A Mathematical Journal
Vol. 7, No 3, (75 - 85). December 2005.

Relations of al Functions over Subvarieties
in a Hyperelliptic Jacobian

Shigeki Matsutani
8-21-1 Higashi-Linkan, Sagamihara, 228-0811, JAPAN

rxb01142@nifty.com

ABSTRACT

The sine-Gordon equation has hyperelliptic al function solutions over a hy-
perelliptic Jacobian for y2 = f (x) of arbitrary genus g. This article gives an
extension of the sine-Gordon equation to that over subvarieties of the hyperellip-
tic Jacobian. We also obtain the condition that the sine-Gordon equation in a
proper subvariety of the Jacobian is defined.

RESUMEN

La ecuación de sine-Gordon tiene soluciones funciones hipereĺıpticas sobre un
Jacobiano hipereĺıptico para y2 = f (x) de género arbitrario g. En este art́ıculo
damos una extensión de la ecuación de Sine-Gordon sobre subvariedades de Ja-
cobiano hipereĺıptico. También obtenemos la condición para que la ecuación de
sine-Gordon esté definida en una subvariedad propia del Jacobiano.

Key words and phrases: sine-Gordon equation, nonlinear integrable
differential equation, hyperelliptic functions,
a subvariety in a Jacobian

Math. Subj. Class.: Primary 14H05, 14K12; Secondary 14H51, 14H70



76 Shigeki Matsutani
7, 3(2005)

1 Introduction

For a hyperelliptic curve Cg given by an affine curve y2 =
∏2g+1

i=1 (x − bi), where bi’s
are complex numbers, we have a Jacobian Jg as a complex torus Cg/Λ by the Abel
map ω [Mu]. Due to the Abelian theorem, we have a natural morphism from the
symmetrical product Symg(Cg) to the Jacobian Jg ≈ ω[Symg(Cg)]/Λ. As zeros of an
appropriate shifted Riemann theta function over Jg, the theta divisor is defined as

Θ := ω[Symg−1(Cg)]/Λ

which is a subvariety of Jg. Similarly, it is natural to introduce a subvariety

Θk := ω[Sym
k(Cg)]/Λ

and a sequence,
Θ0 ⊂ Θ1 ⊂ Θ2 ⊂ ··· ⊂ Θg−1 ⊂ Θg ≡ Jg

Vanhaecke studied the structure of these subvarieties as stratifications of the Jacobian
Jg using the strategies developed in the studies of the infinite dimensional integrable
system [V1]. He showed that these stratifications of the Jacobian are connected with
stratifications of the Sato Grassmannian. Further Vanhaecke investigated Lie-Poisson
structures in the Jacobian in [V2]. He showed that invariant manifolds associated with
Poisson brackets can be identified with these strata; it implies that the strata are char-
acterized by the Lie-Poisson structures. He also showed that the Poisson brackets are
connected with a finite-dimensional integrable system, Henon-Heiles system. Follow-
ing the study, Abenda and Fedorov [AF] investigated these strata and their relations
to Henon-Heiles system and Neumann systems.

On the other hand, functions over the embedded hyperelliptic curve Θ1 in a hyper-
elliptic Jacobian Jg were also studied from viewpoint of number theory in [C, G, Ô].
In [Ô], Ônishi also investigated the sequence of the subvarieties, and explicitly stud-
ied behaviors of functions over them in order to obtain higher genus analog of the
Frobenius-Stickelberger relations for genus one case. Though Vanhaecke, Abenda and
Fedorov found some relations of functions over these subvarieties explicitly using the
infinite universal grassmannians and so-called Mumford’s U V W expressions [Mu],
Ônishi gave more explicit relations on some functions over the subvarieties using the
theory of hyperelliptic functions in the nineteenth century fashion [Ba1, Ba2, Ba3].

In this article, we will also investigate some relations of functions over the subva-
rieties based upon the studies of the hyperelliptic function theory developed in the
nineteenth century [Ba2, Ba3, W]. Especially this article deals with the “sine-Gordon
equation” over there.

Modern expressions of the sine-Gordon equation in terms of Riemann theta func-
tions were given in [[Mu] 3.241],

∂

∂tP

∂

∂tQ
log([2P − 2Q]) = A([2P − 2Q] − [2Q − 2P]), (1.1)

where P and Q are ramified points of Cg, A is a constant number, [D] is a meromorphic
function over Symg(Cg) with a divisor D for each Cg and tP′ is a coordinate in the



7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian 77

Jacobi variety such that it is identified with a local parameter at a ramified point P′

up to constant.
In the previous work [Ma], we also studied (1.1) using the fashion of the nineteenth

century. In [W] Weierstrass defined al function by alr := γr
√

Fg(br) and Fg(z) :=
(x1 − z) · · · (xg − z) over Jg with a constant factor γr. Let γr = 1 in this article.
As Weierstrass implicitly seemed to deal with it, (1.1) is naturally described by al-
functions as [Ma],

∂2

∂v
(g)
1 ∂v

(g)
2

log
alr
als

=
1

(br − bs)

(
f′(bs)

(
alr
als

)2
+ f′(br)

(
als
alr

)2)
. (1.2)

Here f′(x) := df (x)/dx and v(g)’s are defined in (2.4). ((1.2) was obtained in the
previous article [Ma] by more direct computations and will be shown as Corollary 3.3
in this article). We call (1.2) Weierstrass relation in this article.

In this article, we will introduce an “al” function over the subvariety in the Jaco-
bian, al(m)r :=

√
Fm(br) and Fm(z) := (x1 − z) · · · (xm − z) for a point ((x1, y1), · · · ,

(xm, ym)) in the symmetric product of the m curves Sym
mCg (m = 1, · · · , g − 1). In

[Mu], Mumford dealt with Fm function (he denoted it by U ) for 1 ≤ m < g and stud-
ied the properties. Further Abenda and Fedorov also studied some properties of the
al(m)r and Fm functions in [AF] though they did not mention about Weierstrass’s pa-
per nor the relation (1.2). We will consider a variant of the Weierstrass relation (1.2)
to al(m)r over subvariety in non-degenerated and degenerated hyperelliptic Jacobian.

As in our main theorem 3.1, even on the subvarieties, we have a similar relation
to (1.1),

∂

∂v
(m)
r

∂

∂v
(m)
s

log
al(m)r
al(m)s

=
1

(br − bs)


 f′(br)

(Q(2)m (br))2

(
al(m)s
al(m)r

)2
+

f′(bs)

(Q(2)m (bs))2

(
al(m)r
al(m)s

)2
+ reminder terms.

(1.3)

Here Q(2)m is defined in (2.2). We regard (1.3) or (3.1) as a subvariety version of the
Weierstrass relation (1.2). In fact, (1.3) contains the same form as (1.1) up to the
factors (Q(2)m (bt))2 (t = r, s) and the reminder terms. Thus (1.3) or (3.1) should be
regarded as an extension of the sine-Gordon equation (1.2) over the Jacobian to that
over the subvariety of the Jacobian.

Further a certain degenerate curve, the remainders in (1.3) vanishes. Then we
have a relations over subvarieties in the Jacobian, which is formally the same as
the Weierstrass relations (1.2) up to the factors (Q(2)m (bt))2 (t = r, s), which means
that we can find solutions of sine-Gordon equation over subvarieties in hyperelliptic
Jacobian. We expect that our results shed a light on the new field of a relation
between “integrability” and a subvariety in the Jacobian, which was brought off by
[V1, V2, AF].



78 Shigeki Matsutani
7, 3(2005)

The author is grateful to the referee for directing his attensions to the references
[AF] and [V2].

2 Differentials of a Hyperelliptic Curve

In this section, we will give our conventions of hyperelliptic functions of a hyperelliptic
curve Cg of genus g (g > 0) given by an affine equation,

y2 = f (x) = (x − b1)(x − b2) · · · (x − b2g)(x − b2g+1)
= Q(x)P (x),

(2.1)

where bj ’s are complex numbers. Here we use the expressions Q(x) := Q
(1)
m (x)Q

(2)
m (x),

Q(1)m (x) := (x − a1)(x − a2) · · · (x − am),
Q(2)m (x) := (x − am+1)(x − am+2) · · · (x − ag),

P (x) := (x − c1)(x − c2) · · · (x − cg)(x − c),
(2.2)

where ak ≡ bk, ck ≡ bg+k, (k = 1, · · · , g) c ≡ b2g+1.

Definition 2.1 [Ba1, Ba2] For a point (xi, yi) ∈ Cg, we define the following quanti-
ties.

1. The unnormalized differentials of the first kind are defined by,

dv
(g,i)
k :=

Q(xi)dxi
2(xi − ak)Q′(ak)yi

, (k = 1, · · · , g) (2.3)

2. The Abel map for g-th symmetric product of the curve Cg is defined by,

v(g) ≡ (v(g)1 , · · · , v
(g)
g ) : Sym

g(Cg) −→ Cg,

(
v
(g)
k ((x1, y1), · · · , (xg, yg)) :=

g∑
i=1

∫ (xi,yi)
∞

dv
(g,i)
k

)
. (2.4)

3. For v(g) ∈ Cg, we define the subspace,

Ξm := v
(g)(Symm(Cg) × (am+1, 0) ×···× (ag, 0))/ΛΛΛ. (2.5)

Here C is a complex field and ΛΛΛ is a g-dimensional lattice generated by the
related periods or the hyperelliptic integrals of the first kind.



7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian 79

The Jacobi variety Jg are defined as complex torus as Jg := Ξg. As Ξm (m < g)
is embedded in Jg whose complex dimension as subvariety is m, the differential forms
(dv(g)k )k=1,··· ,g are not linearly independent. We select linearly independent bases
such as v(m)k := v

(g)
k ((x1, y1), · · · , (xm, ym), (am+1, 0), · · · , (ag, 0)), (k = 1, · · · , m) at

Ξm.
Ξ0 ⊂ Ξ1 ⊂ Ξ2 ⊂ ··· ⊂ Ξg−1 ⊂ Ξg ≡ Jg

For (x1, · · · , xm) ∈ Symm(Cg), we introduce

Fm(x) := (x − x1) · · · (x − xm), (2.6)

and in terms of Fm(x), a hyperelliptic al-function over (v(m)) ∈ Ξm, [Ba2 p.340, W],

al(m)r (v
(m)) =

√
Fm(br). (2.7)

Further we introduce m × m-matrices,

Mm :=




1
x1 − a1

1
x2 − a1

· · ·
1

xm − a1
1

x1 − a2
1

x2 − a2
· · ·

1
xm − a2

...
...

. . .
...

1
x1 − am

1
x2 − am

· · ·
1

xm − am




,

Qm =




√
Q(x1)
P (x1) √

Q(x2)
P (x2)

. . . √
Q(xm)
P (xm)




,

Am =




Q′(a1)
Q′(a2)

. . .
Q′(am)


 .

Lemma 2.2 1.

det Mm =
(−1)m(m−1)/2P (x1, · · · , xm)P (a1, · · · , am)∏

k,l(xk − al)
,



80 Shigeki Matsutani
7, 3(2005)

where

P (z1, · · · , zm) :=
∏
i<j

(zi − zj ).

2.

M−1m =


( Fm(aj )Q(1)m (xi)

F ′m(xi)Q
(1)′
m (aj )(aj − xi)

)
i,j


 ,

where F ′m(x) := dFm(x)/dx and Q
(1)′
m (x) = dQ

(1)
m (x)/dx.

3.

(MQ)−1A =


( 2yiFm(aj )

F ′m(xi)Q
(2)
m (xi)(aj − xi)

)
i,j


 . (2.8)

Proof. (1) is a well-known result [T]. The zero and singularity in the left hand side
give the right hand side as

CP(x1, · · · , xm)P(a1, · · · , am)/
∏
k,l

(xk − al),

for a certain constant C. In order to determine C, we multiply
∏

k,l(xk − al) both
sides and let x1 = a1, x2 = a2, · · · , and xm = am. Then C is determined as above. (2)
is obtained by the Laplace formula using the minor determinant for the inverse matrix.
On (3) we note that Q(1)m Q

(2)
m = Q(x) in (2.2) and thus Q

(1)
m (x)

√
P (x)/Q(x) = y/Q(2)m .

Then we obtain (3).

Corollary 2.3 Let ∂(r)vi := ∂/∂v
(r)
i , and ∂xi := ∂/∂xi.


v1

∂v2
...

∂vm


 = 2(MQm)−1Am




x1

∂x2
...

∂xm


 . (2.9)

3 Weierstrass relation on Ξm

The hyperelliptic solution of the sine-Gordon equation over Jg related to ramified
points (a1, 0) and (a2, 0) is obtained as (1.1) by Mumford [Mu], whose expression in
an old fashion is the Weierstrass relation (1.2). Let us consider an extension of the
Weierstrass relation (1.2) over Ξm as our main theorem. We will give the theorem as
follows.



7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian 81

Theorem 3.1 al(m)r and al
(m)
s (r, s ∈ {1, 2, · · · , m}) over Ξm in (2.5) obey the rela-

tion,

∂

∂v
(m)
r

∂

∂v
(m)
s

log
al(m)r (v

(m))

al(m)s (v(m))

=
1

(ar − as)


 f′(ar)

(Q(2)m (ar))2

(
al(m)s (v

(m))

al(m)r (v(m))

)2
+

f′(as)

(Q(2)m (as))2

(
al(m)r (v

(m))

al(m)s (v(m))

)2
+

f′(am+1)(al
(m)
r (v

(m)))2(al(m)s (v
(m)))2(ar − as)

(am+1 − ar)(am+1 − as)(al
(m)
m+1(v

(m)))4(Q(2)′m (am+1))2

+ · · · · · ·

+
f′(ag)(al

(m)
r (v

(m)))2(al(m)s (v
(m)))2(ar − as)

(ag − ar)(ag − as)(al(m)g (v(m)))4(Q
(2)
m (ag)′)2

.

(3.1)

Proof. From (2.7), we will consider the following formula instead of (3.1) without
loss of generality,

∂

∂v
(m)
1

∂

∂v
(m)
2

log
Fm(a1)
Fm(a2)

= 2
Fm(a1)Fm(a2)

(a1 − a2)

( f′(a1)
Fm(a1)2(Q

(2)
m (a1))2

+
f′(a2)

Fm(a2)2(Q
(2)
m (a1))2

+
f′(am+1)(a1 − a2)2

(am+1 − a1)(am+1 − a2)Fm(am+1)2(Q
(2)′
m (am+1))2

+ · · ·

+
f′(ag)(a1 − a2)2

(ag − a1)(ag − a2)Fm(ag)2(Q
(2)′
m (ag))2

)
.

(3.2)

Before we start the proof, we will comment on our strategy, which is essentially the
same as [Ba3]. First we translate the words of the Jacobian into those of the curves;
we rewrite the differentials v(m)

(r)
’s in terms of the differentials over curves as in (3.3).

We count the residue of an integration and use a combinatorial trick as in Lemma
3.2. Then we will obtain (3.2).

From (2.8) and (2.9), the derivative v’s over Ξm in (2.5) are expressed by the affine
coordinate xi’s,

∂

∂v
(m)
i

= Fm(ai)Q
(2)
m (ai)

m∑
j=1

2yj
F ′m(xj )Q

(2)
m (xj )(xj − ai)

∂

∂xj
. (3.3)



82 Shigeki Matsutani
7, 3(2005)

The right hand side of (3.2) becomes,

∂2

∂v1∂v2
log

Fm(a1)
Fm(a2)

= Fm(a1)Q
(2)
m (a1)

m∑
i,j=1

2yj
(xi − a1)F ′m(xj )Q

(2)
m (xj )

∂

∂xj

2yiFm(a2)Q
(2)
m (a1)

F ′m(xi)Q
(2)
m (xi)(xi − a2)

(a1 − a2)
(xi − a1)(xi − a2)

.

The right hand side is

Fm(a1)Fm(a2)
( m∑

i=1

1
F ′m(xi)

[
∂

∂x

(
f (x)(a2 − a1)

(x − a1)2(x − a2)2(Q
(2)
m (x))2F ′m(x)

)]
x=xi

−
∑

k,l,k 6=l

2ykyl(a2 − a1)
F ′m(xk)F ′m(xl)(xl − a1)(xl − a2)Q

(2)
m (xl)(xk − a1)(xk − a2)Q

(2)
m (xk)(xl − xk)

)
.

Then the proof of Theorem 3.1 is completely done due to next lemma.

Lemma 3.2 1)

m∑
i=1

1
F ′m(xi)

[
∂

∂x

(
f (x)

(x − a1)2(x − a2)2(Q
(2)
m (x)2F ′m(x)

)]
x=xi

=
2

(a1 − a2)2
( f′(a1)

Fm(a1)2(Q
(2)
m (a1))2

+
f′(a2)

Fm(a2)2(Q
(2)
m (a1))2

+
f′(am+1)(a1 − a2)2

(am+1 − a1)(am+1 − a2)Fm(am+1)2(Q
(2)′
m (am+1))2

+ · · ·

+
f′(ag)(a1 − a2)2

(ag − a1)(ag − a2)Fm(ag)2(Q
(2)′
m (ag))2

)
.

∑
k,l,k 6=l

2ykyl(a2 − a1)
F ′m(xk)F ′m(xl)(xl − a1)(xl − a2)Q

(2)
m (xl)(xk − a1)(xk − a2)Q

(2)
m (xk)(xl − xk)

)
= 0.

Proof. : (1) will be proved by the following residual computations: Let ∂Cog be the
boundary of a polygon representation Cog of Cg,∮

∂Cog

f (x)

(x − a1)2(x − a2)2Fm(x)2(Q
(2)
m (x))2

dx = 0. (3.4)

The divisor of the integrand of (3.4) is

2g+1∑
i=1

(bi, 0) − 4
∑

i=1,2,m+1,m+2,··· ,g
(ai, 0) − 2

m∑
i=1

(xi, yi) − 2
m∑

i=1

(xi, −yi) + 3∞



7, 3(2005)
Relations of al Functions over Subvarieties in a Hyperelliptic Jacobian 83

We check these poles: First we consider the contribution around ∞ point.

res(xk,±yk)
f (x)

(x − a1)2(x − a2)2Fm(x)2(Q
(2)
m (x))2

dx

=
1

F ′m(xk)

[
∂

∂x

(
f (x)

(x − a1)2(x − a2)2(Q
(2)
m (x))2F ′m(x)

)]
x=xk

.

At the point (a1, 0), noting that the local parameter t is given by t =
√

(x − a1)
there, we have

res(a1,0)
f (x)

(x − a1)2(x − a2)2Fm(x)2(Q
(2)
m (x))2

dx =
2f′(a1)

(a1 − a2)2Fm(a1)2(Q
(2)
m (a1))2

.

The residue at (a2, 0) is similarly obtained. For the points (ak, 0) (g ≥ k > m), we
have

res(ak,0)
f (x)

(x − a1)2(x − a2)2Fm(x)2(Q
(2)
m (x))2

dx

=
2f′(ak)

(ak − a1)2(ak − a2)2Fm(a2)2(Q
(2)′
m (ak))2

.

By arranging them, we obtain (1). (2) is obvious.

As a corollary, we have Weierstrass relation (1.2) which was proved in [Ma]:

Corollary 3.3 For m = g case, we have the Weierstrass relation for a general curve
Cg,

∂

∂v
(g)
r

∂

∂v
(g)
s

log
al(g)r
al(g)s

=
1

(ar − as)


f′(ar)

(
al(m)s
al(m)r

)2
+ f′(as)

(
al(m)r
al(m)s

)2 . (3.5)
Now we will give our final proposition as corollary.

Corollary 3.4 For a curve satisfying the relations cj = aj for j = m + 1, · · · , g,
al(m)r and al

(m)
s (r, s ∈ {1, 2, · · · , m}) over Ξm in (2.5) obey the relation,

∂

∂v
(m)
r

∂

∂v
(m)
s

log
al(m)r
al(m)s

=
1

(ar − as)


 f′(ar)

(Q(2)m (ar))2

(
al(m)s
al(m)r

)2
+

f′(as)

(Q(2)m (as))2

(
al(m)r
al(m)s

)2 . (3.6)



84 Shigeki Matsutani
7, 3(2005)

Proof. Since the condition cj = aj for j = m + 1, · · · , g means f′(aj ) = 0 for
j = m + 1, · · · , g, Theorem 3.1 reduces to this one.

Under the same assumption of Corollary 3.4, letting A =
2
√

f′(ar)f′(as)
(ar − as)Qm(ar)Qm(as)

,

and

φ(r,s)m (u) :=
1

√
−1

log

√
f′(ar)
f′(as

Qm(ar)
Qm(as)

Fm(ar)
Fm(as)

,

defined over Ξm, φ
(r,s)
m obeys the sin-Gordon equation,

∂

∂v
(m)
r

∂

∂v
(m)
s

φ(r,s)m = A cos(φ
(r,s)
m ). (3.7)

Received: September 2004. Revised: November 2004.

References

[AF] S. Abenda, Yu. Fedorov, On the Weak Kowalevski-Painleve Property for
Hyperelliptically Separable Systems, Acta Appl. Math., 60 (2000) 137-178.

[Ba1] H. F. Baker, Abelian functions – Abel’s theorem and the allied theory
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