A new approach to separation of variables for the Clebsch integrable system. Part II: Inversion of the Abel--Prym map
aa r X i v : . [ n li n . S I] F e b A new approach to separation of variables for the Clebschintegrable system. Part II: Inversion of the Abel–Prym map
Y. Fedorov , F. Magri , T. Skrypnyk , Polytechnic University of Catalonia, Barcelona, Spain Dipartimento di Matematica e Applicazioni- Universit´a di Milano Bicocca, Milano, Italia Universit´a degli Studi di Torino, via Carlo Alberto 10, 10123, Torino, Italia Bogolyubov Institute for Theoretical Physics, Metrolohichna str.14-b, 03115, Kiev, Ukraine [email protected], [email protected], [email protected]
Abstract
This is the second part of a paper describing a new concept of separation ofvariables applied to the classical Clebsch integrable case. The quadratures obtainedin Part I lead to a new type of the Abel map which contains Abelian integrals ontwo different algebraic curves.Here we show that this map is from the product of the two curves to the Prymvariety of one of them, that it is well defined, although not a bijection. We analyseits properties and formulate a new extention of the Riemann vanishing theorem,which allows to invert the map in terms of theta-functions of higher order.Lastly, we describe how to express the original variables of the Clebsch systemin terms of the preimages of the map. This enables one to obtain theta-functionsolution for the system.
Keywords: algebraic integrable systems, Abelian varieties, generalized Abel–Jacobimap, theta-functions.
As was shown in Part I, to any point ( S α , T α ), α = 1 , , H p , H s , C , K there correspond 8sets of separating variables { x , x } and the corresponding conjugated momenta. Theirevolution with respect to the time parameters t s , t p of the flows X s , X p is described bythe same quadratures, whose integral form reads (see formulas (9), (10) in Part I) Z x x dxw ( w − g ( x )) + Z x x √ dxW ( W − g ( x )) = iC t s , Z x x x dxw ( w − g ( x )) + Z x x √ x dxW ( W − g ( x )) = iC t p , i = √− , (1)1here g ( x ) = K x + H p x + H s and x , x are some initial values. (In both Parts weassume that C = 0.)Here the pairs ( x , w ) and ( x , W ) satisfy equations of algebraic curves C : w − g ( x ) w + g ( x ) − C ( x + j )( x + j )( x + j ) = 0 ,K : W − g ( x ) W + 4 C ( x + j )( x + j )( x + j ) = 0 . (2)As one can check, both curves are non-hyperelliptic of genus 3, and they are not bira-tionally equivalent. The left hand sides of (1) involve two holomorphic differentials onthe curve C and two holomorphic differentials on K .The form of the quadratures reminds the standard Abel–Jacobi map Γ (2) → Jac(Γ),where Γ (2) is the symmetric product of two copies of a regular genus 2 curve Γ and Jac(Γ)is the common notation for the Jacobian variety of Γ (see, e.g., [GH, Mum984]). Sucha map can be inverted: in particular, any symmetric function of coordinates of the twopoints on Γ can be expressed in terms of the Riemann theta-functions of Γ (see, e.g.,[Dub81]).However, the facts that in the quadratures (1) the periods of the Abelian integralson the two curves are distinct and the genus of C and K is higher than the dimensionof the invariant tori (and the number of equations in (1)) raise a natural doubt aboutinvertibility of these quadratures in terms of meromorphic functions of the complex times t s , t p .On the other hand, it is known ([AvM984]) that the Clebsch integrable case is analgebraic integrable system, which implies that all its solutions are meromorphic.In Section 1 of this Part II we will show that the periods of the integrals in (1),although being different, are commensurable, and that the quadratures lead to a well-defined, so called Abel–Prym map from the product C × K to a two-dimensional Prymsubvariety of the Jacobian of one of the curves.In Section 2 we introduce higher order theta-functions and formulate analogs of theRiemann vanishing theorem, which will be a main tool of the inversion of the map. Wewill see that, in contrast to the standard Abel map, the Abel–Prym maps are not one-to-one: a full preimage of a point in the Prym variety consists of 8 pairs of points on C × K .In Section 3 we prove that the coordinates of these 8 pairs of points can be identifiedwith the eight sets of separating variables for the Clebsch system constructed in Sections3,4 of Part I. This enables one to derive a (new) theta-function solution for the originalvariables S α , T α : they can be expressed as symmetric functions of coordinates of theeight points on one of the curves, while the latter functions can be written in terms ofthe higher order theta-functions which solve the problem of inversion of the Abel–Prymmap. (We will only give a sketch of this procedure, as explicit expressions are too longto present.)We start with recalling necessary properties of the genus 3 curves C and K andtheir relation to the complex invariant tori of the Clebsch system, already described in[Hai983, EF17] The analysis in [Hai983] has been made for other pair of genus 3 curves related to the integrableFrahm–Manakov top on so (4), however both pairs of curves are birationally equivament. K and C actually already appeared as the spectralcurves of two different Lax representations of the Clebsch system obtained in [Per81]and, respectively, in [Bob86] (see also [BBEIM, Fed95]). Equations (2) can be writtenin the form C : w = g ( x ) + 2 C p Φ( x ) , Φ = ( x + j )( x + j )( x + j ) , (3) K : W = g ( x ) + p Ψ( x ) , (4)Ψ( x ) = g ( x ) − C Φ( x ) = K ( x − s )( x − s )( x − s )( x − s ) , which makes evident that the curves C, K have the involution σ : ( x , w ) → ( x , − w ),respectively, σ : ( x , W ) → ( x , − W ), and they are 2-fold ramified coverings of ellipticcurves E, E : π : C → E = { y = Φ( x ) } , ¯ π : K → E = { y = Ψ( x ) } . The coverings are ramified at points Q , . . . Q ∈ E , respectively Z , . . . , Z ∈ E , so that σ ( Q j ) = Q j , σ ( Z j ) = Z j (one of Z j is an infinite point of K ). Note that in our case C = 0 these points are not in the hyperelliptic involution ( x, y ) → ( x, − y ) acting on E ,respectively on E .The involution σ : ( x, w ) → ( x, − w ) on C extends to its Jacobian variety, Jac( C ).Thus the latter contains two Abelian subvarieties: the elliptic curve E itself and the 2-dimensional Prym variety denoted as Prym( C, σ ), which is anti-symmetric with respectto the extended involution, whereas E is invariant. Equivalently,Prym( C, σ ) = ker (1 + σ ), see e.g., [Mum974]. One should stress that Prym( C, σ ) is notthe Jacobian variety of a genus 2 curves because it has polarization (1 , K, σ ) ⊂ Jac( K ), which is anti-symmetricwith respect to the involution σ extended to the Jacobian. Theorem 1 (L. Haine [Hai983]) .
1) The complex invariant manifold I H of the Clebschsystem with generic constants of motion H p , H s , C , K is isomorphic to an opensubset of Prym(
C, σ ) . Namely, I H = Prym( C, σ ) \ D , where D is a genus 9curve wrapped around the complex torus Prym(
C, σ ) . The variables S α , T α aremeromorphic functions on Prym(
C, σ ) having a simple pole along D .2)There is a 4-fold unramified covering (an isogeny) Π ∗ : Prym( C, σ ) → Prym(
K, σ ) ,which is associated with the action of discrete group g of 4 elements on Prym(
C, σ ) ,which change signs of some of S α , T α . Then Prym(
C, σ ) / g = Prym( K, σ ) . Thecurve D can be viewed as 4-fold unramified covering Π :
D → C = D / g .Thus the squares S α , T α are meromorphic functions on Prym(
K, σ ) , having a sec-ond order pole along C ⊂ Prym(
K, σ ) .3) Let b E be the 4-fold unramified covering of the elliptic curve E = C/σ , obtained bydoubling its two periods. Then D can be viewed as 2-fold covering of b E ramified atthe 16 preimages of the branch points Q , . . . , Q ∈ E , as described in the diagram Π : D ur −−−−→ C ram. at 16 Π − ( Q i ) y y ram. at 4 Q i b E ur −−−−→ E = C/σ here ur and ram mean ”unramified” and ”ramified” respectively.4) The complex time flows X s , X p of the Clebsch system are straight lines on Prym(
C, σ ) . The curve D will play an important role in the sequel. As a 4-fold covering of C , itcan be described as a spacial curve in C ( x, w, v , v , v ) given by the conditions D : w = g ( x ) + 2 C v v v , v = x + j , v = x + j , v = x + j . (5)They are obtained from the equation (3) of C . Indeed, any point ( x, w ) on C specifiesa value of the product v v v , leaving 4 choices of signs of v , v , v . The projectionΠ : D → C simply reads as: ( x, w, v , v , v ) → ( x, w ).Alternatively, substituting in (5) the expressions (24, Part I) for v i , x in terms of thecoordinates on the big elliptic curve ˆ E = { Y = 4( Z + j )( Z + j )( Z + j ) } (23, Part I),namely v α = Z + 2 j α Z + j α ( j β + j γ ) − j β j γ Y , ( α, β, γ ) = (1 , , , (6) x = ( Z − j j − j j − j j ) − j j j ( j + j + j ) Y , one obtains the equations defining D in C ( Z, Y, W ); it is now viewed as the 2-foldramified covering of ˆ E : G = p ( Z ) + 2 C Y p ( Z ) , Y = 4( Z + j )( Z + j )( Z + j ) , (7)where G = wY and p ( Z ) , p ( Z ) are certain polynomials of degrees 4 and 6.The above results can be completed with the following theorem. Theorem 2 (W. Barth, [Bar985]) . Prym(
K, σ ) ⊂ Jac( K ) contains the curve C asthe intersection Prym(
K, σ ) ∩ Θ K , where Θ K ⊂ Jac( K ) is a translate of the theta-divisor of curve K . Moreover, C ⊂ Prym(
K, σ ) belongs to the pencil { C λ } , λ ∈ P of curves of genus ≤ given by intersections Prym(
K, σ ) ∩ Θ K,λ , where Θ K,λ aretranslations of Θ K along the elliptic curve E ⊂
Jac( K ) . The base points of { C λ } are precisely the branch points Q , . . . , Q of the covering C → E .2) For any curve C λ ⊂ Prym(
K, σ ) , the involution σ : C λ → C λ has the same fixedpoints Q , . . . , Q . Thus a generic curve C λ is a 2-fold covering of an elliptic curve E λ , and the corresponding varieties Prym( C λ , σ ) are all isomorphic.3) Similarly, Prym(
C, σ ) ⊂ Jac( C ) contains the curve K as the intersection Prym(
C, σ ) ∩ Θ C ; K belongs to the pencil { K λ } , λ ∈ P of curves of genus ≤ as intersections Prym(
C, σ ) ∩ Θ C,λ , where Θ C,λ are translations of the theta divisor Θ C along the el-liptic curve E ⊂ Jac( C ) . The base points of { K λ } are the branch points Z , . . . , Z of the covering K → E . he periods of the Prym varieties and the Abel–Prym map. In accordanceto the equations (1), choose the following basis of the holomorphic differentials on thecurves C and K : ω = d x C w p Φ( x ) = d xw ( w − g ( x )) , ω = x d x C w p Φ( x ) = x d xw ( w − g ( x )) , (8) ω = d x p Φ( x ) , and, respectively,¯ ω = √ d xW p Ψ( x ) = √ d xW ( W − g ( x )) , ¯ ω = √ x d xW p Ψ( x ) = √ x d xW ( W − g ( x )) , (9)¯ ω = d x p Ψ( x ) . The differentials ω , ω , ¯ ω , ¯ ω are anti-symmetric with respect to involution σ , whereas ω , ¯ ω are symmetric.Let cycles γ , . . . γ and ¯ γ , . . . ¯ γ form canonical bases of cycles in H ( C, Z ) and,respectively, in H ( K, Z ). Theorem 3.
1) The 6 period vectors V i = H γ i (cid:18) ω ω (cid:19) ∈ C ( u , u ) , i = 1 , . . . , forma lattice Λ of rank 4. The complex torus C / Λ is isomorphic to the Prym variety Prym(
K, σ ) .In the same space C ( u , u ) , the 6 period vectors ¯ V i = H ¯ γ i (cid:18) ¯ ω ¯ ω (cid:19) , i = 1 , . . . , forma lattice ¯Λ of rank 4, and Prym(
C, σ ) = C / ¯Λ .Further, chose { γ , . . . , γ } = { a, A, b, B, ¯ a, ¯ b } such that the cycles ( a, b ) , (¯ a, ¯ b ) and A, B are pairwise conjugated and σ ( A ) = − A, σ ( B ) = − B, σ ( a ) = ¯ a, σ ( b ) = ¯ b, then for the basis { γ , . . . , γ } = { a, A, b, B } the period matrix of Prym(
K, σ ) reads Ω = ( V V V V ) .For the similar choice of canonically conjugated cycles ¯ γ , . . . , ¯ γ on K , the periodmatrix of Prym(
C, σ ) is ¯Ω = ( ¯ V ¯ V ¯ V ¯ V ) .2) The above period matrices are related as follows ¯Ω = ( ¯ V ¯ V ¯ V ¯ V ) = (2 V V V V ) . (10)Thus, the periods of the holomorphic differentials (8), (9) on K, C are conmensu-rable, which implies that the quadratures (1) can be inverted in terms of meromorphicfunctions. 5ote that the relation (10) is consistent with Theorem 1 saying the torus Prym(
C, σ )is a 4-fold unramified covering of Prym(
K, σ ). One should stress however that (10) holdsonly if anti-symmetric differentials on
C, K are both proportional to (8), (9).
Proof of Theorem
3. Item 1) was proven in [Hai983], where it was shown that Prym(
K, σ )and Prym(
C, σ ) are dual Prym subvarieties.Item 2) is a recent observation. Since Prym(
C, σ ) can be regarded as 4-fold coveringof Prym(
K, σ ) (see Theorem 1), the relation (10) between the period matrices must holdfor certain bases of holomorphic anti-symmetric differentials on
K, C . To prove that itholds for the bases (8), (9), we use the geometric fact K = Prym( C, σ ) ∩ Θ C (item 3of Theorem 2). Namely, let, as above u , u , u be the coordinates in C , the universalcovering of Jac( C ), and du , du be the corresponding holomorphic σ -anti-symmetricone-forms on Prym( C, σ ) ⊂ Jac( C ) such that ω = du (cid:12)(cid:12) C , ω = du (cid:12)(cid:12) C . Then relation (10) is geometrically equivalent to du (cid:12)(cid:12) K = ¯ ω , du (cid:12)(cid:12) K = ¯ ω . (11)To prove (11) explicitly, we use the following algebraic description of the theta-divisorΘ C = { P + P − ∞ − ∞ | P i = ( x i , w i ) ∈ C } , ∞ , being the two points on C with x = ∞ , σ ( ∞ ) = ∞ , and of the embedding K ֒ → Θ C , which was described in [Pan986] and made explicit in Appendix 5 of [Aud996]: K ∋ ( x, W ) D = ( x , w ) + ( x , w ) − ∞ − ∞ ∈ Θ C , (12)where now x = x = x and w , w are the solutions of the quadratic equation r ( w ) = w − W √ w + W − g ( x ) = 0 . (13)As one can check, the points ( x, w ) , ( x, w ) belong to C , and { w , w , − w , − w } giveall the solutions of the equation of C in (2) for a given x . Then D + σ ( D ) = ( x, w ) + ( x, w ) + ( x, − w ) + ( x, − w ) − ∞ − ∞ is the divisor of a meromorphic function f on C having 4 zeros over x and double polesat ∞ , ∞ , that is, D + σ ( D ) ≡
0. Thus for any point of K its image D belongs both toPrym( C, σ ) and to Θ C . One can also show that the map (12) is injective and surjective,hence it describes the isomorphism between K and Prym( C, σ ) ∩ Θ C .By the definition of du , du , and of the theta-divisor, on Θ C ⊂ Jac( C ) one has du (cid:12)(cid:12) Θ C = dx w ( w − g ( x )) + dx w ( w − g ( x )) ,du (cid:12)(cid:12) Θ C = x dx w ( w − g ( x )) + x dx w ( w − g ( x )) . K ⊂ Θ C we have x = x and du | K = w + w − g ( x )( w + w ) w w ( w − g ( x ))( w − g ( x )) dx du | K = w + w − g ( x )( w + w ) w w ( w − g ( x ))( w − g ( x )) x dx , (14)where now the coordinates w , w are the roots of the quadratic equation (13), hencetheir symmetric functions become functions of x, W . Then, after simplifications, theright hand sides of (14) yield du | K = √ dxW ( W − g ( x )) = ¯ ω , du | K = √ x dxW ( W − g ( x )) = ¯ ω , which proves (11) and item (2) of the theorem. (cid:3) The Abel–Prym map.
Analytically, the curve C ⊂ Prym(
K, σ ) can be viewed asthe image of the smooth embedding A : P ∈ C Z PP (cid:18) ω ω (cid:19) mod Λ (15)where P ∈ C is any fixed basepoint and Λ is the period lattice described in Theorem 3.For concreteness, we choose P to be one of the branch points of the covering π : C → E ,say P = Q . Then A sends Q to the origin, and the points Q , Q , Q to some half-periods of Prym( K, σ ).By analogy with A , we also define the map¯ A : K Prym(
C, σ ) = C / ¯Λ , (16) R ∈ K Z RZ (cid:18) ¯ ω ¯ ω (cid:19) mod ¯Λgiving a smooth embedding of K to Prym( C, σ ). The image ˆ K = Π ◦ ¯ A ( K ) ⊂ Prym(
K, σ )is a curve with self-intersections.Then the quadratures (1) for the Clebsch system give rise to the map P : C × K → Prym(
K, σ ) (17) P ∈ C, R ∈ K Z PQ (cid:18) ω ω (cid:19) + Z RZ (cid:18) ¯ ω ¯ ω (cid:19) = (cid:18) u u (cid:19) mod Λ . According to Theorem 3, P is well-defined: when the point P goes along a full cycleon C , the first integral in (17) changes by a vector of the lattice Λ; and when the point R goes along a full cycle on K , the second integral in (17) changes by a vector of ¯Λ,which is a sub-lattice of Λ.In the sequel it is natural to call P the Abel–Prym map . As we shall see below,in contrast to the standard Abel–Jacobi map, P is not injective, so, for a generic u =( u , u ) ∈ Prym(
K, σ ), its inversion is not unique. To our best knowledge, such kind ofmap did not appear before, neither in the classical nor in the modern literature. Note that the differentials ω , ω in (9) do not have common zeros on C . Not to be confused with the
Prym map , which has a completely different meaning in algebraicgeometry. he extended Abel–Prym map. The map P cannot be extended to the map C × K → Prym(
C, σ ) as the latter is not well-defined: under the map A , one and thesame point in C yields 4 different points in Prym( C, σ ). In this connection it is naturalto replace the curve C by its 4-fold covering D described in Theorem 1 and (5), (7), andto introduce the map b A : D 7→
Prym(
C, σ ): P ∈ D → b A ( P ) = Z Π( P ) Q (cid:18) ω ω (cid:19) mod ¯Λ , where, as above, Π is the projection D → C .Note that for a general genus 9 curve, the standard Abel map requires integrals of9 holomorphic differentials on it, otherwise it is not correctly defined. However, as wasshown in [Hai983], in case of D the following property holds. Proposition 4.
The map b A is injective and realizes a smooth isomorphism between D and its image b A ( D ) ⊂ Prym(
C, σ ) . It can also be written as P ∈ D → b A ( P ) = Z PQ ∗ (cid:18) Π ∗ ( ω )Π ∗ ( ω ) (cid:19) mod ¯Λ , where Π ∗ ( ω j ) , j = 1 , are the pull-backs of the holomorphic differentials ω j on C , and Q ∗ is one of the preimages Π − ( Q ) . Now introduce extended Abel–Prym map b P : D × K → Prym(
C, σ ): b P : ( P ∈ D , R ∈ K ) u = ( u , u ) t = b A ( P ) + ¯ A ( R ) , (18)¯ A being defined in (16). Like P , the above map b P is well-defined: when P goes along afull cycle on D or when the point R goes along a full cycle on K , the integrals in (18)change by a vector of the lattice ¯Λ.Note (without a proof) that, for any ( P, R ) ∈ D × K ,Π ∗ ◦ b P ( P, R ) = P (Π( P ) , R ) . (19) P , ˆ P by means of theta-functions We first build a set of theta-functions which will be used to invert the maps P , b P .As follows from item 2) of Theorem 3, there exists a unique change of variables( u , u ) → ( z , z ), described by a non-degenerate matrix T ∈ GL (2 , C ), which takes theperiod matrices of Prym( K, σ ), Prym(
C, σ ) to the normalized form T · ( V V V V ) = (cid:18) a b b c (cid:19) , T · ( ¯ V ¯ V ¯ V ¯ V ) = (cid:18) a b b c (cid:19) , (20)with some a, b, c satisfying the Riemann conditions, and by the extra change ( z , z ) → ( z , z / (cid:18) a b b c/ (cid:19) .8s was shown in [HvM989, EF17], Prym( K, σ ) is a 2-fold unramified covering of 3different principally polarized Abelian tori. As follows from the first period matrix in(20), one of them is the Jacobian of a genus 2 curve Γ with the Riemann matrix τ = (cid:18) a bb c (cid:19) . (21)Similarly, Prym( C, σ ) is a 2-fold unramified covering of 3 other principally polarizedAbelian tori, and, following the second relation in (20), one of them is the Jacobian of agenus 2 curve ˜Γ with the period matrix (cid:18) a b b c (cid:19) , giving, by the rescaling z → z , the matrix (cid:18) a/ b/
20 1 b/ c/ (cid:19) . (22)Thus, the Riemann matrix of ˜Γ is τ /
2, and one has the following chain of 2-fold coveringsPrym(
C, σ ) −−−−→ Jac(˜Γ) −−−−→
Prym(
K, σ ) −−−−→ Jac(Γ) . (23)Explicit algebraic equation of Γ in terms of the coefficients of C was given by F.K¨otter in [Kot892] (see also [EF17]), who linearized the Clebsch system on Jac(Γ). Inthe sequel we will need only the matrix τ , and not expressions for Γ , ˜Γ themselves.Now consider the standard Riemann theta-function θ ( z | τ ), z = ( z , z ) t = T ( u , u ) t ,associated with the Riemann period matrix τ in (21), θ ( z | τ ) = X N ∈ Z exp ( π h N, τ N i + 2 π h N, z i ) , = √− α = ( α , α ) , β = ( β , β ) ∈ Z / Z defined by θ (cid:20) αβ (cid:21) ( z | τ ) = exp (cid:0) π ( ατ α t + 2 βα t + 2 β z ) (cid:1) θ ( z + β t + τ α t | τ ) . Thus θ h αβ i ( z | τ ) is the Riemann theta-function with the argument translated by the half-period β t + τ α t and multiplied by an exponent. These functions have the quasiperiodicproperty: for any N, M ∈ Z ,θ (cid:20) αβ (cid:21) ( z + K + τ M ) = exp(2 πǫ ) exp {h M, τ M i / h M, z i} θ (cid:20) αβ (cid:21) ( z ) , (24) ǫ = h α, K i − h β, M i , It is known ([Mum984, BBEIM, Fay84]) that the condition θ ( z | τ ) = 0 defines thetheta-divisor Θ ⊂ Jac(Γ) isomorphic to the curve Γ itself, and θ ( z | τ ) vanishes at 6 oddhalf-periods in Jac(Γ) corresponding to the characteristics∆ = (cid:18) / /
20 1 / (cid:19) , ∆ = (cid:18) /
20 1 / (cid:19) , ∆ = (cid:18) / / / (cid:19) , ∆ = (cid:18) / / / (cid:19) , ∆ = (cid:18) / / (cid:19) , ∆ = (cid:18) / / / (cid:19) . (25)9 he quasi-periodic theta-functions on Prym(
K, σ ) . Let us set θ ( z ) = θ (cid:20) (cid:21) ( z + K | τ ) , θ ( z ) = θ (cid:20) /
20 0 (cid:21) ( z + K | τ ) , (26)where K = (1 / , t + τ (1 / , / t is the vector of Riemann constants, the half-periodcorresponding to the characteristic ∆ (see, e.g., [Mum984]).In view of (24), for a generic quotient ( ν : µ ) ∈ P , the functionΞ ν,µ ( z ) = νθ ( z ) + µθ ( z )is not quasiperiodic on Jac(Γ), but on its 2-fold covering T → Jac(Γ), obtained bydoubling the period vector (0 , t of Jac(Γ). Indeed, θ (cid:0) z + (0 , t (cid:1) = θ ( z ) , θ (cid:0) z + (0 , t (cid:1) = − θ ( z ) , and if z changes by any other period vector, θ , θ are multiplied by the same factor. Inview of the first matrix in (22), T is precisely Prym( K, σ ). Hence the zeros of Ξ ν,µ ( z )on Prym( K, σ ) are well-defined. Following [HvM989], equation Ξ ν,µ ( z ) = 0 defines thecorresponding curve of the pencil { C λ | λ = µ/ν ∈ C } described in Theorem 2.Note that Ξ ν,µ ( z ) can be regarded as a higher-order theta-function (or a Prym theta-function) introduced in [Fay84].Since we changed to the normalized coordinates ( z , z ) t = T ( u , u ) t on the Pryms,we now redefine the maps (15), (16) accordingly: A : P ∈ C Z PQ T (cid:18) ω ω (cid:19) , ¯ A : R ∈ K Z RZ T (cid:18) ¯ ω ¯ ω (cid:19) . Observe that the images A ( Q ) , . . . , A ( Q ) ∈ Prym(
K, σ ) of the basepoints Q , . . . , Q of the pencil { C λ } are solutions of both equations θ ( z ) = 0 , θ ( z ) = 0. On Jac(Γ)they define two points z = , z = τ (0 , t , which give 4 preimages on Prym( K, σ ).For a particular λ ∗ = µ ∗ /ν ∗ , equation Ξ ν ∗ ,µ ∗ ( z ) = 0 defines the curve C ⊂ Prym(
K, σ )itself, that is, Ξ ν ∗ ,µ ∗ ( A ( P )) ≡ P ∈ C . (27)The coefficients ν ∗ , µ ∗ can be found as follows. Let ε be a local parameter near the basepoint Q = ( s ,
0) such that ε ( Q ) = 0. One can chose ε = √ x − s . Then, near Q , A ( P ) = φ ( Q ) ε + O ( ε ) , φ ( Q ) = ( φ , φ ) t = ddε A ( P ) (cid:12)(cid:12)(cid:12) P = Q . In view of the choice of the differentials on C , ( φ , φ ) t is proportional to T · (1 , s ) t .Then, taking derivative of (27) with respect to ε , we get ν ∗ (cid:20) ∂θ ∂z (0 , φ + ∂θ ∂z (0 , φ (cid:21) + µ ∗ (cid:20) ∂θ ∂z (0 , φ + ∂θ ∂z (0 , φ (cid:21) = 0 , or ν ∗ ∂ V θ (0 ,
0) + µ ∗ ∂ V θ (0 ,
0) = 0 , V = T · (1 , s ) t . (28)10he vector V and the derivatives ∂ z j θ (0 , , ∂ z j θ (0 , j = 1 , τ in (21).As a result, for ν ∗ , µ ∗ satisfying (28), the theta-function in (27) takes the formΞ ν ∗ ,µ ∗ ( z ) = ∂ V θ ( ) θ ( z ) − ∂ V θ ( ) θ ( z ) . (29)In the sequel we will denote this function as just Ξ ∗ ( z ). F. K¨otter’s solution in terms of theta-functions.
Using the above notation, thecomplete solution of the Clebsch system given on page 100 of [Kot892] can be writtenas follows S α = d α θ [ η α ]( z + K ) + e α θ [ η α β ]( z + K )Ξ ν ∗ ,µ ∗ ( z ) ,T α = ¯ d α θ [ η α ]( z + K ) + ¯ e α θ [ η α β ]( z + K )Ξ ν ∗ ,µ ∗ ( z ) , α = 1 , , ,, (30) z = T (cid:18) t s t p (cid:19) + z , (31)where η α , η β are some of the characteristics from (25), and η α β = η α + η β mod Z / Z .(We do not give explicit expressions for them and for the constants d α , e α , ¯ d α , e α , α =1 , , t s , t p are the complex times along the flows X s , X p , and z is theinitial phase.Using the quasiperidicity property (24), one can check that the theta-quotients in theright hand sides in (30) are not meromorphic on Prym( K, σ ), but on its 4-fold coveringPrym(
C, σ ), as stated in Theorem 1. According to the above solution, the variables S α , T α have simple poles along the preimage Π − ( C ), which is the curve D ⊂
Prym(
C, σ )described in the same theorem.
The quasi-periodic theta-functions on
Prym(
C, σ ) . By analogy with the theta-function Ξ ν,µ ( z ), introduce the family of functionsΥ ν,µ ( z ) = ν θ (cid:16) z | τ (cid:17) + µ θ (cid:16) z | τ (cid:17) , (32) θ (cid:16) z | τ (cid:17) = θ (cid:20) (cid:21) (cid:16) z | τ (cid:17) , θ (cid:16) z | τ (cid:17) = θ (cid:20) / (cid:21) (cid:16) z | τ (cid:17) ,ν, µ being arbitrary constants. Here each of the theta functions θ , θ is quasi-periodicon the Jacobian of the curve ˜Γ in the diagram (23), but Υ ν,µ ( z ) is quasiperiodic only on2-fold covering of Jac(˜Γ), which is Prym( C, σ ) .Indeed, in view of (22) (20), the period matrix of Prym( C, σ ) is obtained from thatof Jac(˜Γ) by doubling the 3rd period vector. Then, in view of (24), θ (cid:16) z τ , t (cid:17) = exp (cid:16) τ z (cid:17) θ (cid:16) z (cid:17) , θ (cid:16) z τ , t (cid:17) = − exp (cid:16) τ z (cid:17) θ (cid:16) z (cid:17) , The argument z / z / ν,µ ( z ) on Prym( C, σ ) are well-defined. For some particular values ν ∗ , µ ∗ , the equation Υ ν ∗ ,µ ∗ ( z ) = 0 defines the curve K ⊂ Prym(
C, σ ), that isΥ ν ∗ ,µ ∗ ( ¯ A ( R )) ≡ R ∈ K. Similarly to (28), (29), one can show thatΥ ν ∗ ,µ ∗ ( z ) = ∂ v θ ( | τ / θ (cid:16) z | τ (cid:17) − ∂ v θ ( | τ / θ (cid:16) z | τ (cid:17) , (33)where v = T (1 , z ) t is the vector tangent to the base point Z = (0 , z ) of the curve K ⊂ Prym(
C, σ ). In the sequel we denote Υ ν ∗ ,µ ∗ ( z ) as Υ ∗ ( z ).We note that by using addition formulae for theta-functions with characteristics (seee.g., [Fay84]), the functions θ ( z | τ ) , θ ( z | τ ) can be written as sums of products of θ (cid:0) z | τ (cid:1) , θ (cid:0) z | τ (cid:1) . Inversion of the Abel–Prym maps P , b P . For Ξ ∗ ( z ) introduced in (29), considerthe function F ( P ) = Ξ ∗ ( e + A ( P )) , P ∈ C,e ∈ C being any fixed vector. F ( P ) is not single-valued on the curve C as, by itsdefinition, Ξ ∗ is multiplied by an exponent when its argument increases by a vector ofthe lattice Λ. However, zeros of F ( P ) on C are well-defined. Theorem 5. If e = 0 , the function F ( P ) has precisely 4 zeros on C (possibly, withmultiplicity).Proof. Let D ⊂ A be an algebraic curve giving an ample divisor on an Abelian surface A with polarization ( δ , δ ), and L ( kD ) denote the linear space of meromorphic functionson A having poles of degree at most k along D only.To prove the theorem we use the adjunction formula (see, e.g., [GH]):dim L ( D ) = 12 D · D = genus( D ) − δ δ , where D · D is the number of intersections of D with its any translation or with anycurve D ′ ⊂ A linearly equivalent to D .Equation Ξ ∗ ( z ) = 0 defines the curve C ⊂ Prym(
K, σ ), whereas e + A ( P ) , P ∈ C defines its translation by the vector e . Setting in the above formula D = C and ( δ , δ ) =(1 , C in Prym( K, σ ) intersect at 4 points. (cid:3)
Next, for a fixed vector e ∈ C consider the function¯ F ( R ) = Ξ ∗ (cid:0) e + Π ◦ ¯ A ( R ) (cid:1) = an exponent · Ξ ∗ (cid:18) e + Z RZ T (cid:18) ¯ ω ¯ ω (cid:19)(cid:19) , R ∈ K, where, as above, Π is the projection Prym( C, σ ) → Prym(
K, σ ).12 heorem 6. If e = 0 , ¯ F ( R ) has precisely 8 zeros on the curve K (possibly, with multi-plicity).Proof. Observe that, for R ∈ K , the image e + Π ◦ ¯ A ( R ) describes a translation ˆ K e ofΠ( K ) in Prym( K, σ ). Then ¯ F ( R ) vanishes exactly at the intersection points C ∩ ˆ K e .Note that C · Π( K ) = Π − ( C ′ ) · K ′ for any curves C ′ , K ′ linearly equivalent to C ,respectively K . Next, the pullback Π − ( C ′ ) is linearly equivalent to 2 K ⊂ Prym(
C, σ ).Hence, Π − ( C ′ ) · K ′ = 2 K · K = 2( K · K ), which, by the adjunction formula, equals2 · (cid:3) Now return to the Abel–Prym map (17) and observe that it can be written in theform P : ( P ∈ C, R ∈ K ) z = ( z , z ) t = A ( P ) + Π ◦ ¯ A ( R ) . (34) Theorem 7.
Under the map P , any z ∈ Prym(
K, σ ) has precisely 8 preimages ( P , R ) , . . . , ( P , R ) on C × K (possibly, with multiplicity) . The points R , . . . , R arezeros of the function F ( R ) = Ξ ∗ (cid:0) z − ¯ A ( R ) (cid:1) , R ∈ K. For each R i , the corresponding point P i ∈ C is found as the unique solution of thetranscendental equation A ( p ) ≡ Z pQ T (cid:18) ω ω (cid:19) = z − ¯ A ( R i ) , p ∈ C (35) Remark.
In practice, Theorem 7 allows to calculate only symmetric functions of co-ordinates of R , . . . , R , i.e., functions which are invariant with respect to permutationsof the points. Such functions can be expressed in terms of the theta-function Ξ ∗ ( z ) andits derivatives, in the same way as for the case of inversion of the standard Abel map,see e.g., [Dub81]). Proof of Theorem
7. In view of Theorem 6, for any z ∈ C the function F ( Q ) on K has 8zeros, denoted as R , . . . , R . Then, for each R i the vector z − ¯ A ( R i ) belongs to the zerolocus of Ξ ∗ , that is, following (27), to C ⊂ Prym(
K, σ ). Since the map A is injective,there is a unique solution to (35) giving P i ∈ C such that A ( P i ) + Π ◦ ¯ A ( R i ) = z . (cid:3) Although the map P is not one-to-one, it also has an obvious ”injectivity” property:for any z and ˜ z ∈ Prym(
K, σ ) z = ˜ z mod Λ = ⇒ {P − ( z ) } ∩ {P − (˜ z ) } = ∅ . (36)Indeed, assume that a pair ( P ∗ , R ∗ ) ∈ C × K belongs to the intersection {P − ( z ) } ∩{P − (˜ z ) } . This means that P ( P ∗ , R ∗ ) gives both z and ˜ z , which is not possible, because P is well-defined. One can compare this with the standard Abel–Jacobi map G ( g ) → Jac( G ), where G is a smoothgenus g curve and G ( g ) is its symmetric power. A generic point in Jac( G ) has just one pre-image on G ( g ) . nversion of the extended Abel–Prym map. Theorem 7 itself does not allow toexpress symmetric functions of coordinates of P , . . . , P ∈ C in terms of theta-functions.This becomes possible by considering inversion of the map b P : D × K → Prym(
C, σ ).Namely, the following analog of Theorem 7 holds.
Theorem 8.
Under the map b P , any z ∈ Prym(
C, σ ) has precisely 8 preimages ( P ∗ , R ) , . . . , ( P ∗ , R ) on D × K (possibly, with multiplicity). The points P ∗ , . . . , P ∗ ∈ D are zeros of the function F ( P ) = Υ ∗ (cid:16) z − b A ( P ) (cid:17) , P ∈ D , with Υ ∗ ( z ) defined in (33) .For each P ∗ i , the corresponding point R i ∈ K is found as the unique solution of thetranscendental equation ¯ A ( q ) ≡ Z qZ T (cid:18) ¯ ω ¯ ω (cid:19) = z − b A ( P ∗ i ) , q ∈ K. (37)Proof of this theorem goes along the same lines as that of Theorem 7 describing theinversion of the map P . Again, in practice, Theorem 8 allows to calculate only symmetricfunctions of the coordinates of P ∗ , . . . , P ∗ ∈ D in terms of Υ ∗ ( z ) and its derivatives.Similarly to P , the extended map ˆ P enjoys the injectivity property z = ˜ z mod ¯Λ = ⇒ { ˆ P − ( z ) } ∩ { ˆ P − (˜ z ) } = ∅ . (38)Both theorems, together with the relation (19), lead to following property. Proposition 9.
For any z ∈ Prym(
C, σ ) , let b z = Π( z ) ∈ Prym(
K, σ ) . Then, if b P − ( z ) = { ( P ∗ , R ) , . . . , ( P ∗ , R ) } , one has P − ( b z ) = { ( P , R ) , . . . , ( P , R ) } such that P i = Π( P ∗ i ) , i = 1 , . . . , . That is, the following diagram holds { ( P ∗ k , R k ) , k = 1 , . . . , } b P − ←−−−− z ∈ Prym(
C, σ ) Π y Π ∗ y { ( P k , R k ) , k = 1 , . . . , } P − ←−−−− b z ∈ Prym(
K, σ ) As stated in Theorem 1, there is a bijection between the compactified complex invarianttorus I H ∪ D and Prym( C, σ ), and between the compactified factor variety ( I H / g ) ∪ C and Prym( K, σ ). On the other hand, as we recalled in Introduction, each point on I H gives rise to 8 sets { ( x , w ) , ( x , W ) } of the separating variables constructed in Section4 of Part I, i.e., there is a correspondence S : ( S α , T α ) → { ( x , w ) , ( x , W ) } . Its inversion is described by the reconstruction formulas (38, Part I).It is then natural to expect that for any z ∈ Prym(
K, σ ), the coordinates of the eightpreimages on C × K , { ( P , R ) , . . . , ( P , R ) } = P − ( z ) give precisely the above eightsets. Indeed, the following theorem holds. 14 heorem 10.
1) The coordinates ( x , w ) , ( x , W ) of any of the preimages ( P , R ) , . . . , ( P , R ) of z ∈ Prym(
K, σ ) give the same values of the squares S α , T α , α = 1 , , via the reconstruction formulas (38, Part I).Equivalently, under the map P , the 8 sets of the separation variables ( x , w ) ∈ C, ( x , W ) ∈ K , corresponding to a point on I H , give the same point z ∈ Prym(
K, σ ) .2) The coordinates ( x , w, v , v , v ) , ( x , W ) of any of the preimages ( P ∗ , R ) , . . . , ( P ∗ , R ) ∈ D × K of z ∗ ∈ Prym(
C, σ ) give the same values of S α , T α , α = 1 , , via thesame reconstruction formulas.Proof. For any fixed point ( ¯ S α , ¯ T α ) ∈ ( I H / g ) ∪ C , let { ( x ∗ , w ∗ ) , ( x ∗ , W ∗ ) } and { ( x ∗∗ , w ∗∗ ) , ( x ∗∗ , W ∗∗ ) } be any two sets of S ( ¯ S α , ¯ T α ). Let now z ∗ = P (( x ∗ , w ∗ ) , ( x ∗ , W ∗ )) , z ∗∗ = P (( x ∗∗ , w ∗∗ ) , ( x ∗∗ , W ∗∗ )) . Assume that z ∗ = z ∗∗ (mod Λ). Then, by the injectivity property (36), {P − ( z ∗ ) } ∩ {P − ( z ∗∗ ) } = ∅ , hence { ( x ∗∗ , w ∗∗ ) , ( x ∗∗ , W ∗∗ ) } 6 = {P − ( z ∗ ) } , { ( x ∗ , w ∗ ) , ( x ∗ , W ∗ ) } 6 = {P − ( z ∗∗ ) } . It follows that the reconstruction S − applied to {P − ( z ∗ ) } gives different values ( S α , T α ),because otherwise, together with { ( x ∗∗ , w ∗∗ ) , ( x ∗∗ , W ∗∗ ) } , there were 9 different sets giv-ing the same ( ¯ S α , ¯ T α ). But this contradicts the fact that the composition of S andthe map P realizes a bijection between ( I H / g ) ∪ C and Prym( K, σ ). Therefore ourassumption z ∗ = z ∗∗ was wrong. This proves item 1).Item 2) follows from the the same arguments and Proposition 9. (cid:3) Theorems 10 and 8 enables one to obtain theta-function solution to the Clebschsystem in terms of theta-functions. To do this we only need to express the variables S α , T α in terms of symmetric functions of coordinates of the points P ∗ , . . . , P ∗ on D ,the preimages of z ∈ Prym(
C, σ ). According to Theorem 8, the latter functions can bewritten in terms of Υ ∗ ( z ) in (33) and its derivatives.Note however, that the structure of the obtained theta-function solutions will bedifferent from K¨otter’s solutions (30), as the latter are written in terms of Υ ∗ ( z ).Our main tool will be again the constraint equation (22, Part I), defining the sub-manifold ˆ S in the extended phase space ˆ M : F ( x ) = X α =1 c α ( v α S α + v β v γ T α ) = 0 , v α = x + j α , c α = 1( j α − j β )( j α − j γ ) . (39)Here, as above, the radicals v α = √ x + j α , α = 1 , , E . Substituting their parameterization (6) into (39), we obtain F = ( f Z + f Z + f ) Y + f Z + f Z + f Z + f Z + f = 0 , (40)15ith f = 2( ˆ S + ˆ S + ˆ S ) , f = 4( j ˆ S + j ˆ S + j ˆ S ) , f = 2 X ( α,β,γ ) j α j β ( ˆ S α + ˆ S β − ˆ S γ ) ,f = ˆ T + ˆ T + ˆ T , f = 2 X ( α,β,γ ) ( j α + j β ) ˆ T γ , f = 6 X ( α,β,γ ) j α j β ˆ T γ , (41) f = 2 X ( α,β,γ ) [ j γ j β ( j + j + j ) + j j j − j α ( j γ + j β )] ˆ T α ,f = X ( α,β,γ ) ( j α j γ + j β j γ − j α j β )( j α j β + j β j γ − j α j γ ) ˆ T α = 0 , where we set ˆ S α = c α S α , ˆ T α = c α T α , ( α, β, γ ) = (1 , , Z, Y areconstrained by the equation of ˆ E , for each generic T, S the equation (40) has precisely8 solutions ( Z , Y ) , . . . , ( Z , Y ), which are projections of the preimages P ∗ , . . . , P ∗ ∈ D onto b E (see the equation (7) of D ).Substituting these solutions into (40) subsequently, one obtains a system of 8 linearhomogeneous equations for the eight coefficients f , . . . , f . By construction, the rankof the system is 6. Solving it and inverting the linear relations (41), one finds thevariables ˆ S α , ˆ T α and, therefore, S α , T α , as symmetric functions of ( Z , Y ) , . . . , ( Z , Y )up to a common factor κ : S α = Σ α κ , T α = Σ α +3 κ , α = 1 , , . (42)Explicit expressions for the symmetric functions Σ , . . . , Σ are quite long, so we do notgive them here.Next, we substitute (42) into the area integral S T + S T + S T = C and obtain κ as a symmetric function of ( Z , Y ) , . . . , ( Z , Y ) as well. Finally, by using equation (7) ofthe curve D ⊂ C ( Z, Y, G ), this function can be shown to be a full square of a symmetricfunction of ( Z , Y , G ) , . . . , ( Z , Y , G ), the coordinates of the points P ∗ , . . . , P ∗ ∈ D .This, together with (42), provides us with the expresions for S α , T α we needed.The latter can be regarded as a (much more cumbersome) alternative to the recon-struction formulas (38, Part I). There is a more essential difference although: whereasthese formulas give the original variables S α , T α in terms of just one pair of points onboth curves, D and K , the expressions (42) involve 8 points, but on D only. Acknowledgments
Y. Fedorov is grateful to V. Enolski, A. Izosimov, T. Shaska for stimulating discus-sions. His contribution was partially supported by the Spanish MINECO-FEDER GrantMTM2015-65715-P and the Catalan grant 2017SGR1049. The authors also acknowledgethe support and hospitality of the Department of Mathematics of Universit´a di MilanoBicocca, where a part of this work had been made.16 eferences [AvM984] Adler M., van Moerbeke P.: Geodesic flow on so (4) and the intersections ofquadrics. Proc.Natl. Acad. Sci. USA. , (1984), 4613–4616[Aud996] Audin, M. Spinning tops. A course on integrable systems.
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