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
Quasi-abelian functions and varieties in the sense of Severi, II 29
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).
Quasi-abelian functions and varieties in the sense of Severi, II 31
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
Quasi-abelian functions and varieties in the sense of Severi, II 33
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)
.
Quasi-abelian functions and varieties in the sense of Severi, II 35
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
.
Quasi-abelian functions and varieties in the sense of Severi, II 37
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′)
Quasi-abelian functions and varieties in the sense of Severi, II 39
=(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.
Quasi-abelian functions and varieties in the sense of Severi, II 41
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
Quasi-abelian functions and varieties in the sense of Severi, II 43
=
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),
Quasi-abelian functions and varieties in the sense of Severi, II 45
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
.
Quasi-abelian functions and varieties in the sense of Severi, II 47
(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.
Quasi-abelian functions and varieties in the sense of Severi, II 49
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)
Quasi-abelian functions and varieties in the sense of Severi, II 51
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.
Quasi-abelian functions and varieties in the sense of Severi, II 53
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.
Quasi-abelian functions and varieties in the sense of Severi, II 55
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.
Quasi-abelian functions and varieties in the sense of Severi, II 57
References
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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)