Effective solutions of Clebsch and C. Neumann systems

E ective solutions of Clebsch and C. Neumann systems

Angel Zhivkov and Ognyan Christov

^y

Faculty of Mathematics and Informatics, Soa University, 5 J. Bourchier blvd., 1164 Soa, Bulgaria

e-mails: zhivkov@fmi.uni-soa.bg christov@fmi.uni-soa.bg 30.01.1998

Abstract

We solve in Riemann theta functions the classical Clebsch system and its particular case { the C. Neumann system. Going from a new Lax representation (with rational parameter) we work out the solutions. Separately, using relations between theta func- tions, we check that the corresponding expressions in theta functions satisfy Clebsch and Neumann systems.

Table of contents

1 Introduction 1

2 Algebro{geometric integration 7

3 Proof of Theorem 2 15

4 Closed geodesics on the 3{axial ellipsoid 22

1 Introduction

The equations of motion of a rigid body in an ideal uid are 3, 6, 5]

x_ = x @H@p p^_ = x @H@x ⁺ p @H@p (1)

where H is certain quadratic form in x and p. A nontrivial integrable case of equations (1) is the Clebsch case which is characterized with

H = ^12X3

j⁼¹(ajp²_j +cjx²_j) (2)

Partially supported by DFG project number 436 BUL 113/86/5

yPartially supported by Grant MM 523/95 with the Ministry of Science of Republic Bulgaria

1

and c^2;c³

a^{1 +}c^3;c¹

a^{2 +}c^1;c² a^{3 = 0}: (3)

The Clebsch case (2), (3) can be easely reduced to the system with the Hamiltonian H = ^12X3

j⁼¹(p²_j +ajx²_j) = h¹ (4)

and then (3) is automatically satised 5]. So, the Clebsch system takes the form x_¹ = p³x^2;p²x³ p_¹ = (a^3;a²)x³x²

x_² = p¹x^3;p³x¹ p_² = (a^1;a³)x¹x³ (5) x_³ = p²x^1;p¹x² p_³ = (a^2;a¹)x¹x²

where a¹ a² a³ are dierent and nonzero constants. In vector notations the system reads x_ = x p x= (x¹x²x³) p= (p¹p²p³)

p_ = x Ax A= diag(a¹ a² a³): (6)

The additional rst integrals are

h² =a¹p²¹+a²p²²+a³p^{23 ;}a²a³x^21;a³a¹x^22;a¹a²x²³ h³ =x¹p¹+x²p²+x³p³ = ^hx pⁱ

h⁴ =x²¹+x²²+x²³ :

Theorem 1

The Clebsch system (5) is equivalent to the Lax equation L_ = LM] =LM ^;ML

(7) L=

0

B

B

B

@

1=a¹ 0 0 0

0 1=a² 0 0 0 0 1=a³ 0

0 0 0 0

1

C

C

C

A

+

0

B

B

B

@

0 ^;p³=^pa¹a² p²=^pa³a¹ ix¹=^pa¹ p³=^pa¹a² 0 ^;p¹=^pa²a³ ix²=^pa²

;p²=^pa³a¹ p¹=^pa²a³ 0 ix³=^pa³

;ix¹=^pa^{1 ;}ix²=^pa^{2 ;}ix³=^pa³ 0

1

C

C

C

A

M =

0

B

B

@

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1

1

C

C

A+i

0

B

B

@

0 0 0 ^;pa¹x¹

0 0 0 ^;pa²x²

0 0 0 ^;pa³x³

pa¹x^{1 p}a²x^{2 p}a³x³ 0

1

C

C

A satis ed for every ^2C.

The proof is straightforward.

A Lax representation for the Clebsch system, quadratic in the spectral parameter, is found in 15]. Another Lax pair, with parameter varying on an elliptic curve, is found in 1].

The kind of the Lax pair (7) aords to connect in a simple way the Clebsch system and Manakov's geodesic ow on SO(4) 11].

(Z +h)^:= Z +h(Z) +h] Z = (zij)²so(4) 2

= diag(¹²³⁴) = diag(¹²³⁴) (Z) = (ijzij) ij = i ^;j

i ^;j: This connection is well known and an isomorphism between the two cases is established in 2] with the help of above mentioned Lax pair with the spectral parameter varying on an elliptic curve.

In our case, the Clebsch system can be presented as a geodesic ow on SO(4) with the specic metric

¹²=¹³=²³= 0 j⁴ =^;aj j= 123: The explicite change of variables is linear:

;z¹ z² z³ z⁴ z⁵ z⁶ =

p¹

pa²a³ p^pa¹²a³ p^pa³¹a^{2 ;} ix¹

pa^{1 ;} ix²

pa^{2 ;} ix³

pa³

: The aim of this article is to solve the system (5).

Given a Lax pair (LM), there exists an algorithm called algebro{geometric integration, leading to an integration of the system (5), that is to representation of x¹x²x³p¹p²p³ as certain functions depending of the time t. These functions are always theta functions.

Especially convenient and simple formulae are found by Dubrovin 5].

Let us sketch the algorithm of the algebro{geometric integration. Using the Lax equation (7), one constructs the algebraic curve:

C : det^;L()^;=⁴+I¹³+I²²² +I³³+I⁴²+I⁵+I⁶²+I⁷ = 0 (8)

where

I¹ = ^;1=a^1;1=a^2;1=a³ I² = (a¹+a²+a³)(a¹a²a³)^;1 I³ = ^;(a¹a²a³)^;1

I⁴ = ^;I³h² I⁵ = I³h^1;I²h⁴ I⁶ = I³h⁴

I⁷ = I³h²³ :

are rst integrals of the system (5). Therefore, the curve C does not depend on t. Adding four innite points

1j = =¹ = 1aj

j = 123 ¹⁴ ==¹ = 0 we compactify the curve C to a (closed) Riemann surface ; of genus 3.

Next, we x a canonical basis in the one{dimensional homologies H¹(;^Z). This basis denes the matrix of the periods B of ;. B is a symmetric (3 3) matrix with negatively dened real part: B^t=B and ReB <0.

The lattice of the periods = ^f2iN+BM^jNM ^2Z3g has rank 6. The Jacobi variety (Jacobian of ;)

J(;) =^C3= is a six{dimensional real torus.

3

Let us dene the Riemann theta functions of genus 3. Recall that the Riemann theta function of genus g with characteristics mn^2Rg is given by its Fourier series

B _mn ](z) = _mn ](z) = ^X

M^2Zgexp^DB 2

M +m 4

+z+in M + m 2

E (9)

z ^{2 Cg}, B is a symmetric (g g) matrix, ReB < 0 7]. If m and n are integers ¹, they are called semi-periods.

Finally, we write down the solutions of (5) as functions ³, which are ratios of genus 3 theta functions with arguments z = z(t) ^{2 C3}, linearly depending on the time t { it is said that the ow of (5) is linearized on the Jacobian J(;) of ;. The functions ³ depend on certain constants which are Abelian integrals on ;.

In this paper, the term \eective" has been used in the following sense. We consider that eective mathematical operations are summation, subtraction, multiplication, division and in- nite summation when the sum is an entire function. The solutions of certain dynamical system are considered eective, when the general solution of the system is expressed via eectively computing functions.

For example, theta functions are eectively computed if there are no restrictions for the argument z ^2Cg and the matrixB.

The obtained in 5] theta functions ³ are not eective. One of the reasons is the symmetry L(^;) =^;L^t() M(^;) =^;M^t()

of the Lax matricesLandM. This symmetry means that there exists an holomorphic involution , acting on the Riemann surface ;:

: ;^!; () = (^;;) ² = identity:

The involutionhas 4 xed points: ^11121314. Hence, by the Riemann{Hurwitz formula, the factor{surface ;¹ = ;= has genus 1, i.e. is an elliptic curve². The Jacobian J(;¹) of ;¹ is a two{dimensional real torus, dened by a rank two sublattice ² of the lattice . The factor

Prym(;)^def= J(;) J(;¹)

is a two{dimensional complex variety, called Prym variety of ; relative to 7].

The symmetry makes the three{dimensional theta functions factor in two{dimensional and one{dimensional theta functions 7]. Then, the solutions ³ of (5) are reduced to ex- pressions ²¹, including two{dimensional theta functions with arguments on Prym(;) and one{dimensional theta functions with arguments on J(;¹).

On the other hand, the elliptic theta functions onJ(;¹) can be reduced to two{dimensional theta functions exactly on Prym(;). This can be done using the Fay formula 7, p.219]. As a result we get formulae only in terms of theta functions on Prym(;).

In section 2, using two Fay formulae we get rid of the noneectivity in ². The main result of this paper is the following theorem, giving eective solutions of the Clebsch system (5).

1Usually the notationsm=2 andn=2 instead ofmandnare used. We prefer the integer notations.

2This is the curve where the parameterfrom the Lax pair in 1] vary

4

Theorem 2

^Let

is an arbitrary symmetric (2 2) matrix with negatively de ned real part

r ⁶= 0 anda⁰ are arbitrary complex constants

= (¹²) and z⁰ = (z⁰¹z⁰²) are arbitrary vectors in ^C2 . De ne

theta functions ^m_n^11m_n²²](u) =^m_n^11m_n²²](u) , m¹m²n¹n^{2 2Z} u^2C2

the vector U = (U¹U²) =

=

0

@⁰¹⁰⁰]()@¹⁰¹⁰]()

@^{2 ;1010}]()@⁰¹⁰⁰]()

@^{2 1100](})@⁰¹⁰⁰]()

@^{1 ;1000}]()@¹¹⁰⁰]()

@¹

1

A

the dierentiation @U =U¹_@^@1 +U²_@^@2 := ⁰

the constants h =ln⁰¹⁰⁰]()^;ln¹⁰⁰⁰]()()⁰ and K =ln()¹⁰⁰⁰]()⁰

the vector z =z⁰+tU

Then the general solution of the Clebsch system (5) is given by x¹ = r

W

n⁰¹⁰¹]()⁰¹⁰¹](z) +¹⁰¹¹]()¹¹⁰¹](z)^o x² = ir

W

n¹¹¹¹]()¹¹¹¹](z) +⁰¹¹¹]()⁰¹¹¹](z)^o x³ = ^;r

W

n⁰¹¹⁰]()⁰¹¹⁰](z) +¹¹¹⁰]()¹¹¹⁰](z)^o p¹ = ^;i

W

n@U⁰¹⁰¹]()⁰¹⁰¹](z) +@U¹¹⁰¹]()¹⁰¹¹](z)^o+hx¹ p² = 1W

n@U¹¹¹¹]()¹¹¹¹](z) +@U⁰¹¹¹]()⁰¹¹¹](z)^o+hx² p³ = i

W

n@U⁰¹¹⁰]()⁰¹¹⁰](z) +@U¹¹¹⁰]()¹¹¹⁰](z)^o+hx³ (10)

W = ¹⁰⁰⁰]()⁰¹⁰⁰](z) +¹¹⁰⁰]()¹¹⁰⁰](z) a¹ = ^{0 00101}]()^;K ⁰¹⁰⁰¹]()

¹⁰⁰¹]() +a⁰ a² = ^{0 01111}]()^;K ⁰¹¹¹¹]()

¹¹¹¹]() +a⁰ a³ = ^{0 00110}]()^;K ⁰¹⁰¹⁰]()

¹⁰¹⁰]() +a⁰ : 5

Following the above described procedure for algebro{geometric integration, in section 2 we prove Theorem 2. In section 3 we prove Theorem 2 again using only relations between theta functions.

So, the solutions of the Clebsch system (5) are parameterized by the points of 9{dimensional open in ^C9 subset

=

=

^{11 12 12 22}

r a⁰= (¹²) z⁰ = (z⁰¹z⁰²)^j Re<0 r⁶= 0

: The constants ¹¹¹²²²ra⁰¹²z⁰¹z⁰² are called algebro{geometric coordinates of the system (5). The two{dimensional torus, dened by xing ra⁰ is parameterized by the points z^{0 2}Prym(;). The vector U =U() is the winding vector on this invariant under the ow (5) torus.

Non-eective solutions for the Clebsch system are found by K!otter 10], see also 1]. In the K!otter's formulae, the vector is xed and a¹a²a³ are parameters of the problem. As it is seen from Theorem 2, expressing via a^1;a⁰ a^2;a⁰ and a^3;a⁰ is a transcendent problem.

Probably, this is the reason that in 10] and 1] the winding vector U and p¹p²p³ are not explicitly computed.

A requirement for eectivity of the solutions is the presenting the solutions as fast converging series. This is always true for the theta functions due to the condition ReB <0 (or Re<0).

Remark.

The choice of the semi-periods of the theta functions in Theorem 2 is not canon- ical. It depends on the choice of the basis in H¹(;^Z). It turns out that there exist 15 types of similar formulae. Every type corresponds to a nonzero period, equal to the dierence be- tween the rst and the second group of theta periods in the formulae for x¹x²::: p³. In Theorem 2, this dierence is the semi-period ^{10 00}] mod(2). Adding the permutations between x¹Wx²Wx³W and W one obtains 4!15 = 360 relations between theta functions, solving (5).

In 10], K!otter uses one of these variants.

The special case of the Clebsch system, when the integral^hpxⁱ vanishes, is called C. Neu- mann system and describes the motion of a particle on the sphere in the eld of quadratic potential 14, 13]. It turns out that ^hpxⁱ= 0 i

²S =

8

<

:

: @⁰¹⁰⁰]()

@¹ @¹¹⁰⁰]()

@^{2 =} @⁰¹⁰⁰]()

@² @¹¹⁰⁰]()

@¹

9

=

: In addition, the C. Neumann system is invariant under the changes

U ^!e^kU pi ^!e^kpi ai ^!e^;2kai i = 123

for all constants k. Then, instead of to be a point on the one{dimensional variety S, we can x = 0 andk ^2C.

The constants ¹¹¹²²²ka⁰z⁰¹z⁰² are algebro{geometric coordinates of the C. Neu- mann system.

6

Theorem 3

The general solution of the C. Neumann system is x¹ = ⁰⁰¹¹]⁰⁰¹¹](z)

⁰⁰¹⁰]⁰⁰¹⁰](z) p¹ =^;i@U¹¹⁰¹]¹¹⁰¹](z)

⁰¹⁰⁰]⁰¹⁰⁰](z) a¹ = @_U²⁰⁰¹¹] ⁰⁰¹¹] ^;a⁰ x² = i ¹¹¹¹]¹¹¹¹](z)

⁰¹⁰⁰]⁰¹⁰⁰](z) p² = @U⁰¹¹¹]⁰¹¹¹](z)

⁰¹⁰⁰]⁰¹⁰⁰](z) a² = @_U²¹¹¹¹] ¹¹¹¹] ^;a⁰ (11)

x³ = ^;0110]⁰¹¹⁰](z)

⁰¹⁰⁰]⁰⁰¹⁰](z) p³ = i@U¹¹¹⁰]¹¹¹⁰](z)

⁰⁰¹⁰]⁰⁰¹⁰](z) a³ = @_U²⁰¹¹⁰] ⁰¹¹⁰] ^;a⁰

Herez =z⁰+tU, ^ab_cd]^def= ^ab_cd](00), U =k_@^@21100] ^;_@^@11100], k is an arbitrary constant and r = 1.

Of course, there are 360 possible variants of solutions. Two of them are used in 13, 1], where non-eective (according to accepted in this paper denition) are only the vectors U.

Next, well known change of variables (see (72), section 4) brings the ow of C. Neumann system into the geodesic ow on the ellipsoid 12, 13]. In 16], Weierstrass deals with one of the 360 variants of the relations between theta functions, solving the equation of the ellipsoid x²¹=a¹+x²²=a²+x²³=a³ = 1.

In section 4, we answer the question when a geodesic on the three{axial ellipsoid is closed#

the answer requires a check whether some theta constant is a rational number.

2 Algebro{geometric integration of the Clebsch system

In this section we realize in details the sketched in the introduction scheme for algebro{geometric integration of the Clebsch system (5). As a result we obtain the formulae for x¹x²x³ from the Theorem 2. We comment algebro{geometric coordinates for the Clebsch system by the end of the section.

2.1 Lax pair for the Clebsch system

Straightforward calculations show that the Clebsch system is equivalent to the Lax equation L_ = LM] =LM ^;ML, where the matrices L() andM() are dened by (7).

2.2 Riemann surface ; , connected to the Lax equation

Using the matrix L(), we construct the at algebraic curve of degree four

C : det^;L()^;=⁴+I¹³+I²²²+I³³+I⁴²+I⁵+I⁶²+I⁷ = 0: The coe$cients I¹::: I⁷ of C are polynomials of x¹x²x³p¹p²p³a¹a² and a³ and are rst integrals of the Clebsch system. If the curve C is non{singular, its genus is equal to

1

2(4^;1)(4^;2) = 3.

7

The curve C is compactied by adding four innite points

1k =^;=¹ ==ak k= 123 ¹⁴ =^;=¹ =^;I³=I⁶: The Riemann surface ; = C^11121314 has genus 3.

Next, we construct all necessary for the integration of (5) and standard for the theory of Riemann surfaces 7, 8] objects.

Let¹²³¹²³be a basis of cycles in the one{dimensional homologiesH¹(;^Z) with intersection indices

jk =j k = 0 jk =^;kj =jk jk= 123 jk is the Kronecker symbol.

Fix the basis!¹!²!³ of holomorphic on ; dierentials, normed by

Z

j!k = 2ijk jk = 123 i=^p;1 : Denote by B the matrix of the periods for the Riemann surface ; :

B = (Bjk)³_jk⁼¹ Bjk =

Z

j!k :

Let = 2iN +BN ^j NM ^2Z3

be the lattice of the periods of ;. has rank 6 in ^C3 =^R6. The Jacobian of ; J(;) =^C3=

is a six dimensional real torus. Let

Div(;) =ⁿD=^XN

k⁼¹nkPk^jnk ²ZPk ²;^o

be the group of divisors on ;# degD=^PN_k⁼¹nk is the degree of the divisor D and Divp(;) are all divisors of degree p (p^2Z). Then, the JacobianJ(;) is isomorphic to Div⁰(;) modulo the principal divisors, i.e:the divisors of the meromorphic functionsf : ;^!CP1. This isomorphism is realized via Abel's map

A : Div⁰(;) ^! J(;) (12)

D=^XN

k⁼¹(Pk^;Qk) ^{7! A}(D) = ^XN

k⁼¹

Z Pk

Qk (!¹!²!³) :

Remark

. We shall write D instead of ^A(D) for the Abel's map when this is not mislead- ing. The same notations we shall also use for the multi-valued because of dierent possible integration paths between the points Pk and Qk Abel's map (12) ^A : Div⁰(;)^!C3.

LetB(z) z ^2C3, be the theta function on ; and ^⁴be the Riemann theta divisor 8]. The Riemann theta divisor ^⁴possess the fundamental property that for every two pointsP¹P^{2 2};,

B(P¹+P^{2;4^}) = 0 : 8

Let PQ PQ ²;, be the abelian dierential of third kind with simple poles in P and Q, residues (+1) in P, (^;1) in Q and zero {periods.

Finally, dene the prime{formE(PQ) PQ²; on Riemann surface ;, via E²(PQ) =

;f(P)^;f(Q)² df(P)df(Q) exp

Z

;P⁺f^;1(f⁽P⁾⁾

;Q⁺f^;1(f⁽Q⁾⁾ PQ (13)

where f : ;^!CP1 is an arbitrary meromorphic function 7].

2.3 Dubrovin's formulae for the matrix M

When the diagonal elements Mkk and Mjj of the matrix M in the Lax equation are dierent, then there exist simple formulae in theta functions for the entries Mkj and Mjk of M 5]. In the Lax pair (7), M^{44 6}=M¹¹ =M²² =M³³ and Mk⁴ =^;M⁴k,k = 123. Then, fork = 123

M⁴kMk⁴

ak =x²_k = B(¹k^;14+ ^z⁰ +tU^{^})B(^14;1k+ ^z⁰ +tU^{^}) ak_B²(^z⁰+tU^{^})E²(¹k¹⁴)d^;1(¹k)d^;1(¹⁴) # (14)

the integration paths k from¹⁴ to¹k are xed and same for (¹k^;14) =^A(¹k^;14) and E(¹k¹⁴)# ^U ^{2 C3} is the vector of the {periods of the Abelian dierential ^ with a singularity (^;d)+(holomorphic part) in¹⁴ and zero {periods. The vector ^U reads

U^ = (^U¹U^{^2}U^{^3}) = 1 d^;1(¹⁴)

;!¹(¹⁴) !²(¹⁴) !³(¹⁴) : (15)

Now, we intend to reduce the formulae (14) to the formulae in Theorem 2.

2.4 Symmetries on the Riemann surface ^;

The Clebsch system is invariant under the involution

: (x¹x²x³p¹p²p³t) ^{7! ;}(x¹x²x³p¹p²p³t) :

Denote by and the symmetries resulted from on the objects on the Riemann surface ;.

For the Lax matrices L and M the corresponding to symmetry is L^t() =^;L(^;) M^t() =^;M(^;) : The {action on ; is:

: ;^!; () = (^;;) (¹k) =¹k k= 1234: (16)

The basis of cycles in H¹(;Z) can be chosen such, that 7]

(¹²³¹²³) = (^;3;2;1;3;2;1): Then,

!¹ =^;!³ !² =^;!² !³ =^;!^{1 ;}!k(P) = !k(P) # 9

the Abel map ^A possess symmetry ^Z

(!¹!²!³) =

Z

(!¹!²!³) =^;

Z

(!³!²!¹) (17)

where is the{image of the integration path . For :^{C3 !C3}, the symmetry is (z¹z²z³) = ^;(z³z²z¹) (z¹z²z³)^2C3:

When the integral³ I⁷ =^hpxⁱ²(a¹a²a³)^;1 does not vanish, then the holomorphic involution : ; ^! ; has exactly four xed points ^11121314. Therefore, the factor ;= is a well dened closed Riemann surface. According to the Riemann{Hurwitz formula, the genus of ;= is equal to ^;2 + 1 +⁴² = 1.

Let: ;^!;= be the natural projection. The basis inH¹(;=^Z) is spanned by¹ and ¹# a basic dierential in H¹(;=^C) is(!^1;!³). Next, we dene

:J(;=)^!J(;) (D) =^;1(D) D²Div⁰(;=) and :^{C !C3} (z) = (z0^;z) :

Let be the (1 1) matrix of the periods of ;=. The constant and the matrixB of the periods of ; are connected by 7]

B =

0

@ 1

2( + ¹¹) ^{12 12}( ^;11)

²¹ 2^{22 12}

1

2( ^;11) ^{21 12}( + ¹¹)

1

A ¹²= ²¹: (18)

The matrix will be discussed in the next paragraph.

Finally, denote by ⁴ the Riemann theta divisor for the factor{surface ;=.

2.5 Prym variety

The matrix = (jk)²_jk⁼¹ from (18) is called Prym matrix for the pair (;). The two{

dimensional complex variety

Prym(;) =^C2=^{4 4} =2iN + M ^j N M ^2Z2 (19)

is called Prym variety for the pair (;). ⁴ is a rank 4 lattice, is a symmetric matrix with negatively dened real part. Actually, Prym(;) is a four dimensional real torus. The map

p : Prym(;)^!J(;) (z¹z²)^7!(z¹z²z¹) (20)

linearly embed Prym(;) as a codimension one subvariety into the Jacobian J(;). Then, p : J(;) ^! Prym(;) has the form p(z¹z²z³) = (z¹+z³ 2z²). In terms of Abel's map, for PQ²;,

2^A(P ^;Q) =^A(P ^;Q) + pp^A(P ^;Q): (21)

3see (8)

10

The role of a Riemann theta divisor on Prym(;) plays the divisor of degree two D= ^^4;4 2D=¹¹+¹²+¹³+¹⁴ :

As a nal result, we can consider the JacobianJ(;=) and Prym(;) as linearly embedded in the Jacobian J(;) submanifolds:

J(;=) = ⁿ(z0^;z) mod ^j z ^{2C o} (22)

Prym(;) = ⁿ(z¹z²z¹) mod ^j (z¹z²)^2C2o:

The following identities between theta functions onJ(;)J(;=) and Prym(;) hold:

B(z¹z²z³) = ^X

p^=012p0](z^1;z³)^2p000](z¹+z³ 2z²) (23)

^2p000]

Z P¹⁺P²

D (!¹+!³ 2!²)

=c(P¹P²)^2p0]

Z P¹⁺P²

D (!^1;!³)

(24)

where

c(P¹P²) = B(P¹+P^{2 ;}D)

(0)^;(P¹+P^2;D) : (25)

The identities (23) and (24) are true for all zk ^2C, p= 0 or 1 and P¹P^{2 2}; , see 7]⁴.

2.6 Linearization of the ow (5) on the Prym variety

Now, we can reduce the Dubrovin formulae (14), taking into account the involution : ;^!;.

The necessary and su$cient conditions for the existence of the symmetries L^t() = ^;L(^;) and M^t() =^;M(^;) are the existence of and the requirement

z^^{0 2}Prym(;) i:e: z^⁰ = (z⁰¹z⁰²z⁰¹) (26)

in (14), see 4].

It follows from (15) and (17) that ^U ² Prym(;). In order to dene the formulae (14), we choose the integration paths k such that

^1;1 =^{2 2;2} =²+^{2 3;3} =² : Using the symmetry (!¹!²!³) = ^;(!³!²!¹), it is easy to calculate that

Z

11

1

4

(!¹+!³ 2!²) = ¹²

Z

²(!¹+!³ 2!²) = (0i)

Z

1

2

1

4

(!¹+!³ 2!²) = ¹²

Z

²⁺²(!¹+!³ 2!²) = (0i+ ²²) (27)

Z

13

1

4

(!¹+!³ 2!²) = ¹²

Z

²(!¹+!³ 2!²) = (0²²) :

Now we reduce the thetas B from (14) to theta functions² with semi-periods.

4eq. (109), p. 95 and eq. (112), p. 96

11

Lemma 4

^Let

u ^def=

Z Q⁴⁺¹⁴

D p! =

Z Q⁴⁺¹⁴

D (!¹+!³ 2!²) = p(Q⁴+^14;D) (28)

where Q⁴Q⁴ and ¹⁴ are the zeroes of three{sheeted function : ; ^{! CP1}, de ned by (8).

Then for k = 123

B(¹k^;14+ ^z⁰+tU^{^})B(^14;1k+ ^z⁰+tU^{^})

²_B( ^z⁰ +tU^{^}) = c(Q⁴¹⁴)c(Q⁴¹⁴) c(Q⁴¹k)c(Q⁴¹k)Fk F¹ = 1

w

X

p^=012p001](u)^2p001](z) F² = i

w

X

p^=012p011](u)^2p011](z) (29)

F³ = 1 w

X

p^=012p010](u)^2p010](z) w = ^X

p^=012p000](u)^2p000](z) z =p( ^z⁰+tU^{^}):

Proof.

The function has divisor

Div() =Q⁴+Q⁴+^14;11;12;13 : Therefore, on J(;=),

(Q⁴+¹⁴) = ¹²(Q⁴+Q⁴+ 2¹⁴) = ^12X4

k⁼¹

1k =D :

Similarly,(Q⁴+¹⁴) = D. Applying (23) and (24) forP¹ =Q⁴ or Q⁴ and forP² =¹⁴, we get

B(¹k¹⁴ + ^z⁰+tU^{^})

(23)= ^X

p^=012p0](¹k^;14)^2p000](¹k¹⁴+ ^z⁰+tU^{^})

= ^X

p^=012p0](¹k+Q^{4 ;}D)^2p000](¹k¹⁴+z) (30)

(24)= c^;1;1k¹²¹²Q^{4 X}

p^=012p000](¹k¹⁴+u)^2p000](¹k¹⁴+z):

Now, we convert the arguments ¹k¹⁴ =^R141k(!¹+!³2!²) of theta functions into semi-periods of ², using (27), (28), (30) and the formula

²¹¹²²](z¹z²) = ^2;(z¹z²) +i(¹²) + ¹²(¹²)

exp^D161(^{1 2}) (¹²) +i(¹²) + 2(z¹z²)^E : 12