KOWALEVSKI TOP AND GENUS TWO CURVES

DIMITRI MARKUSHEVICH

We studied in [LM] two curves of genus 2, such that the equations of motion of the Kowalevski top are linearized on their Jacobians: Kowalewski’s [Kow],

C_{1} : u^{2} = x((x − H)^{2} − \frac{1}{4}I_{2})((x − H)^{2} − \frac{1}{4}I_{2} + 1),

and the Bobenko–Reyman–Semenov-Tian-Shansky (BRS) curve [BRS]

C_{2} : u^{2} = x(x^{2} + 2Hx + \frac{1}{4}I_{2})(x^{2} + 2Hx + \frac{1}{4}I_{2} − 1),

where H is the Hamiltonian, I_{2} Kowalevski’s integral of motion. Both equations are written for the case when the angular momentum l is orthogonal to the gravity vector g, because only in this case the procedure of Bobenko–Reyman–Semenov-Tian-Shansky leads to a genus 2 curve. We established an isogeny between the Jacobians J(C_{i}) by purely algebraic means, using Richelot’s transformation of a genus 2 curve.

The curve C_{2} arises in a quite natural way when one applies the method of spectral curves to the Lax representation of the Kowalevski top discovered by Reyman–Semenov-Tian-Shansky [R-STS]. The curve C_{1} was found by Kowalevski as a result of an ingenious change of variables, and it is quite surprising that no simple explanation of her trick, putting it in the framework of a more general procedure, has been suggested during the period of more than 100 years. The results of [LM] show how Kowalevski’s curve appears from the spectral curve of the Lax representation of Reyman–Semenov-Tian-Shansky. This gives a partial answer to the above question. Now we will explain the relation of C_{1}, C_{2} to the spectral curve B of another Lax representation: the one of Kuznetsov–Tsiganov [KT]. The latter exists under the same condition as the one for the existence of C_{2}: the scalar product (l, g) should be zero.

This relation is based on the transformation of Kowalevski’s curve into a genus 3 curve introduced by Kuznetsov [K1], [K2], [KK] in his solution to the problem of separation of variables for the Kowalevski top. Remark, that it exists for arbitrary (l, g), but we use it only in the case (l, g) = 0. It yields the following diagram of morphisms of (smooth compact complex) curves:

The paper for the Proceedings of the workshop Mathematical Methods of Regular Dynamics (Leeds, April 11–15, 2000), dedicated to the 150th anniversary of Sophie Kowalevski.

\begin{aligned}C_{1} & ⟵^{f} & B_{1} \\ π & ↓ \\ & & E_{1} \end{aligned},  (1)

where g(C_{1}) = 2, g(B_{1}) = 3, g(E_{1}) = 1, and f, π are double coverings; f is unramified, and π is ramified at 4 points.

We prove that the diagram induces an isogeny P (B_{1}/E_{1})⟶J(C_{1}) with kernel of order 8 and another one J(C_{1})⟶P (B_{1}/E_{1}) in the opposite direction with kernel of order 2. Next, we fulfil a parallel construction for another curve of genus 3, namely, the spectral curve B_{2} = B of the Kuznetsov–Tsiganov L-matrix:

\begin{aligned}C′_{2} & ⟵^{f̃} & B_{2} \\ π̃ & ↓ \\ & & E_{2} \end{aligned},  (2)

where the genera of curves and the properties of the morphisms are the same as in the preceding diagram. It turns out that C′_{2} is nothing else but the BRS curve C_{2}. So, we get another pair of isogenies between P (B_{2}/E_{2}), J(C_{2}) with kernels of orders 2 and 8. Combining them with the Richelot isogeny J(C_{2})⟶J(C_{1}) (with kernel of order 4), we obtain an isogeny between the Prym varieties P (B_{i}/E_{i}) with kernel of order 64. It factors through the multiplication by 2, hence there also exists an isogeny between P (B_{i}/E_{i}) of degree 4.

To summarize, we have the following relations between the above curves. The curves B_{2}, C_{2} are “natural” in the sense that they come from a general technique of integrable systems – method of spectral curves, – but this general technique is applied to different Lax matrices. B_{2}, C_{2} are related to each other by the analog of Kuznetsov’s transformation, which he originally applied to the other pair of curves B_{1}, C_{1}. Now, B_{1}, C_{1} cannot be obtained by the method of spectral curves from the known Lax representations. Nonetheless, C_{1}, C_{2} are connected by Richelot’s transformation, so B_{1}, C_{1} become indirectly related to B_{2}, C_{2}. Remark, that for generic constants of motion H, I_{2}, there are no non-trivial maps between any two curves from different triples B_{i}, C_{i}, E_{i}, i = 1, 2, but there exists a (2, 2)-correspondence C_{1}⟷C_{2}. One can see that the curves E_{1}, E_{2}, as well as the Jacobian threefolds J(B_{1}), J(B_{2}), are not isogeneous, though the latter ones contain isogeneous abelian surfaces P (B_{i}/E_{i}).

One can think of Kuznetsov’s transformation as a way to pass from the Jacobian of a curve of genus 2 to an isogeneous Prym variety of a double covering B⟶E of a genus 3 curve over an elliptic curve. As such, it represents a particular case of Audin’s construction, described in Section 5.3 of her book [Au].

Acknowledgements. The author is grateful to V. B. Kuznetsov and I. V. Komarov for discussions and for supplying formulas for Section 3. The author also acknowledges with pleasure the hospitality of the Max-Planck-Institut für Mathematik at Bonn, where he wrote this paper.

1. Generalities

We will use the notations and the normalization of dimensional parameters from [BRS]. The top represents a solid with a fixed point O in a constant gravity field. There are 6 dynamical variables: 3 components of the angular momentum l = (l_{1}, l_{2}, l_{3}) and 3 components of the gravity vector g = (g_{1}, g_{2}, g_{3}), everything with respect to a moving orthonormal frame, attached to the solid. The motion of the top can be described by the following system:

\{ \begin{matrix}\frac{dl}{dt} = [l, ω] + [c, g] \\ \frac{dg}{dt} = [g, ω] \\ l = Iω \end{matrix} ,  (3)

where c is the constant vector of the center of mass, I the inertia tensor. The system (3) is Hamiltonian with respect to the Poisson brackets

{l_{i}, l_{j}} = ϵ_{ijk}l_{k}, {l_{i}, g_{j}} = ϵ_{ijk}g_{k}, {g_{i}, g_{j}} = 0.

In Kowalevski’s integrable case, the principal inertia moments have the ratio 2m : 2m : m and c lies in the plane, perpendicular to the symmetry axis. One can normalize the parameters so that I = diag(1, 1, 1/2), c = (1, 0, 0) and |g| = 1. The problem possesses a trivial integral of motion, the scalar product (l, g), which we will consider as a parameter. Fixing its value (l, g) = δ, we obtain a 4-dimensional phase space, which is a symplectic manifold. The Hamiltonian

H = \frac{1}{2}(l_{1}^{2} + l_{2}^{2} + 2l_{3}^{2}) − g_{1}

Poisson commutes with another, Kowalevski’s first integral

I_{2} = (l_{1}^{2} − l_{2}^{2} + 2 g_{1})^{2} + 4 (l_{1}l_{2} + g_{2})^{2} ,

and the common level sets of H, I_{2} represent the Liouville tori in our symplectic 4-fold.

The first step of Kowalevski’s solution to the problem is the complexification: she considers x = l_{1} + il_{2}, y = l_{1} − il_{2} as independent complex variables. Next she makes her famous change of variables ξ_{1} = ξ_{1}(x, y), ξ_{2} = ξ_{2}(x, y), which we will not specify here. It brings the system (3) to the form

\frac{dξ_{1}/dt}{η_{1}} + \frac{dξ_{2}/dt}{η_{2}} = 0 , \frac{ξ_{1}dξ_{1}/dt}{η_{1}} + \frac{ξ_{2}dξ_{2}/dt}{η_{2}} = \sqrt{−2}  (4)

where the two points (ξ_{i}, η_{i}), i = 1, 2, belong to Kowalevski’s curve C_{1} of genus 2

η^{2} = [ξ((ξ − H)^{2} + 1 − \frac{1}{4}I_{2}) − δ^{2}]((ξ − H)^{2} − \frac{1}{4}I_{2}).

The equations (4) represent thus a system of ODE on the symmetric square Sym^{2}(C_{1}) of C_{1}. The Abel–Jacobi map

\begin{matrix}AJ : Sym^{2} C_{1} & ⟶ & J(C_{1}) = Ω_{C_{1}}^{∗}/H_{1}(C_{1}, ℤ) \\ (P_{1}, P_{2}) & ↦ & ∫_{P_{0}}^{P_{1}} + ∫_{P_{0}}^{P_{2}} mod H_{1}(C_{1}, ℤ), \end{matrix}

maps birationally the symmetric square onto the Jacobian J(C_{1}); here the integrals are considered as linear functionals on the space Ω_{C_{1}} of holomorphic 1-forms, defined modulo the period lattice H_{1}(C_{1}, ℤ). This allows to consider (4) as the linearized system Ẋ = V_{0} = const on the Jacobian, where V_{0} = (0, \sqrt{−2}) in the basis of T_{0}J(C_{1}) = Ω_{C_{1}}^{∗}, dual to the basis (dξ/η, ξdξ/η) of Ω_{C_{1}}.

The method of spectral curves, applied to the Lax representation of Reyman–Semenov-Tian-Shansky, brings us to a similar linearized equation Ẏ = W_{0} = const on the Prym variety P (C_{2}/E_{2}) of a double unramified covering of curves C_{2} ⟶^{2:1} E_{2}. Generically, the genera of C_{2}, E_{2} are equal resp. to 3, 1. However, if we restrict ourselves to the symplectic leaf δ = 0, we have g(C_{2}) = 2, g(E_{2}) = 0, so the Prymian P (C_{2}/E_{2}) becomes simply the Jacobian J(C_{2}). In this case we find two different curves of genus 2 with the same property that the Hamiltonian flow of the Kowalevski top is linearized on their Jacobians: C_{1} and C_{2}. The authors of [BRS] wrote out the equation of C_{2} and raised the question about the relation between the two curves. We will call C_{2} the Bobenko–Reyman–Semenov-Tian-Shansky (BRS) curve. Remark that none of the Jacobians is isomorphic to the complex Liouville torus, but only isogeneous to it.

The following theorem is the main result of [LM]:

Theorem 1. Assume δ = 0, that is l ⊥ g. Then C_{1}, C_{2} are related by the Richelot transformation which induces an isogeny ψ :

J(C_{2})⟶J(C_{1}) with kernel ℤ/2ℤ ⊕ ℤ/2ℤ. This isogeny transforms the flow of W_{0} into that of V_{0}.

We will describe briefly the way in which appears the curve C_{2} and postpone the definition of the Richelot isogeny until the next section.

The Lax pair of Reyman–Semenov-Tyan-Shansky yields a more general system, called Kowalevsky gyrostat. If we put the gyrostatic terms to 0, we will obtain the equation

\frac{dL}{dt} = [L, M]  (5)

with Lax matrix

L(λ) = \frac{1}{λ} [ \begin{matrix}g_{1} & g_{2} & g_{3} & 0 \\ g_{2} & −g_{1} & 0 & −g_{3} \\ g_{3} & 0 & −g_{1} & g_{2} \\ 0 & −g_{3} & g_{2} & g_{1} \end{matrix} ] + [ \begin{matrix}0 & 0 & −l_{2} & −l_{1} \\ 0 & 0 & l_{1} & −l_{2} \\ l_{2} & −l_{1} & 0 & −2l_{3} \\ l_{1} & l_{2} & 2l_{3} & 0 \end{matrix} ]

+ λ [ \begin{matrix}0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & −2 & 0 \\ 0 & 0 & 0 & −2 \end{matrix} ]

where [L, M] = LM − ML for some matrix M, which we will not explicitize here. The functions H, δ^{2} and I_{2} belong to the algebra generated by the coefficients of λ^{−2} and λ^{0} in the Laurent expansions of Tr(L(λ)^{2}) and Tr(L(λ)^{4}). Since these coefficients are invariant under the flow of (5), the spectral curve P (λ, μ) = 0 is also invariant, where

P (λ, μ) = det (L(λ) − μ).

Let Γ be the non-singular compactification of the spectral curve, and L(t) a solution of (5). Then we have the line bundle E_{t} of eigenvectors of L(t) on Γ. It is defined a priori on a Zariski open subset of Γ, but it is uniquely extended to all of Γ as a line subbundle of the trivial vector bundle ℂ^{4} ×Γ. According to [Au], Section III.2.3, the degree of E_{t} is 8, so that the isomorphism class [E_{t}] is an element of Pic^{8}(Γ). Choosing a reference point P_{0} ∈ C_{2}, we can write Pic^{8}(Γ) = 8[P_{0}] + J(Γ), which identifies Pic^{8}(Γ) with J(Γ), hence the evolution equation for [E_{t}] can be written on J(Γ). The following theorem is proved in [BRS].

Theorem 2 (Bobenko–Reyman–Semenov-Tyan-Shansky). The evolution equation for [E_{t}] has the form

\frac{d[E_{t}]}{dt} = W_{0} = const,  (6)

where the vector W_{0} ∈ Ω_{Γ}^{∗} is given via its values on the elements ω of the dual space:

<ω|W_{0} >= ∑_{P:λ(P)=∞} Res_{P} (\frac{1}{2}μω) .

Further, Γ possesses 2 commuting involutions, changing the signs of coordinates:

τ_{1} : (λ, μ) ↦ (−λ, μ), τ_{2} : (λ, μ) ↦ (λ, −μ).

Let C_{2} = Γ/ < τ_{1} >, E = Γ/ < τ_{1}, τ_{2} > be the quotients. Then the flow of (6) is costraint to the image π^{∗} Pic^{4}(C_{2}) ⊂ Pic^{8}(Γ) and tangent to P (C_{2}/E), where π : Γ ⟶^{2:1} C_{2} is the quotient map.

From now on, we assume δ = 0. Then, as we mentioned above, the genus of C_{2} is 2 and P (C_{2}/E) = J(C_{2}). By an appropriate change of variables (λ^{2}, μ) → (x, y), the authors of [BRS] bring the equation of C_{2} to the standard form

y^{2} = x(x^{2} + 2Hx + \frac{1}{4}I_{2})(x^{2} + 2Hx + \frac{1}{4}I_{2} − 1)  (7)

The application of the formula for W_{0} from the above Theorem yields W_{0} = (0, −\sqrt{2}) in the basis of Ω_{C_{2}}^{∗}, dual to (dx/y, xdx/y).

2. Richelot’s transformation of the BRS curve

Richelot’s transformation was first introduced in 1836 by Richelot [R1], [R2] in the problem of the approximate calculation of hyperelliptic integrals of genus 2. By an appropriate iteration of this transformation, he obtained a sequence of genus 2 curves converging very rapidly to a rational curve, which allowed to approximate the hyperelliptic integrals by integrals of rational functions. Later the Richelot transform reappeared in 1901, in the work of Humbert [H], where he studied its action on 2-dimensional theta-functions. See [BM] and [CF] for more details.

Let C be a genus 2 curve defined by an equation

y^{2} = P_{1}(x)P_{2}(x)P_{3}(x),

where P_{j}(x) are quadratic polynomials without multiple or common roots in ℙ^{1}(ℂ). The fact that we are looking at the roots in ℙ^{1}(ℂ) rather than those in ℂ means that we admit the case when one of the quadratic polynomials is in fact linear, and then ∞ is one of its roots. Let C′ be the curve defined by the equation

Δy_{1}^{2} = U_{1}(z)U_{2}(z)U_{3}(z),

where

U_{i+2}(z) = [P_{i}, P_{i+1}] = P′_{i}(z)P_{i+1}(z) − P_{i}(z)P′_{i+1}(z)

(addition of subscripts modulo 3) and Δ = det(P_{1}P_{2}P_{3}) is the determinant of the 3 × 3 matrix of coefficients of the P_{i}. C′ is called the Richelot transform of C. It depends on the partition of the 6 roots of the degree 6 polynomial defining C into three pairs. If one reapplies the Richelot transform to C′ with the natural partition into pairs, one will get again C up to the scaling of the variable y, so that the Richelot transform is an involution in this case. But if one changes the partition on each step, one can get an infinite sequence of nonisomorphic curves.

The ubiquity of the Richelot transform is in the possibility to transport the abelian differentials from one curve to the other. The standard way to transport the 1-forms from one variety to another is the pullback with respect to a map between them. In our case, this does not work: there are no non-constant maps between C and C′. However, there is a nice substitute for a map – a (2, 2)-correspondence. It is a curve Z ⊂ C × C′ with double projections to both factors (one can think of it as the graph of a bivalued map from C to C′, or in the opposite direction, from C′ to C). It is defined by the explicit equations

Z = \{ \begin{matrix}P_{1}(x)U_{1}(z) + P_{2}(x)U_{2}(z) = 0, \\ yy_{1} = P_{1}(x)U_{1}(z)(x − z). \end{matrix}

Some other equations satisfied on the same set Z

\begin{matrix}y_{1}P_{2}P_{3} = U_{1}y(x − z) & , & y_{1}P_{1}P_{3} = −U_{2}y(x − z) \\ yU_{2}U_{3} = ΔP_{1}y_{1}(x − z) & , & yU_{1}U_{3} = −ΔP_{2}y_{1}(x − z) \end{matrix}

and the identity

P_{1}(x)U_{1}(z) + P_{2}(x)U_{2}(z) + P_{3}(x)U_{3}(z) + (x − z)^{2}Δ ≡ 0 ∀ (x, z) ∈ ℂ^{2}

are useful in the proofs.

The induced map δ_{Z} : Ω_{C′}⟶Ω_{C} between the 2-dimensional vector spaces of 1-forms is defined by δ_{Z} = p_{1∗}p_{2}^{∗}, where p_{1}, p_{2} are the natural projections from Z to C, resp. C′, p_{2}^{∗} is the usual pullback of exterior forms and p_{1∗} the trace map for the unramified double covering p_{1}, a particular case of the integration over the fibers of a smooth map. This map δ_{Z} is calculated in [BM]: δ_{Z} (S(z)\frac{dz}{y_{1}}) = S(x)\frac{dx}{y} for S a polynomial of degree ≤ 1.

Besides the transformation of Abelian differentials, the correspondence induces an isogeny ψ_{Z} : J(C)⟶J(C′) between the Jacobians of the two curves. For the definition, represent J(C) as the abelian group of divisor classes [∑ n_{i}P_{i}] modulo divisors of meromorphic functions, where P_{i} ∈ C are points of the curve, and n_{i} are integers such that ∑ n_{i} = 0. Then define

ψ_{Z}([∑ n_{i}P_{i}]) = [∑ n_{i}p_{2}p_{1}^{−1}P_{i}].

The Abel-Jacobi isomorphism

\begin{matrix}AJ : J(C) & ⟶ & Ω_{C}^{∗}/H_{1}(C, ℤ) \\ [∑ n_{i}P_{i}] & ↦ & ∑ n_{i} ∫_{P_{0}}^{P_{i}} mod H_{1}(C, ℤ), \end{matrix}

together with the Newton–Leibnitz rule of derivation of the integral with respect to the upper limit implies that δ_{Z} = (d_{0}ψ_{Z})^{∗}, the adjoint of the differential of ψ_{Z}. This proves the assertion of Theorem 1 about the transformation of the flows of V_{0}, W_{0}.

The following lemma gives the kernel of ψ_{Z} which will be used in Section 4:

Lemma 1. Let a_{i}, a′_{i} be the roots of the polynomial P_{i}(x), i = 1, 2, 3. Then the kernel of ψ_{Z} is the subgroup {0, [(a_{1}, 0) − (a′_{1}, 0)], [(a_{2}, 0) − (a′_{2}, 0)], [(a_{3}, 0) − (a′_{3}, 0)]} ≃ (ℤ/2ℤ)^{⊕2} of the group J(C)_{2} ≃ (ℤ/2ℤ)^{⊕4} of points of order two in J(C).

Proof. The inclusion “⊃” is obvious. Indeed, p_{2}p_{1}^{−1}((a_{i}, 0)) = p_{2}p_{1}^{−1}((a′_{i}, 0)) = {(u_{i}, 0), (u′_{i}, 0)}, where u_{i}, u′_{i} are the roots of U_{i}(z), hence p_{2}p_{1}^{−1}((a_{i}, 0) − (a′_{i}, 0)) = 0. The opposite inclusion follows from the observation that the iteration of the Richelot isogeny J(C) ⟶^{ψ} J(C′) ⟶^{ψ′} J(C′′) produces the map of multiplication by 2 on the Jacobian of C′′ ≃ C, and its kernel J(C)_{2} contains 16 elements. Hence | ker ψ| = | ker ψ′| = 4.

Apply now the Richelot transform to the BRS curve C = C_{2} with P_{1}(x) = x, P_{2}(x) = x^{2} + 2Hx + \frac{1}{4}I_{2}, P_{3}(x) = x^{2} + 2Hx + \frac{1}{4}I_{2} − 1. We will obtain the curve

y_{1}^{2} = −2(z + H)(z^{2} + 1 − \frac{1}{4}I_{2})(z^{2} − \frac{1}{4}I_{2})

which differs from Kowalevski’s curve for δ = 0 only by transformation η = \sqrt{−2}y_{1}, ξ = z + H.

3. Kuznetsov–Tsiganov Lax representation

The authors of [KT] start from the (2 × 2) Lax matrix for the Neumann system with one spectral parameter u

L_{N} = [ \begin{matrix}u^{2} − 2 l_{3} u − l_{1}^{2} − l_{2}^{2} − \frac{2α}{g_{3}^{2}} & ib ((g_{1} + ig_{2}) u − g_{3} (l_{1} + il_{2})) \\ ib ((g_{1} − ig_{2}) u − g_{3} (l_{1} − il_{2})) & b^{2}g_{3}^{2} \end{matrix} ]

having for Hamiltonian

H_{N} = \frac{1}{2}(l_{1}^{2} + l_{2}^{2} − b^{2}g_{3}^{2}) + \frac{α}{g_{3}^{2}} .

They construct from it a Lax matrix for the Kowalevsky– Goryachev– Chaplygin top (as before, δ = (l, g) = 0) by the following formula:

L_{KGCT} = K_{+}(u + 2iκ)L_{N}(u)K_{−}(u − 2iκ)σ_{2}L_{N}^{t}(−u)σ_{2},

where

κ = H^{2} − \frac{1}{4}I_{2} + \frac{1}{2},

K_{−}(u) = [ \begin{matrix}α_{1} & u \\ −β_{1}u & α_{1} \end{matrix} ] , K_{+}(u) = [ \begin{matrix}α_{2} & β_{2}u \\ −u & α_{2} \end{matrix} ] , α_{i}, β_{i} ∈ ℂ.

The Hamiltonian has the form

H_{KGCT} = \frac{1}{2}(l_{1}^{2} + l_{2}^{2} + 2l_{3}^{2}) + c_{1}g_{1} + c_{2}g_{2} + c_{3}(g_{1}^{2} − g_{2}^{2}) + c_{4}g_{1}g_{2} + \frac{c_{5}}{g_{3}^{2}} ,

where

c_{1} = \frac{ib}{2}(α_{2} − α_{1}), c_{2} = \frac{b}{2}(α_{1} + α_{2}), c_{3} = −\frac{b^{2}}{4} (β_{1} + β_{2}),

c_{4} = \frac{ib^{2}}{2} (β_{2} − β_{1}), c_{5} = α.

One obtains the Kowalevski top from this by putting

α_{1} = i, α_{2} = −i, b = \frac{1}{2}, β_{1} = β_{2} = α = 0.

It yields the following spectral curve:

B = \{λ + \frac{1}{λ} = −4(u^{4} − 2Hu^{2} + κ̃)\}, κ̃ = \frac{1}{4}I_{2} − \frac{1}{2}.  (8)

In the next section, we will relate it to the curves of genus 2 C_{1}, C_{2} introduced above.

4. Kuznetsov’s transformation

Let

C_{1} = {η^{2} = P_{5}(ξ)},

P_{5}(ξ) = ξ((ξ − H)^{2} + 1 − \frac{1}{4}I_{2})((ξ − H)^{2} − \frac{1}{4}I_{2}),

be the the Kowalevski curve for the case (l, g) = 0. Kowalevski’s equations (4) represent a system of two ODE in two independent variables ξ_{i}, and η_{i} can be eliminated by η_{i} = \sqrt{P_{5}(ξ_{i})}. Kuznetsov introduces the new unknown functions – the canonically conjugated momenta p_{i} = 2 ∫ \frac{∂H}{∂ξ̇_{i}} \frac{dξ̇_{i}}{ξ̇_{i}}, which allow to separate variables in (4). The relation between p_{i}, ξ_{i} is transcendental, but if we introduce λ_{i} = exp(±2\sqrt{−2ξ_{i}}p_{i}), then the pair λ_{i}, \sqrt{±ξ_{i}} will satisfy an algebraic relation which is an equation of a genus 3 curve. Let us fix the choices of signs, which determine the change of variables and our new genus 3 curve B_{1}:

\{ \begin{matrix}u = \sqrt{ξ} \\ λ = ^{\frac{2η+\sqrt{4η^{2}+ξ}}{ξ}} \end{matrix} ,

B_{1} = \{λ + \frac{1}{λ} = 4(u^{4} − 2Hu^{2} + κ)\},

where κ = H^{2} + \frac{1}{2} − \frac{1}{4}I_{2}. We solve the quadratic equation in λ to write down the equation of B_{1} in the standard hyperelliptic form μ^{2} = P_{8}(u), where μ = λ − 2(u^{4} − 2Hu^{2} + κ) and 4P_{8}(u) is the discriminant. We have:

\{ \begin{matrix}u = \sqrt{ξ} \\ μ = \frac{2η}{\sqrt{ξ}} \end{matrix}  ,  \{ \begin{matrix}ξ = u^{2} \\ η = \frac{μu}{2} \end{matrix} ,  (9)

B_{1} = \{μ^{2} = 4(u^{4} − 2Hu^{2} + κ − \frac{1}{2})(u^{4} − 2Hu^{2} + κ + \frac{1}{2})\}  (10)

The elliptic curve E_{1} and the map π from the diagram (1) are obtained by substituting ξ = u^{2} in (10), whereas formulas (9) define the map f. One can immediately verify that f is unramified and π is ramified at 4 points (0, ±μ_{0}), (∞, ±∞), where μ_{0} = \sqrt{4κ^{2} − 1}. The Prym variety P (B_{1}/E_{1}), by definition, is the connected component of 0 in the kernel of the map π_{∗} : J(B_{1})⟶J(E_{1}) and can be understood as the subvariety of J(B_{1}) of divisor classes [∑_{k}((u_{k}, μ_{k})−(−u_{k}, μ_{k}))]. As dim P (B_{1}/E_{1}) = 2 in our case, the sums with k taking only two values 1, 2 will suffice. One can easily see that P (B_{1}/E_{1}) coincides with the image of f^{∗} : J(C_{1})⟶J(B_{1}). Indeed, the inclusion im f^{∗} ⊂ ker π_{∗} follows from the observation that the class of the divisor π_{∗} ∘ f^{∗}((ξ, η) + (ξ′, η′)) is constant in Pic^{4}(E_{1}), namely, it is the lift of the class of degree 2 from ℙ^{1} via the double covering map E_{1}⟶ℙ^{1}, (ξ, μ) ↦ ξ. The opposite inclusion follows from the fact that the kernel of f^{∗} is finite, which we will now prove.

Let f_{∗} : J(B_{1})⟶J(C_{1}) be the natural map and φ = f_{∗}|_{P (B_{1}/E_{1})}. Then φ ∘ f^{∗} = f_{∗} ∘ f^{∗} = 2 id_{J(C_{1})}, hence the kernels of φ, f^{∗} are sums of a number of copies of ℤ/2ℤ, so that | ker φ|·| ker f^{∗}| = | ker 2 id_{J(C_{1})} | = 16.

It is easy to determine directly the kernel of f^{∗}. We have f^{∗}(K_{C_{1}}) = K_{B_{1}}, where K stands for the canonical divisor class of a curve. In order to deal with positive divisors, we will look at (f^{∗})^{−1}K_{B_{1}}, rather than (f^{∗})^{−1}(0). We have (f^{∗})^{−1}K_{B_{1}} ⊂ \frac{1}{2}(2K_{C_{1}}), where the last expression is a notation for the set of all the 16 halves of the divisor class 2K_{C_{1}}. They can be represented by K_{C_{1}} itself and 15 other divisors (ξ_{i}, 0) + (ξ_{j}, 0), 1 ≤ i, j ≤ 6, i ≠ j, where ξ_{i} runs over 5 roots of P_{5}(ξ) and ∞; let us number them so that ξ_{5} = 0, ξ_{6} = ∞. If we apply f^{∗} to any of these 15 divisors, we will get a divisor of degree 4, and we have to determine whether it lies in the canonical linear system |K_{B_{1}}| or not. Since any 1-form on B_{1} can be written as (a + bu + cu^{2})\frac{du}{μ}, the positive canonical divisors are exactly the sums of 4 points of the form (u_{1}, μ_{1}) + (u_{2}, μ_{2}) + (u_{1}, −μ_{1}) + (u_{2}, −μ_{2}). Only (0, 0) + (∞, 0), among the 15 nontrivial halves of 2K_{C_{1}}, provides a divisor of such a form. Hence ker f^{∗} ≃ ℤ/2ℤ and | ker φ| = 8.

One can identify the kernel of f^{∗} in a different way. The following assertion reproduces Lemma 5.3.2 from the Appendix to [Au]:

Lemma 2. Let Z be a nonsingular curve, D an element of order 2 in Pic^{0}(Z). Let f : Y ⟶Z be the unramified double covering defined by D. Then the kernel of the map

f^{∗} : Pic(Z)⟶ Pic(Y)

is the subgroup of order 2, generated by D.

This lemma implies that our double covering f : B_{1}⟶C_{1} is defined by the 2-torsion element D = [(0, 0) − (∞, 0)] ∈ J(C_{1}). Remark, that Audin mentions Kuznetsov’s curve B_{1} in the last lines of her book, but does not identify it as such (her notation for B_{1} is X̃). Audin provides also a coordinate free construction of B_{1}: it is the normalization of the cartesian product E_{1} ×_{ℙ^{1}} C_{1} (see loc.cit., Proposition 5.3.1).

Corollary 1. The Prym variety P (B_{1}/E_{1}) is isomorphic to the quotient of J(C_{1}) by the subgroup generated by the element D of order 2 which defines the unramified covering map f.

The kernel of φ can be identified as the set of divisor classes [f^{∗}((ξ_{i}, 0)− (ξ_{j}, 0))], 1 ≤ i, j ≤ 6, i ≠ j, among which there are only 8 distinct ones. They form a subgroup (ℤ/2ℤ)^{⊕3} of the group P (B_{1}/E_{1})_{2} ≃ (ℤ/2ℤ)^{⊕4} of points of order 2 in P (B_{1}/E_{1}). Remark, that φ(P (B_{1}/E_{1})_{2}) = {0, D}, and that the Richelot isogeny R : J(C_{1})⟶J(C_{2}) transforms D into 0. Hence P (B_{1}/E_{1})_{2} ⊂ ker(R ∘ φ). The composition R ∘ φ thus factors through the multiplication by 2 on P (B_{1}/E_{1}).

Now, similarly, consider the spectral curve B = B_{2} from the previous section and complete it to the diagram (2) in using formulas, analogous to (9), (10). We are using the same characters for the notation of variables, but mark them with tildes. We obtain:

C′_{2} = \{η̃^{2} = ξ̃(ξ̃^{2} − 2Hξ̃+ \frac{1}{4}I_{2})(ξ̃^{2} − 2Hξ̃+ \frac{1}{4}I_{2} − 1)\} ,  (11)

\{ \begin{matrix}ũ = \sqrt{ξ̃} \\ μ̃ = \frac{2η̃}{\sqrt{ξ̃}} \end{matrix}  ,  \{ \begin{matrix}ξ̃ = ũ^{2} \\ η̃ = \frac{μ̃ũ}{2} \end{matrix} ,  (12)

B_{2} = \{μ̃^{2} = 4(ũ^{4} − 2Hũ^{2} + κ̃ − \frac{1}{2})(ũ^{4} − 2Hũ^{2} + κ̃ + \frac{1}{2})\} , (13)

The change of the equation of B_{2} from the form (8) to (13) is given by μ̃ = λ + 2(u^{4} − 2Hu^{2} + κ̃), ũ = u. The equation (11) differs from the BRS equation (7) only by the signs of terms with H. We can transform one to the other by the change of variables y = iη̃, x = −ξ̃; thus C′_{2} ≃ C_{2}. The equation of E_{2} is obtained by the substitution of ũ^{2} = ξ̃ in (13). We can repeat all the above arguments in replacing (1) (9), (10) by (2), (11)-(13).

We summarize the relations between the above curves in the following statement.

Theorem 3. Let the curves B_{i}, C_{i}, E_{i} and the maps f, f̃, φ, φ̃ be defined as above. Then we have the following sequence of isogenies

P (B_{1}/E_{1}) ⟶^{f_{∗}} J(C_{1}) ⟶^{Richelot} J(C_{2}) ⟶^{f̃^{∗}} P (B_{2}/E_{2})

of degrees respectively 8, 4, 2, their kernels being direct sums of copies of ℤ/2ℤ. The following assertions are verified:

(i) The composition P (B_{1}/E_{1}) ⟶^{f_{∗}} J(C_{1}) ⟶^{Richelot} J(C_{2}) is an isogeny of degree 32 whose kernel contains all the points of order 2 of P (B_{1}/E_{1}). Hence it factors through the multiplication by 2 on P (B_{1}/E_{1}), and there exists an isogeny P (B_{1}/E_{1})⟶J(C_{2}) of degree 2.

(ii) The composition of all the three isogenies is an isogeny of degree 64 which factors through the multiplication by 2, hence there exists also an isogeny of order 4 P (B_{1}/E_{1}) ⟶ P (B_{2}/E_{2}) with kernel ℤ/2ℤ ⊕ ℤ/2ℤ.

The same assertions hold for the maps in the opposite direction.

(iii) The elliptic curves E_{1}, E_{2} are non-isogeneous for generic constants of motion I_{2} and H.

The fact that the elliptic curves are non-isogeneous for generic constants of motion follows from the observation that if we fix the value of I_{2} and vary H, then the j-invariant of E_{1} will be constant, but that of E_{2} will vary. We can obtain a generic curve E_{2} by such a variation in keeping E_{1} fixed, hence they are non-isogeneous.

References

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[BM] J.-B. Bost, and J.-F. Mestre, Moyenne arithmético-géométrique et période des courbes de genre 1 et 2, Gaz. Math. Soc. France 38 (1988), 36-64.

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[LM] Leprévost, F., Markushevich, D.: A tower of genus two curves related to the Kowalewski top, J. reine u. angew. Math. 514 (1999), 103-111.

[R1] F. Richelot, Essai sur une méthode générale pour déterminer la valeur des intégrales ultra-elliptiques, fondée sur des transformations remarquables de ces transcendantes, C. R. Acad. Sci. Paris 2 (1836), 622-627.

[R2] F. Richelot, De transformatione Integralium Abelianorum primiordinis commentation, J. reine angew. Math. 16 (1837), 221-341.

[R-STS] A. G. Reyman, M. A. Semenov-Tian-Shansky Lax representation with a spectral parameter for the Kowalewski top and its generalizations Lett. Math.Phys. 1987, 14, No.1, 55-61.

D. M.: Mathématiques - bât. M2, Université Lille 1, F-59655 Villeneuve d’Ascq Cedex, France

E-mail address: markushe@agat.univ-lille1.fr