Sigma Functions for Telescopic Curves

0.23 MB·English

by  Takanori Ayano

Sigma Functions for Telescopic Curves
∗
2
Takanori Ayano
1
0
2

n
a Abstract
J
In this paper, we consider a symplectic basis of the first cohomology
3
group andthesigma functionsforalgebraic curvesexpressedbyacanon-

] ical form using a finite sequence (a1,··· ,at) of positive integers whose
G greatest common divisor is equal to one (Miura [13]). The idea is to ex-
pressanon-singularalgebraiccurvebyaffineequationsoftvariableswhose
A
orders at infinityare (a1,··· ,at). Weconstruct asymplectic basis of the
.
h firstcohomology groupandthesigma functionsfortelescopiccurves,i.e.,
t the curves such that the number of defining equations is exactly t−1 in
a
theMiuracanonical form. Thelargest classofcurvesforwhichsuchcon-
m
struction has been obtained thus far is (n,s)-curves ([3][15]), which are
[ telescopic because they are expressed in the Miura canonical form with

1 t=2, a1=n, and a2 =s,and thenumberof definingequations is one.
v
4
1 Introduction
4
6
0 In recent years, the theory of Abelian functions is attracting increasing
1. interest in mathematical physics and applied mathematics. In particular, the
0 sigma functions associated with algebraic curves with higher genus have been
2 studied actively. In this paper, we construct the sigma functions for a class of
1
algebraic curves for which such construction has not been obtained thus far.
:
v Let C be a compact Riemann surface with genus g, and H1(C,C) the first
i cohomologygroup,which is defined by the linear space of second kind differen-
X
tials modulo meromorphic exact forms. We say a meromorphic differential on
r
a C to be second kind if it is locally exact.
We consider a basis of H1(C,C) consisting of dimCH1(C,C)=2g elements
(cf. [11], pp.29-31, Theorem 8.1,8.2) . In particular, in order to construct the
sigmafunctions, wewishto constructabasis(symplectic basis) du ,dr g of
{ i i}i=1
H1(C,C) such that
1. du is holomorphic on C for each i, and
i
2. du du =dr dr =0 and du dr =δ for each i,j,
i j i j i j ij
◦ ◦ ◦
∗2010Mathematics SubjectClassification. Primary14H55;Secondary14H42,14H50.
1
where the operator is the intersection form on H1(C,C) defined by
◦
p
η η′ = Res( η)η′(p)
◦
p Z
X
for second kind differentials η,η′ (the summation is over the singular points of
η and η′, and Res denotes the residue at the point).
In order to express defining equations of C, we use a canonical form for
expressingnon-singularalgebraiccurves proposedby Miura [13]. Given a finite
sequence(a ,...,a )ofpositiveintegerswhosegreatestcommondivisorisequal
1 t
to one, Miura [13] proposed a non-singular algebraic curve determined by the
sequence (a ,...,a ), which is called (a ,...,a )-curve. The idea is to express
1 t 1 t
a non-singular algebraic curve by affine equations of t variables whose orders
at infinity are (a , ,a ). Any non-singular algebraic curve is birationally
1 t
···
equivalent to some (a ,...,a )-curve (cf. [13]).
1 t
Klein [9][10] constructed a symplectic basis of the first cohomology group
and the sigma functions for the hyperelliptic curves with genus g, which are
expressed in the Miura canonical form with t = 2, a = 2, and a = 2g+1.
1 2
Buchstaber et al. [3] and Nakayashiki [15] constructed them for more general
plane algebraic curves called (n,s)-curves, which are expressed in the Miura
canonical form with t = 2, a = n, and a = s. In this paper, we construct
1 2
themforthe telescopic(a ,...,a )-curves,i.e., the (a ,...,a )-curvessuchthat
1 t 1 t
for any i (2 i t)
≤ ≤
a a a
i 1 N+ + i−1N, d :=GCD a , ,a .
i 1 i
d ∈ d ··· d { ··· }
i i−1 i−1
Theclassofthetelescopic(a ,...,a )-curvesisequaltothatofthe(a ,...,a )-
1 t 1 t
curves such that the number of the defining equations in the Miura canoni-
cal form is exactly t 1 (cf. [18]). Since the (n,s)-curves are telescopic, our
−
curvescontainthem strictly. Recently,Matsutani[12] constructedthem for the
(3,4,5)-curves,which are not telescopic.
The plan of this paper is as follows. In section 2, we define the (a ,...,a )-
1 t
curves. Because there exists no English paper for the (a ,...,a )-curves, we
1 t
give complete proofs in Appendix. In section 3, we construct holomorphic 1-
forms du g for the telescopic (a ,...,a )-curves. In section 4, we construct
second{kinid}id=i1fferentials dr g for1theteletscopic(a ,...,a )-curves,andshow
that the set du ,dr g { isi}ai=s1ymplectic basis of th1e first tcohomology group.
{ i i}i=1
In section 5, we construct the sigma functions associated with the telescopic
(a ,...,a )-curves.
1 t
Throughout this paper, N,N ,Z, and C denote the non-negative integers,
+
the positive integers, the integers, and the complex numbers, respectively.
2 (a ,...,a )-Curves
1 t
Miura [13] proposed a canonical form for expressing non-singular algebraic
curves. Let t 2, and a ,...,a positive integers such that GCD a ,...,a =
1 t 1 t
≥ { }
2
1. We denote A := (a ,...,a ) Nt and A := a N+ +a N, assuming
t 1 t ∈ + h ti 1 ··· t
that the order of a ,...,a is fixed. For the map Ψ : Nt A defined by
1 t t
Ψ((m ,...,m ))= t a m , we define the order< in Nt s→o thhatiM <M′ for
1 t i=1 i i
M =(m ,...,m ) and M′ =(m′,...,m′) if
1 t P 1 t
1. Ψ(M)<Ψ(M′) or
2. Ψ(M) = Ψ(M′) and m = m′,...,m = m′ ,m > m′ for some
1 1 i−1 i−1 i i
i(1 i t).
≤ ≤
Let M(a)be the minimum elementwith respectto the order < in Nt satisfying
Ψ(M)=a A . We define B(A ) Nt and V(A ) Nt B(A ) by
t t t t
∈h i ⊆ ⊆ \
B(A )= M(a) a A
t t
{ | ∈h i}
and
V(A )= L Nt B(A ) L=M+N,M Nt B(A ),N Nt N =(0,...,0) ,
t t t
{ ∈ \ | ∈ \ ∈ ⇒ }
respectively.
Hereafter, C[X] := C[X ,...,X ] denotes the polynomial ring over C of t-
1 t
variables X ,...,X . For A C[X], Span A and (A) denote the linear space
1 t
⊂ { }
overCgeneratedbyAandtheidealinC[X]generatedbyA,respectively. Also,
XM,M =(m ,...,m ), denotes XM =Xm1 Xmt for simplicity.
1 t 1 ··· t
For M V(A ), we define the polynomial F (X) C[X] by
t M
∈ ∈
F (X)=XM XL λ XN, λ C, (1)
M N N
− − ∈
{N∈B(At)X|Ψ(N)<Ψ(M)}
where L is the element of B(A ) such that Ψ(L)=Ψ(M). We assume that the
t
set of the polynomials F M V(A ) satisfies the following condition:
M t
{ | ∈ }
Span XN N B(A ) ( F M V(A ) )= 0 . (2)
t M t
{ | ∈ }∩ { | ∈ } { }
Let I := ( F M V(A ) ), R := C[X]/I, x the image of X for the
M t i i
{ | ∈ }
projection C[X] R, and K the total quotient ring of R. Then, we have the
→
following three propositions. For complete proofs, see Appendix.
Proposition 2.1 (Miura [13]).
(i) The set xN N B(A ) is a basis of R over C, where x:=(x ,...,x ).
t 1 t
{ | ∈ }
(ii) The ring R is an integral domain, hence, K is the quotient field of R.
(iii) The field K is an algebraic function field of one variable over C.
(iv) There exists a discrete valuation v of K such that (x ) =a v for any
∞ i ∞ i ∞
i, where (x ) denotes the pole divisor of x (cf.[19] p.19).
i ∞ i
LetCaff := (z ,...,z ) Ct f(z ,...,z )=0, f I . FromProposition
1 t 1 t
{ ∈ | ∀ ∈ }
2.1(ii)(iii),CaffisanaffinealgebraiccurveinCt. Hereafter,weassumethatCaff
3
isnon-singular. Fork N,wedefineL(kv ):= f K (f)+kv 0 0 ,
∞ ∞
∈ { ∈ | ≥ }∪{ }
where (f) denotes the divisor of f, i.e., (f)= v(f) v.
v ·
Proposition 2.2 (Miura [13]). (i) R= ∞P L(kv ).
k=0 ∞
(ii) The map φ
S
Caff discrete valuation of K v
∞
→{ }\{ }
p v
p
→
is bijective, where v is the discrete valuation corresponding to p Caff (cf.
p
∈
[17], p.21,22).
Let C be the compactRiemannsurface correspondingto Caff. FromPropo-
sition2.2(ii),wecanobtainC byaddingonepoint toCaff,wherethediscrete
∞
valuationcorrespondingto isv . Itisknownthatanynon-singularalgebraic
∞
∞
curve is birationally equivalent to such C for some A (cf. [13]). Hereafter, we
t
represent each curve C by the sequence A = (a ,...,a ) and call (a ,...,a )-
t 1 t 1 t
curve for short.
The sequence A =(a ,...,a ) is called telescopic if for any i (2 i t)
t 1 t
≤ ≤
a a a
i 1 N+ + i−1N, d :=GCD a , ,a .
i 1 i
d ∈ d ··· d { ··· }
i i−1 i−1
Note that A =(a ,a ) is always telescopic.
2 1 2
Proposition 2.3 (Miura [13]). If A is telescopic, the condition (2) is met,
t
and we have the followings.
(i) B(A )= (m ,...,m ) Nt 0 m d /d 1, 2 i t .
t 1 t i i−1 i
{ ∈ | ≤ ≤ − ≤ ≤ }
(ii) V(A ) = (0,...,0,d /d ,0,...,0) 2 i t , where d /d is in the
t i−1 i i−1 i
{ | ≤ ≤ }
i-th component.
(iii) The genus g of C is
t
1 d
i−1
g = (1 a )+ 1 a . (3)
1 i
2( − i=2(cid:18) di − (cid:19) )
X
Note that ♯V(A ) is the number of the defining equations, where ♯ denotes
t
thenumberofelements. FromLemmaC.1(iv)inAppendix,weobtain♯V(A )
t
≥
t 1. IfA istelescopic,thenfromProposition2.3(ii)weobtain♯V(A )=t 1.
t t
− −
On the other hand, Suzuki [18] proved that if ♯V(A ) = t 1, then A is
t t
−
telescopic.
If A = (a ,...,a ) is telescopic, we call the (a ,...,a )-curve telescopic
t 1 t 1 t
(a ,...,a )-curve. From Proposition 2.3, the defining equations of a telescopic
1 t
(a ,...,a )-curve is given as follows: for 2 i t,
1 t
≤ ≤
t
F (X ,...,X )=Xdi−1/di Xmij λ(i) Xj1 Xjt,
i 1 t i − j − j1,...,jt 1 ··· t
jY=1 j1X,...,jt
4
where (m ,...,m ) B(A ), t a m = a d /d , λ(i) C, and
i1 it ∈ t j=1 j ij i i−1 i j1,...,jt ∈
(j ,...,j ) runs over (j ,...,j ) B(A ) with t a j <a d /d .
1 t 1 t ∈P t k=1 k k i i−1 i
EXAMPLE 1. A =(n,s) for relatively primPe positive integers n and s.
2
Since A = (n,s) is telescopic, from Proposition 2.3 (ii), we have V(A ) =
2 2
(0,n) . Hence, the (n,s)-curves (cf. [15], p.182) are given by the following
{ }
defining equation:
F (X ,X )=Xn Xs λ(2) Xj1Xj2.
2 1 2 2 − 1 − j1,j2 1 2
nj1+Xsj2<ns
In particular, we obtain the elliptic curves if n=2 and s=3, and the hyperel-
liptic curves with genus g if n=2 and s=2g+1.
EXAMPLE 2. A =(4,6,5).
3
Since A = (4,6,5) is telescopic, from Proposition 2.3 (ii), we have V(A ) =
3 3
(0,2,0),(0,0,2) . Hence,the(4,6,5)-curvesaregivenbythefollowingdefining
{ }
equations:
F (X ,X ,X )=X2 X3 λ(2) X X λ(2) X X λ(2) X X λ(2) X2
2 1 2 3 2− 1− 0,1,1 2 3− 1,1,0 1 2− 1,0,1 1 3− 2,0,0 1
λ(2) X λ(2) X λ(2) X λ(2)
− 0,1,0 2− 0,0,1 3− 1,0,0 1− 0,0,0
and
F (X ,X ,X )=X2 X X λ(3) X X λ(3) X2 λ(3) X λ(3) X
3 1 2 3 3 − 1 2− 1,0,1 1 3− 2,0,0 1 − 0,1,0 2− 0,0,1 3
λ(3) X λ(3) .
− 1,0,0 1− 0,0,0
3 Holomorphic 1-Forms for the Telescopic Curves
Let C be a telescopic (a ,...,a )-curve,and Γ(C,Ω1) the linear space con-
1 t C
sisting of holomorphic 1-forms on C. In this section, we construct a basis of
Γ(C,Ω1). Let G be the matrix defined by
C
∂F ∂F
2 2
...
∂X ∂X
1 t
G:=.................,
∂F ∂F
 t ... t
∂X ∂X 
 1 t
 
and G the matrix obtained by removing the i-th column from G. Then, we
i
have the following theorem.
Theorem 3.1. The set
xk1 xkt t
P := 1 ··· t dx (k ,...,k ) B(A ), 0 a k 2g 2
1 1 t t i i
(detG1(x) | ∈ ≤ ≤ − )
i=1
X
5
is a basis of Γ(C,Ω1) over C, where detG (x) denotes detG (X =x).
C 1 1
We rearrange the elements of P in the ascending order with respect to the
order at , and write du ,...,du .
1 g
∞ { }
In order to prove Theorem 3.1, we need some lemmas.
Lemma 3.1. If detG (p) = 0 for p = (p ,...,p ) Caff and 1 i t,
i 1 t
6 ∈ ≤ ≤
then v (x p )=1.
p i i
−
Proof. Without loss of generality, we assume i= 1. Suppose v (x p )
p 1 1
− ≥
2. Then, there exists k (2 k t) such that v (x p ) = 1. In fact, if
p k k
≤ ≤ −
v (x p ) 2 for any k, then v (f) 2 or v (f) = 0 for any f R. Then,
p k k p p
− ≥ ≥ ∈
v (g) 2 or v (g) =0 for any g R , where R be the localization of R at p.
p p p p
≥ ∈
This contradicts that R is a discrete valuation ring.
p
There exist γ ,δ(i) C such that for 2 i t
{ ij j1,...,jt}∈ ≤ ≤
t
F (X ,...,X )= γ (X p )+ δ(i) (X p )j1 (X p )jt,
i 1 t ij j− j j1,...,jt 1− 1 ··· t− t
Xj=1 j1+·X··+jt≥2
where γ = ∂Fi(p). Since F (x ,...,x ) = 0 and v (x p ) 2, we have
ij ∂Xj i 1 t p 1 − 1 ≥
v t γ (x p ) = v (x p )( t γ xj−pj) 2. Since v (x
p j=2 ij j − j p k− k j=2 ijxk−pk ≥ p k −
p )(cid:16)=P1, we have t (cid:17)γ b =(cid:16)0, where bP= xj−pj (p)(cid:17). Hence, we obtain
k j=2 ij j j xk−pk
(cid:16) (cid:17)
P
b 0
2
G1(p)· =·.
· ·
bt 0
   
Since b = 1(= 0), we have detG (p) = 0. This contradicts the assumption of
k 1
6
Lemma 3.1. Hence, we obtain v (x p )=1.
p 1 1
− (cid:3)
Lemma 3.2. (i) As an element of K, we have detG (x)=0.
1
6
dx
1
(ii) div =(2g 2) .
detG (x) − ∞
(cid:18) 1 (cid:19)
Proof. Since the differential d(F (x ,...,x ))=0 for any i, we have
i 1 t
dx 0
1
G(x) · =· .
· ·
dx  0
 t  
   
By multiplying some elementary matrices on the left, the above equation be-
6
comes
w z z z dx 0
2 22 23 2t 1
···
w 0 z z
 3 33··· 3t · =·.
...
· ·
wt 0 zttdxt 0
 ···    
    
SinceCaff isnon-singularforanyp Caff,thereexistsisuchthatdetG (p)=0.
i
∈ 6
Hence, we have w = 0 or z = 0 as elements of K. Since v (x ) = a , we
t tt ∞ j j
6 6 −
have x / C, hence, dx = 0 for any j. Since w dx = z dx , we have w = 0
j j t 1 tt t t
∈ 6 6
and z = 0. Hence, by multiplying some elementary matrices on the left, the
tt
6
above equation becomes
w′ z z 0 dx 0
2 22 23··· 1
w′ 0 z 0
 3 33···  · =·.
...
· ·
wt 0 zttdxt 0
 ··    
    
Similarly, we obtain
w′′ z 0 0 dx 0
2 22 ··· 1
w′′ 0 z 0
 3 33···  · =·,
...
· ·
wt′′ 0 ··· zttdxt 0
    
where w′′,...,w′′,z ,...,z K are non-zero. Hence, we obtain detG (x)=
2 t 22 tt ∈ 1
z z =0, which complete the proof of (i).
22 tt
± ··· 6
Next, we prove that the 1-form dx /detG (x) is both holomorphic and
1 1
non-vanishing on Caff. When detG (p) = 0 for p Caff, from Lemma 3.1,
1
6 ∈
dx /detG (x)isbothholomorphicandnon-vanishingatp. SupposedetG (p)=
1 1 1
0 for p Caff. Since Caff is non-singular at p, there exists i (2 i t) such
∈ ≤ ≤
that detG (p)=0. Since w′′dx +z dx =0, we have w′′z z z dx +
i 6 i 1 ii i i 22··· ii··· tt 1
z z dx = 0, i.e., detG (x)dx detG (x)dx = 0, where z denotes to
22 tt i i 1 1 i ii
··· ±
remove zii. Since detG1(x) = 0 and detGi(x) = 0, we have dx1c/detG1(x) =
6 6
dx /detG (x). Hence, from detG (p) = 0 and Lemma 3.1, dx /detG (x) is
i i i 1c 1
± 6
holomorphic and non-vanishing at p. On the other hand, by Riemann-Roch’s
theorem, we have degdiv(dx /detG (x)) = 2g 2, which complete the proof
1 1
−
of (ii).
(cid:3)
ProofofTheorem3.1. FromLemma3.2andProposition2.1(i),wehaveP
⊂
Γ(C,Ω1C),andtheelementsofP arelinearlyindependent. SincedimCΓ(C,Ω1C)=
g, it is sufficient to prove ♯P = g. It is well-known that there are g gap values
at from 0 to 2g 1. Since dimCL((2g 1)v∞) = dimCL((2g 2)v∞) = g
∞ − − −
(Riemann-Roch’stheorem),2g 1isagapvalueat . Hence,fromProposition
2.1(i)andProposition2.2(i),w−ehave♯ (k ,...,k )∞ B(A ) 0 t a k
{ 1 t ∈ t | ≤ i=1 i i ≤
2g 2 =g, which complete the proof of Theorem 3.1.
− } P (cid:3)
7
4 Second Kind Differentials for the Telescopic
Curves
In this section, we construct dr for a telescopic (a ,...,a )-curve C. For
i 1 t
2 i t and 1 j t, we divide F (Y ,...,Y ,X ,...,X ) by X Y ,
i 1 j−1 j t j j
≤ ≤ ≤ ≤ −
where we regard X as a variable. Let h C[X ,...,X ,Y ,...,Y ] be the
j ij 1 t 1 t
∈
quotient, i.e.,
F (Y ,...,Y ,X ,...,X )=h (X Y )+F (Y ,...,Y ,X ,...,X ), (4)
i 1 j−1 j t ij j j i 1 j j+1 t
· −
and H the matrix defined by
h ... h
22 2t
H := .............. .
 
h ... h
t2 tt
 
We consider the 1-form
detH(x,y)
Ω(x,y):= dx
1
(x y )detG (x)
1 1 1
−
and the bilinear form (cf. [15], p.181, 2.4)
xi1 xityj1 yjt
ωˆ(x,y):=d Ω(x,y)+ c 1 ··· t 1 ··· t dx dy (5)
y i1,...,it,j1,...,jtdetG (x)detG (y) 1 1
1 1
X
onC C,wherex=(x ,...,x ),y =(y ,...,y ),c C,(i ...,i )
× 1 t 1 t i1,...,it,j1,...,jt ∈ i t ∈
B(A ) with 0 t a i 2g 2, and (j ...,j ) B(A ).
Wte take a≤basisk=α1 ,kβk g≤ of−the homoliogy grtou∈p H (Ct,Z) such that their
P { i i}i=1 1
intersection numbers are α α =β β =0 and α β =δ .
i j i j i j ij
◦ ◦ ◦
DEFINITION 4.1. (cf. [15], p.181, 2.4) Let ∆ := (p,p) p C . A
{ | ∈ }
meromorphic symmetric bilinear form ω(x,y) on C C is called a normalized
×
fundamental form if the following conditions are satisfied.
(i) ω(x,y) is holomorphic except ∆ where it has a double pole. For p C, we
∈
takealocalcoordinatesaroundp. Then,the expansionins(x)ats(y)isofthe
form
1
ω(x,y)= +regular ds(x)ds(y).
(s(x) s(y))2
(cid:18) − (cid:19)
(ii) ω =0 for any i, where the integration is with respect to any one of the
Zαi
variables.
Normalizedfundamentalformexistsandunique(cf. [15]p.182). Then,wehave
the following theorem.
Theorem 4.1.
8
(i) There exists a set of c such that ωˆ(x,y)=ωˆ(y,x).
i1,...,it,j1,...,jt
Wetakec suchthatωˆ(x,y)=ωˆ(y,x). Then,wehavethefollowings.
i1,...,it,j1,...,jt
(ii) The bilinear form ωˆ satisfies the condition (i) of Definition 4.1.
(iii) For du :=(xki,1 xki,t/detG (x))dx , we define
i 1 ··· t 1 1
yj1 yjt
dr := c 1 ··· t dy .
i ki,1,...,ki,t,j1,...,jtdetG (y) 1
1
j1X,...,jt
Then, dr is a second kind differential for any i, and the set du ,dr g is a
i { i i}i=1
symplectic basis of H1(C,C).
In order to proveTheorem 4.1, we generalize Lemma 2,3,4,5,6 in [15] to the
telescopic (a ,...,a )-curves.
1 t
Let B be the set of branch points for the map x : C P1,(x ,...,x )
1 1 t
→ →
[x : 1] (cf. [17], p.24, Example 2.2). Since the ramification index of the map
1
x at is a , we have degx =a (cf. [17], p.28, Proposition 2.6). For p C,
1 1 1 1
we set∞x−11(x1(p)) = {p(0),p(1),...,p(a1−1)} with p = p(0), where the same∈p(i)
is listed according to its ramification index.
Lemma 4.1. Let U be a domain in C, f(z ,z ) a holomorphic function on
1 2
U U, and g(z) := f(z,z). If g 0 on U, then there exists a holomorphic
× ≡
function h(z ,z ) on U U such that f(z ,z )=(z z )h(z ,z ).
1 2 1 2 1 2 1 2
× −
Proof. Leth(z ,z ):=f(z ,z )/(z z ). Givenz ,h(z , )hasasingularity
1 2 1 2 1 2 1 1
− ·
onlyatz , where its singularityis removable,i.e., h(z , )is holomorphiconU.
1 1
·
Similarly, h( ,z ) is holomorphic on U. Hence, h is holomorphic on U U.
2
· × (cid:3)
Lemma 4.2. The 1-form Ω(x,y) is holomorphic except ∆ (p(i),p) i =
∪{ | 6
0, p B or p(i) B C C.
∈ ∈ }∪ ×{∞}∪{∞}×
Proof. Since dx /detG (x) is holomorphic on C (cf. Lemma 3.2), Ω(x,y)
1 1
is holomorphic except ∆ (p(i),p) p C,i=0 C C. We
∪{ | ∈ 6 }∪ ×{∞}∪{∞}×
prove that Ω(x,y) is holomorphic on (p(i),p) i = 0,p / B,p(i) / B . From
{ | 6 ∈ ∈ }
(4), we obtain
t
F (X ,...,X )= h (X Y )+F (Y ,...,Y ). (6)
i 1 t ij j j i 1 t
· −
j=1
X
Set X =x and Y =y, then we have
t
h (x,y) (x y )=0.
ij j j
· −
j=1
X
9
Take (p(i),p) C C such that i=0,p / B, and p(i) / B, then we have
∈ × 6 ∈ ∈
h ... h p(i) p 0
21 2t 1 − 1
.............. = .
   ·  ·
ht1 ... htt X=p(i),Y=ppt(i)−pt 0
     
Since p(i) p =0, we have
1 − 1
p(i) p 0
2 − 2
H(p(i),p) = .
 ·  ·
p(i) p 0
 t − t
   
Since (p(i) p ,...,p(i) p ) = (0,...,0), we have detH(p(i),p) = 0. Since
2 − 2 t − t 6
p / B and p(i) / B, we can take (x ,y ) as a local coordinate around (p(i),p).
1 1
∈ ∈
Hence, from Lemma 4.1, there exists a holomorphic function h(x ,y ) around
1 1
(p(i),p)suchthatdetH(x,y)=(x y )h(x ,y ). Hence,Ω(x,y)isholomorphic
1 1 1 1
−
at (p(i),p).
(cid:3)
Lemma 4.3. Let p / B, s a local coordinate around p. Then, the expansion
∈
of Ω(x,y) in s(y) at s(x) is of the form
1
Ω(x,y)= − +regular ds(x).
s(y) s(x)
(cid:18) − (cid:19)
Proof. Set Y =y in (6), then we have
t
F (X ,...,X )= h (X,y) (X y ).
i 1 t ij j j
· −
j=1
X
Hence, we obtain
t
∂F ∂h
i ij
(x ,...,x )= (x,y) (x y )+h (x,y).
1 t j j ik
∂X ∂X · −
k k
j=1
X
Set x=y, then we have
∂F
i
(x ,...,x )=h (x,x).
1 t ik
∂X
k
Hence, we obtain detG (x) = detH(x,x). On the other hand, since p / B,
1
∈
we can take (x ,y ) as a local coordinate around (p,p). Since p / B, we have
1 1
∈
detG (p)=0. In fact, if detG (p)=0, then dx /detG (x) is notholomorphic
1 1 1 1
6
at p, which contradicts Lemma 3.2 (ii). Hence, detH(x,y)/detG (x) is holo-
1
morphic at (p,p). Hence, from Lemma 4.1, there exists a holomorphic function
10