quasi-abelian functions and varieties in the sense of ... · quasi-abelian functions and varieties...

Toyama Math. J.Vol. 29(2006), 25-58

Quasi-abelian functions and varieties in the sense of Severi,

II.

Degeneration of compact Riemann surfaces

Yukitaka Abe

1. Introduction.

In the previous paper [2], we constructed a moduli space

Mg = I(Tg)/ ∼

=g∪

k=0

k∪g′=0

({Ik} × S∗g′)

of quasi-abelian varieties of dimension g in the sense of Severi, where Ik isthe unit matrix of degree k and S∗

g′ = Sg′/Sp(g′, Z) is the moduli spaceof principally polarized abelian varieties of dimension g′. For any point(Ik, [τ ′]) ∈ Mg with [τ ′] ∈ S∗

g′ (g′ 5 k), there exists a family {(Mt, τt)}t inTg such that

Mt −→ M in (Cg)g

for some M ∈ Mg−k(k) and

[τt] −→ [τ ′] in S∗g

with [(I(M), [τ ])] = (Ik, [τ ′]), where Cg is the one point compactification ofCg and S∗

g is the Satake compactification of S∗g (see the next section for

definitions of Tg, Mg−k(k) etc.).

25

26 Y. Abe

Every (M, τ) ∈ GL(g, C)×Sg corresponds to an abelian variety A withperiod matrix P = M(Ig τ). Let R be a compact Riemann surface of genusg, and let J(R) be its Jacobi variety. There exists a point (M, [τ ]) in Tg

corresponding to J(R). We study the following problem.Problem. When (M, [τ ]) tends to (M ′, [τ ′]) in Tg, how does the corre-sponding compact Riemann surface R degenerate?

There is a series of papers by Lebowitz ([5, 6, 7]), in which he studiedan admissible splitting degeneration of Riemann surfaces and a part of ourinvestigation. Our problem is of course related to the studies concerningwith the compactification of moduli spaces of abelian varieties (for exam-ple [3], [8] and [9]). However our interpretation of limit points of abelianvarieties is different from those in algebraic geometry (see [2]).

2. Moduli space of quasi-abelian varieties

We summarize results in the previous paper [2] in order to give readersa clear outline.

Let M(g, C) be the linear space of all square matrices with complexcoefficients of degree g. Considering M ∈ M(g, C) as

M = (m1,m2, . . .mg), mi ∈ Cg (i = 1, . . . , g),

we identify M(g, C) with (Cg)g. We denote by Cg = Cg∪{∞} the one pointcompactification of Cg.

Definition 2.1. For any M = (m1, m2, . . . , mg) ∈ (Cg)g, we say the num-ber of mi’s with mi = ∞ the divergent rank of M and denote it by dr(M).

Let M ∈ (Cg)g with dr(M) = ℓ. We suppose

mi1 = · · · = miℓ = ∞, i1 < · · · < iℓ.

We consider the matrix

M = (m1, . . . , 0, . . . , 0, . . . , mg)

replaced the ℓ vectors mik (k = 1, . . . , ℓ) with 0 from M. We call it theinterpretation matrix of M, and write M = i(M). For any ℓ = 0, . . . , g, wedefine

Mℓ := {M ∈ (Cg)g; dr(M) = ℓ, rank M = g − ℓ}.

Quasi-abelian functions and varieties in the sense of Severi, II 27

We note M0 = GL(g, C) and Mg = {(∞, . . . ,∞)}.

Definition 2.2. Let ℓ = 1, . . . , g. For any i with 0 5 i 5 g we define

Mℓ(i) :=

{M = (m1, . . . , mg) ∈ Mℓ; mi1 = · · · = miℓ = ∞, iµ > i (µ = 1, . . . , ℓ)}.

If ℓ = 0, then we define M0(i) = M0 for any i with 0 5 i 5 g.

Let Sg be the Siegel upper half space of degree g, and let S∗g := Sg/Sp(g, Z).

The Satake compactification S∗g of S∗

g is the disjoint union

S∗g = S∗

g ⊔ S∗g−1 ⊔ · · ·S∗

1 ⊔ S∗0.

Every pair (M, τ) of M ∈ GL(g, C) and τ ∈ Sg corresponds to an abelianvariety A = Cg/Γ, where Γ = M(Ig τ)Z2g. Let x ∈ S∗

g. If x ∈ S∗g, then we

write x = [τ ], τ ∈ Sg, where [τ ] is the equivalent class of τ in S∗g. In the

case x ∈ S∗g′ (g′ < g), x = [τ ] means that

τ =

(τ ′ 00 0

), τ ′ ∈ Sg′

and [τ ] is the equivalent class [τ ′] of τ ′ in S∗g′ . The topological space (Cg)g×

S∗g is a compactification of GL(g, C) × S∗

g.

We define

Tg :=g∪

g′=0

∪g′5i

ℓ5g−g′

(Mℓ(i) × S∗

g′) ,

which has the relative topology as a subset of (Cg)g ×S∗g. Let I(Mℓ(i)) :=

{i(M); M ∈ Mℓ(i)} and

I(Tg) :=g∪

g′=0

∪g′5i

ℓ5g−g′

(I(Mℓ(i)) × S∗

g′) .

We identify Tg with I(Tg).

28 Y. Abe

Definition 2.3. Two elements (M, [τ ]), (M ′, [τ ′]) ∈ I(Tg) are said to beequivalent if there exist non-negative integers g′, i and ℓ with g′ 5 i andℓ 5 g − g′ such that

(M, [τ ]), (M ′, [τ ′]) ∈ I(Mℓ(i)) × S∗g′ and [τ ] = [τ ′].

In this case we write (M, [τ ]) ∼ (M ′, [τ ′]).

We define a moduli space of quasi-abelian varieties of dimension g in thesense of Severi by

Mg := I(Tg)/ ∼ .

Here we mean that a moduli space is a topological space correspondingone-to-one to the isomorphic classes of quasi-abelian varieties in the senseof Severi. It has a concrete expression

Mg =g∪

g′=0

∪g′5k5g

({Ik} × S∗g′)

=

g∪g′=0

({Ig} × S∗g′)

∪

g−1∪g′=0

({Ig−1} × S∗g′)

∪ · · · ∪ ({I0} × S∗0))

=({Ig} × S∗

g

)∪

({Ig−1} × S∗

g−1

)∪ · · · ∪

({I0} × S∗

0

).

Since we have a homeomorphism

Mg∼= S∗

g ⊔ S∗g−1 ⊔ · · · ⊔ S∗

1 ⊔ S∗0,

Mg is projective algebraic.

3. Jacobi varieties

Let R be a compact Riemann surface of genus g. We take a canonicalhomology basis α1, β1, . . . , αg, βg, i.e. KI(αi, βj) = δij and KI(αi, αj) =KI(βi, βj) = 0, where KI is the intersection number. Let ω1, . . . , ωg be abasis of abelian differentials of first kind. Then

P :=

∫α1

ω1 . . .∫αg

ω1

∫β1

ω1 . . .∫βg

ω1

......

......∫

α1ωg . . .

∫αg

ωg

∫β1

ωg . . .∫βg

ωg


is a period matrix of a lattice Γ in Cg. We know that Cg/Γ is an abelianvariety. We write J(R) = Cn/Γ and call it a Jacobi variety of R. Let Mg

be the moduli space of compact Riemann surfaces of genus g. By the map-ping J : R 7−→ J(R) we obtain an immersion i : Mg

∼−→ i(Mg) ⊂ S∗g

(Torelli-Oort-Steenbrincke). We can write P = M(Ig τ), where τ ∈ Sg.

This period matrix P corresponds to a point (M, [τ ]) in Tg. We state theproblem again.Problem. When (M, [τ ]) tends to (M ′, [τ ′]) in Tg, how does the corre-sponding compact Riemann surface R degenerate?

We study the above problem in the following sections. If R is hyperel-liptic, then it is a branch cover on the one dimensional complex projectivespace P1. Therefore, we can see the behaviour of R when J(R) tends toa limit. Although our method is valid for any genus g in theory, we treatthe cases g = 1 and 2 for theta functions are very complicated in the largegenus.

4. Theta functions

We devote this section to definitions and basic properties of general thetafunctions, which are used in our arguments. We refer to [11] for details.

Definition 4.1. Let g be a positive integer. A g-characteristic is a 2 × g

matrix[ ε

ε′

]of integers, where ε = (ε1, . . . , εg), ε′ = (ε′1, . . . , ε

′g). We say

that the characteristic[ ε

ε′

]is even or odd depending on whether

∑gi=1 εiε

′i

is even or odd. A reduced characteristic is a characteristic each of whichentries is zero or one.

Definition 4.2. Let u ∈ Cg and τ ∈ Sg. The first order g-theta functionwith characteristic

[ ε

ε′

]is

θ[ ε

ε′

](u, τ) =

∑n∈Zg

exp πi

g∑

i,j=1

τij

(ni +

εi

2

)(nj +

εj

2

)

+2g∑

i=1

(ni +

εi

2

) (ui +

ε′i2

)}.

30 Y. Abe

The series in the above definition converges absolutely and uniformly oncompact subsets of Cg × Sg. The theta constant with g-characteristic

[εε′

]at τ is

θ[ ε

ε′

](0, τ).

We write it θ[ ε

ε′

]where no confusion can arise. We use

θi

[ ε

ε′

]=

∂

∂ui

(θ[ ε

ε′

])(0, τ)

with the same meaning.Let P = (Ig τ) be a period matrix. We write Ig = (e(1), . . . , e(g)) and τ =

(τ (1), . . . , τ (g)). For integral vectors µ = (µ1, . . . , µg) and µ′ = (µ′1, . . . , µ

′g),

we define a period by{µ

µ′

}:= µ′

1e(1) + · · ·µ′

ge(g) + µ1τ

(1) + · · · + µgτ(g).

A half-period(

µ

µ′

)is half a period, i.e.

(µ

µ′

):=

12

{µ

µ′

}.

The following lemmas are well-known.

Lemma 4.3 (Reduction formula) Let[ ε

ε′

]and

[ε

ε′

]be g-characteristics

with ε = ε + 2ν and ε′ = ε′ + 2ν ′, where ν = (ν1, . . . , νg), ν ′ = (ν ′1, . . . , ν

′g),

νi, ν′i ∈ Z. Then we have

θ[ ε

ε′

](u, τ) = (−1)

P

i εiν′iθ

[ε

ε′

](u, τ).

Lemma 4.4 (Functional equation)

θ[ ε

ε′

] (u +

{µ

µ′

}, τ

)= exp πi

{∑i

(εiµ′i − ε′iµi) − 2

∑i

µiui

−∑i,j

τijµiµi

× θ[ ε

ε′

](u, τ).


Lemma 4.5 (Substitution formula)

θ[ ε

ε′

] (u +

(µ

µ′

), τ

)= exp πi

−14

∑i,j

τijµiµj −12

∑i

µi(ε′i + µ′i)

−∑

i

µiui

}× θ

[ε + µ

ε′ + µ′

](u, τ).

Let R be a compact Riemann surface of genus g (= 1). Take a canonicalhomology basis α1, β1, . . . , αg, βg on R. Let du1, . . . , dug be the normal basisof abelian differentials of first kind on R with respect to the given homologybasis, i.e. ∫

αj

dui = δij , i, j = 1, . . . , g.

Then the matrix τ = (τij) belongs to Sg, where τij =∫βj

dui.

Definition 4.6. The Riemann theta function with characteristic[

εε′

]as-

sociated with R, α1, β1, . . . , αg, βg is θ[ ε

ε′

](u, τ).

Let P0 be a fixed point on R. We define a map u : R −→ Cg by

u(P ) :=(∫ P

P0

du1, · · · ,∫ P

P0

dug

).

This map has a natural extension to a map on the divisors on R. We notethat θ

[ ε

ε′

](u(P ), τ) is a multivalued function on R.

Proposition 4.7 ([11], Chapter V, Section 1, Theorem 1) For a giveng-characteristic

[ ε

ε′

], θ

[ ε

ε′

](u(P ), τ) is either identically zero on R or has

exactly g zeros P1, . . . , Pg such that

u(P1 · · ·Pg) + K(P0) =( ε

ε′

)+

{µ

µ′

}

for some period{

µ

µ′

}, where K(P0) is the vector of Riemann constants

depending only on R, the canonical homology basis and P0.

32 Y. Abe

5. The case g = 1

In this case we have

T1 =(C∗ × S∗

1

)∪

({∞} × S∗

0

).

A point (ω, [τ ]) ∈ C∗ × S∗1 corresponds to a period matrix ω(1 τ). The

following are the cases which we should consider.(I) ω ∈ C∗, Im τ −→ ∞.

(II) Im τ −→ ∞, ω −→ ∞.

Every compact Riemann surface of genus 1 has a representation as theRiemann surface of

w2 = (z − λ1)(z − λ2)(z − λ3).

It has a concrete realization as a double covering of the Riemann sphereC ∪ {∞} with distinct branch points λ1, λ2, λ3,∞. We may assume

Re λ1 5 Re λ2 5 Re λ3.

We take a homology basis γ, δ of R as shown in Fig.1.

λ1

λ2

λ3∞

γ

δ

Fig.1


Let du be the normal differential with respect to γ, δ, i.e.∫γdu = 1,

∫δdu = τ.

We consider a mapping u : R −→ C defined by

u(P ) :=∫ P

λ1

du

with the base point λ1. We know that u(P ) is a half-period when P is abranch point (cf. [11]). In fact we obtain Table I.

Table I

u(λ1) =(

00

)u(λ2) =

(01

)where the path of integration is taken along the topsheet on the top of the cut λ1 → λ2.

u(λ3) =(

11

)where we proceed from λ2 on the top sheet to λ3.

u(∞) =(

10

)where we proceed along the top sheet on the top ofthe cut from λ3 to ∞.

Lemma 5.1. The point P = λ1 is the only zero of θ

[11

](u(P ), τ).

Proof. Since a theta function with odd characteristic is odd function, wehave

θ

[11

](u(λ1), τ) = θ

[11

](0, τ) = 0.

Then, it suffices to show that θ

[11

](u(P ), τ) is not identically zero by

Proposition 4.7. Using Substitution formula and Reduction formula, weobtain

θ

[11

](u + u(λ3), τ) = θ

[11

](u +

(11

), τ

)= exp πi

(−1

4− 1 − u

)θ

[00

](u, τ).

Since

θ

[00

](u, τ) =

∞∑n=−∞

exp 2πi

(12τn2 + nu

)

34 Y. Abe

is not identically zero and u(P ) is not a constant mapping, we get theconclusion. ¤

By the same way as Lemma 5.1, we obtain the following.

θ

[00

](u(P ), τ) has the only zero at P = λ3.

θ

[10

](u(P ), τ) has the only zero at P = λ2.

θ

[01

](u(P ), τ) has the only zero at P = ∞.

Letdv =

dz

2√

(z − λ1)(z − λ2)(z − λ3).

Then dv is also an abelian differential of first kind on R. The relationbetween du and dv is as follows

du =1ω1

dv, τ =ω2

ω1,

where ∫γdv = ω1 and

∫δdv = ω2.

The function z−λ1 is meromorphic on R, and has the zero of order 2 atP = λ1 and the pole of order 2 at P = ∞ without any other zeros and poles.

Although θ

[11

](u(P ), τ) is not single-valued, θ2

[11

](u(P ), τ)

/θ2

[01

](u(P ), τ)

is a single-valued meromorphic function on R by Functional equation. Thenwe have

(5.1) z − λ1 = Cθ2

[11

](u(P ), τ)

θ2[

01

](u(P ), τ)

,

where C is some constant.Next we determine the constant C. Take a local coordinate t1 around

P = λ1, i.e. z − λ1 = t21 in a some neighbourhood of λ1. Then we have

u(P ) =1ω1

∫ t1(P )

0

dt1√{t21 + (λ1 − λ2)

}{t21 + (λ1 − λ3)

} .

Dividing (5.1) by t21, we obtain

(5.2) 1 = C

(θ[

11

](u(P ), τ)

t1(P )

)21

θ2[

01

](u(P ), τ)

.


Here we have

limP→λ1

θ[

11

](u(P ), τ)

t1(P )

= θ′[11

](0, τ)

du(P )dt1

∣∣∣∣t1=0

= θ′[11

](0, τ)

1ω1

1√(λ1 − λ2)(λ1 − λ3)

.

Then we obtain by (5.2)

1 = C1

ω21(λ1 − λ2)(λ1 − λ3)

(θ′

[11

]θ[

01

] )2

,

hence

C = ω21(λ1 − λ2)(λ1 − λ3)

(θ[

01

]θ′

[11

])2

.

From (5.1) it follows that

(5.3) z − λ1 = ω21(λ1 − λ2)(λ1 − λ3)

(θ[

01

]θ′

[11

])2θ2

[11

](u(P ), τ)

θ2[

01

](u(P ), τ)

.

We have the following equalities by Substitution formula and Reductionformula

θ

[11

](u(λ2), τ) = θ

[11

]((01

), τ

)= − θ

[10

],

θ

[01

](u(λ2), τ) = θ

[01

]((01

), τ

)= θ

[00

].

Letting P = λ2 in (5.3), we obtain

λ2 − λ1 = ω21(λ1 − λ2)(λ1 − λ3)

(θ[

01

]θ[

10

]θ′

[11

]θ[

00

])2

.

Therefore we have

λ1 − λ3 = − 1ω2

1

(θ′

[11

]θ[

00

]θ[

01

]θ[

10

] )2

.

36 Y. Abe

Let P = λ3 in (5.3). Since

θ

[11

](u(λ3), τ) = θ

[11

]((11

), τ

)= exp πi

(−1

4τ − 1

)θ

[22

]= exp πi

(−1

4τ − 1

)θ

[00

]and

θ

[01

](u(λ3), τ) = θ

[01

]((11

), τ

)= exp πi

(−1

4τ − 1

)θ

[12

]= − exp πi

(−1

4τ − 1

)θ

[10

],

we have

λ3 − λ1 = ω21(λ1 − λ2)(λ1 − λ3)

(θ[

01

]θ[

00

]θ′

[11

]θ[

10

])2

.

Then we obtain

λ1 − λ2 = − 1ω2

1

(θ′

[11

]θ[

10

]θ[

01

]θ[

00

] )2

.

We next consider a meromorphic function z − λ2 on R. We have thefollowing equality by the same reason as above

(5.4) z − λ2 = C ′ θ2[

10

](u(P ), τ)

θ2[

01

](u(P ), τ)

,

where C ′ is a constant. We can determine C ′ by the same way as in thecase z − λ1. Consequently we obtain

(5.5) z − λ2 = ω21(λ2 − λ1)(λ2 − λ3)

(θ[

01

] ((01

), τ

)θ′

[10

] ((01

), τ

))2θ2

[10

](u(P ), τ)

θ2[

01

](u(P ), τ)

.

Letting P = λ1, we obtain

λ1 − λ2 = ω21(λ2 − λ1)(λ2 − λ3)

(θ[

01

] ((01

), τ

)θ[

10

]θ′

[10

] ((01

), τ

)θ[

01

])2

.


We noteθ

[01

]((01

), τ

)= θ

[02

]= θ

[00

].

Then we have

λ2 − λ3 = − 1ω2

1

(θ′

[10

] ((01

), τ

)θ[

01

]θ[

00

]θ[

10

] )2

.

Now we have

θ′[10

](u, τ) =

∞∑n=−∞

2πi

(n +

12

)exp 2πi

{12τ

(n +

12

)2

+(

n +12

)u

}.

Since(

01

)=

12, we obtain

θ′[10

]((01

), τ

)=

∞∑n=−∞

2πi

(n +

12

)exp 2πi

{12τ

(n +

12

)2

+12

(n +

12

)}

= θ′[11

].

Therefore we finally obtain

λ2 − λ3 = − 1ω2

1

(θ′

[11

]θ[

01

]θ[

00

]θ[

10

] )2

.

We collect the above results in the following.

(5.6)

λ1 − λ2 = − 1ω2

1

(θ′[ 1

1 ]θ[10 ]

θ[ 01 ]θ[

00 ]

)2

,

λ1 − λ3 = − 1ω2

1

(θ′[ 1

1 ]θ[00 ]

θ[ 01 ]θ[

10 ]

)2

,

λ2 − λ3 = − 1ω2

1

(θ′[ 1

1 ]θ[01 ]

θ[ 00 ]θ[

10 ]

)2

.

Theta constants appeared in the above equalities are as follows

θ

[00

]=

∞∑n=−∞

exp 2πi

(12τn2

),

38 Y. Abe

θ

[01

]=

∞∑n=−∞

exp 2πi

(12τn2 +

n

2

),

θ

[10

]=

∞∑n=−∞

exp 2πi

{12τ

(n +

12

)2}

.

Now we are ready to calculate limits in each case.(I) Let ω1 be fixed. As Im τ −→ ∞, we obtain

θ

[00

]−→ 1, θ

[01

]−→ 1, θ

[10

]−→ 0, θ′

[11

]−→ 0.

Furthermore we have

limIm τ→∞

θ′[

11

]θ[

10

] = −π.

It follows from (5.6) that if Im τ −→ ∞, then

λ1 − λ2 −→ 0, λ1 − λ3 −→ −(

π

ω1

)2

and λ2 − λ3 −→ −(

π

ω1

)2

.

In this case, λ1 and λ2 tend to the same point λ and λ3 tends to λ +(

π

ω1

)2

.

The limit curve is

w2 = (z − λ)2(

z −

(λ +

(π

ω1

)2))

.

(II) We further assume ω1 −→ ∞ in (I). Then we have

λ1 − λ2 −→ 0, λ1 − λ3 −→ 0 and λ2 − λ3 −→ 0.

Therefore λ1, λ2 and λ3 tend to the same point λ, and we obtain the limitcurve

w2 = (z − λ)3.

6. The case g = 2

The total space T2 to consider is

T2 =2∪

g′=0

∪g′5i

ℓ52−g′

(Mℓ(i) × S∗

g′)


=(GL(2, C) × S∗

2

)∪

(M1(1) × S∗

1

)∪

[(M1(0) ∪M2(0)) × S∗

0

].

Let M(I2 τ) be a period matrix with M =

(M11 M12

M21 M22

)∈ GL(2, C)

and τ ∈ S2. The following are all the cases to investigate.(I) The case [τ ] −→ [τ11] in S∗

2.

(I-1) M ∈ GL(2, C) is fixed.(I-2)

(M12

M22

)−→ ∞ in C2.

(II) The case [τ ] −→ 0 in S∗2.

(II-1) M ∈ GL(2, C) is fixed.(II-2)

(M12

M22

)−→ ∞ in C2.

(II-2’)(

M11

M21

)−→ ∞ in C2.

(II-3) M −→ (∞,∞) in (C2)2.

Any compact Riemann surface of genus 2 can be regarded as the Riemannsurface R of

w2 = (z − λ1)(z − λ2)(z − λ3)(z − λ4)(z − λ5).

It is a double covering of the Riemann sphere C∪{∞} with distinct branchpoints λ1, λ2, λ3, λ4, λ5 and ∞ with

Re λ1 5 Re λ2 5 Re λ3 5 Re λ4 5 Re λ5.

We assume that the concrete realization of R is as shown in Fig.2.

40 Y. Abe

λ1

λ2

λ4

∞

γ1

γ2

δ2

δ1

λ5

λ3

Fig.2

Let (Γ, ∆) = (γ1, γ2, δ1, δ2) be a homology basis of R with

KI(γi, δj) = δij , KI(γi, γj) = KI(δi, δj) = 0.

We take the normal basis of abelian defferentials of first kind du1, du2 withrespect to (Γ, ∆). Let

τij :=∫

δj

dui.

Then τ = (τij) ∈ S2.

We define a mapping u : R −→ C2, u(P ) = (u1(P ), u2(P )) by

ui(P ) :=∫ P

λ1

dui.

We obtain Table II by a straight calculation.


Table II

u(λ1) =(

0 00 0

)u(λ2) =

(0 01 0

)where the path of integration is taken along the topsheet on the top of the cut λ1 → λ2.

u(λ3) =(

1 −11 0

)where we proceed from λ2 on the top sheet to λ3.

u(λ4) =(

1 −11 1

)where we proceed from λ3 on the top sheet on the leftof the branch cut as we go to λ4.

u(λ5) =(

1 01 1

)where we proceed on the top sheet from λ4 to λ5.

u(∞) =(

1 00 0

)where we proceed along the top sheet on the top ofthe cut from λ5 to ∞.

The number of reduced 2-characteristics is 16. First we consider a thetafunction θ

[0 00 0

](u(P ), τ). Using Substitution formula and Reduction for-

mula, we can verify that λ3 and λ5 are zeros of it. The vector of Riemannconstants K(λ1) with the base point λ1 is the sum of u(λj)’s with oddsymbols ([11], Chapter V, Section 3, Theorem 9). Then

K(λ1) = u(λ3) + u(λ5) =(

2 − 12 − 1

).

Let a := λ3λ5. Then a is an integral divisor of degree 2 on R. It holds that

u(a) + K(λ1) +(

0 00 0

)=

{µ

µ′

}for some period

{µµ′

}, and ind a = 0. Therefore θ

[0 00 0

](u(P ), τ) is not

identically zero on R (see [11], Chapter V, Section 1, Theorem 2). Henceit has the only zeros λ3 and λ5.

We obtain a similar result for θ[

εε′

](u(P ), τ) except

[εε′

]=

[0 10 1

]. We can

see that θ[

0 10 1

](u(P ), τ) vanishes at all branch points. Then it is identically

zero.We sum up the above results in Table III.

42 Y. Abe

Table III

Theta Function Zero

θ[

0 00 0

](u(P ), τ) λ3, λ5

θ[

1 00 0

](u(P ), τ) λ2, λ4

θ[

0 10 0

](u(P ), τ) λ3, λ4

θ[

0 01 0

](u(P ), τ) λ4,∞

θ[

0 00 1

](u(P ), τ) λ4, λ5

θ[

1 01 0

](u(P ), τ) λ1, λ4

θ[

1 10 0

](u(P ), τ) λ2, λ5

θ[

1 00 1

](u(P ), τ) λ2, λ3

θ[

0 11 0

](u(P ), τ) λ5,∞

θ[

0 10 1

](u(P ), τ) all points

θ[

0 01 1

](u(P ), τ) λ3,∞

θ[

1 11 0

](u(P ), τ) λ1, λ5

θ[

1 10 1

](u(P ), τ) λ1,∞

θ[

1 01 1

](u(P ), τ) λ1, λ3

θ[

0 11 1

](u(P ), τ) λ1, λ2

θ[

1 11 1

](u(P ), τ) λ2,∞

Lemma 6.1. A function θ[

εε′

](u(P ), τ)

θ[

εε′

](u(P ), τ)

2

is a single-valued meromorphic function on R, where[

εε′] and

[εε′

]are

reduced 2-characteristics with[

εε′

]=

[0 10 1

].

Proof. Let{

µµ′

}be a period. By Functional equation we obtain

θ[

εε′

] (u +

{µµ′

}, τ

)θ[

εε′

] (u +

{µµ′

}, τ

)2

=

exp πi

[∑i

{(εi − εi)µ′

i − (ε′i − ε′i)µi

}]θ[

εε′

](u, τ)

θ[

εε′

](u, τ)

2

=

±θ[

εε′

](u, τ)

θ[

εε′

](u, τ)

2


=

θ[

εε′

](u, τ)

θ[

εε′

](u, τ)

2

.

This shows the assertion. ¤We consider a meromorphic function z − λi on R which has the zero of

order 2 at P = λi and the pole of order 2 at P = ∞. We have the followingequalities by Lemma 6.1 and Table III.

z − λ1 = C1

(θ[

1 01 0

](u(P ), τ)

θ[

0 01 0

](u(P ), τ)

)2

.(6.1)

z − λ2 = C2

(θ[

1 00 0

](u(P ), τ)

θ[

0 01 0

](u(P ), τ)

)2

.(6.2)

z − λ3 = C3

(θ[

0 10 0

](u(P ), τ)

θ[

0 01 0

](u(P ), τ)

)2

.(6.3)

z − λ4 = C4

(θ[

0 00 1

](u(P ), τ)

θ[

0 11 0

](u(P ), τ)

)2

.(6.4)

z − λ5 = C5

(θ[

0 00 1

](u(P ), τ)

θ[

0 01 0

](u(P ), τ)

)2

.(6.5)

Here C1, . . . , C5 are some constants.We have another basis of abelian differentials of first kind

dv1 =dz

2w, dv2 =

zdz

2w,

wherew =

√(z − λ1)(z − λ2)(z − λ3)(z − λ4)(z − λ5).

We define v(P ) = (v1(P ), v2(P )) by

vi(P ) :=∫ P

λ1

dvi.

44 Y. Abe

We consider a period matrix with respect to dv1, dv2

(A B) =

(A11 A12 B11 B12

A21 A22 B21 B22

),

whereAij =

∫γj

dvi, Bij =∫

δj

dvi.

We set

C =

(C11 C12

C21 C22

):= A−1.

Then we have(I τ) = C(A B).

The relation between du1, du2 and dv1, dv2 is(du1

du2

)=

(C11 C12

C21 C22

)(dv1

dv2

).

Next we determine the constants C1, . . . , C5. We take a local coordinatet1 around P = λ1 with the origin at λ1. Then we have in a neighbourhoodof λ1 (

u1(P )u2(P )

)= C

∫ t1(P )0

dt1q

Q5i=2{t21+(λ1−λi)}∫ t1(P )

0(λ1+t21)dt1

q

Q5i=2{t21+(λ1−λi)}

.

Dividing the both sides of (6.1) by t21, we obtain

1 = C1

(θ[

1 01 0

](u(P ), τ)

t1(P )1

θ[

0 01 0

](u(P ), τ)

)2

.(6.6)

Further we have

limp→λ1

θ[

1 01 0

](u(P ), τ)

t1(P )= θ1

[1 01 0

]du1

dt1(0) + θ2

[1 01 0

]du2

dt1(0).(6.7)

Since

duj

dt1(0) = Cj1

dv1

dt1(0) + Cj2

dv2

dt1(0)

=1√∏5

i=2(λ1 − λi)(Cj1 + λ1Cj2),


we obtain by (6.6) and (6.7)

1 =C1∏5

i=2(λ1 − λi)

2∑j=1

θj

[1 01 0

](Cj1 + λ1Cj2)

2

1θ2

[0 01 0

] ,

hence

C1 =5∏

i=2

(λ1 − λi)

(θ[

0 01 0

]∑2j=1 θj

[1 01 0

](Cj1 + λ1Cj2)

)2

.(6.8)

Therefore it follows from (6.1) that

z − λ1 =5∏

i=2

(λ1 − λi)

(θ[

0 01 0

]∑2j=1 θj

[1 01 0

](Cj1 + λ1Cj2)

)2

(6.9)

×

(θ[

1 01 0

](u(P ), τ)

θ[

0 01 0

](u(P ), τ)

)2

.

Substituting P = λ2, λ3, λ4, λ5 in (6.9), we obtain the following equalities.

(λ1 − λ3)(λ1 − λ4)(λ1 − λ5)(6.10)

= −

(∑2j=1 θj

[1 01 0

](Cj1 + λ1Cj2)

θ[

0 01 0

] )2 (θ[

0 01 0

](u(λ2), τ)

θ[

1 01 0

](u(λ2), τ)

)2

.

(λ1 − λ2)(λ1 − λ4)(λ1 − λ5)(6.11)

= −

(∑2j=1 θj

[1 01 0

](Cj1 + λ1Cj2)

θ[

0 01 0

] )2 (θ[

0 01 0

](u(λ3), τ)

θ[

1 01 0

](u(λ3), τ)

)2

.

(λ1 − λ2)(λ1 − λ3)(λ1 − λ5)(6.12)

= −

(∑2j=1 θj

[1 01 0

](Cj1 + λ1Cj2)

θ[

0 01 0

] )2 (θ[

0 01 0

](u(λ4), τ)

θ[

1 01 0

](u(λ4), τ)

)2

.

(λ1 − λ2)(λ1 − λ3)(λ1 − λ4)(6.13)

= −

(∑2j=1 θj

[1 01 0

](Cj1 + λ1Cj2)

θ[

0 01 0

] )2 (θ[

0 01 0

](u(λ5), τ)

θ[

1 01 0

](u(λ5), τ)

)2

.

46 Y. Abe

We also obtain the following equalities by (6.2), (6.3), (6.4) and (6.5).We omit details, for the argument is the same as the above.

(λ2 − λ3)(λ2 − λ4)(λ2 − λ5)(6.14)

= −

(∑2j=1 θj

[1 00 0

](u(λ2), τ)(Cj1 + λ2Cj2)

θ[

0 01 0

](u(λ2), τ)

)2 (θ[

0 01 0

]θ[

1 00 0

])2

.

(6.15)

(λ2 − λ1)(λ2 − λ4)(λ2 − λ5)

= −

(∑2j=1 θj

[1 00 0

](u(λ2), τ)(Cj1 + λ2Cj2)

θ[

0 01 0

](u(λ2), τ)

)2 (θ[

0 01 0

](u(λ3), τ)

θ[

1 00 0

](u(λ3), τ)

)2

.

(6.16)

(λ2 − λ1)(λ2 − λ3)(λ2 − λ5)

= −

(∑2j=1 θj

[1 00 0

](u(λ2), τ)(Cj1 + λ2Cj2)

θ[

0 01 0

](u(λ2), τ)

)2 (θ[

0 01 0

](u(λ4), τ)

θ[

1 00 0

](u(λ4), τ)

)2

.

(6.17)

(λ2 − λ1)(λ2 − λ3)(λ2 − λ5)

= −

(∑2j=1 θj

[1 00 0

](u(λ2), τ)(Cj1 + λ2Cj2)

θ[

0 01 0

](u(λ2), τ)

)2 (θ[

0 01 0

](u(λ5), τ)

θ[

1 00 0

](u(λ5), τ)

)2

.

(λ3 − λ2)(λ3 − λ4)(λ3 − λ5)(6.18)

= −

(∑2j=1 θj

[0 10 0

](u(λ3), τ)(Cj1 + λ3Cj2)

θ[

0 01 0

](u(λ3), τ)

)2 (θ[

0 01 0

]θ[

0 10 0

])2

.

(6.19)

(λ3 − λ1)(λ3 − λ4)(λ3 − λ5)

= −

(∑2j=1 θj

[0 10 0

](u(λ3), τ)(Cj1 + λ3Cj2)

θ[

0 01 0

](u(λ3), τ)

)2 (θ[

0 01 0

](u(λ2), τ)

θ[

0 10 0

](u(λ2), τ)

)2

.


(6.20)

(λ3 − λ1)(λ3 − λ2)(λ3 − λ5)

= −

(∑2j=1 θj

[0 10 0

](u(λ3), τ)(Cj1 + λ3Cj2)

θ[

0 01 0

](u(λ3), τ)

)2 (θ[

0 01 0

](u(λ4), τ)

θ[

0 10 0

](u(λ4), τ)

)2

.

(6.21)

(λ3 − λ1)(λ3 − λ2)(λ3 − λ4)

= −

(∑2j=1 θj

[0 10 0

](u(λ3), τ)(Cj1 + λ3Cj2)

θ[

0 01 0

](u(λ3), τ)

)2 (θ[

0 01 0

](u(λ5), τ)

θ[

0 10 0

](u(λ5), τ)

)2

.

(λ4 − λ2)(λ4 − λ3)(λ4 − λ5)(6.22)

= −

(∑2j=1 θj

[0 00 1

](u(λ4), τ)(Cj1 + λ4Cj2)

θ[

0 11 0

](u(λ4), τ)

)2 (θ[

0 11 0

]θ[

0 00 1

])2

.

(6.23)

(λ4 − λ1)(λ4 − λ3)(λ4 − λ5)

= −

(∑2j=1 θj

[0 00 1

](u(λ4), τ)(Cj1 + λ4Cj2)

θ[

0 11 0

](u(λ4), τ)

)2 (θ[

0 11 0

](u(λ2), τ)

θ[

0 00 1

](u(λ2), τ)

)2

.

(6.24)

(λ4 − λ1)(λ4 − λ2)(λ4 − λ5)

= −

(∑2j=1 θj

[0 00 1

](u(λ4), τ)(Cj1 + λ4Cj2)

θ[

0 11 0

](u(λ4), τ)

)2 (θ[

0 11 0

](u(λ3), τ)

θ[

0 00 1

](u(λ3), τ)

)2

.

(6.25)

(λ4 − λ1)(λ4 − λ2)(λ4 − λ3)

= −

(∑2j=1 θj

[0 00 1

](u(λ4), τ)(Cj1 + λ4Cj2)

θ[

0 11 0

](u(λ4), τ)

)2 (θ[

0 11 0

](u(λ5), τ)

θ[

0 00 1

](u(λ5), τ)

)2

.

(λ5 − λ2)(λ5 − λ3)(λ5 − λ4)(6.26)

= −

(∑2j=1 θj

[0 00 1

](u(λ5), τ)(Cj1 + λ5Cj2)

θ[

0 01 0

](u(λ5), τ)

)2 (θ[

0 01 0

]θ[

0 00 1

])2

.

48 Y. Abe

(6.27)

(λ5 − λ1)(λ5 − λ3)(λ5 − λ4)

= −

(∑2j=1 θj

[0 00 1

](u(λ5), τ)(Cj1 + λ5Cj2)

θ[

0 01 0

](u(λ5), τ)

)2 (θ[

0 01 0

](u(λ2), τ)

θ[

0 00 1

](u(λ2), τ)

)2

.

(6.28)

(λ5 − λ1)(λ5 − λ2)(λ5 − λ3)

= −

(∑2j=1 θj

[0 00 1

](u(λ5), τ)(Cj1 + λ5Cj2)

θ[

0 01 0

](u(λ5), τ)

)2 (θ[

0 01 0

](u(λ4), τ)

θ[

0 00 1

](u(λ4), τ)

)2

.

From (6.10) ∼ (6.28) we derive Table IV, from which we can constructall ratios of differences of branch points up to a reciprocal and a differenceof sign.

Table IV

λ1−λiλ1−λj

=(

θ[ 1 01 0 ](u(λi),τ) θ[ 0 0

1 0 ](u(λj),τ)

θ[ 1 01 0 ](u(λj),τ) θ[ 0 0

1 0 ](u(λi),τ)

)2

, i, j = 1

λ2−λiλ2−λj

=(

θ[ 1 00 0 ](u(λi),τ) θ[ 0 0

1 0 ](u(λj),τ)

θ[ 1 00 0 ](u(λj),τ) θ[ 0 0

1 0 ](u(λi),τ)

)2

, i, j = 2

λ3−λiλ3−λj

=(

θ[ 0 10 0 ](u(λi),τ) θ[ 0 0

1 0 ](u(λj),τ)

θ[ 0 10 0 ](u(λj),τ) θ[ 0 0

1 0 ](u(λi),τ)

)2

, i, j = 3

λ4−λiλ4−λj

=(

θ[ 0 00 1 ](u(λi),τ) θ[ 0 1

1 0 ](u(λj),τ)

θ[ 0 00 1 ](u(λj),τ) θ[ 0 1

1 0 ](u(λi),τ)

)2

, i, j = 4

λ5−λiλ5−λj

=(

θ[ 0 00 1 ](u(λi),τ) θ[ 0 0

1 0 ](u(λj),τ)

θ[ 0 00 1 ](u(λj),τ) θ[ 0 0

1 0 ](u(λi),τ)

)2

, i, j = 5

7. The case (I)

In this section we deal with the case that [τ ] −→ [τ11] in S∗2. We fix τ11

and τ12 = τ21. And we consider limits when Im τ22 −→ ∞. By representa-tion of theta functions we obtain Table V.


Table V

Value of Theta Function Limit

θ[

1 01 0

]0

θ[

1 01 0

](u(λ2), τ) −θ

[10

](0, τ11)

θ[

1 01 0

](u(λ3), τ) A1(τ11, τ12)

θ[

1 01 0

](u(λ4), τ) ε1(τ11, τ12)

θ[

1 01 0

](u(λ5), τ) − expπi

(−1

4τ11

)θ[

00

](0, τ11)

θ[

0 01 0

]θ[

01

](0, τ11)

θ[

0 01 0

](u(λ2), τ) θ

[00

](0, τ11)

θ[

0 01 0

](u(λ3), τ) A2(τ11, τ12)

θ[

0 01 0

](u(λ4), τ) ε2(τ11, τ12)

θ[

0 01 0

](u(λ5), τ) expπi

(−1

4τ11

)θ[

10

](0, τ11)

θ[

1 00 0

]θ[

10

](0, τ11)

θ[

1 00 0

](u(λ2), τ) 0

θ[

1 00 0

](u(λ3), τ) A3(τ11, τ12)

θ[

1 00 0

](u(λ4), τ) ε3(τ11, τ12)

θ[

1 00 0

](u(λ5), τ) −i expπi

(−1

4τ11

)θ[

01

](0, τ11)

θ[

0 10 0

]0

θ[

0 10 0

](u(λ2), τ) 0

θ[

0 10 0

](u(λ3), τ) 0

θ[

0 10 0

](u(λ4), τ) 0

θ[

0 10 0

](u(λ5), τ) 0

θ[

0 00 1

]θ[

00

](0, τ11)

θ[

0 00 1

](u(λ2), τ) θ

[01

](0, τ11)

θ[

0 00 1

](u(λ3), τ) ε4(τ11, τ12)

θ[

0 00 1

](u(λ4), τ) ε5(τ11, τ12)

θ[

0 00 1

](u(λ5), τ) 0

θ[

0 11 0

]0

θ[

0 11 0

](u(λ2), τ) 0

θ[

0 11 0

](u(λ3), τ) ∞

θ[

0 11 0

](u(λ4), τ) ∞

θ[

0 11 0

](u(λ5), τ) 0

50 Y. Abe

Here we have

Ai(τ11, τ12) −→ Ai(τ11) ≡0 (τ12 → 0) for i = 1, 2, 3,

εi(τ11, τ12) −→ 0 (τ12 → 0) for 1 = 1, . . . , 5.

We obtain the following limits by Tables IV and V. Other limits of ratiosin Table IV are indefinite.

λ1 − λ3

λ1 − λ2−→

(A1(τ11, τ12)θ

[00

](0, τ11)

θ[

10

](0, τ11)A2(τ11, τ12)

)2

.(7.1)

λ1 − λ5

λ1 − λ2−→

(θ[

00

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.2)

λ1 − λ5

λ1 − λ3−→

(θ[

00

](0, τ11)A2(τ11, τ12)

A1(τ11, τ12)θ[

10

](0, τ11)

)2

.(7.3)

λ2 − λ3

λ2 − λ1−→

(A3(τ11, τ12)θ

[01

](0, τ11)

θ[

10

](0, τ11)A2(τ11, τ12)

)2

.(7.4)

λ2 − λ5

λ2 − λ1−→ −

(θ[

01

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.5)

λ2 − λ5

λ2 − λ3−→ −

(θ[

00

](0, τ11)A2(τ11, τ12)

A3(τ11, τ12)θ[

10

](0, τ11)

)2

.(7.6)

λ4 − λ3

λ4 − λ1−→ 0.(7.7)

λ4 − λ3

λ4 − λ2−→ 0.(7.8)

λ5 − λ2

λ5 − λ1−→

(θ[

01

](0, τ11)

θ[

00

](0, τ11)

)4

.(7.9)


λ5 − λ3

λ5 − λ1−→

(ε4(τ11, τ12)θ

[01

](0, τ11)

θ[

00

](0, τ11)A2(τ11, τ12)

)2

.(7.10)

λ5 − λ4

λ5 − λ1−→

(ε5(τ11, τ12)θ

[01

](0, τ11)

θ[

00

](0, τ11)ε2(τ11, τ12)

)2

.(7.11)

λ5 − λ3

λ5 − λ2−→

(ε4(τ11, τ12)θ

[00

](0, τ11)

θ[

01

](0, τ11)A2(τ11, τ12)

)2

.(7.12)

λ5 − λ4

λ5 − λ2−→

(ε5(τ11, τ12)θ

[00

](0, τ11)

θ[

01

](0, τ11)ε2(τ11, τ12)

)2

.(7.13)

λ5 − λ4

λ5 − λ3−→

(ε5(τ11, τ12)A2(τ11, τ12)ε2(τ11, τ12)ε2(τ11, τ12)

)2

.(7.14)

We note that

A1(τ11, τ12)A2(τ11, τ12)

−→ −θ[

00

](0, τ11)

θ[

10

](0, τ11)

(τ12 → 0),

A2(τ11, τ12)A3(τ11, τ12)

−→ iθ[

10

](0, τ11)

θ[

01

](0, τ11)

(τ12 → 0).

We may assume that λi’s have finite limits. Because the limit of curvesis five projective lines as at least one of λi (i = 1, . . . , 5) tends to infinity.

First we consider the case λ1−λ4 −→ 0. If λ1−λ2 −→ 0, then λ1−λ3, λ1−λ5, λ2 − λ3 −→ 0 by (7.1), (7.2) and (7.4). In this case λ1, λ2, λ3, λ4, λ5

have the same limit point µ. Then the limit curve is

w2 = (z − µ)5.

We assume that λ1 − λ4 −→ 0. If λ1 − λ2 −→ 0, then

λ4 − λ3

λ4 − λ1=

(λ4 − λ1) + (λ1 − λ3)λ4 − λ1

−→ 1.

52 Y. Abe

This contradicts (7.7). Hence λ1 − λ2 −→ 0. From (7.1) ∼ (7.5) it followsthat

λ1 − λ3 −→ 0, λ1 − λ5 −→ 0, λ2 − λ3 −→ 0, λ2 − λ5 −→ 0.

By (7.7) we have λ3 − λ4 −→ 0. Then it also holds that λ2 − λ4 −→ 0.

Therefore λ3 and λ4 have the same limit point µ3 and

λ1 −→ µ1, λ2 −→ µ2, λ5 −→ µ4,

where µ1, µ2, µ3, µ4 are distinct complex numbers in general. We obtainthe limit curve

w2 = (z − µ1)(z − µ2)(z − µ3)2(z − µ4)

in this case.Next we consider the condition(

A12

A22

)−→ ∞ in C2

in addition to [τ ] −→ [τ11] in S∗2. We assume that A12, A22 −→ ∞ and

A12/A22 −→ α with A11 − αA21 = 0.1 Then the inverse matrix C of A

tends to (1

A11−αA21

αA11−αA21

0 0

).

However, we can not get further information from (6.10) ∼ (6.28).Remark. We fix τ11 and τ22. Letting τ12 −→ 0, we obtain a result ofLebowitz ([5]). In fact, we have the following limits in this case.

λ1 − λ3

λ1 − λ2−→

(θ[

00

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.15)

λ1 − λ4

λ1 − λ2−→

(θ[

00

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.16)

1We shall discuss a new concept of generalized Jacobi varieties in a forthcoming paper.

After that we shall be able to understand the reason why we may consider only this case.


λ1 − λ5

λ1 − λ2−→

(θ[

00

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.17)

λ1 − λ4

λ1 − λ3−→ 1.(7.18)

λ1 − λ5

λ1 − λ3−→ 1.(7.19)

λ1 − λ5

λ1 − λ4−→ 1.(7.20)

λ2 − λ3

λ2 − λ1−→ −

(θ[

01

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.21)

λ2 − λ4

λ2 − λ1−→ −

(θ[

01

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.22)

λ2 − λ5

λ2 − λ1−→ −

(θ[

01

](0, τ11)

θ[

10

](0, τ11)

)4

.(7.23)

λ2 − λ4

λ2 − λ3−→ 1.(7.24)

λ2 − λ5

λ2 − λ3−→ 1.(7.25)

λ2 − λ5

λ2 − λ4−→ 1.(7.26)

λ3 − λ2

λ3 − λ1−→

(θ[

01

](0, τ11)

θ[

00

](0, τ11)

)4

.(7.27)

54 Y. Abe

λ3 − λ5

λ3 − λ1−→ 0.(7.28)

λ3 − λ5

λ3 − λ2−→ 0.(7.29)

λ3 − λ5

λ3 − λ4−→ 0.(7.30)

λ4 − λ2

λ4 − λ1−→

(θ[

01

](0, τ11)

θ[

00

](0, τ11)

)4

.(7.31)

λ4 − λ3

λ4 − λ1−→ 0.(7.32)

λ4 − λ3

λ4 − λ2−→ 0.(7.33)

λ4 − λ5

λ4 − λ3−→ ∞.(7.34)

λ5 − λ2

λ5 − λ1−→

(θ[

01

](0, τ11)

θ[

00

](0, τ11)

)4

.(7.35)

λ5 − λ3

λ5 − λ1−→ 0.(7.36)

λ5 − λ3

λ5 − λ2−→ 0.(7.37)

λ5 − λ4

λ5 − λ3−→ ∞.(7.38)

Other limits of ratios in Table IV are indefinite.


If λ1−λ2 −→ 0, then λ1−λ3, λ1−λ4, λ1−λ5 −→ 0 by (7.15), (7.16) and(7.17). Therefore λ1, λ2, λ3, λ4, λ5 tend to the same limit point µ. Hencethe limit curve is

w2 = (z − µ)5.

Assume that λ1 − λ2 −→ 0 (this is the case which Lebowitz consideredin [5]). By (7.15), (7.16) and (7.17) we have

λ1 − λ3 −→ 0, λ1 − λ4 −→ 0, λ1 − λ5 −→ 0.

We also obtain by (7.21), (7.22) and (7.23) that

λ2 − λ3 −→ 0, λ2 − λ4 −→ 0, λ2 − λ5 −→ 0.

It follows from (7.28) and (7.32) that

λ3 − λ4 −→ 0, λ3 − λ5 −→ 0.

From the above observation we obtain

λ1 −→ µ1, λ2 −→ µ2

andλ3, λ4, λ5 −→ µ3,

where µ1, µ2, µ3 are distinct complex numbers. In this case, the limit curveis

w2 = (z − µ1)(z − µ2)(z − µ3)3.

We note that λ3 − λ5 tends to zero more rapidly than λ3 − λ4 (see (7.30)).We also know from (7.34) that λ3 −λ4 vanishes more rapidly than λ4 −λ5.

Notes 1. The reason why the limits of ratiosλk − λℓ

λi − λjdepend only on τ11

but not on τ22 will be made clear in a forthcoming paper.2. By (7.15) and (7.21) we have

θ4[

00

](0, τ11)

θ4[

10

](0, τ11)

−θ4

[01

](0, τ11)

θ4[

10

](0, τ11)

= 1.

Then we obtain a well-known theta constant identity

θ4

[00

](0, τ11) − θ4

[01

](0, τ11) − θ4

[10

](0, τ11) = 0.

56 Y. Abe

8. The case (II)

We consider the case that [τ ] −→ 0 in S∗2. This means that Im τ −→ ∞

as symmetric matrix. We obtain the following limits by Tables IV and V.Other limits of ratios in Table IV are indefinite.

λ1 − λ5

λ1 − λ2−→ ∞.(8.1)

λ2 − λ5

λ2 − λ1−→ ∞.(8.2)

λ3 − λ2

λ3 − λ1−→ 1.(8.3)

λ4 − λ2

λ4 − λ1−→ 1.(8.4)

λ5 − λ2

λ5 − λ1−→ 1.(8.5)

Then λ1 and λ2 have the same limit point µ1. Letting λ3 −→ µ2, λ4 −→ µ3

and λ5 −→ µ4, we obtain the limit curve

w2 = (z − µ1)2(z − µ2)(z − µ3)(z − µ4).

There is a possibility that µ1, µ2, µ3 and µ4 have various relations.Remark. We can not get further information when A tends to a limit in(C2)2 in the case (II), too.

9. Cocluding remarks

Let Mg be the moduli space of compact Riemann surfaces of genus g.Deligne and Mumford gave a compactification of Mg using stable curves ofgenus g ([4]). In the preceding sections, we have investigated degenerationsof compact Riemann surfaces of genera 1 and 2 from our viewpoint of thelimits of abelian varieties. We have seen that a limit curve may have amore complicated singularity than an ordinary double point. Then it is notnecessarily a stable curve. We think that there is another compactificationof Mg which is suitable for our moduli space of quasi-abelian varieties inthe sense of Severi.


References

[1] Y. Abe, A statement of Weierstrass on meromorphic functions whichadmit an algebraic addition theorem, J. Math. Soc. Japan., 57(2005),709–723.

[2] Y. Abe, Quasi-abelian functions and varieties in the sense of Severi, I,Limits of abelian varieties, to appear in Far East J. Math. Sci..

[3] A. Ash, D. Mumford, M. Rapoport and Y. Tai, Smooth Compacti-fication of Locally Symmetric Varieties, Math. Sci. Press, Brookline,1975.

[4] P. Deligne and D. Mumford, The irreducibility of the space of curvesof given genus, Publ. math. I.H.E.S., 36(1969), 75–110.

[5] A. Lebowitz, Degeneration of a compact Riemann surface of genus 2,Israel J. Math., 12(1972), 223–236.

[6] A. Lebowitz, Handle removal on a compact Riemann surface of genus2, Israel J. Math., 15(1973), 189–192.

[7] A. Lebowitz, A remark on degeneration of a compact Riemann surfaceof genus 2, Israel J. Math., 18(1974), 349–351.

[8] Y. Namikawa, A new compactification of the Siegel space and degen-eration of abelian varieties, I, Math. Ann., 221(1976), 97–141.

[9] Y. Namikawa, A new compactification of the Siegel space and degen-eration of abelian varieties, II, Math. Ann., 221(1976), 201–241.

[10] P. Painleve, Sur les fonctions qui admettent un theoreme d’addition,Acta Math., 27(1903), 1–54.

[11] H. E. Rauch and H. M. Farkas, Theta Functions with Applicationsto Riemann Surfaces, The Williams & Wilkins Company, Baltimore,1974.

[12] I. Satake, On the compactification of the Siegel space, J. Indian Math.Soc., 20(1956), 259–281.

58 Y. Abe

[13] F. Severi, Fonctions et varietes quasi-abeliennes, in Proceedings of theInternational Congress of Mathematicians, Amsterdam 1954, Vol.III,pp.521–528.

[14] F. Severi, Funzioni Quasi Abeliane, Seconda Edizione Ampliata, Pont.Acad. Scient. Scripta Varia 20, Vatican, 1961.

Yukitaka AbeUniversity of ToyamaGofuku, Toyama 930-8555, JAPANe-mail: [email protected]

(Received June 21, 2006)

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