Quasi-abelian functions and varieties in the sense of Severi, II.
Degeneration of compact Riemann surfaces Yukitaka A be
1. Introduction.
In the previous paper [2], we constructed a moduli space M
g= I ( T
g)/ ∼
=
∪
g k=0
∪
kg′=0
({I
k} × S
∗g′)
of quasi-abelian varieties of dimension g in the sense of Severi, where I
kis the unit matrix of degree k and S
∗g′= S
g′/Sp(g
′, Z) is the moduli space of principally polarized abelian varieties of dimension g
′. For any point (I
k, [τ
′]) ∈ M
gwith [τ
′] ∈ S
∗g′(g
′5 k), there exists a family { (M
t, τ
t) }
tin T
gsuch that
M
t−→ M in ( ˙ C
g)
gfor some M ∈ M
g−k(k) and
[τ
t] −→ [τ
′] in S
∗gwith [(I (M ), [τ ])] = (I
k, [τ
′]), where ˙ C
gis the one point compactification of C
gand S
∗gis the Satake compactification of S
∗g(see the next section for definitions of T
g, M
g−k(k) etc.).
25
Every (M, τ ) ∈ GL(g, C ) × S
gcorresponds to an abelian variety A with period matrix P = M (I
gτ ). Let R be a compact Riemann surface of genus g, and let J(R) be its Jacobi variety. There exists a point (M, [τ ]) in T
gcorresponding to J(R). We study the following problem.
Problem. When (M, [τ ]) tends to (M
′, [τ
′]) in T
g, how does the corre- sponding compact Riemann surface R degenerate?
There is a series of papers by Lebowitz ([5, 6, 7]), in which he studied an admissible splitting degeneration of Riemann surfaces and a part of our investigation. Our problem is of course related to the studies concerning with the compactification of moduli spaces of abelian varieties (for exam- ple [3], [8] and [9]). However our interpretation of limit points of abelian varieties 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 readers a clear outline.
Let M (g, C ) be the linear space of all square matrices with complex coefficients of degree g. Considering M ∈ M (g, C) as
M = (m
1, m
2, . . . m
g), m
i∈ C
g(i = 1, . . . , g),
we identify M (g, C ) with ( C
g)
g. We denote by ˙ C
g= C
g∪{∞} the one point compactification of C
g.
Definition 2.1. For any M = (m
1, m
2, . . . , m
g) ∈ ( ˙ C
g)
g, we say the num- ber of m
i’s with m
i= ∞ the divergent rank of M and denote it by dr(M ).
Let M ∈ ( ˙ C
g)
gwith dr(M ) = ℓ. We suppose
m
i1= · · · = m
iℓ= ∞ , i
1< · · · < i
ℓ. We consider the matrix
M = (m
1, . . . , 0, . . . , 0, . . . , m
g)
replaced the ℓ vectors m
ik(k = 1, . . . , ℓ) with 0 from M . We call it the interpretation matrix of M , and write M = i(M ). For any ℓ = 0, . . . , g, we define
M
ℓ:= { M ∈ ( ˙ C
g)
g; dr(M) = ℓ, rank M = g − ℓ } .
We note M
0= GL(g, C ) and M
g= { ( ∞ , . . . , ∞ ) } .
Definition 2.2. Let ℓ = 1, . . . , g. For any i with 0 5 i 5 g we define M
ℓ(i) :=
{ M = (m
1, . . . , m
g) ∈ M
ℓ; m
i1= · · · = m
iℓ= ∞ , i
µ> i (µ = 1, . . . , ℓ) } . If ℓ = 0, then we define M
0(i) = M
0for any i with 0 5 i 5 g.
Let S
gbe the Siegel upper half space of degree g, and let S
∗g:= S
g/Sp(g, Z ).
The Satake compactification S
∗gof S
∗gis 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 ∈ S
gcorresponds to an abelian variety A = C
g/Γ, where Γ = M (I
gτ ) Z
2g. Let x ∈ S
∗g. If x ∈ S
∗g, then we write x = [τ ], τ ∈ S
g, 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
′0 0 0
)
, τ
′∈ S
g′and [τ ] is the equivalent class [τ
′] of τ
′in S
∗g′. The topological space ( ˙ C
g)
g× S
∗gis a compactification of GL(g, C ) × S
∗g.
We define
T
g:=
∪
g g′=0
∪
g′5i ℓ5g−g′
( M
ℓ(i) × S
∗g′)
,
which has the relative topology as a subset of ( ˙ C
g)
g× S
∗g. Let I ( M
ℓ(i)) :=
{i(M); M ∈ M
ℓ(i)} and
I ( T
g) :=
∪
g g′=0
∪
g′5i ℓ5g−g′
( I( M
ℓ(i)) × S
∗g′)
.
We identify T
gwith I ( T
g).
Definition 2.3. Two elements (M, [τ ]), (M
′, [τ
′]) ∈ I( T
g) are said to be equivalent if there exist non-negative integers g
′, i and ℓ with g
′5 i and ℓ 5 g − g
′such that
(M, [τ ]), (M
′, [τ
′]) ∈ I ( M
ℓ(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 the sense of Severi by
M
g:= I( T
g)/ ∼ .
Here we mean that a moduli space is a topological space corresponding one-to-one to the isomorphic classes of quasi-abelian varieties in the sense of Severi. It has a concrete expression
M
g=
∪
g g′=0
∪
g′5k5g
({I
k} × S
∗g′)
=
∪
gg′=0
({I
g} × S
∗g′)
∪
g∪
−1g′=0
({I
g−1} × S
∗g′)
∪ · · · ∪ ({I
0} × S
∗0))
= (
{ I
g} × S
∗g)
∪ (
{ I
g−1} × S
∗g−1) ∪ · · · ∪ (
{ I
0} × S
∗0) . Since we have a homeomorphism
M
g∼ = S
∗g⊔ S
∗g−1⊔ · · · ⊔ S
∗1⊔ S
∗0, M
gis projective algebraic.
3. Jacobi varieties
Let R be a compact Riemann surface of genus g. We take a canonical homology basis α
1, β
1, . . . , α
g, β
g, i.e. KI(α
i, β
j) = δ
ijand KI(α
i, α
j) = KI(β
i, β
j) = 0, where KI is the intersection number. Let ω
1, . . . , ω
gbe a basis of abelian differentials of first kind. Then
P :=
∫
α1
ω
1. . . ∫
αg
ω
1∫
β1
ω
1. . . ∫
βg
ω
1.. . .. . .. . .. .
∫
α1
ω
g. . . ∫
αg
ω
g∫
β1
ω
g. . . ∫
βg
ω
g
is a period matrix of a lattice Γ in C
g. We know that C
g/Γ is an abelian variety. We write J (R) = C
n/Γ and call it a Jacobi variety of R. Let M
gbe the moduli space of compact Riemann surfaces of genus g. By the map- ping J : R 7−→ J(R) we obtain an immersion i : M
g−→
∼i(M
g) ⊂ S
∗g(Torelli-Oort-Steenbrincke). We can write P = M(I
gτ ), where τ ∈ S
g. This period matrix P corresponds to a point (M, [τ ]) in T
g. We state the problem again.
Problem. When (M, [τ ]) tends to (M
′, [τ
′]) in T
g, 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 projective space P
1. Therefore, we can see the behaviour of R when J(R) tends to a limit. Although our method is valid for any genus g in theory, we treat the cases g = 1 and 2 for theta functions are very complicated in the large genus.
4. Theta functions
We devote this section to definitions and basic properties of general theta functions, 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 = (ε
1, . . . , ε
g), ε
′= (ε
′1, . . . , ε
′g). We say that the characteristic
[ ε ε
′]
is even or odd depending on whether ∑
g i=1ε
iε
′iis even or odd. A reduced characteristic is a characteristic each of which entries is zero or one.
Definition 4.2. Let u ∈ C
gand τ ∈ S
g. The first order g-theta function with characteristic
[ ε ε
′]
is
θ [ ε
ε
′]
(u, τ) = ∑
n∈Zg
exp πi
∑
g i,j=1τ
ij(
n
i+ ε
i2
) (
n
j+ ε
j2
)
+2
∑
g i=1( n
i+ ε
i2 ) (
u
i+ ε
′i2
)}
.
The series in the above definition converges absolutely and uniformly on compact subsets of C
g× S
g. 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 θ
i[ ε ε
′]
= ∂
∂u
i(
θ [ ε
ε
′])
(0, τ ) with the same meaning.
Let P = (I
gτ ) be a period matrix. We write I
g= (e
(1), . . . , e
(g)) and τ = (τ
(1), . . . , τ
(g)). For integral vectors µ = (µ
1, . . . , µ
g) and µ
′= (µ
′1, . . . , µ
′g), we define a period by
{ µ µ
′}
:= µ
′1e
(1)+ · · · µ
′ge
(g)+ µ
1τ
(1)+ · · · + µ
gτ
(g).
A half-period ( µ
µ
′)
is half a period, i.e.
( µ µ
′) := 1
2 { µ
µ
′}
. The following lemmas are well-known.
Lemma 4.3 (Reduction formula) Let [ ε
ε
′]
and [ ε ˆ
ˆ ε
′]
be g-characteristics with ε = ˆ ε + 2ν and ε
′= ˆ ε
′+ 2ν
′, where ν = (ν
1, . . . , ν
g), ν
′= (ν
1′, . . . , ν
g′), ν
i, ν
i′∈ Z . Then we have
θ [ ε
ε
′]
(u, τ ) = ( − 1)
Piεiν′iθ [ ε ˆ
ˆ ε
′] (u, τ).
Lemma 4.4 (Functional equation)
θ [ ε
ε
′] (
u + { µ
µ
′}
, τ )
= exp πi { ∑
i
(ε
iµ
′i− ε
′iµ
i) − 2 ∑
i
µ
iu
i− ∑
i,j
τ
ijµ
iµ
i
× θ [ ε
ε
′]
(u, τ ).
Lemma 4.5 (Substitution formula)
θ [ ε
ε
′] (
u + ( µ
µ
′)
, τ )
= exp πi
− 1 4
∑
i,j
τ
ijµ
iµ
j− 1 2
∑
i
µ
i(ε
′i+ µ
′i)
− ∑
i
µ
iu
i}
× θ
[ ε + µ ε
′+ µ
′] (u, τ ).
Let R be a compact Riemann surface of genus g ( = 1). Take a canonical homology basis α
1, β
1, . . . , α
g, β
gon R. Let du
1, . . . , du
gbe the normal basis of abelian differentials of first kind on R with respect to the given homology
basis, i.e. ∫
αj
du
i= δ
ij, i, j = 1, . . . , g.
Then the matrix τ = (τ
ij) belongs to S
g, where τ
ij= ∫
βj
du
i.
Definition 4.6. The Riemann theta function with characteristic [
εε′
] as- sociated with R, α
1, β
1, . . . , α
g, β
gis θ
[ ε ε
′]
(u, τ).
Let P
0be a fixed point on R. We define a map u : R −→ C
gby u(P ) :=
(∫
PP0
du
1, · · · ,
∫
PP0
du
g)
.
This map has a natural extension to a map on the divisors on R. We note that θ
[ ε ε
′]
(u(P), τ ) is a multivalued function on R.
Proposition 4.7 ([11], Chapter V, Section 1, Theorem 1) For a given g-characteristic
[ ε ε
′]
, θ [ ε
ε
′]
(u(P), τ ) is either identically zero on R or has exactly g zeros P
1, . . . , P
gsuch that
u(P
1· · · P
g) + K(P
0) = ( ε
ε
′)
+ { µ
µ
′}
for some period { µ
µ
′}
, where K (P
0) is the vector of Riemann constants
depending only on R, the canonical homology basis and P
0.
5. The case g = 1 In this case we have
T
1= (
C
∗× S
∗1)
∪ (
{∞} × S
∗0) .
A point (ω, [τ ]) ∈ C
∗× S
∗1corresponds 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 the Riemann surface of
w
2= (z − λ
1)(z − λ
2)(z − λ
3).
It has a concrete realization as a double covering of the Riemann sphere C ∪ {∞} with distinct branch points λ
1, λ
2, λ
3, ∞ . We may assume
Re λ
15 Re λ
25 Re λ
3. 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.
∫
γ
du = 1,
∫
δ
du = τ.
We consider a mapping u : R −→ C defined by u(P) :=
∫
Pλ1
du
with the base point λ
1. We know that u(P) is a half-period when P is a branch point (cf. [11]). In fact we obtain Table I.
Table I u(λ
1) = (
00
) u(λ
2) = (
01
) where the path of integration is taken along the top sheet on the top of the cut λ
1→ λ
2.
u(λ
3) = (
11
) where we proceed from λ
2on the top sheet to λ
3. u(∞) = (
10
) where we proceed along the top sheet on the top of the cut from λ
3to ∞ .
Lemma 5.1. The point P = λ
1is the only zero of θ [ 1
1 ]
(u(P ), τ ).
Proof. Since a theta function with odd characteristic is odd function, we have
θ [ 1
1 ]
(u(λ
1), τ ) = θ [ 1
1 ]
(0, τ ) = 0.
Then, it suffices to show that θ [ 1
1 ]
(u(P ), τ ) is not identically zero by Proposition 4.7. Using Substitution formula and Reduction formula, we obtain
θ [ 1
1 ]
(u + u(λ
3), τ ) = θ [ 1
1 ] (
u + ( 1
1 )
, τ )
= exp πi (
− 1
4 − 1 − u )
θ [ 0
0 ]
(u, τ).
Since
θ [ 0
0 ]
(u, τ ) =
∑
∞ n=−∞exp 2πi ( 1
2 τ n
2+ nu
)
is not identically zero and u(P ) is not a constant mapping, we get the
conclusion. ¤
By the same way as Lemma 5.1, we obtain the following.
θ [ 0
0 ]
(u(P ), τ ) has the only zero at P = λ
3. θ
[ 1 0 ]
(u(P ), τ ) has the only zero at P = λ
2. θ
[ 0 1 ]
(u(P ), τ ) has the only zero at P = ∞ . Let
dv = dz
2 √
(z − λ
1)(z − λ
2)(z − λ
3) .
Then dv is also an abelian differential of first kind on R. The relation between du and dv is as follows
du = 1
ω
1dv, τ = ω
2ω
1,
where ∫
γ
dv = ω
1and
∫
δ
dv = ω
2.
The function z − λ
1is meromorphic on R, and has the zero of order 2 at P = λ
1and the pole of order 2 at P = ∞ without any other zeros and poles.
Although θ [ 1
1 ]
(u(P), τ ) is not single-valued, θ
2[ 1
1 ]
(u(P ), τ ) /
θ
2[ 0
1 ]
(u(P ), τ ) is a single-valued meromorphic function on R by Functional equation. Then we have
(5.1) z − λ
1= C θ
2[
11
] (u(P ), τ ) θ
2[
01
] (u(P ), τ ) , where C is some constant.
Next we determine the constant C. Take a local coordinate t
1around P = λ
1, i.e. z − λ
1= t
21in a some neighbourhood of λ
1. Then we have
u(P ) = 1 ω
1∫
t1(P)0
dt
1√{ t
21+ (λ
1− λ
2) } {
t
21+ (λ
1− λ
3) } . Dividing (5.1) by t
21, we obtain
(5.2) 1 = C
( θ [
11
] (u(P ), τ ) t
1(P )
)
21 θ
2[
01
] (u(P ), τ ) .
Here we have
P
lim
→λ1θ [
11
] (u(P ), τ ) t
1(P)
= θ
′[ 1
1 ]
(0, τ ) du(P ) dt
1¯¯ ¯¯
t1=0
= θ
′[ 1
1 ]
(0, τ ) 1 ω
1√ 1
(λ
1− λ
2)(λ
1− λ
3) . Then we obtain by (5.2)
1 = C 1
ω
12(λ
1− λ
2)(λ
1− λ
3) ( θ
′[
11
] θ [
01
] )
2,
hence
C = ω
12(λ
1− λ
2)(λ
1− λ
3) ( θ [
01
] θ
′[
11
] )
2. From (5.1) it follows that
(5.3) z − λ
1= ω
12(λ
1− λ
2)(λ
1− λ
3) ( θ [
01
] θ
′[
11
] )
2θ
2[
11
] (u(P ), τ ) θ
2[
01
] (u(P ), τ ) .
We have the following equalities by Substitution formula and Reduction formula
θ [ 1
1 ]
(u(λ
2), τ ) = θ [ 1
1 ] (( 0
1 )
, τ )
= − θ [ 1
0 ]
,
θ [ 0
1 ]
(u(λ
2), τ ) = θ [ 0
1 ] (( 0
1 )
, τ )
= θ [ 0
0 ]
. Letting P = λ
2in (5.3), we obtain
λ
2− λ
1= ω
21(λ
1− λ
2)(λ
1− λ
3) ( θ [
01
] θ [
10
] θ
′[
11
] θ [
00
] )
2.
Therefore we have
λ
1− λ
3= − 1 ω
12( θ
′[
11
] θ [
00
] θ [
01
] θ [
10
] )
2.
Let P = λ
3in (5.3). Since θ
[ 1 1 ]
(u(λ
3), τ ) = θ [ 1
1 ] (( 1
1 )
, τ )
= exp πi (
− 1 4 τ − 1
) θ
[ 2 2 ]
= exp πi (
− 1 4 τ − 1
) θ
[ 0 0 ]
and
θ [ 0
1 ]
(u(λ
3), τ ) = θ [ 0
1 ] (( 1
1 )
, τ )
= exp πi (
− 1 4 τ − 1
) θ
[ 1 2 ]
= − exp πi (
− 1 4 τ − 1
) θ
[ 1 0 ]
, we have
λ
3− λ
1= ω
21(λ
1− λ
2)(λ
1− λ
3) ( θ [
01
] θ [
00
] θ
′[
11
] θ [
10
] )
2. Then we obtain
λ
1− λ
2= − 1 ω
12( θ
′[
11
] θ [
10
] θ [
01
] θ [
00
] )
2.
We next consider a meromorphic function z − λ
2on R. We have the following equality by the same reason as above
(5.4) z − λ
2= C
′θ
2[
10
] (u(P ), τ ) θ
2[
01
] (u(P ), τ ) ,
where C
′is a constant. We can determine C
′by the same way as in the case z − λ
1. Consequently we obtain
(5.5) z − λ
2= ω
12(λ
2− λ
1)(λ
2− λ
3) ( θ [
01
] ((
01
) , τ ) θ
′[
10
] ((
01
) , τ ) )
2θ
2[
10
] (u(P), τ ) θ
2[
01
] (u(P), τ ) . Letting P = λ
1, we obtain
λ
1− λ
2= ω
12(λ
2− λ
1)(λ
2− λ
3) ( θ [
01
] ((
01
) , τ ) θ [
10
] θ
′[
10
] ((
01
) , τ ) θ [
01
] )
2.
We note
θ [ 0
1 ] (( 0
1 )
, τ )
= θ [ 0
2 ]
= θ [ 0
0 ]
. Then we have
λ
2− λ
3= − 1 ω
12( θ
′[
10
] ((
01
) , τ ) θ [
01
] θ [
00
] θ [
10
]
)
2.
Now we have θ
′[ 1 0 ]
(u, τ) =
∑
∞ n=−∞2πi (
n + 1 2
)
exp 2πi {
1 2 τ
( n + 1
2 )
2+ (
n + 1 2
) u
} .
Since ( 0
1 )
= 1
2 , we obtain θ
′[ 1 0
] (( 0 1
) , τ
)
=
∑
∞ n=−∞2πi (
n + 1 2
)
exp 2πi {
1 2 τ
( n + 1
2 )
2+ 1 2
( n + 1
2 )}
= θ
′[ 1
1 ]
.
Therefore we finally obtain
λ
2− λ
3= − 1 ω
12( θ
′[
11
] θ [
01
] θ [
00
] θ [
10
] )
2. We collect the above results in the following.
(5.6)
λ
1− λ
2= −
ω12 1(
θ′[
11]
θ[
10]
θ
[
01]
θ[
00] )
2, λ
1− λ
3= −
ω121
(
θ′[
11]
θ[
00]
θ
[
01]
θ[
10] )
2, λ
2− λ
3= −
ω121
(
θ′[
11]
θ[
01]
θ
[
00]
θ[
10] )
2.
Theta constants appeared in the above equalities are as follows θ
[ 0 0 ]
=
∑
∞ n=−∞exp 2πi ( 1
2 τ n
2)
,
θ [ 0
1 ]
=
∑
∞ n=−∞exp 2πi ( 1
2 τ n
2+ n 2
) ,
θ [ 1
0 ]
=
∑
∞ n=−∞exp 2πi {
1 2 τ
( n + 1
2 )
2}
. Now we are ready to calculate limits in each case.
(I) Let ω
1be fixed. As Im τ −→ ∞, we obtain θ
[ 0 0 ]
−→ 1, θ [ 0
1 ]
−→ 1, θ [ 1
0 ]
−→ 0, θ
′[ 1
1 ]
−→ 0.
Furthermore we have
Im
lim
τ→∞θ
′[
11
] θ [
10
] = − π.
It follows from (5.6) that if Im τ −→ ∞ , then λ
1− λ
2−→ 0, λ
1− λ
3−→ −
( π ω
1)
2and λ
2− λ
3−→ − ( π
ω
1)
2.
In this case, λ
1and λ
2tend to the same point λ and λ
3tends to λ + ( π
ω
1)
2. The limit curve is
w
2= (z − λ)
2(
z − (
λ + ( π
ω
1)
2)) .
(II) We further assume ω
1−→ ∞ in (I). Then we have
λ
1− λ
2−→ 0, λ
1− λ
3−→ 0 and λ
2− λ
3−→ 0.
Therefore λ
1, λ
2and λ
3tend to the same point λ, and we obtain the limit curve
w
2= (z − λ)
3. 6. The case g = 2
The total space T
2to consider is
T
2=
∪
2 g′=0
∪
g′5i ℓ52−g′
( M
ℓ(i) × S
∗g′)
= (
GL(2, C ) × S
∗2)
∪ (
M
1(1) × S
∗1)
∪ [
(M
1(0) ∪ M
2(0)) × S
∗0] .
Let M (I
2τ ) be a period matrix with M = (
M
11M
12M
21M
22)
∈ GL(2, C ) and τ ∈ S
2. The following are all the cases to investigate.
(I) The case [τ ] −→ [τ
11] in S
∗2. (I-1) M ∈ GL(2, C ) is fixed.
(I-2) (
M12M22
) −→ ∞ in ˙ C
2. (II) The case [τ ] −→ 0 in S
∗2. (II-1) M ∈ GL(2, C ) is fixed.
(II-2) (
M12M22
) −→ ∞ in ˙ C
2. (II-2’)
(
M11M21
) −→ ∞ in ˙ C
2. (II-3) M −→ (∞, ∞) in ( ˙ C
2)
2.
Any compact Riemann surface of genus 2 can be regarded as the Riemann surface R of
w
2= (z − λ
1)(z − λ
2)(z − λ
3)(z − λ
4)(z − λ
5).
It is a double covering of the Riemann sphere C∪{∞} with distinct branch points λ
1, λ
2, λ
3, λ
4, λ
5and ∞ with
Re λ
15 Re λ
25 Re λ
35 Re λ
45 Re λ
5.
We assume that the concrete realization of R is as shown in Fig.2.
λ1
λ2
λ4
∞
γ1γ2
δ2
δ1
λ5
λ3
Fig.2
Let (Γ, ∆) = (γ
1, γ
2, δ
1, δ
2) be a homology basis of R with KI(γ
i, δ
j) = δ
ij, KI (γ
i, γ
j) = KI (δ
i, δ
j) = 0.
We take the normal basis of abelian defferentials of first kind du
1, du
2with respect to (Γ, ∆). Let
τ
ij:=
∫
δj
du
i. Then τ = (τ
ij) ∈ S
2.
We define a mapping u : R −→ C
2, u(P ) = (u
1(P), u
2(P )) by u
i(P ) :=
∫
Pλ1
du
i.
We obtain Table II by a straight calculation.
Table II u(λ
1) = (
0 00 0
) u(λ
2) = (
0 01 0
) where the path of integration is taken along the top sheet on the top of the cut λ
1→ λ
2.
u(λ
3) = (
1 −11 0
) where we proceed from λ
2on the top sheet to λ
3. u(λ
4) = (
1 −11 1
) where we proceed from λ
3on the top sheet on the left of the branch cut as we go to λ
4.
u(λ
5) = (
1 01 1
) where we proceed on the top sheet from λ
4to λ
5. u(∞) = (
1 00 0
) where we proceed along the top sheet on the top of the cut from λ
5to ∞ .
The number of reduced 2-characteristics is 16. First we consider a theta function θ [
0 00 0
] (u(P ), τ ). Using Substitution formula and Reduction for- mula, we can verify that λ
3and λ
5are zeros of it. The vector of Riemann constants K(λ
1) with the base point λ
1is the sum of u(λ
j)’s with odd symbols ([11], Chapter V, Section 3, Theorem 9). Then
K(λ
1) = u(λ
3) + u(λ
5) =
( 2 − 1 2 − 1
) .
Let a := λ
3λ
5. Then a is an integral divisor of degree 2 on R. It holds that u(a) + K(λ
1) +
( 0 0 0 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). Hence it has the only zeros λ
3and λ
5.
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.
Table III Theta Function Zero θ [
0 00 0
] (u(P ), τ ) λ
3, λ
5θ [
1 00 0
] (u(P ), τ ) λ
2, λ
4θ [
0 10 0
] (u(P ), τ ) λ
3, λ
4θ [
0 01 0
] (u(P ), τ ) λ
4, ∞ θ [
0 00 1
] (u(P ), τ ) λ
4, λ
5θ [
1 01 0
] (u(P ), τ ) λ
1, λ
4θ [
1 10 0
] (u(P ), τ ) λ
2, λ
5θ [
1 00 1
] (u(P ), τ ) λ
2, λ
3θ [
0 11 0
] (u(P ), τ ) λ
5, ∞ θ [
0 10 1
] (u(P ), τ ) all points θ [
0 01 1
] (u(P ), τ ) λ
3, ∞ θ [
1 11 0
] (u(P ), τ ) λ
1, λ
5θ [
1 10 1
] (u(P ), τ ) λ
1, ∞ θ [
1 01 1
] (u(P ), τ ) λ
1, λ
3θ [
0 11 1
] (u(P ), τ ) λ
1, λ
2θ [
1 11 1
] (u(P ), τ ) λ
2, ∞ Lemma 6.1. A function
θ [
εε′
] (u(P ), τ ) θ
[
εˆ ˆ ε′]
(u(P ), τ )
2
is a single-valued meromorphic function on R, where [
εε
′
] and
[
ˆε ˆ ε′] are reduced 2-characteristics with
[
εˆ ˆ ε′] ̸ = [
0 10 1
] .
Proof. Let {
µµ′
}
be a period. By Functional equation we obtain
θ [
εε′
] ( u + {
µµ′
} , τ
) θ
[
εˆ ˆ ε′] ( u +
{
µ µ′} , τ
)
2
=
exp πi [ ∑
i
{ (ε
i− ε ˆ
i)µ
′i− (ε
′i− ε ˆ
′i)µ
i} ] θ [
εε′
] (u, τ ) θ
[
εˆ ˆ ε′] (u, τ)
2
=
± θ [
εε′
] (u, τ) θ
[
εˆ ˆ ε′] (u, τ)
2
=
θ [
εε′
] (u, τ ) θ
[
εˆ ˆ ε′] (u, τ )
2
.
This shows the assertion. ¤
We consider a meromorphic function z − λ
ion R which has the zero of order 2 at P = λ
iand the pole of order 2 at P = ∞ . We have the following equalities by Lemma 6.1 and Table III.
z − λ
1= C
1( θ [
1 01 0
] (u(P ), τ ) θ [
0 01 0
] (u(P ), τ ) )
2. (6.1)
z − λ
2= C
2( θ [
1 00 0
] (u(P ), τ ) θ [
0 01 0
] (u(P ), τ ) )
2. (6.2)
z − λ
3= C
3( θ [
0 10 0
] (u(P ), τ ) θ [
0 01 0
] (u(P ), τ ) )
2. (6.3)
z − λ
4= C
4( θ [
0 00 1
] (u(P ), τ ) θ [
0 11 0
] (u(P ), τ ) )
2. (6.4)
z − λ
5= C
5( θ [
0 00 1
] (u(P ), τ ) θ [
0 01 0
] (u(P ), τ ) )
2. (6.5)
Here C
1, . . . , C
5are some constants.
We have another basis of abelian differentials of first kind dv
1= dz
2w , dv
2= zdz 2w , where
w = √
(z − λ
1)(z − λ
2)(z − λ
3)(z − λ
4)(z − λ
5).
We define v(P ) = (v
1(P), v
2(P )) by v
i(P ) :=
∫
Pλ1
dv
i.
We consider a period matrix with respect to dv
1, dv
2(A B) =
( A
11A
12B
11B
12A
21A
22B
21B
22) , where
A
ij=
∫
γj
dv
i, B
ij=
∫
δj
dv
i. We set
C =
( C
11C
12C
21C
22)
:= A
−1. Then we have
(I τ ) = C(A B).
The relation between du
1, du
2and dv
1, dv
2is (
du
1du
2)
= (
C
11C
12C
21C
22) ( dv
1dv
2) .
Next we determine the constants C
1, . . . , C
5. We take a local coordinate t
1around P = λ
1with the origin at λ
1. Then we have in a neighbourhood of λ
1( u
1(P ) u
2(P )
)
= C
∫
t1(P)0
dt1
qQ5
i=2
{
t21+(λ1−λi)}
∫
t1(P)0
(λ1+t21)dt1
qQ5
i=2
{
t21+(λ1−λi)}
.
Dividing the both sides of (6.1) by t
21, we obtain 1 = C
1( θ [
1 01 0
] (u(P ), τ ) t
1(P )
1 θ [
0 01 0
] (u(P ), τ ) )
2. (6.6)
Further we have
p
lim
→λ1θ [
1 01 0
] (u(P ), τ ) t
1(P) = θ
1[ 1 0 1 0
] du
1dt
1(0) + θ
2[ 1 0
1 0 ] du
2dt
1(0).
(6.7) Since
du
jdt
1(0) = C
j1dv
1dt
1(0) + C
j2dv
2dt
1(0)
= 1
√∏
5i=2
(λ
1− λ
i)
(C
j1+ λ
1C
j2),
we obtain by (6.6) and (6.7)
1 = C
1∏
5i=2
(λ
1− λ
i)
∑
2j=1
θ
j[ 1 0
1 0 ]
(C
j1+ λ
1C
j2)
2
1 θ
2[
0 01 0
] ,
hence
C
1=
∏
5 i=2(λ
1− λ
i) (
θ [
0 01 0
]
∑
2j=1
θ
j[
1 01 0
] (C
j1+ λ
1C
j2) )
2. (6.8)
Therefore it follows from (6.1) that z − λ
1=
∏
5 i=2(λ
1− λ
i)
( θ [
0 01 0
]
∑
2j=1
θ
j[
1 01 0
] (C
j1+ λ
1C
j2) )
2(6.9)
×
( θ [
1 01 0
] (u(P), τ ) θ [
0 01 0
] (u(P), τ ) )
2.
Substituting P = λ
2, λ
3, λ
4, λ
5in (6.9), we obtain the following equalities.
(λ
1− λ
3)(λ
1− λ
4)(λ
1− λ
5) (6.10)
= − (∑
2j=1
θ
j[
1 01 0
] (C
j1+ λ
1C
j2) θ [
0 01 0
]
)
2( θ [
0 01 0
] (u(λ
2), τ ) θ [
1 01 0
] (u(λ
2), τ ) )
2.
(λ
1− λ
2)(λ
1− λ
4)(λ
1− λ
5) (6.11)
= − (∑
2j=1
θ
j[
1 01 0
] (C
j1+ λ
1C
j2) θ [
0 01 0
]
)
2( θ [
0 01 0
] (u(λ
3), τ ) θ [
1 01 0
] (u(λ
3), τ ) )
2.
(λ
1− λ
2)(λ
1− λ
3)(λ
1− λ
5) (6.12)
= − (∑
2j=1
θ
j[
1 01 0
] (C
j1+ λ
1C
j2) θ [
0 01 0
]
)
2( θ [
0 01 0
] (u(λ
4), τ ) θ [
1 01 0
] (u(λ
4), τ ) )
2.
(λ
1− λ
2)(λ
1− λ
3)(λ
1− λ
4) (6.13)
= − (∑
2j=1
θ
j[
1 01 0
] (C
j1+ λ
1C
j2) θ [
0 01 0
]
)
2( θ [
0 01 0
] (u(λ
5), τ ) θ [
1 01 0
![Table III Theta Function Zero θ [ 0 0 0 0 ] (u(P ), τ ) λ 3 , λ 5 θ [ 1 0 0 0 ] (u(P ), τ ) λ 2 , λ 4 θ [ 0 1 0 0 ] (u(P ), τ ) λ 3 , λ 4 θ [ 0 0 1 0 ] (u(P ), τ ) λ 4 , ∞ θ [ 0 0 0 1 ] (u(P ), τ ) λ 4 , λ 5 θ [ 1 0 1 0 ] (u(P ), τ ) λ 1 , λ 4 θ [ 1 1 0 0](https://thumb-ap.123doks.com/thumbv2/123deta/10102645.1937422/18.892.180.753.111.1115/table-iii-theta-function-zero-θ-p-τ.webp)
