi cients of y n + or x n + may vanish, the iterates may become unbounded andwe compactify this system by embedding it in the projective space P × P .Compactification is not enough to avoid all problems. Let φ and φ − be respectively the forwardand backward time evolution of Equation (1.1). We denote the action of these mappings φ by φ : ( x , y ; γ e , γ o , z ) (cid:0) ˜ x , ˜ y ; γ e , γ o , z + γ e + γ o ) (cid:1) , (3.1a) φ − : ( x , y ; γ e , γ o , z ) (cid:0) x ˜ , y ˜; γ e , γ o , z − γ e + γ o ) (cid:1) . (3.1b)(We obtain Equation (1.1) by writing x n = φ n ( x ), y n = φ n ( y ), z n = φ n ( z ).) Then there exist pointswhere the image values of φ or φ − approach 0 /
0. We refer to such points as base points becausethey are equivalent to the usual definition used in the case of algebraic curves [16]. Below, wedescribe the base points of the RCG equation explicitly.In general, the process of blowing up a finite sequence of points p i , possibly infinitely near eachother, in P × P , leads to a variety X , called the initial value space. Assuming there are 8 blow-ups,then let the sequence of blow-ups be π i : X i X i − of p i in X i − , with X = X → . . . → X = P × P . Each blow-up replaces p i by an exceptional line E i . We refer to the total sequence of blowups by ǫ : X → P × P , and moreover, denote the linear equivalence classes of the total transformof vertical and horizontal lines in P × P respectively by H and H .To find base points of Equation (1.1), we need to find simultaneous zeroes of pairs of polynomi-als. For example, base points arising from the component ˜ y in the action of φ lie at the simultaneoussolutions of (1 − k sz )cg e dg e xy − (cg − cz )cz dz − (1 − k sg sz )cz dz x = , k (cg − cz )cz dz x y − (1 − k sz )cg e dg e x + (1 − k sg sz )cz dz y = , where the dependence on n has been suppressed. These polynomial equations can be solved ex-plicitly. For example, the first equation in the above pair can be solved for y in terms of x , andsubstituting into the second equation leads to a quartic equation for x . The four solutions of thisequation can be expressed explicitly in terms of the coe ffi cient elliptic functions by using ellipticfunction identities. A similar argument gives us four more base points arising from the remainingequations; for ˜ x in the mapping φ and for x ˜ and y ˜ in the mapping φ − . REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 9
These lead us to the eight base points listed below: p : ( x , y ) = (cid:0) cd ( γ o + κ ) , cd ( z − γ e − γ o + κ ) (cid:1) , (3.2a) p : ( x , y ) = (cid:0) cd (cid:0) γ o + i K ′ (cid:1) , cd (cid:0) z − γ e − γ o + i K ′ (cid:1) (cid:1) , (3.2b) p : ( x , y ) = (cid:0) cd ( γ o + K ) , cd ( z − γ e − γ o + K ) (cid:1) , (3.2c) p : ( x , y ) = (cid:0) cd ( γ o ) , cd ( z − γ e − γ o ) (cid:1) , (3.2d) p : ( x , y ) = (cid:0) cd ( z + κ ) , cd ( γ e + κ ) (cid:1) , (3.2e) p : ( x , y ) = (cid:0) cd (cid:0) z + i K ′ (cid:1) , cd (cid:0) γ e + i K ′ (cid:1) (cid:1) , (3.2f) p : ( x , y ) = (cid:0) cd ( z + K ) , cd ( γ e + K ) (cid:1) , (3.2g) p : ( x , y ) = (cid:0) cd ( z ) , cd ( γ e ) (cid:1) , (3.2h)where K = K ( k ) and K ′ = K ′ ( k ) are complete elliptic integrals and κ = K + i K ′ , (3.3)which lie on the elliptic curvesn ( z − γ e ) (1 + k x y ) + z − γ e ) dn ( z − γ e ) xy − ( x + y ) = . (3.4)The base points (3.2) can be generalized to p i : ( x , y ) = (cid:0) cd ( c i + η ) , cd ( η − c i ) (cid:1) , i = , . . . , , (3.5)where c i , i = , . . . ,
8, and η are non-zero complex parameters. These points lie on the elliptic curvesn (2 η ) (1 + k x y ) + η ) dn (2 η ) xy − ( x + y ) = . (3.6)The generalized base points (3.5) and elliptic curve (3.6) can be respectively reduced to the points(3.2) and curve (3.4) by taking c = c + K , c = c + i K ′ , c = c + κ, c = c + K , (3.7a) c = c + i K ′ , c = c + κ, (3.7b)and letting z = η + c + κ, γ e = c − η + κ, γ o = η + c + κ. (3.7c)Note that the two biquadratic curves (3.4) and (3.6) are non-singular, for k , , ±
1. It follows thateach curve is parametrized by elliptic functions. The resulting initial value space obtained afterresolution is of type A (1)0 as shown in the following section.4. C remona isometries As explained in §
3, we now investigate a variety X , obtained after a sequence of blow-ups. Wefocus on surfaces X defined by blowing up base points on biquadratic curves, such as Equations(3.5) and (3.6). The resulting structure contains equivalence classes of lines and information abouttheir intersections. In this section, we construct Cremona isometries, which are roughly speaking,mappings of X that preserve this structure. As a result, they provide symmetries of the dynamicalsystem iterated on X .An important object in this framework is given by the Picard lattice of X , or Pic( X ), which isdefined by Pic( X ) = Z H + Z H + Z E + · · · + Z E , (4.1)where E i = ǫ − ( p i ), i = , . . . ,
8, are exceptional divisors obtained from blow-up of the base points(3.5). We define a symmetric bilinear form, called the intersection form, on Pic( X ) by( H i | H j ) = − δ i j , ( H i | E j ) = , ( E i | E j ) = − δ i j , (4.2) where δ i j =
0, if i , j , or 1, if i = j . The anti-canonical divisor of X is given by − K X = H + H − X i = E i . (4.3)For later convenience, let δ = − K X . The anti-canonical divisor δ corresponds to the curve of bi-degree (2 ,
2) passing through the base points p i , i = , . . . ,
8, with multiplicity 1, that is, the curve(3.6). Since this curve is non-singular, for k , , ±
1, the anti-canonical divisor cannot be decom-posed. Therefore, we can identify the surface X as being of type A (1)0 in Sakai’s classification [38].We define the root lattice Q ( A (1) ⊥ ) = X k = Z β k (4.4)in Pic(X) that are orthogonal to the anti-canonical divisor δ . The simple roots β i , i = , . . . ,
8, aregiven by β = H − H , β = H − E − E , β i = E i − − E i , i = , . . . , ,β = E − E , β = E − E , (4.5)where δ = β + β + β + β + β + β + β + β + β . (4.6)We can easily verify that( β i | β j ) = − , i = j , i = j − j = , . . . , , or if ( i , j ) = (3 , , (7 , , otherwise . (4.7)Representing intersecting β i and β j by a line between nodes i and j , we obtain the Dynkin diagramof E (1)8 shown in Figure 1. Remark 4.1.
From
Pic( X ) we can obtain the coroot lattice, weight lattice and root lattice in § ∨ is given from Pic( X ) by α ∨ i = β i , i = , . . . , , (4.8) the weight lattice P is given from Pic( X ) / ( Z δ ) byh ≡ −E , h ≡ − H , h ≡ − H − H , (4.9a) h ≡ −E − E − E − E − E − E , (4.9b) h ≡ −E − E − E − E − E , h ≡ −E − E − E − E , (4.9c) h ≡ −E − E − E , h ≡ −E − E , (4.9d) h ≡ − H − H + E , (4.9e) and the root lattice Q is given from Pic( X ) / ( Z δ ) by α i ≡ β i , i = , . . . , . (4.10) Note that in this case we define the bilinear pairing h· , ·i : Q ∨ × P → Z for arbitrary α ∨ ∈ Q ∨ andh ∈ P by h α ∨ , h i = − ( α ∨ | h ) . (4.11)Therefore, in a similar manner as in §
2, we can define the transformation group W ( E (1)8 ) asfollows. Definition 4.2 ( [10]) . An automorphism of
Pic (X) is called a Cremona isometry if it preserves (i) the intersection form ( | ) on Pic (X); (ii) the canonical divisor K X ; (iii) e ff ectiveness of each e ff ective divisor of Pic (X).
REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 11
It is well-known that the reflections are Cremona isometries. In this case we define the reflections s i , i = , . . . ,
8, by the following linear actions: s i ( v ) = v + ( v | β i ) β i , (4.12)for all v ∈ Pic( X ). They collectively form an a ffi ne Weyl group of type E (1)8 , denoted by W (cid:0) E (1)8 (cid:1) .Namely, we can easily verify that under the actions (4.12) the fundamental relations (2.13) hold.Moreover, for each root β ∈ Q ( A (1) ⊥ ), we can define the Kac translation T β on the Picard lattice by T β ( λ ) = λ + ( δ | λ ) β − ( β | β )( δ | λ )2 + ( β | λ ) ! δ, λ ∈ Pic( X ) . (4.13)5. B irational actions of the C remona isometries for the J acobi ’ s setting In this section, we give the birational actions of the Cremona isometries on the coordinates andparameters of the base points (3.5). By using these birational actions, we reconstruct Equation (1.1).We focus on a particular example first to explain how to deduce such birational actions. Recall H and H are given by the linear equivalence classes of vertical lines x = constant and horizontallines y = constant, respectively. Applying the reflection operator s given by (4.12) to H , we find s ( H ) = H + H − E − E , which means that s ( y ) can be described by the curve of bi-degree (1 , p and p with multiplicity 1. (See [25] for for more detail.) This resultleads us to the birational action given below in Equation (5.1b). Similarly, from the linear actionsof s i , i = , . . . ,
8, we obtain their birational actions on the coordinates and parameters of the basepoints (3.5) as follows. The actions of the generators of W (cid:0) E (1)8 (cid:1) on the coordinates ( x , y ) are givenby s ( x ) = y , s ( y ) = x , (5.1a) s ( y ) − cd (cid:16) η − c − c (cid:17) s ( y ) − cd (cid:16) η + c − c (cid:17) x − cd ( η + c ) x − cd ( η + c ) ! y − cd ( η − c ) y − cd ( η − c ) ! = − cd ( η − c )cd ( η )1 − cd ( η − c )cd ( η ) − cd ( η + c )cd ( η )1 − cd ( η + c )cd ( η ) − cd (cid:16) η − c − c (cid:17) cd (cid:16) c + c (cid:17) − cd (cid:16) η + c − c (cid:17) cd (cid:16) c + c (cid:17) , (5.1b)while those on the parameters c i , i = , . . . ,
8, and η are given by s ( c ) = c , s ( c ) = c , s ( η ) = − η, s ( η ) = η − η + c + c , (5.1c) s ( c i ) = c i − η + c + c )4 , i = , , c i + η + c + c , i = , . . . , , (5.1d) s k ( c k − ) = c k , s k ( c k ) = c k − , k = , . . . , , (5.1e) s ( c ) = c , s ( c ) = c . (5.1f)Note that λ = P i = c i is invariant under the action of W (cid:0) E (1)8 (cid:1) .For Jacobi’s elliptic function cd ( u ) it is well known that shifts by half periods give the followingrelations: cd ( u + K ) = − cd ( u ) , cd (cid:0) u + i K ′ (cid:1) = k cd ( u ) . (5.2) These identities motivate our search for the transformations that are identity mappings on the Pic( X ).Indeed, we define such transformations ι i , i = , . . . ,
4, by the following actions: ι : ( c , . . . , c , η, x , y ) c − i K ′ , . . . , c − i K ′ , η − i K ′ , kx , y ! , (5.3a) ι : ( c , . . . , c , η, x , y ) c − i K ′ , . . . , c − i K ′ , η + i K ′ , x , ky ! , (5.3b) ι : ( c , . . . , c , η, x , y ) ( c − K , . . . , c − K , η − K , − x , y ) , (5.3c) ι : ( c , . . . , c , η, x , y ) ( c − K , . . . , c − K , η + K , x , − y ) . (5.3d)Adding the transformations ι i , we extend W (cid:0) E (1)8 (cid:1) to e W (cid:0) E (1)8 (cid:1) = h ι , ι , ι , ι i ⋊ W (cid:0) E (1)8 (cid:1) . (5.4)In general, for a function F = F ( c i , η, x , y ), we let an element w ∈ e W (cid:0) E (1)8 (cid:1) act as w . F = F ( w . c i , w .η, w . x , w . y ),that is, w acts on the arguments from the left. Under the birational actions (5.1) and (5.3), the gen-erators of e W (cid:0) E (1)8 (cid:1) satisfy the fundamental relations of type E (1)8 (2.13) and the following relations:( ι i ι j ) m ij = , i , j = , , , , ι k s l = s l ι k , k = , , , , l , , , (5.5a) ι { , , , } s = s ι { , , , } , ι s = s ι ι , ι s = s ι , (5.5b) ι s = s ι ι , ι s = s ι , (5.5c)where m i j = , i = j , otherwise . (5.6)Now we are in a position to derive Equation (1.1) from the Cremona transformations. Note thatfor convenience we use the notation (2.25) for the composition of the reflections s i and the notation c j ··· j n = c j + · · · + c j n , j · · · j n ∈ { , . . . , } , (5.7)for the summation of the parameters c i . Let R J , = s ι ι ι ι . (5.8)The action of R J , on the root lattice Q ( A (1) ⊥ ): R J , : α α α α α α α α α − − − − − − − − − −
30 0 1 2 1 0 0 0 10 0 0 − − − − − α α α α α α α α α , (5.9)is not translational, and that on the parameter space: R J , ( c i ) = − c i + c − c − κ, i = , . . . , , (5.10) R J , ( c j ) = − c j + c + c − κ, j = , . . . , , R J , ( η ) = η + λ , (5.11)where κ is defined by (3.3), is also not translational. However, when the parameters take specialvalues (3.7), the action of R J , becomes the translational motion in the parameter subspace: R J , : ( γ e , γ o , z ) ( γ e , γ o , z + γ e + γ o ) − κ ) , (5.12) REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 13 and then the action on the coordinates ( x , y ) with x n = R nJ , ( x ), y n = R nJ , ( y ), z n = R nJ , ( z ), gives thetime evolution of Equation (1.1), that is, R J , = φ . Note that we can without loss of generality ignore“2 κ ” in R J , ( z ) = γ e + γ o ) − κ, (5.13)since of the form of Equation (1.1) and the following relations:sn ( u + κ ) = sn ( u ) , cn ( u + κ ) = − cn ( u ) , dn ( u + κ ) = − dn ( u ) . (5.14)6. S pecial solutions of the RCG equation
In this section, we show the special solutions of the RCG equation.Let us consider Equation (1.1) under the following condition: γ o = i K ′ , (6.1)which gives sg o = i k / , cg o = (1 + k ) / k / , dg o = (1 + k ) / . (6.2)Then, the base points p i , i = , , ,
4, in (3.2) can be expressed by p : ( x , y ) = − k / , − cd z − γ e + i K ′ !! , (6.3a) p : ( x , y ) = k / , cd z − γ e + i K ′ !! , (6.3b) p : ( x , y ) = − k / , − cd z − γ e − i K ′ !! , (6.3c) p : ( x , y ) = k / , cd z − γ e − i K ′ !! . (6.3d)This means that there exist the bi-degree (1 ,
0) curve x = − k − / , passing through p and p , and thebi-degree (1 ,
0) curve x = k − / , passing through p and p , which correspond to H − E − E and H − E − E , respectively. Moreover, the action φ : H − E − E ↔ H − E − E , (6.4)implies that there exist the special solutions when x n = ± ( − n k / . (6.5)Therefore, we obtain the following lemma. Lemma 6.1.
The following are special solutions of Equation (1.1):( x n , y n ) = ( − n k / , ( − n k / ! , ( x n , y n ) = ( − n k / , ( − n + k / ! , (6.6a)( x n , y n ) = ( − n + k / , ( − n k / ! , ( x n , y n ) = ( − n + k / , ( − n + k / ! , (6.6b) and ( x n , y n ) = ( − n k / , i tan( u n ) k / ! , ( x n , y n ) = ( − n + k / , − i tan( u n ) k / ! , (6.7) where i = √− and u n is the solution of the following linear equation:u n + + u n = tan − − i (1 − k sg )cz n dz n cg e dg e (1 − k sz n ) ! . (6.8) Proof.
Under the condition x n = ( − n k / , (6.9)Equation (1.1) are reduced to the following discrete Riccati equation: y n + + y n = i A n k / (1 + k y n + y n ) , (6.10)where A n is given by A n = − i (1 − k sg )cz n dz n cg e dg e (1 − k sz n ) . (6.11)Note that under the condition x n = ( − n + k / , (6.12)Equation (1.1) can be reduced to y n + + y n = − i A n k / (1 + k y n + y n ) , (6.13)which can be rewritten as the discrete Riccati equation (6.10) by the transformation y n
7→ − y n .Therefore, it is enough for us to just consider the case (6.9).Let us consider the solutions of the discrete Riccati equation (6.10). If1 + k y n + y n = , (6.14)then we obtain y n + + y n = . (6.15)Therefore, we obtain y n = ( − n k / , ( − n + k / , (6.16)which gives the special solutions (6.6). In the following we assume1 + k y n + y n , . (6.17)Then, the discrete Riccati equation (6.10) can be rewritten as the following: y n + + y n + k y n + y n = i A n k / . (6.18)By letting y n = i tan( u n ) k / , (6.19)and using the tangent addition formula, the discrete Riccati equation (6.18) can be rewritten astan( u n + + u n ) = A n , (6.20)which gives the linear equation (6.8). Therefore, we have completed the proof. (cid:3) A cknowledgments This research was supported by an Australian Laureate Fellowship
REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 15 A ppendix A. A (1)4 - lattice In this section, we give a more detailed description of the weight lattice and a ffi ne Weyl groupby using the lattice of type A (1)4 as an example.We consider the following Z -modules: Q ∨ = X k = Z α ∨ k , P = X k = Z h k , (A.1)with the bilinear pairing h· , ·i : Q ∨ × P → Z defined by h α ∨ i , h j i = δ i j , ≤ i , j ≤ . (A.2)We also define the submodule of P by Q = P k = Z α k , where α i , i = , . . . ,
4, are defined by α α α α α = ( A i j ) i , j = h h h h h , (A.3)and satisfy h α ∨ i , α j i = A i j . (A.4)Here, ( A i j ) i , j = is the Generalized Cartan matrix of type A (1)4 :( A i j ) i , j = = − − − − − − − − − − . (A.5)Then, the generators { α ∨ , . . . , α ∨ } , { h , . . . , h } and { α , . . . , α } are identified with simple coroots,fundamental weights and simple roots of type A (1)4 , respectively. We note that the following relationholds: α + α + α + α + α = , (A.6)and we call the corresponding coroot as null-coroot denoted by δ ∨ : δ ∨ = α ∨ + α ∨ + α ∨ + α ∨ + α ∨ . (A.7)In the following subsections, we consider the transformation group acting on theses lattices.A.1. A ffi ne Weyl group of type A (1)4 . In this section, we consider the transformations which col-lectively form an a ffi ne Weyl group of type A (1)4 .We define the transformations s i , i = , . . . ,
4, by the reflections for the roots { α , . . . , α } : s i ( λ ) = λ − h α ∨ i , λ i α i , i = , . . . , , λ ∈ P , (A.8)which give s ( h ) = − h + h + h , s ( h ) = h − h + h , s ( h ) = h − h + h , s ( h ) = h − h + h , s ( h ) = h + h − h . (A.9)From definitions (A.2), (A.3), (A.4) and (A.8), we can compute actions on the simple roots α i , i = , . . . ,
4, as the following: s i ( α j ) = − α j , i = j α j + α i , i = j ± α j , i = j ± . (A.10) Under the linear actions on the weight lattice (A.9), W ( A (1)4 ) = h s , s , s , s , s i forms an a ffi neWeyl group of type A (1)4 , that is, the following fundamental relations hold:( s i s j ) l ij = , where l i j = , i = j , i = j ± , i = j ± . (A.11)Note that representing the simple reflections s i by nodes and connecting i -th and j -th nodes by( l i j −
2) lines, we obtain the Dynkin diagram of type A (1)4 shown in Figure 2.Figure 2. Dynkin diagram of type A (1)4 . The transformation ι is the reflection with respect to thedotted line, and the transformation σ is the rotation symmetry with respect to an angle of 2 π/ Remark A.1.
We can also define the action of W ( A (1)4 ) on the coroot lattice Q ∨ by replacing α i with α ∨ i in the actions (A.10) . Note that δ ∨ is invariant under the action of W ( A (1)4 ) . Then, thetransformations in W ( A (1)4 ) preserve the form h· , ·i . Let M i , i = , . . . ,
4, be the orbits of h i , i = , . . . ,
4, defined by M i = n w ( h i ) (cid:12)(cid:12)(cid:12) w ∈ W ( A (1)4 ) o , (A.12)and T i , i = , . . . ,
4, be the transformations defined by T = s , T = s , T = s , (A.13a) T = s , T = s , (A.13b)whose actions on the fundamental weights and simple roots are given by T i ( h j ) = h j − α i , T i ( α j ) = α j , i , j = , . . . , . (A.14)Note that T ◦ T ◦ T ◦ T ◦ T =
1. Then, the following lemma holds.
Lemma A.2.
The following hold:M i = { h i + α | α ∈ Q } , i = , . . . , . (A.15) Proof.
The relation M ⊂ { h + α | α ∈ Q } is obvious from the definition (A.8), and the relation M ⊃ { h + α | α ∈ Q } follows from h + X i = k i α i = T k ◦ T k ◦ T k ◦ T k ◦ T k ( h ) , (A.16)where k i ∈ Z . Therefore, the case i = M i i = , . . . ,
4. Therefore, we have completed the proof. (cid:3)
REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 17
Since of Lemma A.2, we can express the lattices M i by M i = { T ( h i ) | T ∈ h T , . . . , T i} , i = , . . . , . (A.17)In general, the translations on the orbits M i , i = , . . . ,
4, are given by the Kac translations definedon the weight lattice P : T α ( λ ) = λ − h δ ∨ , λ i α, λ ∈ P , α ∈ Q . (A.18)The translations T i , i = , . . . ,
4, can be expressed by the Kac translations as the following: T i = T α i , i = , . . . , . (A.19)Note that the translations T i , i = , . . . ,
4, are called the fundamental translations on the weightlattice P or in a ffi ne Weyl group W ( A (1)4 ). Indeed, since of the following property of the Kac trans-lations: T α ◦ T β = T α + β , (A.20)where α, β ∈ Q , all Kac translations in W ( A (1)4 ) can be expressed by the compositions of T i , i = , . . . , h i to another fundamentalweight h j . However, by extending W ( A (1)4 ) with the automorphisms of the Dynkin diagram, we candefine such translations as shown in the following subsection.A.2. Extended a ffi ne Weyl group of type A (1)4 . In this section, we consider the extended a ffi neWeyl group of type A (1)4 .We define the transformations σ and ι by σ : h → h , h → h , h → h , h → h , h → h , (A.21a) ι : h ↔ h , h ↔ h , (A.21b)whose actions on the root lattice are given by σ : α → α , α → α , α → α , α → α , α → α , (A.22a) ι : α ↔ α , α ↔ α . (A.22b)Moreover, we also define their actions on the coroot lattice Q ∨ by replacing α i with α ∨ i in the actions(A.22). Then, the transformations σ and ι satisfy σ = ι = , σ ◦ ι = ι ◦ σ − , (A.23)and the relations with W ( A (1)4 ) = h s , s , s , s , s i are given by σ ◦ s i = s i + ◦ σ, ι ◦ s i = s − i ◦ ι, (A.24)that is, the transformations σ and ι are automorphisms of the Dynkin diagram of type A (1)4 (seeFigure 2). Therefore, we call e W ( A (1)4 ) = W ( A (1)4 ) ⋊ h σ, ι i (A.25)as the extended a ffi ne Weyl group of type A (1)4 . Remark A.3.
We can easily verify that δ ∨ is also invariant under the action of e W ( A (1)4 ) , and thetransformations in e W ( A (1)4 ) preserve the form h· , ·i . Let M be the orbits of h defined by M = n w ( h ) (cid:12)(cid:12)(cid:12) w ∈ e W ( A (1)4 ) o . (A.26)Moreover, we also define the following transformations: T = σ s , T = σ s , T = σ s , (A.27a) T = σ s , T = σ s , (A.27b) whose actions on the fundamental weights are translational as the following: T i i + ( h j ) = h j + v i i + , T i i + ( v j j + ) = v j j + , i , j ∈ Z / (5 Z ) , (A.28)where v = h − h , v = h − h , v = h − h , v = h − h , v = h − h . (A.29)Note that α α α α α = − − − − − . v v v v v , (A.30) v v v v v = . α α α α α , (A.31) v + v + v + v + v = , (A.32) T ◦ T ◦ T ◦ T ◦ T = . (A.33)In a similar manner as the proof of Lemma A.2, by using the translations T i j we can prove thefollowing: M = { h + v | v ∈ V } = { T ( h ) | T ∈ h T , T , T , T , T i} , (A.34)where V = Z v + Z v + Z v + Z v + Z v . Note that since T v ii + : h i + h i , the following hold: h i ∈ M , i = , . . . , . (A.35)The translations on the weight lattice P spanned by V is given by T v ( λ ) = λ − h δ ∨ , λ i v , λ ∈ P , v ∈ V . (A.36)In this case, the translations T i i + , i ∈ Z / (5 Z ), can be expressed by T = T v , T = T v , T = T v , T = T v , T = T v , (A.37)and are called the fundamental translations in e W ( A (1)4 ), that is, all translations in e W ( A (1)4 ) can be ex-pressed as the compositions of these translations. Note that the fundamental translations in W ( A (1)4 )can be expressed as the compositions of the fundamental translations in e W ( A (1)4 ) as the following: T i = T i i + ◦ T i − , i − , i ∈ Z / (5 Z ) . (A.38)For any v = P i = c i α i ∈ V , where c i ∈ R , we define the squared length of T v by | T v | : = h v ∨ , v i = ( c − c ) + ( c − c ) + ( c − c ) + ( c − c ) + ( c − c ) , (A.39)where v ∨ = P i = c i α ∨ i . Note that we here extended the domain of the bilinear pairing h· , ·i from Q ∨ × P to Q ∨ × P , where Q ∨ = P k = R α ∨ k . We can easily verify that the squared length of fundamentaltranslations in W ( A (1)4 ): T i is 2, while the squared length of fundamental translations in e W ( A (1)4 ): T i i + is 4 / REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 19 A ppendix B. G eneral E lliptic D ifference E quations In this section, we provide two generic elliptic di ff erence equations. The first is Sakai’s A (1)0 -surface equation. This was re-expressed by Murata [29] as follows: T ( M ) : ( f , g ; t , b , . . . , b ) (cid:18) f , g ; t + δ , b , . . . , b (cid:19) , (B.1)where f and g are given bydet (cid:16) v ( f , g ) , v , . . . , v , v c (cid:17) det (cid:16) v ( f , g ) , q v , . . . , q v , q v c (cid:17) = P + ( f − f c )( f − f c ) Y i = ( g − g i ) , (B.2a)det (cid:16) v ( g , f ) , b u , . . . , b u , b u c (cid:17) det (cid:16) v ( g , f ) , u , . . . , u , u c (cid:17) = P − ( g − g c )( g − g c ) Y i = ( f − f i ) . (B.2b)Here, δ = X k = b k , v i = v ( f i , g i ) , q v i = v ( f i , g i ) , b u i = v ( g i , f i ) , u i = v ( g i , f i ) , (B.3)for i = f i = ℘ ( t − b i ) , g i = ℘ ( t + b i ) , i = , . . . , , f c = ℘ t + t δ ! , g c = ℘ t − t δ ! . (B.4)Moreover, v ( a , b ) and P ± are given by v ( a , b ) = ( ab , ab , ab , ab , a , b , b , b , b , T , P ± = σ (4 t ) σ (4 t ± δ ) σ (cid:16) t ∓ t δ (cid:17) Y ≤ i < j ≤ σ ( b i − b j ) Y i = σ (cid:16) t δ − b i (cid:17) σ (cid:16) t ± t δ ± b i (cid:17) σ ( t ± b i ) σ ( t ∓ b i ) σ (cid:16) t ∓ b i ± δ (cid:17) , where σ is the Weierstrass sigma function; see [9, Chapter 23]. Note that P − = P − | t → t . The abovesystem was obtained by deducing translations on the lattice of type E (1)8 . We note that this translationcorresponds to NVs in the lattice.The second case of an elliptic di ff erence equation was found by Joshi and Nakazono [21]. It hasa projective reduction to the RCG equation (1.1). The generic equation is given by T ( JN ) :( x , y ; c , . . . , c , c , . . . , c , η ) ( x , y ; c − λ, . . . , c − λ, c + λ, . . . , c + λ, η + λ ) , (B.5) where x and y are given by k cd ( η − c + κ ) y + k cd ( η − c + κ ) y + ! ˜ x − cd (cid:16) η − c + c + λ + κ (cid:17) ˜ x − cd (cid:16) η − c + c + λ + κ (cid:17) = G c − c + λ , c − c + λ , c − c + λ , c − c + λ ,η + λ + κ P c − c + λ , c − c + λ , c − c + λ ,η + λ + κ ( ˜ x , ˜ y ) P c − c + λ , c − c + λ , c − c + λ ,η + λ + κ ( ˜ x , ˜ y ) , (B.6a) k cd ( η + c + κ ) x + k cd ( η + c + κ ) x + ! k cd ( η − c + λ + κ ) y + k cd ( η − c + λ + κ ) y + ! = G η − c + c + λ,η − c + c + λ,η − c + c + λ,η − c + c + λ, c + λ + κ P η − c + c + λ,η − c + c + λ,η − c + c + λ, c + λ + κ (cid:16) − ky , ˜ x (cid:17) P η − c + c + λ,η − c + c + λ,η − c + c + λ, c + λ + κ (cid:16) − ky , ˜ x (cid:17) , (B.6b)and ˜ x and ˜ y are given by k cd (cid:16) η + c − c (cid:17) ˜ y + k cd (cid:16) η + c − c (cid:17) ˜ y + x − cd ( η + c ) x − cd ( η + c ) ! = G c , c , c , c ,η P c , c , c ,η ( x , y ) P c , c , c ,η ( x , y ) , (B.6c) k cd (cid:16) η − c + c (cid:17) ˜ x + k cd (cid:16) η − c + c (cid:17) ˜ x + k cd (cid:16) η + c + c (cid:17) ˜ y + k cd (cid:16) η + c + c (cid:17) ˜ y + = G η + c + c ,η + c + c ,η + c + c ,η + c + c , c P η + c + c ,η + c + c ,η + c + c , c (cid:16) − k ˜ y , x (cid:17) P η + c + c ,η + c + c ,η + c + c , c (cid:16) − k ˜ y , x (cid:17) . (B.6d)Here, λ = P k = c k , κ is defined by (3.3), c j ··· j n is the summation of the parameters c i (see (5.7)), andthe functions G a , a , a , a , b , Q a , a , a , a , a , b ( X ) and P a , a , a , b ( X , Y ) are given by G a , a , a , a , b = − cd (cid:16) a + a + a (cid:17) cd (cid:16) a + a + a (cid:17) − cd (cid:16) a + a + a (cid:17) cd (cid:16) a + a + a (cid:17) − cd( b − a )cd( b − a ) − cd( b − a )cd( b − a ) − cd (cid:16) b + a − a + a + a + a (cid:17) cd (cid:16) b + a + a + a + a + a (cid:17) − cd (cid:16) b + a − a + a + a + a (cid:17) cd (cid:16) b + a + a + a + a + a (cid:17) − cd (cid:16) a + a + a (cid:17) cd (cid:16) b + a − a + a (cid:17) − cd (cid:16) a + a + a (cid:17) cd (cid:16) b + a − a + a (cid:17) , (B.7) Q a , a , a , a , a , b ( X ) = (cid:18) cd (cid:18) b + a − a (cid:19) − cd (cid:18) b + a + a (cid:19)(cid:19) (cid:18) cd (cid:18) b + a + a (cid:19) − cd (cid:18) b + a + a (cid:19)(cid:19)(cid:16) cd ( b + a ) cd ( b + a ) + cd ( b + a ) X (cid:17) + (cid:18) cd (cid:18) b + a − a (cid:19) − cd (cid:18) b + a + a (cid:19)(cid:19)(cid:18) cd (cid:18) b + a + a (cid:19) − cd (cid:18) b + a + a (cid:19)(cid:19) (cid:16) cd ( b + a ) cd ( b + a ) + cd ( b + a ) X (cid:17) − (cid:18) cd (cid:18) b + a − a (cid:19) − cd (cid:18) b + a + a (cid:19)(cid:19) (cid:18) cd (cid:18) b + a + a (cid:19) − cd (cid:18) b + a + a (cid:19)(cid:19) REVIEW OF ELLIPTIC DIFFERENCE PAINLEV´E EQUATIONS 21 (cid:16) cd ( b + a ) cd ( b + a ) + cd ( b + a ) X (cid:17) , (B.8) P a , a , a , b ( X , Y ) = C XY + C X + C Y + C , (B.9)where C = (cid:16) cd ( b − a ) − cd ( b − a ) (cid:17) cd ( b + a ) + (cid:16) cd ( b − a ) − cd ( b − a ) (cid:17) cd ( b + a ) + (cid:16) cd ( b − a ) − cd ( b − a ) (cid:17) cd ( b + a ) , C = (cid:16) cd ( b − a ) − cd ( b − a ) (cid:17) cd ( b − a ) cd ( b + a ) + (cid:16) cd ( b − a ) − cd ( b − a ) (cid:17) cd ( b − a ) cd ( b + a ) + (cid:16) cd ( b − a ) − cd ( b − a ) (cid:17) cd ( b − a ) cd ( b + a ) , C = (cid:16) cd ( b + a ) − cd ( b + a ) (cid:17) cd ( b − a ) cd ( b + a ) + (cid:16) cd ( b + a ) − cd ( b + a ) (cid:17) cd ( b − a ) cd ( b + a ) + (cid:16) cd ( b + a ) − cd ( b + a ) (cid:17) cd ( b − a ) cd ( b + a ) , C = (cid:16) cd ( b + a ) cd ( b − a ) − cd ( b − a ) cd ( b + a ) (cid:17) cd ( b − a ) cd ( b + a ) + (cid:16) cd ( b + a ) cd ( b − a ) − cd ( b − a ) cd ( b + a ) (cid:17) cd ( b − a ) cd ( b + a ) + (cid:16) cd ( b + a ) cd ( b − a ) − cd ( b − a ) cd ( b + a ) (cid:17) cd ( b − a ) cd ( b + a ) . Similarly to the earlier case of Sakai, this equation also arises from a translation on the lattice oftype E (1)8 . We note that this translation corresponds to NNVs in the lattice.R eferences [1] Abramowitz M and Stegun I. Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables .Dover Books on Mathematics. Dover Publications, 2012.[2] Adler V E. B¨acklund transformation for the Krichever-Novikov equation.
Internat. Math. Res. Notices , (1):1–4, 1998.[3] Adler V E, Bobenko A I, and Suris Y B. Classification of integrable equations on quad-graphs. The consistency ap-proach.
Comm. Math. Phys. , 233(3):513–543, 2003.[4] Atkinson J, Howes P, Joshi N, and Nakazono N. Geometry of an elliptic di ff erence equation related to Q4. J. Lond.Math. Soc. (2) , 93(3):763–784, 2016.[5] Bourbaki, N.
Groupes et alg`ebres de Lie: Chapitres 4, 5 et 6 . Bourbaki, Nicolas. Springer Berlin Heidelberg, 2007.[6] Carstea A S, Dzhamay A, and Takenawa T. Fiber-dependent deautonomization of integrable 2D mappings and discretePainlev´e equations.
J. Phys. A , 50(40):405202, 41, 2017.[7] Conway J, Sloane N, Bannai E, Borcherds R, Leech J, Norton S, Odlyzko A, Parker R, Queen L, and Venkov B.
SpherePackings, Lattices and Groups . Grundlehren der mathematischen Wissenschaften. Springer New York, 2013.[8] Cresswell C, and Joshi N. The discrete first, second and thirty-fourth Painlev´e hierarchies.
Journal of Physics A: Math-ematical and General , 32(4):655, 1999.[9]
NIST Digital Library of Mathematical Functions . http: // dlmf.nist.gov / , Release 1.0.14 of 2016-12-21. Olver F W J,Olde Daalhuis A B, Lozier D W, Schneider B I, Boisvert R F, Clark C W, Miller B R and Saunders B V, eds.[10] Dolgachev I V. Classical algebraic geometry . Cambridge University Press, Cambridge, 2012. A modern view; Dol-gachev I and Ortland D. Point sets in projective spaces and theta functions.
Ast´erisque , (165):210 pp. (1989), 1988;Looijenga E. Rational surfaces with an anticanonical cycle.
Ann. of Math. (2) , 114(2):267–322, 1981.[11] Fokas A S., Grammaticos B, and Ramani A. From continuous to discrete Painlev´e equations,
Journal of mathematicalanalysis and applications , 180(2):342–360, 1993.[12] Fokas A S., Its A R., and Kitaev A V. Discrete Painlev´e equations and their appearance in quantum gravity.
Communi-cations in Mathematical Physics , 142(2):313–344, 1991.[13] Fuchs R. Sur quelques ´equations di ff ´erentielles lin´eaires du second ordre. Comptes Rendus de l’Academie des SciencesParis , 141:555–558, 1905.[14] Gambier B. Sur les ´equations di ff ´erentielles du second ordre et du premier degr´e dont l’int´egrale g´en´erale est `a pointscritiques fixes. Acta Mathematica , 33(1):1–55, 1910.[15] Grammaticos B and Ramani A. Discrete Painlev´e equations: a review. In
Discrete integrable systems , volume 644 of
Lecture Notes in Phys. , pages 245–321. Springer, Berlin, 2004.[16] P. Gri ffi ths. Introduction to algebraic curves , volume 76. American Mathematical Soc., 1989. [17] Hietarinta J. Searching for CAC-maps.
J. Nonlinear Math. Phys. , 12(suppl. 2):223–230, 2005.[18] Hietarinta J, Joshi N, and Nijho ff F W.
Discrete systems and integrability . Cambridge Texts in Applied Mathematics.Cambridge University Press, Cambridge, 2016.[19] Humphreys J.
Reflection Groups and Coxeter Groups . Cambridge Studies in Advanced Mathematics. Cambridge Uni-versity Press, 1992.[20] Joshi N, Grammaticos B, Tamizhmani T, and Ramani A. From integrable lattices to non-QRT mappings.
Lett. Math.Phys. , 78(1):27–37, 2006.[21] Joshi N and Nakazono N. Elliptic Painlev´e equations from next-nearest-neighbor translations on the E (1)8 lattice. J.Phys. A , 50(30):305205, 17, 2017.[22] Kajiwara K, Masuda T, Noumi M, Ohta Y, and Yamada Y. E solution to the elliptic Painlev´e equation. J. Phys. A ,36(17):L263–L272, 2003.[23] Kajiwara K and Nakazono N. Hypergeometric solutions to the symmetric q -Painlev´e equations. Int. Math. Res. Not.IMRN , (4):1101–1140, 2015.[24] Kajiwara K, Nakazono N, and Tsuda T. Projective reduction of the discrete Painlev´e system of type ( A + A ) (1) . Int.Math. Res. Not. IMRN , (4):930–966, 2011.[25] Kajiwara K, Noumi M, and Yamada Y. Geometric aspects of Painlev´e equations.
J. Phys. A , 50(7):073001, 164, 2017.[26] Laguerre E. Sur la r´eduction en fractions continues d’une fraction qui satisfait `a une ´equation di ff ´erentielle lin´eaire dupremier ordre dont les coe ffi cients sont rationnels. Journal de Math´ematiques Pures et Appliqu´ees , 1:135–166, 1885.[27] Magnus, A. Painlev´e-type di ff erential equations for the recurrence coe ffi cients of semi-classical orthogonal polynomi-als. J. of computational and applied mathematics . 57(1):215–237, 1995.[28] Milnor, J. Foliations and foliated vector bundles, mimeographed notes.
Institute for Advanced Study, Princeton , 1970.[29] Murata, M. New expressions for discrete Painlev´e equations.
Funkcial. Ekvac. , 47(2):291–305, 2004.[30] Murata M, Sakai H, and Yoneda J. Riccati solutions of discrete Painlev´e equations with Weyl group symmetry of type E (1)8 . J. Math. Phys. , 44(3):1396–1414, 2003.[31] M. Noumi.
Painlev´e Equations Through Symmetry . Translations of mathematical monographs. American MathematicalSociety, 2004.[32] Ohta Y, Ramani A, and Grammaticos B. An a ffi ne Weyl group approach to the eight-parameter discrete Painlev´eequation. J. Phys. A , 34(48):10523–10532, 2001. Symmetries and integrability of di ff erence equations (Tokyo, 2000).[33] Okamoto K. Sur les feuilletages associ´es aux ´equations du second ordre `a points critiques fixes de P. Painlev´e. Japan.J. Math. (N.S.) , 5(1):1–79, 1979.[34] Painlev´e P. Sur les ´equations di ff ´erentielles du second ordre et d’ordre sup´erieur dont l’int´egrale g´en´erale est uniforme. Acta Mathematica , 25(1):1–85, 1902.[35] Periwal V and Shevitz D. Unitary-matrix models as exactly solvable string theories.
Phys. Rev. Lett. , 64(12):1326–1329,1990.[36] Ramani A, Carstea A S, and Grammaticos B. On the non-autonomous form of the Q mapping and its relation to ellipticPainlev´e equations. J. Phys. A , 42(32):322003, 8, 2009.[37] Ramani A, Grammaticos B, and Hietarinta J. Discrete versions of the Painlev´e equations.
Phys. Rev. Lett. , 67(14):1829–1832, 1991.[38] Sakai H. Rational surfaces associated with a ffi ne root systems and geometry of the Painlev´e equations. Comm. Math.Phys. , 220(1):165–229, 2001.[39] Shohat J. A di ff erential equation for orthogonal polynomials. Duke Mathematical Journal , 5(2):401–417, 1939.[40] Whittaker E and Watson G.
A Course of Modern Analysis . A Course of Modern Analysis: An Introduction to theGeneral Theory of Infinite Processes and of Analytic Functions, with an Account of the Principal TranscendentalFunctions. Cambridge University Press, 1996.S chool of M athematics and S tatistics , T he U niversity of S ydney , N ew S outh W ales ustralia . E-mail address : [email protected] D epartment of P hysics and M athematics , A oyama G akuin U niversity , S agamihara , K anagawa apan . E-mail address ::