Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, ZLirich Series: Dept. of Mathematics, Univ. of Maryland, College Park Adviser: J. K. Goldhaber
352 JohnD.Fay University of Maryland, College Park, MD/USA
Theta Functions on Riemann Surfaces IIIIIIII
I
II
II
I
Springer-Verlag Berlin.Heidelberg • New York 1973
A M S Subject Classifications (1970) : 30-02, 30 A 4 8 , 30 A 58
I S B N 3-540-06517-2 S p f i n g e r - V e r l a g Berlin • H e i d e l b e r g - N e w Y o r k I S B N 0-387-06517-2 S p r i n g e r - V e r l a g N e w Y o r k - H e i d e l b e r g - B e r l i n
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is. payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1973. Library of Congress Catalog Card Number 73-15292. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr,
Preface
These notes theory
of theta
interest
between
functions
surfaces Riemann
Mumford
new as well as c l a s s i c a l on R i e m a n n - [5],
theta
moduli,
[i0].
functions
on d e g e n e r a t e
of special
Riemann
surfaces, Topics
results
a subject discussed
and A b e l i a n surfaces,
of r e n e w e d here
include:
differentials,
Schottky
and theta functions
from the
relations
for
on finite b o r d e r e d
surfaces.
I wish con s t a n t
functions
in recent years
the relations theta
present
to express
sincere
thanks
h e l p and e n c o u r a g e m e n t for generous
Research Foundation.
assistance
for these notes
to Prof.
Lars
over many years, at several
was
supported
points
V. Ahlfors
and to Prof.
for his David
in this work.
by the N a t i o n a l
Science
Table of Contents
I.
Riemann's T h e t a Function
i
The Prime Form
16
III.
D e g e n e r a t e Riemann Surfaces
37
IV.
Cyclic U n r a m i f i e d Coverings
61
V.
Double Ramified Coverings
85
VI.
B o r d e r e d Riemann Surfaces
108
Notational
134
II.
References
Index
135
I.
A variation these notes: dimension
of the classical
a principally
g will be written
{g generated identity a point
Riemann's
by the columns
g x g
matrix
in the Siegel
Theta Function
Krazer notation
polarized as
complex Abelian
O~
= ~g/F,
of the
g × 2g
and T a symmetric
(left)
[19] will be used in
half-plane
where matrix
.
~
of
F is the lattice
g × g
~
variety
(2~ii,T)
with
matrix with
Any point
in
I the
Re T < 0,
e e {g
can
0
be written
uniquely
characteristics
as
of e;
e : (e,6)(2~lI)-the notation
where
e =
6~e e [g
are the
will be used for the T
point
e e ~g
function
with characteristics
is defined @(z)
then for any
[el : [~].
If Riemann's
theta
by
= 0[00] (z) = m~egg exp { ½mTmt + mzt}
e =
,
z e ~g
~ {g, T
(i)
exp
{½6T6 t + (z+ 2~is)@ t}@(z+e)
= ~
= 0T[@S] (z)
exp {½(m+6) T(m+6) t + (z + 2~ie) (m+6) t }
meg g where
@ [~](z)
teristics %n[~
(z)
satisfying
*
exp(W)
[~
for
is called the first order theta-function 6,e
~ ~g.
with characteristics the identity
= (2.718..) W
In general,
with charac-
an n th order theta-function
[2] is any holomorphic
function
on ~g
[]
6 (z+KT+2~iX) n g
for
K,I [ ~g;
(2)
= exp{-½nKTKt-nzKt+
that is, for
(Zl'
6 g
n
•
.
2~i(6~ t
sKt)}~n
[I e
j = l,...,g
• zj+2~i, '
. .' zg ) = e
"
(z)
0 n g(~
and (2)'
(Zl+Tjl,..
n e
e[
Any function
n
by the point
1{6} , g T
where
verified
that for
J
n s
eo
of
translated
@
on
~
(z).
defined by the
for each characteristic
functions
[~]
0 n g ,
with
p e (g/ng) g
[19, p. 40].
p,o • zg;
"~
are all n th order theta-funotions n = 2,
a section
[33];
@nTu s j(nz)
n~k~ JCn~), %
when
]
L@ is the line bundle
independent
given by the functions
In particular,
= e
theta function
are n g linearly
It is easily
Zg+Tjg)
can be considered
e
divisor of Ri@mann's there
"'
(~)
on/Ll~z)
and
Lnj
with characteristics
02[el(z)
is a second-order
[2] [19, p. 39]. theta-function
TLBJ
with characteristics
[0] 0 for
called
and is said to be even
a half-period
an even
(resp.
mod 2.
The second order theta-functions
[17, p. 139]:
odd)
function
2~,26 E gg;
such a point (resp.
of z which holds
iff
satisfy
{~} 6 e ~g
odd)iff 4~-6
@[<]
~ 0 (resp.
an addition
is is i)
theorem
(3)
ir ~ ~,'1'- j
and its inversion
(4)
for any
Zl,Z 2 e ~g
Now suppose
and h a l f - i n t e g e r
C is a compact
a canonical basis
characteristics
Riemann surface of genus
AI,...,Ag,BI,... ,Bg
of HI(C,g)
-~ ) 0 , and let
v I,... ,Vg
tials H0(C,~ I) normalized with respect to I : identity Re T < 0.
Then
so that the period matrix of
J(C)
= {g/(2~il,T),
divisors J(C).
the divisor
holomorphic
equivalence
if
classes
X: ~i (C) ÷ ~,2g
sponding to
D : ~ -~
x(A i) = i
two different morphically
with
~
where
matrix with
and
0 modulo princi~
to the point
positive (]~
v )
in
J(C) will often be identified with the group of
fining a flat line bundle
(5)
- J~
o o ,Vg
variety of C, is
on C of degree
of the same degree corresponds
Equivalently,
follows:
D = ~
Vl,.
g × g
the Jacobian
identified with the group of divisors pal divisors:
differen-
has the form (2~iI,T)
matrix and T is a symmetric
Fix
has the form
be a basis of the holomorphic
AI,...,Ag,BI,...,Bg
g x g
g > 0.
such that the inter-
section matrix defined by the cup product on HI(c,z) (0 I
~,6,y,8,e.
of line bundles
is the characteristic
[13, p. 186], the bundle
~ J(C)
X(B i) = exp
bundles,
r (]
v.)
from
~
0 as
homomorphism
of degree
will have characteristic
paths of integration
equivalent
on C of degree
de-
0 corre-
homomorphism
i : l,...,g;
to ~
give rise to holo-
since two flat line bundles
L and
are h o l o m o r p h i c a l l y their characteristic V Y ¢ ~I(C) bundle
equivalent
homomorphisms
[14, p. 238].
characteristic
homomorphism
(6)
= e
if
-2~i~. ]
X(Y)X-I(y) = exp I w, -y last fact also implies that the line
equivalent
are the characteristics g
2~iE. ]
line bundle with
j = i,
~
of the point
(
• ° •
,g
v )
T
For any tions
to the unitary
x(B j ) : e
'
w c H0(~C)7 ,
X and X satisfy
This
L is h o l o m o r p h i c a l l y
x(Aj)
if and only if, for some
0(
a ¢ C
and
e • {g,
s2,, s,7 ,'s? -
v - e) 0 ( v - e )
are, by (2), meromorphic homomorphisms sections follows:
the multiplicative and
sections
0[-
if
D = E
m D is the unique
niP i,
Pi
differential
s; s2v ,'f v](
- e) 0(
func-
v- e)
of L as given by the characteristic
(5) and (6) respectively.
of L can be expressed
meromorphic
Alternatively,
in terms of the Abelian e C,
meromorphic integrals
is any divisor of degree
as
0 and
of the third kind on C with simple poles
of residue n i at Pi and zero A-periods,
the Riemann bilinear relation
gives
f
(7)
where
~
=
%
:
Bj
E niP i n.>0
v. J
A
and
~
=
e
¢
j : 1 ..... g
~ niP i. n.<0
l
Here,
taken within C cut along its homology contain points of D.
Ix
~D
(5).
~D(X)
Similarly,
= ~D(X)
between
divisors
are
basis, which we assume does not
By (7), the m u l t i p l i c a t i v e
is thus a meromorphic
ing to D under
we
i
make the convention that the paths of integration
exp
and henceforth,
meromorphic
section of the line bundle if
g - i,]~=l vi(x)(Re
-i Re ( I~~ vj) T)ij
function correspond-
is the unique differential
of the third kind with simple poles of
residue n i at Pi and purely imaginary A and B-periods, is a meromorphic morphism
(6).
exp
section of the line bundle with characteristic
Finally,
whenever the bundle
we have Abel's Theorem: phie function
then
if
D = ~ -~
is holomorphically
fx
~D
homotrivial,
is the divisor of a meromor-
f on C, g
d in f : ~ @ - A
-
~ i
mjvj
g
(8)
I~
I IAj. d in f 2~i
mj =
For
v k : 2~in k +
classes of divisors
acts on Jn(C) by addition, is an isomorphism • Jn(C).
is,
~ ~
n]. = ~ i
• ~,
IB
dlnf
n • g~ let Jn(C) be the principal homogeneous
equivalence
write
~. mjTj j:l k
: ~
- ~
Then
and for any fixed divisor
~@ : Jn(C)
For two divisors whenever
of degree n on C.
A
~ J0(C) ~
given by
and ~
~
~ (~)
J(C)
= JG(C)
• Jn(C) : ~
- ~
there for
are the same point in J (C) - that n is the divisor of a meromorphic function on C. Frequently
classes of holomorphic
and ~
line bundles
{U i} of C, associate
line bundle of degree n given by the cocycle d i, dj are meromorphie
functions
at least -D, there is an isomorphism ~ H0(~).
~
functions
H0(D)
the
If
on C with divisors
= H0(~)
Alternatively,
to
• HI(c, 0 n ) , where
on Ui, Uj with divisor D there.
is the sheaf of germs of meromorphic
to
equivalence
on C of degree n as follows:
any divisor D of degree n and open covering
(s i) • H0(D)
space of linear
of the same degree n, we will
Jn(C) will be identified with the space of (holomorphic)
~
•~.
by sending
holomorphic
sections
of D will often be considered h o l o m o r p h i c
functions
cover
to
U
C
by lifting
(s i)
H0(D)
on the universal for
H0(U, i -
some t r i v i a l i z a t i o n
(o i) ¢ H0(U,~*(D))
of the induced bundle
~*(D)
on U. Pn c- Jn(C )
If degree
n > 0
n times,
I- i
speciality
i(D)
= g - n;
inversion
-~
D of degree g-2
and,
D =
Jg = J0~
divisor
in J0"
The map ~g fails to be
g with
i(D)
dimensional
> 0;
g -2
D • Pg-i
d i v c e ( ~ -x-A)
for any positive
while
i ( ~ ) > 0.
classical
and
@ ( ~ -x-A)
and divisor g with
for those divisors theorem,
this is
K C -D', where D'
K C • J2g_2(C)
is
on C [32, p. 73]. Pg-i c
the
The follow-
Jg_l(C)
theta-function
is multD_A(@)
is a
(0) c J0(C)
~ 0
divisor ~ of degree
on C if ~
in Lewittes
cone
= i(D); in particular, g on C with
is any divisor of degree
For a proof of this important
presentation
from
and that the degree of the tangent
of Pg-i at a point = ~
of degree
of Jg described by
of the divisor of the classical A e Jg_l(C)
~
by the Riemann-Roch
subvariety
which means,
e • J0(C)
I- i
t h e o r e m of Riemann says that
by a fixed class
i ( ~ ) : 0,
for which the index of
that for any point
class of the divisor of any differential
translate
i
it can be shown [22] that n (Xl,...,x n) • C (n) is equal to g -i( ~ xi). The i
g, there is a positive
ing fundamental
n ~ x. • P i i n
---~
n ~x i in general,
is any positive divisor of degree
with
of
product of C with itself
t h e o r e m says that ~g is onto Jg(C)
the i s o m o r p h i s m of degree
(Xl,...,x n) • C (n)
at those points
the rank of ~n at
the
divisors
the mapping
onto Pn is
e = ~
of positive
on C, and C (n) is the symmetric
On:
Jacobi
is the set of all classes
theorem,
[21], or Mayer's
g
see either the
thesis
[22].
T h e o r e m i.i (Riemann).
There is a d i v i s o r class
A e Jg_l(C)
with
(9)
2A = K C ~ J2g_2(C)
such that for any i) with
If
a c C
O(e) ~ 0,
i(~ ) : 0
If
@(e) : 0,
is p o s i t i v e of degree g
=
-a -A
in J0(C).
then for some p o s i t i v e divisor { of degree e = ~ -A
i(~)
degree j d-l.
0(~
- ~
In case
in all other cases,
Generally,
in J0(C).
-e)
i({)
~ 0 = I
9(x-a-e)
z 0
for all p o s i t i v e divisors ~ and
i(~+a) : 0,
0(0)
,~
of
divc@(x-a-e) : a+~
;
on C.
A is not the class of a positive
(ii) would imply that
= 0,
divisor on C; other-
w h i c h is possible only for
curves of special moduli since the variety contain,
g - i,
is the m u l t i p l i c i t y of (@) at e and is the smallest inte-
ger d such that
wise
divcg(X-a-e)
e = ~
(ii) Here
then
e ( {g:
and
(i0)
ii)
and
9(0)
= 0
in ~ g
does not
say, period m a t r i c e s of curves of genus g w h i c h are nearly
products of g e l l i p t i c curves
(see Cor.
3.2).
However,
for any
a e C,
A is the class of the divisor of a h a l f - o r d e r d i f f e r e n t i a l with only a simple pole at a, called the S z e g o - k e r n e l of C with e h a r a c t e r i s t i e (Cot.
2.12).
A l t h o u g h A depends on the choice of canonical h o m o l o g y
basis d e f i n i n g O, if the h o m o l o g y is changed to
= b
with
"(a \c
[17, p.
[~]
b ~ E Sp(2g,~), d ] 84] says that
the t r a n s f o r m a t i o n
B
law of theta functions
where
M = (cT +2~id),
K is some constant
independent d
"Diag" denoting As a result
of
-c)(g)
and
T ~ ~g
½Diag (edt)£
of the
to
even) h a l f - p e r i o d
z E {g +
column vector
A is t r a n s f o r m e d
(necessarily
½
the
} = 2~i(aT + 2 ~ i b ) ( c T + 2~id) -I,
2~iz : zM ~ {g,
A ~ J in
diagonal
g-i
J0(C)
(C)
and
~2g,
{ )
entries
where
to \abt
A- A
= {g/(2~il,T)
.
is the
given by
[c d t Diag
Classically,
c :
u/r with
A was constructed
U : {z ~ ¢ [ Izl < l}
first kind with compact can be lifted to paths fixed base point and
fundamental
and r a domain,
gAi 6
j = 1,2 ..... g, the sum
E i __-/
i~j
F. gAi
Then for any (z 0 )
J
(13)
]
vj )
k b - k a = (g-l)
ruchsian group of the
then the cycles A i on C
la v
a E C
z0 E U
some
and
is independent
of the
z0 define
+ 2~± i:l : IA i~j
Since
and
7a
of Riemann constants
2~i 2
C > i
x
| (vi(x)I
z 0 so that we can unambiguously
k a = (k I ..... kg) £ {g
if genus
from z 0 to gAi(Z 0) in U with
g
basepoint
as follows:
~ ~g
the vector
with basepoint
(vi(x)
a 6 C
by
la: vj )
]
for any two points
a,b E C,
the
b divisor class quence
k a + (g-l)a
is independent
of
a E C;
it is a conse-
of the classical theory that:
(14)
More precisely,
A = (g-l)a + k a e Jg_l(C).
the relation in ~g, giving
(i0) under the projection
{g ÷ J0(C), Theorem
is a special case of the following
(i0) for differentials
Cauchy's
residue
P0 ~ C
to the cycles
Proposition for some point
of the second kind,
theorem on the simply connected
by fixing a basepoint homologous
1.2.
and dissecting
AI,BI,...,Ag,Bg
Let
a c C.
e ~ Cg
fA ~
=
~ +
k
gp
surface
p e C
"
~ Res ( j=l x:bj
C O obtained
C along loops in ~l(C,P0)
and suppose
Then for any
proved by applying
[14, p. 151].
the second kind on C with at most poles at
2 - ~ k~_lek
version of Riemann's
divc(9(x-a-e) : ~
and any differential
~ C w of
bl,...,b N e C~
fx .d in(9( fxv -
e))
PO (15)
+ 2-~
k=l
L
2
g
+
+
k
f
E [(mk-l) ~ - (nk-l) k=l Bk
~Ak
LkJ~
in
where mk = ~
d in@(
v -e)
E
k
(16)
1 nk = ~ and where
[ Tkk + L 2
all integrals
vk - e k +
j = l,...,g,
or the points
when ~ is the normalized and when
v - e)
are taken within C dissected
which we assume does not contain ~ particular,
d in (9(
mk,n k 6 Z
ej: fg~aVj
Vk
holomorphic
property
along ~l(C,P0),
differential
In
vj,
are defined by (16),
a j) +
(14) and the relation
From the fundamental
Z ,
a,p,bl,... ,b N.
3J2
+
= (mk-l)Tkj
k~j which gives
(
Bk
(i0) in J0(C). (9) of A, we have
- 2~i(nj-l)
i0
Corollary
1.3.
@(x-a+e)@(x-b-e) third
kind
on C w i t h
latter
a,b
vanishes
of a d i f f e r e n t i a l This
If
e ¢ {g
2g zeroes
at a and b if
second
differential
and
at the
poles
of the
e C
will
kind
have
with
~ 0,
of a d i f f e r e n t i a l a ~ b,
with
@(e)
and
a double
g double
zeroes
of the
at the
pole
then
2g zeroes
at a if
a = b.
if e is a h a l f -
period.
Corollary either
1.4.
vanishes
and at the g
zeroes
~ . ~--~.~O(f)vi(x), i:l 1
point
and
i(A+a+f)
Proof.
Let
@(x-a-f)@(x-b+f)
= b+~
(ii),
f = ~-A
points
a, b since
then
with
x and
symmetry
zero
taking
a Taylor
conclude V
that
a e C If we
@non-sing
the
= {%-A
the
To
diagonal
and
latter
f E (@)
x = b
Hf(x)
:
situation
holds,
is a n o n - s i n g u l a r
and
~ and
assume
that
= a+{
and
of d e g r e e
{ are
for
find
x = a
and w i t h near
dimensional
I { positive
in the
on
C x C
no o t h e r
~i
so
differa = b;
variables with
a
zeroes.
point
is h o l o m o rgp h i c
of the
to
is,
the
From
= 2A = K C
specialize
bundle
up to a c o n s t a n t , g-i
~+~
some h o l o m o r p h i e
Hf(x),
section
g-l.
independent
Furthermore,
~ ' (if ) v i ( a )
open
e (@)
The
divisors
of a line
of this
is,
@(x-a-f)@(x-b+f)
x = a
@ ( x -H af -( fx )) @H (f x( -a a) + f )
a,
section
Hf(x)
at
divc@(x-a-f)
= i.
of a, b.
g ~I
H f ~i
define
if
so that
= i(~)
expansion
so that
only
~,~ p o s i t i v e
in x and
along
£ C.
= a + b + divcHf(x)
a, a h o l o m o r p h i c
double
a,b
then
differential
be n o n - s i n g u l a r
and -f = ~-A i(%)
= 0
= 0.
on C; t h e n
Hf i n d e p e n d e n t
by
if and
E (@)
divc@(X-a-f)@(x-b+f) ential
of
= i(A+b-f)
~ 0
@(f)
of the h o l o m o r p h i c
(ii),
f
with
on C or v a n i s h e s
independent
Theorem
divc@(X-b+f)
f ~ ~g
identically
2g-2
by R i e m a n n ' s
If
By
x = a, we
and n o n - z e r o
~@(f)vi(x).
subvariety
of d e g r e e
g-i
with
i(6)
= i},
ii
then the map ~(f)v i
0non_sing ÷ ~(H0(~C] )) -~ ~g_l({)
i
sending
f ÷ Hf =
and is ramified at the points
of degree g-lJ
{~- A C (@)
I ~ is positive
in ~ the same};
of degree
g-i
with at least two points
the image of this set in ~(H0(~C))
consists
of differ-
entials Hf with at least one double zero, which is the dual curve to the canonical Andreotti
imbedding of a n o n - h y p e r e l l i p t i c
and Mayer
of (0), is
g-4
[5] have shown that
for curves
Pi-A
¢ (0)
dim @sing'
the singular
with the
g-3
(p. 13
Pi ~ C
below);
distinct
and
i(
pi ) = 2
@sing
and
i
220 =i Szi~zj(f)z'z'z] = 0
surface of rank _< 4 containing
(See Cor.
is the equation of a
the canonical
curve in ~g_l(~).
2.18.)
The group of 4 g half-periods of 4 g h o l o m o r p h i e a l l y
inequivalent
in J0(C)
acts effectively
line bundles
We will denote by L 0 the bundle
corresponding
visor class
A ¢ Jg_l(C)
be that bundle
L] 1
on p. 7 and let L
is the h a l f - p e r i o d
for half-periods Corollary
~ ~ J0(C).
on the set
L c Jg_l(C)
L Q L = K C.
®
is
i
dense in
L
set
the set of all points
I
quadric
[4, p. 820].
of genus _> 4 except in the hyperelliptic
case, where it has dimension f =
C in ~g_l({)
Then Riemann's
with
to the difor which
Theorem I.i
implies
I. S .~
dim H0(L
) : dim HILL' ) : mult @
is even for
even and odd for e odd. For generic Riemann even e and
i
section of L
surfaces,
dim H0(L
for the ½(4 g - 2 g) odd ~.
By Cor.
for ~ odd and n o n - s i n g u l a r
For a discussion see [6].
of this corollary
)
is
0
for the ½(4g+ 2 g)
1.4, the holomorphic
is, up to a constant,
given
in terms of spin-structures,
12
by the h a l f - o r d e r d i f f e r e n t i a l g E I
~O[a](0)vi ( x ) ~ z . l
.
h a e H0(La)
A meromorphic
singular, with a simple pole at O[B](x-a) ha(x). O[a](x-a)
=
section of L 6 for B even and non-
x = a • C, will then be given by
("root")
functions of degree n is, by the Riemann-
Roch formula,
<
I if n=l, a I odd; or n=2,
n
La.) 1
For example, when
= (n-1)(g-l)
+
quartic equation
g = 3,
n : 2
and
a I ~ a 2 ~ any linear r e l a t i o n
(sections of
for n o n - h y p e r e l l i p t i c
of q u a d r a t i c p o l y n o m i a l s
g-i
K C (9 (a I + a 2)) gives the
C as a sum of three square roots
in the h o m o g e n e o u s
80 and [7, p. 387].
and degree n, the
al:~ 2
0 otherwise.
among three P r y m d i f f e r e n t i a l s
see p.
h~(x)
In general, the number of linearly independent
h o l o m o r p h i c Abelian
d i m H 0 (-[-[ 1
satisfying
coordinates of ~2(~)
In terms of a plane model of C of genus g
points
an adjoint curve of degree
divch a
are the points of tangency of
n-3, while the g points
are the points of tangency of an adjoint of degree
divcO[6](x-a) n-2
through the
points of i n t e r s e c t i o n of C with a line tangent to C at a.
An algo-
r i t h m for c o m p u t i n g these adjoints from the coefficients of the equation for C is given in [30]. 2g+2 ]~ (z-z(Qi)) i given as a t w o - s h e e t e d covering of the Riemann sphere by the function Example.
Let C be the h y p e r e l l i p t i c
z with r a m i f i c a t i o n at the W e i e r s t r a s s
curve
points
s2 =
QI"'"Q2g+2
is the involution on C i n t e r c h a n g i n g the two sheets, D : x + %(x) ~ J2(C)
is i n d e p e n d e n t of x E C
(g+l)D =
Qi
and
e C.
the divisor class
and satisfies
(g-l)D = K C = 2A.
i For any
Xl'" " " 'Xg+l
C,
0(
I
x. - D m
A)
If
vanishes exactly w h e n
13
x i = ¢(xj)
for some
Weierstrass
gap theorem
which
~ vanishes
of points plicity
i ~ j,
particular, spondence of
x i = ¢(xj),
m on the subvariety with
A
for integers
i ~ j.
Thus
x i ~ ¢(xj)
[~],
with the 4 g partitions
i(
@ has multi-
of J0 given by all points
xi ~ C
the 4g half-periods
{1,...,2g+2}
i.i and the
x. - D) to i is the total number of disjoint pairs
at such a point
g+£~m~ o x. + ( m - l ) D i l
Theorem
[21, p. 60], the order
(xi,x j) for which
exactly
and by Riemann's
2B,2~
e ~g,
{il,...,ig+l_2 m}
m > 0
as follows:
for
i ~ j;
are in
i-i
while
for
odd half-periods g+l-2m for
m even
explicitly: i
and for
singular (odd).
corresponding
m : i,
for
m = O,
In the
~ = ~I i
9z i (0)vi(x)
ease
are the
of of
multiplicity
the
non-singular
differentials - A
mentioned
dz(x) ~
On the other hand,
if
m which
on
non-singular are the are
even
half-periods,
(odd) the
on p. 12 can be given
is an odd half-period
then
h2(x)
=
Qik
is, up to a constant,
g-i TT (z(x)- z(Qik)) i
g-l/
g+l-2m k=l~ Qik + (m-l)D - A
m > i,
half-periods
half-order if
k:l Qi k - A
corre-
U {jl,...,Jg+l+2m }
e
periods;
in
with
g-I
given by the differential
double
zeroes at
Qil,...,Qi
. g-i
= ~
Qik
-
D
A -
then
(17)
L
is an even half-period
and
14
is a b i l i n e a r
half-order
and y and w i t h
differential
a simple
pole
only
on
along
C x C, a n t i s y m m e t r i c y = x;
in x
specializing
x = Qi i
say,
divc
vanishes period
mB(Qi~Y)
at the 6.
=
i
g zeroes
If we set
- 2Qil=
of m B ( x , y ) , the
E(x,y)
: @[~](
is i n d e p e n d e n t
of the
non-singular
phic
a zero
of first
with
in turn
only
construct
on C -
kind
see
the
k a for
point
QI and
ical
homology
as in the and
Vl,...,Vg
The
some
~ C:
This
along
y = x;
differentials
(
for the
v) half-
then
~ and
is h o l o m o r -
f r o m E(x,y)
of the
E(x,y)
second
and
we
can
third
explicit
value take
of the
Riemann
con-
a to be the W e i e r s t r a s s
canon-
AI,BI,...,Ag,Bg
with
% ( A i) = -A i
in H I ( C , ~ ) .
normalized
The
differentials
%*v i : -vi,
constants
i = l,...,g
for the b a s e p o i n t
so that
QI are
therefore
~k
A - (g-l)Ql
= k
Q1
=
{½ ½ ½ ½ ½
which
kernel
half-period
for s i m p l i c i t y ,
( 18 )
and
even
the
a symmetric
basis
satisfy
Riemann
to h a v e
a
choose
= -Bj
corresponding
Szego
~[~]
§2.
picture
%(Bj)
so that
v)/~[~](0)m~(x,y),
order
normalized
It is c o n v e n i e n t stants
~ + A,
is odd fact,
for
g = I or
together
with
0
½
2 rood 4
the
"}
0
and
relations
¢ J0(C),
a half-period
T
even
for
g --- 0 or
3 rood 4.
15
{oo... QI- Q2 :
½½
{o...o_,o... (k)
'
½
Q 2 k + l - Q2 =
½
oo
o
(k)
and
Q2k+2
in J0(C),
determine
described
above.
Further
survey article
special
_-
Q2k+l
{Oo...o...o1
divisors
0 T
contains
theory. in J0(C),
found in the recent notes
,
i -< k - <
corresponding
of hyperelliptic
[20], a general
An excellent including on Jacobi
account
as
can be found in
of §2; and in the reference
for the
of the varieties
the hyperelliptic varieties
g
to any partition
0-functions
some addition-theorems
by Krazer-Wirtinger
classical
½
the half-period
discussion
[7, Ch. XI], which
entire
_
case,
by R. Gunning
of
can be [37].
II.
The P r i m e - F o r m
In order to give the relations Abelian
differentials
Definition
fying
h2(x)
g ~ i=l
=
(19)
Let ~ be a n o n - s i n g u l a r L
~[z?](0)vi(x). l
= h0[~](y-x) (x)h (y)
The existence
definition
is provided
corre-
section h
Then the p r i m e - f o r m
satis-
is given by
~ x,y e C.
of a non-singular
from the fact that the h 2 actually alternate
odd half-period
as on p. ii with a holomorphic
E(x,y)
Remark.
and
on C, we need
2.1.
sponding to a bundle
between theta-functions
odd half-period
(Cor. 4.21).
span H 0 ( C , ~ )
of the prime-form,
follows
independent
An
of O-functions,
by (v) below.
From the properties
of Riemann's
theta-function
given in §i, it
is easily seen that: i)
E(x,y)
holomorphie ~ L o I (9 ~ L 0 1
is independent
C xC
of ~ and is a
Z
section of the bundle (9 6*(0)
on
C x C, where
and ~2 are the projections
of
C × C
~C
~i
(x,y)
Thus,
is a multiplicative
x ~ C,
E(x,y)
ential in y with multipliers
~C
onto its first and second
tors, and ~ is the map sending for fixed
,,,~ ~(ct
¢ C × C
into
y-x
fac-
c J(C).
-½ order differ-
along the A i and Bj cycles
in y given by:
Y (20)
I
and
exp (-
3~ 2
vj)
respectively.
x ii)
E(x,y)
= -E(y,x)
order along the diagonal
V x,y e C y = x
in
and E(x,y) C × C
vanishes
to first
and is otherwise
non-zero.
17
n
iii)
If
~
=
n
~ ai 1
n d In ~ i
(21)
n E(x,bi) ~]~ E(x'ai)
so that
corresponding classical
and
~ =
E(x,b i ) E'x~ai~,z
=
~ bi 1
with
~-A
(x)
is a meromorphic
to the divisor
"interchange"
~ - ~
V
section under
a i,b i 6 C ,
x 6 C
of the line bundle
(5).
This gives
the
law: Y
(22)
I
for any divisors iv)
Though
A ,~
~
:
~Y-X
, X and Y of degree
E(x,y)
depends
E(x,y)
n on C.
on the choice =
in (12), transforms
I
(a t(At a
b
B
of homology
~
d
basis
defin-
C Sp(2g~Z)
as
into
u~.,} v)
If C is realized
meromorphic
function
as a covering
z: C ÷ PI(@)
sphere by a
9
I!
=
l J[ijd
mj = ~-~
within
C cut along its homology
proved
as follows:
by (22) and Abel's exp
arg
z(y)
where
fact that
of the Riemann
for
~......... z - z(x)
p,q
Theorem
and p a t h s basis.
near
integration
This
classical
are taken formula
is
x,y E C
(8); now let
Ip~W E(y,q)E(x,p) y-x : E(x,q)E(y,p)
of
q ÷ y
by (21).
and
p ÷ x,
using the
18
The holomorphic
Prym differentials
[14, p. 160] with
g-i
double
zeroes on C are given by: Proposition
g Hf(x)
:
2.2.
For any n o n - s i n g u l a r
30
¢
~ ~--~7.(f)vi(x) i=l i
Then for all
x,y
point
and
Qf(x)
:
f e (0),
let
32@
~'~ i,j:l
3z.~z.(f)vi(x)vj(x) l ]
e C,
9(y-x-f) E(x,y) ) 2
(23)
= Hf(x)Hf(y)
j ~Qf f .
exp
Y Both sides of this identity tions of
K C @ (2f) e J2g_2(C)
Proof. meromorphic setting (24)
For
and y; letting ~ .
equation that
[ C
with
fixed,
g-i
double
@(y-x-f) @(y-a-f)
sec-
zeroes.
E(a,y) E(x,y)
is, by (20), a 1.4; so
y : b e C,
@(y-x-f) E(x,y)
-Hf(x)
x,a
in the variable y, h o l o m o r p h i c
function of y with no zeroes or poles by Cor.
O(y-x-f) E(x,y)
Thus
are,
near
-d2- E ( x , y ) dy 2
: @(y-a-f) E(a,y)
- ¢(x)~(y) y + x
O(b-x-f) O(b-a-f)
where
we find
Now compute y = x,
E(a,b) E(x,b)
V
x,y,a,b
¢ and ~ are h o l o m o r p h i e ¢(x)~(x)
= -Hf(x)
the Taylor expansion
sections in x O(y-x-f) so that E(x,y) :
of both sides of this
taking y as local coordinate,
y=x = 0
e c.
from the definition
and use the fact
(19) to conclude
that
X
~(x)
= (Hf(x)) ½ exp (-½
Corollary
(25)
2.3.
For
I ~---) Qf x,y
O(y-x-f)@(y-x+f)
,
e C
which gives
and
(23).
f e (@)
= E(x,y)E(y,x)
non-singular,
= -E(x,y) 2.
Hf(x)Hf(y) When f is singular, tically by Riemann's
numerator Theorem.
and denominator
of (25) will vanish
iden-
19
Corollarx
Let % and ~ be positive
2.4.
f = {-A
such that
and
-f = ~-A
divisors
are non-singular
of degree
points
g-I
on (@).
Then In X
where
m~_~
X
is the normalized
poles of residue
+I,-i
at
X
differential
$,~
of the third kind with
and
is the lattice .....o
r
point
in {g given by
-2f + |
v.
J
If ~ and ~ are distinct ential on C; equivalently,
divisors, Qf(x)/Hf(x)
~_{
is a meromorphic
differ-
will be a holomorphic
differ-
ential on C if and only if f is an odd half-period. a singular a Taylor
point,
expansion
0(y-x-f) near
E 0
on
y = x
C × C
shows that
When
by Riemann's Qf(x)
~ 0
f 6 (e) Theorem,
is and
on C (Corollary
2.18). Corollary
2.5.
Suppose
x,y 6 C
have
local coordinates
x,y
in
a neighborhood of a point p ( C, and let Tf(p) be the cubic differg ~3@ ential ~ (f)vi(P)vj(P)Vk(p) for non-singular f E (@). i, =i ~zi~zj~z k Then near
y = x, E(x,y) d x / ~
(26)
(x-y) 2 =
S(p) + higher
I
y-x
order terms
12
where + 3 ( Qfl 2
Tf
(27)
is a holomorphic projective singular
point
operator
- see [13, p. 164].
connection
f E (@); here
QfCp) :
on C independent
{ , } is the Schwarzian At a zero of Hf(p),
and
of the nondifferential
it is seen that
rf(p) = -Hf(p) ± } Q}(p)
20
with the sign (~) chosen according as
@(x-p~f)
z 0
for all
x ~ C
by (25); here the derivatives
' and " have meaning in any local coordi-
nate because of the condition
on the point p.
For example,
the construction 2
hyperelliptic
curve
s
=
on p. 13 of the p r i m e - f o r m
2g+2 T ~ ( z - z(Qi)) i
gives
1 for a n y n o n - s i n g u l a r
even h a l f - p e r i o d
{Qi ,...,Q i } u {QJl ..... QJ } I g+l g+l Differentiating differential
ta)=l
6 corresponding
of
{1,2 .... ,2g+2}.
(23) or (24), we obtain the fundamental
2.6.
For
-
is a w e l l - d e f i n e d
with
d2 in E(x,y)dxdy
dxdy
-
bilinear meromorphic
of the n o n - s i n g u l a r
point
dxdy
in O(y-x-f)dxdy
differential
f 6 (0).
on
C xC,
Equivalently,
for
indee ~ {g
@(e) # 0,
g (29)
~(P'q)
for all
i)
2.12.) m(x,y)
y = x
neighborhood
In addition, is h o l o m o r p h i c
where, of
~(x,y)
a21n e
: - i,]~=l v~(p)vj(q)
satisfying
(p,q) E C x C
(See Cor.
along
normalized
x,y 6 C,
d2 m(x,y)
pendent
to a partition
of the second kind on C:
Corollary (28)
for the
either
~zi~z------~(e)
@(p-q-e) = 0
or
@(p-q+e) = 0.
w(x,y) has the properties: everywhere
except for a double pole
if x and y have local coordinates
x,q in a
p E C, =
I + ~ S(p) (x-y) 2
with S(p) the projective
connection
+ higher order terms
(27).
dxdy
21
ii)
For any fixed
x 6 C,
r
(30)
I . m(x,y) = 0 ]
and
I
fl
~ = ~b i
and
(31)
-A(x)
for generic
j = l,...,g.
n
= E ai
1
= vj(x)
]
n
If
m(x,y)
B.
JA
i~
:
non-singular
a r e two p o s i t i v e
~(x,y)
: d In
fi 6 (0);
divisors
n
O(x_bi_fi )
1
O(x-ai-f i )
in particular
d =
md-c
for all
a,b,c,d
iii)
~(x,y)
:
E(b,d) E(a,d)
: in
~b-a
E(a,c) E(b,c)
6 C.
The indefinite
Weierstrass Zp(X)
c
on C,
~-function:
is a meromorphic
integral if
Zp(X)
affine
of m(x,y)
is an analogue
d = ~-~inE(x,p)
connection
for
of the
x,p 6 C~
for the bundle
; 6 (0)
then @
1
on C, where cochain
~ (x) = x-p e J(C) for x E C; that is, Z (x) is a P P in the sheaf of germs of meromorphic differentials on C with
coboundary
the element
tive of the transition residue ~b_a(X)
=
~(x,y)
defined
in some neighborhood
at
and
U of p,
Z (x) has a pole of P Zb(X) - Za(X) =
For any function
f(P)
deriva-
= ~I
f on C
I~U f(x)Zp(X)
so
"Cauchy-kernel". basis
~
=
b
B
as on
becomes
~(x,y) = m ( x , y ) - ½
x,y £ C
x : p a,b ~ C.
Under a change of homology
p. 17 (iv), m(x,y)
by the logarithmic
for the bundle.
for distinct
that Z (x) is a local P
for all
functions
+i : deg [ 6~(@) @ L~I]_
holomorphic
iv)
of H I ( c , ~ )
E [vi(x)vj(y) + vi(Y)Vj(X)] i!j
so that by (12) again,
~--~-77..I n d e t ( c m + 2~id) i]
22
~(x,y) + 2 i vi(x)vj(y) ~ 22 in ~ @[B](z) z-O 4g+2 g i,j:l ~zi~zj B even is a symmetric differential on homology defining 0 and ~.
C ×C
independent of the choice of
Such differentials were considered by
Klein in his work on Abelian functions and invariant theory [18]. By Corollary 2.12 below, this differential is also given by 2 E @[B]2(y-x) 4g+2 g 8 even O[812(0)E2(x,y) The theta-funetions are expressed in terms of Abelian integrals by Lemma 2.7. on C with
Let
g = ~ a. i 1
~
i(/~) : O, and set
be a positive divisor of degree g
e = ~-a-A
=
v a
and k a given by (13).
Then
ka
6
cg
with
a ~ C
V x,p 6 C
(32)
d__dxin O(x-a-e) =
gp~(X'Y) - ~ i
(33)
0(x-a-e) _ exp O(e)
~ gP x-a
(34)
g @(x-a-e) _ ~i O(e)
A k Vk( q )
k:l
vk(q) ~
k:l
E(x'ai) { i ~a--~i) exp - 2-~T
~(x,y)
~x-a
k
IA Vk(q) in E(a,q--------~I E(x~q) k
and 22 g ~x----~In@(x-a-e) = i,j=iZ ~(x'ai)v(~)-lij vj(a)
(35)
where v ( ~ ) -I is the inverse matrix to the non-singular
g xg
matrix
(vi(aj)). Proof.
Take
~ = e(x,y)
in Prop. 1.2 (15) to get (32); (33)
comes from integrating (32) between x and a, and the third equation comes from (33) and (21).
Finally,
(35) is obtained either by differ-
entiating (32) or by solving the system of equations f = ak-a-e C (0):
(28) with
23
g : - i,3 ~:I
m(x,ak)
~21n 0, ~'~z-~X-a-e)vi(x)vj(ak)i j
k : l,...,g.
From (33), we get an extension of Prop. 2.2 for the divisor of zeroes of meromorphic double pole at
Prym differentials
with double zeroes and a
y = x:
Proposition
2.8.
For
e 6 ~g
with
O(e) ~ 0,
(y) be positive divisors of degree g such that -e : ~ (y)-y-A
in J0(C) for any
x,y £ C.
(36)
~(x)
and
e : ~(x)-x-A
and
Then
A(x)
82(y-x-e) 02(e)
let
g
y
]
= exp [ I my_x ~(y)
+
~__imj f vj J: x
the
i n (~g g i v e n by
y+ A(x) where
is
g
lattice
point
2e- I
Proof.
By (33) ,
O(y-x-e I) @(y-x+e 2) I ~ (x) = exp @(el) O(e2) (Y)my_x
(x) el : I
v
.
x+ ~ (y)
T
if
A (y)
v - kx
and
e2 : I
gx
v - ky
in { g.
On the other hand,
gy
O
where
- exp
my-x
(y)
+
~ ~i 1
x
vi
is the lattice point in ~g given by
2e I +
v
T A(Y) since, for fixed y, both sides have the same zeroes and poles and are sections of the bundle -2e I in x. the lattice point
Multiplying these two equations, in (36) is given by
T
24
Let v ( ~ ) be the non-singular g x g matrix g : ~ a i with a i ~ C and i( ~ ) = 0. Then (vi(aj)) for a divisor i p,x,y ~ C Proposition
2.9.
&%
I i
j f
is a w e l l - d e f i n e d
differential
divcx C Ap(x,y)
in x and meromorphio
> C x {p} + ~
xC - C x ~
function of y with
- {p} x C - Diagonal
and - Ay(x,p) for all
x,y,p,q
(37)'
6 C.
If
Ap(x,y)
For any smooth curve ~(X)Ap(x,y) visor > p - ~ region f(Y)
=
= Ap(x,y) e 6 {g
with
y C C- p
e = ~ -p-A
and continuous
is a meromorphic
"Cauchy-kernel"
in J0(C), then
V
function
y 6 U.
x,y 6 c.
¢(x) on y,
C-y
with di-
on the closure of some
the divisor
for C associated
V
function of y on
; and if f(x) is h o l o m o r p h i c
I~uf(X)Ap(x,y )
+ Aq(x,y)
O(y-x-e)@(x-p-e)E(p,y) 8(y-p-e)@(e)E(x,y)E(x,p)
U C C, not containing
= ~i
= Ap(x,q)
p +~
,
then
We can thus call Ap (x,y) a
to the divisor
p + ~
- see [16,
p. 651]. Proof. properties x £ C~
Corollary of Ap(x,y)
2.6 and Cauchy's directly,
A (x,y)@(y-p-e)E(y,x) E(y,p)@(y-x-e)
most g zeroes at
~
integral
formula give all
except for (37)'. is a meromorphic
= divcO(y-p-e) ;
since
However,
for fixed
function of y with at
i ( ~ ) = 0,
this function
25
must be a constant
c(x,p)
c(x,p) where,
= lim O(y-p-e) y÷x O(y-x-e)E(y,p)
The identity expression
by (26) and (31), O(x-p-e) • lim E(y,X)Ap(X,y)
.i.
O(e)E(x,p)
(37)' can also be proved
for the Cauchy kernel,
volved in Cors.
=
y÷x
from (35) and the following
which gives the differentials
in-
1.3-1.4.
Proposition
2.10.
For
a,b,x 6 C
and
e ~ {g:
(38) Equivalently:
(38)'
-~ m, O&-..-q E~.,,~/ '~ O~,.-b-4 mc,,q Proof.
ea,b(X)
=
For fixed a and b, there
depending
is a m u l t i p l i c a t i v e
on a and b such that
V
differential
z ~ ~g~
L=I
since both sides of this equation are holomorphic same bundle on J(C), containing c a b(X)
and the left-hand
{x-b+f} u {a-x+f},
giving (38), take
%,b
(x)
V
f 6 (@)
z = f 6 (@)
by (31).
=
then by (24)
=
E(",~/Et'S,,J
From (38)' we see that for generic
of ~i(~)
To evaluate
non-singular;
(38)' is derived by substituting
O(2x-a-b-e)
in z of the
side has a divisor of zeroes
O(~t~-x) e(+-~.,~) Finally
sections
on C are the ramification
defined by the meromorphic
x-b-e
for e in (38).
e E ~g, the 4g zeroes of
points
function
of the
(g+l)-sheeted
cover
@(x-a-e)E(x,b)/@(x-b-e)E(x,a).
26
Corollary !.11.
For generic
a,b E C,
the 4 g differentials with
2g double zeroes and simple poles of residue -i,+i at a,b are given by
I2x
O[e]2(½
v) +b E(a,b) 0[e]2(½ b v) E(x,a)E(x,b) "a
g ~ In e[e] b : ~b_a(X) + 2 E (½1 v)vi(x) i:l ~z.i a
for any half-period e (see §5, pp. 100-103). such that
e[e](½
v ) : 0, then O[e](½ a entiating the above identity at b yields
(i__~l ~O[e]" /2x v)vi(b)) 2 ~--'~--i(~]a+b
f
When a and b ~ a are v) - 0
on C and differ-
+b
30[e](½ b
E(a,b) : ~
g i
~20[e] (½ =i ~zi~zj -a
)vi(b)vj (b) E(x,a)E(b,x)
This is a holomorphic differential
double zeroes which are common zeroes of
O[e](
s; e[e](
and by
v)
for any fixed
v + ½
Cor.
v )E(x,a)E(y,b)
P E C;
@[e](
thus V
-a
v- ½
v)
and
x,y E c
= -OEe](
v + ½
v )E(y,a)E(x,b)
= ~a_b(X)
g ~O[e] ~ --(½ i=l ~zi
2.4:
g
~28[e]( Ib z i ~ z j ½ v )vi(x)vj(x) i,j=l a
Corollary 2.12. (39)
~z i
on C vanishing at a,b and with
g-2
v +½
v )vi(x).
i:±:
For all
O(y-x-e)@(y-x+e) O2(e)E(x,y) 2
For e an even half-period,
x,y e C
g = m(x,y) + i~
"a
v)vi(x).
e E ~g,
~2 In@(e)vi(x)vj(y). Zzi~zJ
the left hand side is a differential with
2g double zeroes and a double pole at half-order differential
and
[b
y = x;
it is the square of the
Q[e](y-x) , a meromorphic section of L e @[e](0)E(x,y)
called the Szego kernel of C with characteristics
[e].
When C is the
27
double
of a finite bordered
basic relation
connecting
for R - see Prop. Corollary
is independent
Riemann
R, (39) then becomes
surface
the Szego and Bergman
reproducing
the
kernels
6.14 and pp. 125-6.
2.13.
The holomorphie
of the point
e ~ ~g.
quartie
Here
differential
S is the projective
connec-
tion
(27) and v: v'.' are derivatives in some local coordinate; the 3' 3 connection S transforms in such a way [13, p. 170] that the third term
above is actually Proof.
a quartic
By computing
opment of (39) near given by
6a23(x) +
coefficient pressed case,
in terms
on C.
the second order terms
y = x
the quartie
6 ~a4(x)
of E(x,y)
differential
at
- 20aS(x)
y = x.
differential where
an(X)
The differential
of any fixed non-singular
in question
is
is the n th Taylor can thus be ex-
see p. 35.
the even
O-functions
Proposition
2.14.
For
g
a,b,c,d
£ C
~4 In 0
~ ~j
= V(a,b,e,d)
=
for
[ @ 2 ( d _ c ) @ 2 ( b _ a ) _ 02(0)O(d_c+b_a)O(d_c_b+a) 1
= - ½04(0)
V(a,b,c,d)
"addition-theorem"
on C:
i E2(a,b)E2(c,d)
where
devel-
f 6 (0); for the elliptic
From (38) and (39) we can give a special
(40)
in the Laurent
(0)vi(a)vj(b)Vk(e)v~(d) :! ~zi~zj ~Zk~Z~
+ V(a,c,b,d)
+ V(a,d,b,c)
O(a-c)O(a-d)@(b-c)O(b-d) E(a,c)E(a,d)E(b,c)E(b,d)
In these
identities,
28
0 may be r e p l a o e d by O[e] for any even h a l f - p e r i o d Proof.
By Prop.
2.10 and the fact that
e E J0"
~d-c + ~c-b : ~d-b'
the
third t e r m in (40) is:
h~= I
which gives the first term of (40) by setting in (39).
e = d-c
and
x = b
On the other hand,
so using Cor.
2.12 again, we get the second t e r m of (40).
The s y m m e t r i c h o l o m o r p h i o d i f f e r e n t i a l in (40) figures into the Schiffer variation of the Szego kernel - see [15]. and
d
=
b,
Letting
c = a
we get
C o r o l l a r y 2.15.
For any even half period
e ~ J0(C)
and
x,y ~ C,
C o m p a r i n g the s e c o n d - o r d e r terms in the Laurent d e v e l o p m e n t of this identity near nection of Cor.
y = x
gives an e x p r e s s i o n for the p r o j e c t i v e con-
2.5 in terms of any even theta function - for the
elliptic case, see p. 36.
Though there is no analogue of this corol-
lary for odd half-periods,
Cor.
@(2y-2x-f) E2(x,y)
_ 02(y-x-f) Hf(p)
2.12 does imply g
i,j~:l
23 in 0 ~zi~zj~zk ( y - x - f ) v i ( x ) v j ( y ) v k ( p )
29
for all n o n - s i n g u l a r
f E (0)
"duplication
for generic
(41)
formula"
(42)-(43)
C with
are all special
2.16.*
dim H0(D)
for the h o l o m o r p h i c on p. 5.
e e {g: ~ x,y e C.
cases of a general addition theorem
0
Let D be a divisor of degree
= N ~ n, sections
Then for any
)
i(D+
(39) gives a
which we now take up.
Proposition on
Likewise
p,x,y 6 C.
~2 In 0(y-x-e) = -E2(x,y)@(y-x-e) 2 - E(x,y) ~x~y
@(2y-2x-e)@(e)
These identities
and
and for
and suppose
g+n-i ~ g-i
~I,...,gN
form a basis
of the line bundle
(generic)
corresponding to D as N-n divisor ~ = ~ bi with i
positive
(Xl,...,x N) ( C x C ×... × C = C N,
N det(~i(x'))3 N N-n diVcN@( l ~ X i + A - D - ~ ) = diVcN ]~ ~ E(xi,b4).j i<j E(x i ,xj ) i=l j=l
(42)
Proof.
The divisors
variety of codimension for any basis
~
with
i(D + ~
) > 0
are a sub-
one in C N-n given by the zeroes of
VI,...,VN_ n
D; so the divisors
N-n ~ b. 1 i
=
~
of the Abelian
with
i(D + ~ ) = 0
differentials
det
(Vi(bj))
with divisor
are generic in C N-n and
N
for such ~
,
e( ~ x . + A - D - ~ i l
) ~ 0
on C N since otherwise
the con-
N
dition
dim H0(D + ~ - ~ x i) > 0 I
dim H0(D + ~
) > N.
Now since
tive divisor
~ of degree
such that
+ Z : D
~
for all
dim H0(D)
N-I,
in Jg+n-l"
Xl,...,x N 6 C : N,
there is, for any posi-
a positive divisor Consequently,
would imply
A
of degree
g+n-N
for generic
N
x2,...,x N E C, where
A
divc@( ~ x i + A - D - ~ ) = ~ + A
is the unique positive divisor
of degree
as a function g+n-N
of Xl,
for which
N
+ ~ x. = D. 2 i *
So the section in x I given by
This t h e o r e m has appeared
[18],
[20]
or
[30].
in various
forms classically
- see [7],
30
0( ~ x. + A - D - ~ I l is a holomorphic
) 7]- E(xi,x j) i<j
7[ E(xi,b j) i=l j :i
section of a line bundle holomorphioally
equivalent
N
to
D 6 Jg+n-i
with zeroes at
~
+
~ xi, 2
the same divisor of zeroes as det(gi(xj)),
and must therefore have a holomorphic
section in
N
x I of
D 6 Jg+n-i
determinant
with zeroes at
are symmetric in
For example
(Riemann),
for the holomorphic
@(
Since both the section and
Xl,...,XN, if
~ £ J0
sections of
diVc2g_2
~ x i. 2
this argument proves and
~i,...,~2g_2
K C ~ L 0 ~ ~ £ J3g_3(C),
(42).
are a basis then
2g-2 det(~.(x.)) ~ x i - KC ~) = . ...... l 3 i dlVc2g-2 7"]" E(xi,x j)
i<j
Likewise holomorphic
if
0 ~ ~ £ J0
sections of
and
%l,...,%g_l
are a basis for the
KC ~ [ & J2g-2'
g-i det(%i(xj)) diVcg_l @( ~ x . - A - ~) = diVcg_l i l ~ E(xi,xj) i<j while if
%l,...,#g
K C 8 ~ 6 J2g-2
div
When
are a basis for the meromorphic
sections of
with at most a simple pole at some fixed point g @ ( ~ xi-p Cg 1
6 = 0,
det(~i(xj)) - A - ~) = div C g -[[ E(xi,x j) i<j
these last equations
Corollary 2.17.
For all
change to:
P,Xl,...,Xg 6 C,
g
0(~xi-P-A) i g
]~ Z(xi'p)
~(p)
= c det(vi(x$))" g
i<j77mx i,~j) 7I°(Xi)l
p ~ C,
g ~E(xi,P) i
31
where
c is a constant
O(p)
independent
exp
=
f
of
P,Xl,...,Xg
5i
- ~
j=l
and
•
]
is a h o l o m o r p h i o section of a trivial bundle on C. In particular, g-i f : ~ a i - A E (e) with A given by (13)-(14), then i
if
/~'~'lv'L~"'' ~{~"~/,
Proof.
Prop.
2.16 implies
that
~
P,Xl,...,Xg
E C,
@
g I
~<j
I
g where =
c ( ~ ,~ ) depends only on chosen fixed divisors g ~ b i. On the other hand, from (34) of Lemma 2.7, i g g 0( ~ x i - p -A) = S(Xl,...,Xg)~(p) IT E(xi, p) i i=l
where S(Xl,...,Xg) on
C × ... x C
identities
is a symmetric h o l o m o r p h i c
of degree
g-i
:
Z ai 1
and
~ p 6 C
section of a line bundle
in each variable.
Combining these two
gives the corollary with the universal
constant
c = e ( ~ ,~ ) s ( a l , . . . , a g ) [ S ( b l , . . . , b g ] - g x I = ... = Xg = x ~ C,
By specializing that
divcO(gx-p-A)
choice of homology of Weierstrass
= gp + W defining
points
where W, independent O, is of degree
on C [13, p. 123]
in J0'
Weierstrass Wronskian
d i v c @ ( ( g - l ) x - A - ~)
points
for
KC @ ~
g3
of g
P E C
for
and the
and is the divisor
counted with their
given by the order of the zero of O(gx-p-~) ~ 0
this corollary tells us
x £ W.
"weights",
Likewise
if
is of degree g(g-l) 2 and gives the
consisting
for any basis of H0(Kc ® ~).
of the zeroes of the
32
Corollary 2.18.
Let
be a singular point of order m.
f E (0)
m
Then for any two positive divisors
m
= Z ai 1
and
~
: ~ bi 1
of
degree m on C, m
m
m
°(~ixi- A
+ f)@( ~-x ii
E(x i
~ -f)
= (_i) ½ m ( m + l )
i,3:i
)E(xi,b j) ,aj
m
E(ai'bj)~jE2(xi'xj
0 ( ~ - ~ - f)Hf(x I .... ,xm)
where, for any
i,]=l
Xl,...,x m ( C, g
Hf(x l,...,x m) = il,.., im=l
Thus for all
~zi I
~m 0 " ~Zim(f)vil(Xl) " " "Vim (xm) "
xi,Y j E C,
m
m
m
0( E (xi-Y i) + f ) O ( ~ (xi-Y i) - f) 1 1
=
(-i) m
Hf(x l,...,xm)Hf(yl,..- ,ym )
E2(xi,Yj ) i,j=l ]~ E2(xi,xj)E2(yi,Y j) i<j
and the differential Hf(Xl,...,x m) vanishes identically to the second order when
x.i = x.]
Proof.
Setting
for some D = A+ f,
i ~ j. N = m
there is a constant c depending on ~ all
and and
n = 0 Xl,...,x m
in Prop. 2.16 such that for
yl,...,y m 6 C, m m m ~ E(xi,Y j ) 0( ~ Yi - ~i xi + f) : c@( ~. Yi - ~ + f) i I i,j:l E(ai,Y j)
Differentiating with respect to g i ~Zll
sm 0 ~z.
yl,...,y m
and setting
Yi : xi'
m m ~E(xi,x j ) (f)vil(Xl)'''vi (xm) = cO( ~ x i- ~ + f) ~ iCJm im m i j=l ~ E(ai,xj ) i=l
* This is implicitly used in classical proofs of Riemann's Theorem see [19, p. 434].
33
m
which gives the corollary by evaluating c using the divisor From Prop. 2.16 we can prove an "addition-theorem"
~ Yi : ~ i
for Abelian
functions: Corollary 2.19.
If
e E {g
with
@(e) ~ 0,
n ~ E(x i )E(yj,Y i) @3) @( ~ X i - ~ Yi - e)@(e)n-i i<j ,xj = det 1 1 ]~E(xi,Yj)
i,j for all
xi,...,Xn,Yl,...,y n ( C.
If
f E {g
is a non-singular point
of (@), then n 8( ~ x i-
n ~i y i - f)Hf(p) n-I
i
=
~ @(xi_Yi_f)][E(xi,Yj) i:l i~j (-i) n-I ~
In particular,
(44)
]~E(xi,xj)E(yj,Y i) i<j
det
specializing
n 0( ~. (xi-a)-f) i
where, for
j = l,...,n,
If
Yl = "'" = Yn = a 6 C,
Hf(p)n-l]~E(xii<j ,xj ) (_l)½(n_l)(n_2) n = n-i det(~i(xj)) ]~ O(xi-a-f)E(xi,a)n-i ]~ k~ i:l k=l
~l(Xj) = i
~i-i ~i(xj) : ~
Proof.
Vpco
t + ~ Din @(xi_Yj_f)vk(P ) k=l ~z k t:0
O(e) ~ G,
( k=l
take
and
~ in @(xj_y_f)vk(P) ) ~z k y=a n D = A + e + Z Y~, I
N: n
and sections
n ~k(X) take
@(x-Yk-e) E(x,Y k)
e = f+p-q
] ' - [ E ( x , y i ) in (42).
i:l
in (43) for
p,q E C
the factorization of Prop. 2.2.
For a n o n - s i n g u l a r and let
q + p,
f 6 (0),
making use of
34
Many relations specializations (45)
discussed
of the case
in this chapter can be considered n = 2
of (43):
@(x-a-e)@(y-b-e)E(x,b)E(a,y)
+ @(x-b-e)@(y-a-e)E(x,a)E(y,b)
= @(x+y-a-b-e)@(e)E(x,y)E(a,b)
V x,y,a,b ~ C
and this in turn can be used to obtain the addition-theorem arbitrary n.
In the elliptic
case,
(44) is the classical
form of the addition theorem for Weierstrass' Example.
If C has genus
be taken to be the z-plane
(2~i,T),
Re T < 0.
acteristics:
i, the universal
v(x) = dz(x),
determinant
~ -function: cover of
~, and the normalized
ential will then be given by
(43) for
C -- J(C)
holomorphic
There are four @-functions
is the only point on (@).
(z), ~(z) and o(z) constructed
differ-
x E C, with period matrix with half-integer
½ and [~] even and [½1 ½ odd; the class 100], [0]
2~i 2 + T E J0(C)
can
The Weierstrass
from the lattice
A
charI ½~
functions
in { generated by
2~i and • satisfy the relations:
-~(z)
(46)
~(z)
;
= -
(z+2~i) o(z)
= -e
- ~dz
in
~(z)
= ~
o(z)
-
in
-2~i~ (z+~i)
dz 2 in
o(z)
and
= ~
@L½j(z)
in
o(z+T) ~
+ rl
@ ½
= -e
-(Tn+l) (z+½T)
where 0'"
(0)
i
i O'
The connection
(0)
~(z)dz
S(p) of Corollary
-6~(dz) 2, and for
=
~(z)- ~(z+2~i)
A
2.5 is the quadratic
x,y E C, the prime-form
is given by
differential
35
E(x,y)#dz(x)dz(y)
=
~J
0r½1'(0) The differential
of the second kind on
~o(x,y) : ~
in (9
I
= ~( Y d z ) exp ~- ( Ix~ d z )2. ~X
C ×C
(y-x)dxdy
is
= [ ~(
v)-
n]v(x)v(y)
,
X
so that the "invariant"
differential
=
of Cor.
~",~)"-
2.6 (iv) is given by
7 ~'~-0
Ill
II;i.
I IIZ
by virtue of Jacobi's
identity
[19, p. 334] and the "heat equation"
452) below:
For any even h a l f - p e r i o d
e, Corollary
(~(z)
is a differential
and
-q+O"[e](0)
I dz
O[e](O)
double
zero
at
z = e + A ;
e +
formula:
- ~ ( e +A)
:
i
3
2.12 implies that
½
with a double pole at
; this
0,.p]
implies
the
½ (0)
e " [ e ] (0)
o, [½](o)
OEel(O)
Z
:
0
well-known
L½J where
e + A
is a non-zero h a l f - p e r i o d
ential in Corollary
for even e.
The quartie differ-
2.13 is (dz) 4 times the constant ^M P a ,
L '/~J (o) for all
z E ~ •
The special addition t h e o r e m
(40) is, for 6 an even
36
half-period
and
z,w 6 {,
(40)'
while
Cor.
period c4 ~ 0
6 as
2.15
gives
S(p)
the
connection
S(p)
= -(4c 2 + 2~4)dz(p)e2
for any T by
in
where
(40)'
The d u p l i c a t i o n
= a4(z)
d3 -in a(z) dz 3
terms
of
any e v e n h a l f -
c n = -d n l ndz n916] (0) formula
at the b o t t o m
p. 28 b e c o m e s : a(2z)
and the a d d i t i o n
theorem
(44)
= -o4(z) ~ ' ( z )
is:
k=j
.... ~
for all
z I, .... z n 6 5-
c.~,,,/
i
and of
III.
The versal tions
variational
constants between
Desenerate
method
surface.
In this
for the
two b a s i c
types
either
a zero
surfaces
structed
over
as follows:
g l ' g2 e a c h w i t h z2:
U2 $ D
For
let
Then
S be the
~
= WI u
(xl,t)
unit
take
a point
two
disc
¢
D = {t •
Riemann
P l ' P2 r e m o v e d
and
in n e i g h b o r h o o d s
obtained
by
{
out
"pinching"
surface.
be a f a m i l y
I Itl
of
< i}
con-
C I and
C 2 of
genus
Zl:
UI ~ D
and
let UI,
on a
are w o r k e d
+ D
surfaces
rela-
formulas
on a R i e m a n n
Let
uni-
U 2 of t h e s e
points.
set
w k : {(xk,t) and
the
to Zero.
generating
differentials
moduli
cycle
for e v a l u a t i n g
as f o r
variational
homology
be c o o r d i n a t e s
k = 1,2
as w e l l
of d e g e n e r a t i n g
Homolo$ous
tool
and A b e i i a n
chapter,
or n o n - z e r o
Pinchin [ a Cycle Riemann
space
theta-functions
Riemann
Surfaces
is an i m p o r t a n t
on the m o d u ! i
the
Riemann
~ WI n
I t~n,
x k~ Ck-U k
non-singular S u W2
or
surface
where,
x k~ U k
with
{XY= t
Ez~(xk)r
I (X,Y,t)
e
t ,
> Itl}
D × D × D}.
in the o v e r l a p s ,
U I xD
is i d e n t i f i e d
with
(Zl(X I) , ~
U 2 ×D
is i d e n t i f i e d
with
(
t) e
S
t) E
S.
and (x2,t) ~
W2 a
Choose
eoordinates
x = ½(X+Y)
fibers
C t of
Riemann
the
"pinched
~
regions"
of a neighborhood t = 0, the
are
fiber
x = y = t = 0,
of
Ct n x = 0
C O crosses and
the
two
and
y = ½(X-Y)
surfaces S
of genus
are r a m i f i e d with
itself
~
branch
, z 2 (x 2)
on S so that g = gl+g2
double
points
at t h e p o i n t
components
t
C I and
at
for w h i c h
coverings x = ±~.
p corresponding
C 2 of the
the
Y = x2~-t For to
normalization
38
of C O have,
on
C 0 ~ S, the equations
ly; the corresponding x : ½z 2
local uniformizing
will be called the p i n c h i n g
p : pl,P2
respectively.
homology bases C2-U 2
y = -x
variables
coordinates
of Hl(Ct,~)
(CI-U I) × D
and
respective-
x : ½z I
and
for C I and C 2 at
~
can be chosen
(C2-U2) x D
for C I and C 2 lying in
CI-U I
C,
+TJ. I
the
and
respectively. Proposition
phic 2-forms
on
a sufficiently ,2,...,g I
3.1. ~
are a normalized
i :
and
For each t, a canonical basis
Al(t),Bl(t),...,Ag(t),Bg(t) by extending across
y : x
There are
whose residues
gl+g2
linearly
Ul(X,t),...,ugl+g2(X,t)
basis of the h o l o m o r p h i e
small disc D and
of radius
differentials
s about
j : gl+l,...,gl+g2
t = 0.
Ii
=
(47) u.(x,t) ]
basis
for
+ O(t 2)
x E CI-U I
~tVj(P2)~l(X,Pl)
+ O(t 2)
x E CI-U I
:
i ~ gl
(resp.
are the normalized
CI × CI
and
C 2 ×C2,
on W I or W2, for which
+ 9 t v j ( P 2 ) ~ 2 ( x , p 2) + O(t 2)
vj(x)
for
differentials differentials
j > gl )
on C I (resp.
C2) , ~l(x,y)
and each term O(t 2) is a holomorphic lim ~i O(t 2) t÷0
and ~k(X,Pk)
is a m e r o m o r p h i c
are all evaluated
x £ C2-U 2
are a normalized
of the second kind
C I or C 2 with at most a pole of order 4 at p.
coordinates.
For
x E C2-U 2
~2(x,y)
vj(P2)
on C t for t in
+ O(t 2)
for the holomorphic
vi(Pl),
along C t
tvi(Pl)~2(x,P2)
j(x)
where vi(x)
holomor-
,
i x) + 9tvi(Pl)Wl(X,Pl)
u.(x,t) i
independent
and
(28) on differential
differential
on
The differentials in terms of the pinching
39
Proof. *
If ~
2
is the sheaf of holomorphic
2-forms on ~
and ~C a
is the sheaf of holomorphic
differentials
on C a for
a 6 D
then on
there is the exact sequence of sheaves ma 0
>
~
~
>
where m a is multiplication
r a
f~
>
> 0
~C a
by (t-a) and r a is the residue mapping.
This in turn gives rise to the exact sequence on D: -
is locally free,
Since ~ k _> 0,
H'
~k
the direct image sheaves
are all coherent on D by a theorem of Grauert
implies that in any sufficiently space H 0 ( ~ k l u ) that ~ k
small neighborhood
.)
= TL~Hk(~2~ ) ,
[12, p. 59]; this U ¢_ D,
is a finitely generated H 0 ( O I U ) - m o d u l e
the vector
[14, p. 27] so
will be a locally free sheaf on U if and only if
ma: H 0 ( ~ k I u )
+ H0(~klu)
exact sequence, jective
-
V
~,H2(~
a; thus
is injective
V a g U.
Now in the above
) is the zero sheaf since m a cannot be sur-
~I/ma~l
-- ~,HI(~c
),
a skyscraper sheaf on D
a with stalk 0 at is arbitrary, D and
ra: ~ * ~
t ~ a E D
[i
and
{ : HI(Ca,~ C ) a
at
is a locally free sheaf of rank
-+ ~*~Ca
free sheaf of rank
is surjective
g : dim H0(Ca,~C
).
~ a £ D,
t : a.
Since a
1 = dim HI(~ C ) on a with ~,~
a locally
There exist, then, holomor-
a phie two forms along
t = 0
Ul(X,t),...,Ugl+g2(X,t)
for t near 0 whose residues
are a given normalized basis
°f the differentials
°n C0"
Since
Vl,...,vgl,vgl+l,...,vgl+g 2
( -I-~ R e s 2 ~]A. l
3 *
Suggested by D. Mumford.
ctUi(x,t) ) iSi,j_
is a
40
holomorphio vertible {Un,
matrix which is the identity matrix at
in a neighborhood
n = l,...,g}
D e of
entials on C t. neighborhood coordinate
x = 0
x by
basis
Ul(X,t),...,ug(x,t)
in
Ct n
S
ui(x,t)
along C t
for the differ-
has an expansion
in a
given in terms of the pinching
:
ui(x,t)
where a
and taking the residues
Now each differential
of
it is in-
t = 0; changing the basis
by this matrix,
we then get the normalized
t = 0,
and b
=
Z ~0
a (t)×~d× + ~ ~ ~0
are h o l o m o r p h i c
functions
b (t)
x
near
dx
t : 0; for
i ~ gl'
: vi(x),
x £Cln
S
x ~ C2~
S
ui(x,o) : boCO) -Id + Z (%(0)+ ~ZO and ui(x,O)
= - b o (O)x-ldx +
so that
b0(0)
x G CI ~
S ,
= 0
and
E (a U~O bl(0)
(0)- b +l(O)]x~dx
= a0(0)
= ½vi(Pl).
{
is a differential at
x = Pl
normalized nates,
differentials
and which,
see that this differential
since ui(x,t)
V'~
: 9vi(Pl)
of m(x,y)
proved,
given in Cor.
and the assertion
the terms 0(t 2) comes from the second-order at
t : 0.
are
in its Laurent
must actually be ~vi(Pl)~l(X,Pl).
are similarly
of ui(x,t)
and ui(x,0)
in terms of the pinching coordi-
½bl(0)
at Pl; from the properties
developments
for
/
which has zero A-periods
other expansions
Hence,
of the second kind on C I with at most a double pole
has leading coefficient
velopment
: O,
de-
2.6, we The
concerning
terms of the Taylor-
41
Corollary
3.2.
The Riemann
T(t)
(48)
=
-
matrix
-
T(t)
for C t has an expansion
+ kt R-R + O(t 2)
{ where
T I and ~2 are the Riemann
matrices
for C I and C2,
R = (vl(Pl) ..... Vgl(Pl),Vgl+l(P2)
and 0(t 2) is a matrix The canonical
of constants
differential
mt(x,y)
{
~l(x,y)
(49)
mt(x,y)
=
ml(x,y)
and the o(t) i lim ~ o(t) t+0
are bilinear
Proof. Now if
~
÷ D
be the quotient with
bilinear
~t
~
÷ D
-
of
for all
for which
~g x D
= J(Ct)
Likewise,
for instance,
,
for
periods
3.1 and (30). then
t ~ 0 as
x,y ~ CI-U I.
and
t ÷ 0, so The remaining
under the identification for any
family of g-dimensional = ~-l(t)
J(C o) = J(C I) × J(C2). of
differential
(z +
is an analytic
which the fibers
p(t)
on C I and C 2 ,
from (47) of Prop.
and, by (30) and (47), has bounded
(z,t) E ~g x D
y & C2-U 2 x,y E C2-U 2
differentials
= ~ t ( x , y ) - Wl(x,y) -%t~l(x,P)~l(y,p)
lim ~ ( x , y , t ) ~ 0 t+0 are similarly proved.
~
matrix.
x,y 6 CI-U I
+ o(t)
differentials
(48) comes
that actually
Let
+ o(t) x E CI-UI,
+ %t~2(x,P)~2(y,p)
holomorphic
is a holomorphic
expansions
a finite
for C t has an expansion
are the bilinear
The expansion
x,y E CI-U I
2)
in x and y.
~(x,y,t)
(x,y,t)
lim ~i0 ( t t÷0
%t~l(x,P)~2(y, p) + o(t)
and m2(x,y)
E 0
with
+ ¼t~l(x,P)~l(y,p)
~2(x,y)
where
, .... vgl+g2(P2)]~ E cg
for
choosing
K,~ E ~g;
varieties
t ~ 0
and
then
over D for ~ 0 = 7-1(0) =
any fixed holomorphic
a fixed point
in
CI-U I
of
or
secticn C2-U 2
-
42
the spaces Jn(Ct) isomorphic to ~
for any
n E Z
under the mapping
Since @T(z) is a holomorphic subvariety ~ ÷ D for
C
~
t ~ 0
and
~
~t
~t
= ~-l(t)
{g × ~ g ,
the theta divisor
g2-1
~ = ~ + ~
with support in
the portion of (0 t) near Proposition
3.3.
Then
~ = 9~ + ~
of degree CI-UI,
(@i) ×J(C2).
~t
Let A(t),
g-i
and ~
and set
a positive divisor of degree
positive of
leads to:
and
%(t) = ~ - A ( t ) g-i
e = ~
-p-A
I E
J(C I)
and
f =
~
on Ct, as above.
lim %(t) = (e,f) E J(C 0) -A 2 C (e2).
implies two identities
face C of genus g : for any basepoint
The
valid on any Riemann sur-
b E C
and
A = (g-l)b + k b
-
e = A -p-
~zi
~-A
e) =
A E {g,
d inO(p-b-f) dp
f =
condition
given by (13):
- i=l~vi(P)
for
• (@t)
t÷0
@t (¢(t)) I :0 t=0
for
on C t with
A I and A 2 be the Riemann divisor
lim A(t) = A l + A 2 +p £ Jg_l(C0)
E Jg_l(C)
in J(Co).
This observation
t÷0 where
J(C t)
C2-U2; a similar statement holds for
classes for Ct, C i and C 2 respectively, with
(0 t) c
J(CI) x (02) , say, is given by
positive of degree gl with support in degree
we can form the
0 = (00) = J(CI) × (02) U (~i) × J(C 2)
for positive divisors
g J0(Ct).
i, which is an analytic family
That part of the divisor of et near -A(t)
~ Jn(Ct ) ÷ A t - n p ( t )
function on
of codimension
with fibers
n ÷ D
form an analytic family
& (@)
See Lemma 2.7 (32).
~
gbm(P'Y)
positive
=
~(p,y)
~
i=l
of degree g on C; and
- ~ (g-l)b
non-singular,
i"
j=l
~
I
v.(x) A. ] ]
~o(p,y)
positive of degree
g-i
on C.
43
Proof. constants
*
Let
bl
6 ~
gl
and
k
b2
e ~
for C I and C 2 with basepoints
respectively. has,
k
by Cot.
3.2, an expansion
be the vector of Riemann
b I E C I- U I
bl ~g kt &
Then the vector
g2
and
of Riemann
constants
kt
=
k b l +g2
u(x,t)
,k
)
+ (g2-1)
I
where
d
=
(dl,...,dg)
~>~
by
~(t) =
6 {g
has components
Theorem
applied
dj for
to
J
>
gl
~(SP)
given by
b
3.2 again,
-
k
i s2 -
u(x,t) ,
v
bI
+ ~t
IIs v(p)
3, a similar
- k
b2)
(g2-1)b2
~l(p,y)
glbl
In genus in [24].
td +
~bl
Av
(13) and Stokes'
By (50) and Cor.
for C t
of the form
b2 (50)
b 2 ~ C 2- U 2
analysis
+
~2(p,y) (g2-1)b2
has been carried
I 1 - 4d
+ o(t)
out by Poincar~
44
and therefore
where (J(CI)
t÷O
t÷O bl
glbl
From the expression
" (p)vj2(P) J>gl v31
for d. and the fact that 3 ~2e2 v[(p) ~z. ~z. (f) - Z 3 31 32 J>gl
~
(f) 3
-
E(p,b 2 ) in
@2(P-b2-f)
3
2.2, we conclude
Z vi(P) ijgl
= -
~In@ I ~(e) ~
i~
~l(p,y) + ~ i
-
vi(x ) i
i-gl
glbl
in 82(P-b2-f) +
<~ ~ ]
Since t
e E (@i)
b2
whenever
p E ~ ,
I v~x~
t÷0 Jbl
;bl
~or ~ < g ~
(consider the case
e = ~-p-
by (14) and (50).
w~
'
j
A I ~ J(C I)
r~P
~m i u~x~
lira A(t) = A I + A 2 + p t÷0
~l(p,y ) i
2
1
and both sides of this equation vanish identically gl = 0).
~x
~2 (p'y) - 2--~ .
(g2-1)b 2
o~
d -dp
v.(p) TFT(f)
J>gl ] by Prop.
.
(g2-1)b2
=
2
x (@2)
~m~e~
45
Let us consider some further applications of the variational formulas (47)-(49). Example 3.4.
A generalization of Poincar$'s asymptotic period
relation in genus 4 [24, p. 291] is found by repeated use of the expansions of Cor. 3.2: Ctl,...,tg
ql,...,qg
3
Let
be a Riemann surface of genus g ob-
tained by attaching g torii at g points
/~C~/~_.,xt
pl,...,pg
CI,...,Cg
punctured
to ~i(~) punctured at
with pinching parameters
Then the Riemann matrix
~-~C,
tl,-..,tg E D
T(tl,...,tg)
for
k-~C~
respectively.
Ctl ' ...,tg
has the off-
diagonal expansion for
i ~ j:
where
is the differential of the second kind on
w(x,y) -
dxdy 2 (x'y)
PI({) with a double pole at
x = y.
order or higher, the expressions g ~ 4
Thus, neglecting terms of third1 4(qi-qj) , ....... 13 (titjvi(Pi)vj(Pj))½
for
satisfy the asymptotic relations i
for all distlnct Example 3.5.
+
i
i
+
-
0
i,j,k,~ E 1,2,...,g. When
studied [25] , the tangent
g2 = 0,
a case which has been extensively
( vl(P),Vl(P)v2(P),...,Vg( 2 2 p)) e {½g(g+l)
to a curve of period matrices i n ~
g
obtained by attaching a sphere
at a point p in a non-hyperelliptic Riemann surface C will always lie in a 3g-3 dimensional subspaee for any
p E C, since VlVl,VlV2,...,VgVg
span the space of quadratic differentials of the
g(g+l)2 x (3g-3)
H0(~Cg2) ~ ~3g-3
The rank
matrix ~llVivj(Pk)] is 3g-3 for generic points
46
pl,...,P3g_3
E C,
and we can obtain all period matrices of Riemann
surfaces near T by attaching spheres to C at points with pinching parameters locus
~-g C ~ g
hyperelliptic
tl,...,t3g_3.
pl,...,P3g_3
The cotangent space to the
of period matrices of Riemann surfaces at a non-
Riemann surface C is thus isomorphic to H 0 ( ~ 2) via the
identification
(51)
~_~
a..dT..
l
l]
<
>
E
a..v.v.
so that if f is any function O n ~ g df = Tg
E H0(~C 2)
l
vanishing on ~ g ,
'
aij
E {
the differential
Z ~f dTij on ~ g vanishes identically when restricted to iSj 13 - that is to say, the quadratic differential E ~--f v.v. i~j ~Tij 1 3
vanishes identically on C. It should be mentioned that the same variational
formulas given
in this chapter also arise from a family of curves obtained by varying branch points in a realization of the Riemann surface as a branched covering of the sphere; this approach has been worked out in [25]. The following result [31] illustrates the power of this method in the hyperelliptic
case.
Proposition
3.6 (Thomae).
Let C be the hyperelliptic
Riemann
2
2g+2 surface of genus g defined by s : 7~ (z-Pk)' Pk : Z(Qk) 6 ~, and i suppose the normalized differentials Vl,...,Vg on C are given by g v = ~ 6w6 where wB - zB-idZs for B = l,...,g, and (a ~) is a non-singular matrix depending on C. peri°d
e E J0(C)'
let
{Qil
For any non-singular even half-
Qi { g~ +u
Q31 .... ,Qj t h~ e gbe+
partition of the Weierstrass points such that on p. 13.
Then
e : g~± Qik - D -A, as i
47
@[e]8(O) = (det o) -4
Proof. let
For any Weierstrass point
q ~ C
with
z(q) = p E {,
be the family of hyperelliptic Riemann surfaces C t over 2g+2 a t-disc defined by the equations s2(t) = (z-p-}) -IF (z-pk) for i Pk~P each
$
g+l 2 ]~ ( P i k - P i ) (Pjk-Pj~) k,~=l k<~
÷ D
t E D.
series at
By expanding the differentials
t = 0
as in the proof of Prop.
z$-Idz/s(t)
in a Taylor
3.1, it can be seen that
the normalized differentials and period matrix on the curves Ct satisfy exactly the same variational formulas as those given in (47)-(49). [r
Now for any fixed
e E {g
with
O(e) ~ 0, g
let
~
= ~(0)
= ~ ai = I
divcO(x-q-e)
and, in general, ~ (t) = ~ a i ( t ) = divctOt(x-q(t)-e) i where q(t), for t E D, is the Weierstrass point in C t with z(q(t)) = P + ~t ~ ~. (52)
Then since 0 satisfies the heat equation:
$2@[e](z) - Z@[e](z), ~zi~zj ~ij
i # j
and
V
z ~ ~g
~2~[e--------~](z)= 2 ~@[e](z), ~zi~zi ~Tii
(35) implies that for a n y family of Riemann surfaces satisfying the variational formulas
(47)-(49):
(°)
L-_o
L:j-i
If we specialize e to a non-singular even half-period and let q be, say, the point Qig+l in the partition corresponding to e, then g ~i ak =
g l~Qik,
v!(a i)]
= 0
~ i,j
and the equation reduces to
48
d_dt in @t[e](0)Vdet
det o(t)
det l_<j ,k_
vt(~(t))
t:0 : 0;
since
det v t ( ~ ( t ) )
:
( )
, where a(t) is the matrix ~ for Ct, this
st (Qik)
condition becomes
7F (N~=1 A similar equation holds for P3g+l. and ~+]
PJI'''''PJg
e = ~
Pi
QJk- D -A
also;
interchanging
i
(resp. pj g+l
pjl,...,pjg)
since ) with any g+l
Pil'''''Pig
(resp.
we conclude that the function
08[e](0)(det
°)41~k<~g+l~ (Pik-Pi~)-2(pjk-pj£)-2
U =
has zero derivatives
with respect to all branch points and hence is a universal constant on the moduii space of hyperellipti'c Riemann surfaces. To evaluate U , g+l let C s be the hyperelliptic curve s 2 = ]~ (z-P2k_l)(z-P2k_l-g) and k=l g+l C O the rational curve s = ] ~ (z-P2k_l): then as ~ ÷ 0, the holo1 morphic differentials Vk(S) , k : l,...,g, normalized on C with respect to the homology basis on p. 14, become differentials third kind (
dz z-P2k+l
dz I on CO; and z-Pl
lim @ s+O
period matrix T(S) for C s has an expansion identity Also, for
g xg
matrix and
~ = l,...,g
(0) = i
of the
since the
T(S) (in s)l + 0(e)
with I the
lim 0(e) a finite matrix - see p. 53. c÷0 and 6 sufficiently small~
so that if ~(e) is the a matrix for CE,
49
~ ~7~
6-->o
~0
/~
~-'
TI- IT (G,,- f~,-,l
and thus
L
,~-t~-~ = a~ ( ~ , .
-IT (~,~,-r~.)
=
71 (p~,-r~,~,/
,,~,~.o, ~, However, e =0
from the computation
corresponds
to the partition
so we finally can conclude
u = ~
(o/
~
As a corollary, corresponding
(18) of the class
A, the h a l f - p e r i o d
{QI,Q3,...,Q2g+I } u {Q2,Q4,...,Q2g+2 };
that
TT
(.p:~_,-F~,-,F(a~_,~-f~,.,-q -~ - - I .
we see that for any two half-periods
to partitions
'' . . . . ~g+l "' }u {j~,.. {~l'
., Jg+l ' }
{ii,...,ig+l} U
{jl,...,Jg+l }
e and e' and
respectively
O[q"(o)
0 5s']" (q which gives,
in the elliptic
r',q"(o')
These classical of rational
can be proved directly
on h y p e r e l l i p t i c
0 lb,~- ~-a) o 12,~- ctb~-,~) t
(p. 34),
-~
formulas
functions
case
I
--
Riemann
from the factorization surfaces:
e%b1l- ,(,~a- ,t,q
50
where
% is the
Xl,...,Xg
involution
E C;
see,
for i n s t a n c e ,
Pinc h i n [ a N o n - Z e r o of R i e m a n n
surfaces
ing m a n n e r :
on C and Ca, b is a c o n s t a n t
coordinates
Za:
and
U a and U b of two p o i n t s
w : {(p,t)
Here
o v e r the unit t - d i s c
Ua ~ D
Riemann
with
surface
÷ D
w i l l be a f a m i l y in the
of genus
in d i s j o i n t
follow-
g and c h o o s e
neighborhoods
Set
p 6 C-Ua-U b
IZa(p) I > Itl
and let S be the s u r f a c e
~
D constructed
Zb: U b ~ D
a,b 6 C.
I t E D,
of
[31].
Homolo~z Cycle.
let C be a c o m p a c t
independent
{XY = t
or
p ~ Ua
(resp.
1 (X,Y,t)
(resp.
U b)
Izb(p) I > Itl)}
E D × D xD}.
Then
define
= W u S
where,
in the o v e r l a p ,
(Pa,t) &
W n Ua x D
is i d e n t i f i e d
with
(Za(Pa) , ~
(Pb,t) ~ W ~ U b × D
is i d e n t i f i e d
with
t (Zb--~-~b), zb(Pb) , t) E S.
t ,
t) e S
and
Again each
x = ½(X÷Y) fiber
for which
C t of
~
y = ½(X-Y)
for
the p i n c h e d
y = ~ The
and
t ~ 0
region
f i b e r C O is a curve
responding branches y = x,
to the p o i n t s
of
CO n
y = -x
S
is a R i e m a n n Ct n
of a n e i g h b o r h o o d
of
of genus a,b
S
surface
on S so that
of genus
g+l
is a r a m i f i e d
double
covering
x = 0
with branch
points
at
g with
an o r d i n a r y
double
point
in the n o r m a l i z a t i o n
corresponding
with
w i l l be c o o r d i n a t e s
local pinching
cor-
C of CO; the
to n e i g h b o r h o o d s coordinates
x = ±/~.
of
a,b E C
are
x = ½z a
and
x = ½Zb,
respectively. To c h o o s e
some c a n o n i c a l
...,Ag(t),Bg(t)
homology
s i m p l y be a c a n o n i c a l
l y i n g in
C-Ua-U b
DU b × {t}
and,
for
extended Itl < ~,
across
basis
for Ct,
base
AI,BI,...,Ag,Bg
( C - U a - U b) × D.
Bg+l(t)
= y × {t} u
let
Set Yat ~
Al(t),Bl(t), for C
Ag+l(t) Ybt c W
= where
51
y is any fixed path from z-l(½)a to zbl(½) homology za-l(v~) and
basis,
and Yat and Ybt are continuously
to z-l(½) a
I~l
lying within
< IZbl
and from z{l(½) < i
to z b l ( ~ )
respectively,
so that Bg+l(0)
As t goes once around the origin,
termination
of Bg+l(t)
thus a well-defined
choice
varying
lying in
to b in C.
increases
C cut along its from
Iv~l < IZal
< I
is a path from a
any fixed continuous
by a cycle homologous
of Bg+l(t)
paths
de-
to ±Ag+l(t),and
can be given only in the t-disc
out along some path from the origin. Proposition phic 2-forms
3.7.
on ¢
while,
basis for
normalized
disc D
basis
ui(x,t)
at
= vi(x)
e about
differentials
t = 0
on C and Ug+l(X,0)
For
x E C-Ua-U b
and
+ 9t(vi(a) - v i ( b ) ) ( ~ ( x , a ) -
C t for
are a nort ~ 0;
i = l,...,g
of the third kind on C with
a,b.
holomor-
along
on C t for
u.(x,0), I
for the differentials
-i,+I
independent
Ul(X,t),...,Ug+l(x,t)
the differentials
differential
linearly
of radius
for the holomorphie
the normalized
(53)
small
t = 0,
of residue
g+l
whose residues
t in a sufficiently malized
There are
are a
is ~b_a(X),
simple poles
i = 1,2,...,g,
~(x,b))
+ 0(t2),
and Ug+l(x,t)
where
Vl(X),...,Vg(X)
is the differential holomorphic differential
= ~b_a(X)
+ tUg+l(X)
are the n o r ~ l i z e d
+ O(t 2)
differentials
of the second kind on C, the expressions
differentials there,
on
C-Ua-U b
and ~g+l(X)
with
t÷01im~ 0 ( t 2)
is a normalized
second kind on C with only poles
of order
Laurent
of the pinching
±
expansions _ i 2x 3
Proof.
+
on C, m(x,y)
are, + ...
in terms dx
with
46
O(t 2) are
a finite
differential
of the
3 at a and b, where
~in
coordinates
E(b,a)
the :
+ ~inE(a,b).
x
If ~
under the residue
is the sheaf of holomorphic map,~
2-forms
IC t is the sheaf of holomorphic
on ~
, then
differentials
52
on C with simple poles of opposite
residue
dim H0(Ct , ~
ICt ) = g+l ~ t
as in Prop.
again implies
that ~
and,
is a locally
on
~
whose residues
ly, a n o r m a l i z e d basis
Vl(X),...,Vg(X)
~b_a(X),
differential
a,b.
the normalized
The holomorphie
tity matrix at basis
{Ui,
matrix
t = 0
~i
neighborhood
along
on C t.
so
Theorem g+l.
Ul(X,t),..., t : 0
give, respective-
of the differentials
on C and
of the third kind with poles at
IA.(t) ] ResctUi(x,t) ) near
is the iden-
t = 0; by changing
the
by this matrix and taking the residues
along C t we then get a normalized the differentials
3.1, Grauert's
forms
and is invertible
i = l,...,g+l}
t = 0;
free sheaf on D of rank
Thus for t near 0, there are h o l o m o r p h i e Ug(x,t),Ug+l(x,t)
at a and b for
Now for
of the double point,
basis
Ul(X,t),...,Ug+l(x,t)
i < g+l
and
x E Ct ~
S
for in a
let M
ui(x,t)
in the pinching near
t = 0.
=
~a 0
coordinate
(t)xPdx + P
0
x
dx
~x
2_t
x, with a p and b
holomorphic
functions
Then
ui(x,o) : La
d
± Lb
0
so that
b (t)
b0(0)
= 0,
0
vi(a)
: v.(x) ~
1
= a0(0) +bl(0) ,
vi(b)
= a0(0) - bl(0)
and
ui(x,t) - vi(x) lim t÷0
t
is a normalized
p>O
differential
v>O
of the second kind on C with only double
poles of zero residue
at a and b where the Laurent expansions
leading coefficients
±½bl(0)
pinching
coordinates.
Ug+l(x,t)
=
= ± % ( v i ( a ) - vi(b))
On the other hand, ~ (t)xPdx +
0
6 (t) 0
have
in terms of the
if ~
~x~-t
dx
x E Ct n S
53
for holomorphic
e , 6v, then
60(0)
= -i
since
Ug+l(X,0)
= ~b_a(X).
Thus lim
t÷O
U~+l(x't)-~b-a(X)
f a]~(O)x]~dz-+ @(½6v(O)xV-3 + t@'(O)xV-1)dz 0 0
=
t
is a normalized
differential
of the second kind with only triple
at a and b where the Laurent with,
from
developments
of the pinching
Corollary
3.8.
for some constants ai" =
±(-
i
+ B +holom.)dx
2x3 7
(21):
4B : 261(0) = lim{-w.6+0 ~ o-a(b+8) +~b_a(a+6)
in terms
begin
poles
b a vi'
°ij
9(v~(b)-v[(a))~
=
in E(b,a) + ~ I n E ( a , b )
coordinates.
The Riemann
ci,c2,
where
matrix
(Tij)
for C t has
and
an expansion
is the Riemann
= ~(vi(a)-vi(b ))(vj(a) -vj( b)),
+ 6(vi(a)-vi(b))
The differential
+ 2
matrix
for C,
Oig = ag I• =
t÷01imt~O(t2)
is a finite matrix.
of the second kind on C t has an expansion
for all
x,y ~ C-Ua-Ub: ~t(x,y)
with ~(x,y) morphic
entry
Prop.
differential
- in t
for C and
+ O(t 2)
lim ~-~'00(t2) t+0 t ~
a mero-
on C. 3.7 and the general
of the second kind
Tg+l,g+i(t).
~g+l,g+l
+ ~(~(x,a)-~(x,b))(~(y,a)-~(y,b))
the bilinear
differential
Proof. entials
= ~(x,y)
[14, p. 176]
bilinear
give everything
But from the statement is a well-defined
relation
analytic
preceding function
for differexcept
Prop.
the
3.7,
of t in the
54
punctured
disc
D e - {0}, which must actually be analytic
disc D e since otherwise
Re T(t)
would not be negative
in the entire definite
as
t ÷ 0. As an example, which
let
Ctl,...,tg
is being pinched along
so that
C0,0,...,0
Ctl,...,tg
AI,...,Ag
is of genus
to g pairs of points
be a Riemann
surface
with parameters
0 with g double points
al,bl,...,ag,bg 6 ~i(~).
of genus g tl,...,tg 6 D
corresponding
The Riemann matrix for
has an expansion
• ii(tl,...,tg) = in t i + c o n s t a n t I
+ higher order terms in
t I ,... ,tg
Tij(t I ..... tg) : (ai,bi;a j,bj)
where
( ; ) is the cross ratio of four points From Cor.
3.8 we see that two points
point in the lattice the matrix lattice
~I
Ft of rank
(2~il,T(t))
F 0 of rank
i
(g+l)-dimensional
must differ,
2g+l
I :
manifolds
variety
compact Abelian
group
~ + D p. 30].
by the c o l u m s
~
= ~ n i a i + n(a+b)
of the matrix
be the family of
for
and with 7-1(0)
~0'
: ~n.a. %~p
i
given
the non-
and that the projection in [35,
observe
that if p is the double
+np
on C O can be lifted to a
i
on the Riemann
of the same degree
is the divisor of a meromorphic
t ~ 0
; then it can be shown that
- see the lecture by J a ~ o i s
the fiber ~
~ D
of
by a point in the
over D with fiber ~-l(t)
~ 0 : ~g+i/F0
are any two divisors
t ÷ 0,
therefore,
is an analytic mapping
~
as
J(C t) : {g+i/Ft,
point on CO, any divisor divisor
generated
of a complex manifold
To describe
in ~g+l which differ by a
generated by the c o l u m s
We let,
by the Jacobian
has the structure
2g+2
in ~I"
function
su~ace
C; so if ~
on C O such that
~
and
-~
f on C O lifted to C, Abel's
55
Theorem (8) gives -f(b) -
=
f(a)
exp { l Wb_a -
~
I mivi a
6 **
'
=
m *
V
E
,g
which holds even if f has a zero or pole at a since it must also have the same zero or pole at b.
Thus if we let the divisor
~
- }[
of
degree 0 on C O correspond to the equivalence class of ~ ~8v I, • • • , ~ V g ,
~ ~°b_
a
~ E
Cg+l
modulo FO, the variety ~ 0
becomes
the group of divisor classes of degree 0 on C O with two divisors D and D' identified if on C O
D - D'
is the divisor of a meromorphic function
that is, a function f on C satisfying
f(b)/f(a) = i.
There
is an exact sequence of groups
(55)
o
> ¢*
¢
>
~
0
>
Jo(C)
>
0
where ~ is induced from the identity on divisors of C O lifted to C, and ¢(r) for
r E {* is the class in 9 0
morphic function f on C satisfying Z = (Zl,...,Zg,Zg+l) ~ {g+l of ~ 0
f(b) - r. f(a)
¢(r) is the class of (0,...,0,1n r)
modulo F 0 and ~(Z) is the class of
[~ ... 0 ½~T ~ ~g+l 0 0 (t) "
z : (Zl,...,Zg) E ~g
Let 6(t) be the half-period
for
t = 0
in Jo(C).
9~g+l(t) =
Then there is an analytic subvariety ~ 6
of eodimension i which is a family ~ 6 ÷ D over D with fibers at
We will let
denote a point in the universal cover
so that with this notation,
Proposition 3.9.
of the divisor of any mero-
t ~ 0
given by
of g-dimensional varieties
divj(ct)St(Z- ~(t)),
the fiber is the subvariety of ~ 0
defined by
b O(z -½1a v ) (56)
e zg+l +
=
O,
e(z + ½ 1 ~ v )
where 8 is the theta function for C.
C ~
Zg+l E ~
and
z & ~g
while
0
56
Proof.
The eigenva!ues
away from 0 by
2~ < 0, say; and thus the expansion
~gn . n . Re T..(t) i 1 ] 13 Z E ~g+l
of the Riemann matrix T of C are bounded
< ~ ~g n 2 i 1
and expanding
for t near 0 and
~(t) by Cot.
_< ~
n. 6 JR. 1
(it]aS(t)) ~ m
,
g
-m)emC]-[@l(Bi(t ) + m Y i ( t ) )
m~Z where
9k(w)
I
= ~ exp(½n2k +nw), n~Z
are the real parts of analytic this we conclude
c : Re z
and ~(t)
Bi(t) and Yi(t)
g+l'
functions
that for t sufficiently
function of Z and t for t near 0.
'
bounded near
t = 0.
From
near 0, the above series con-
verges by the ratio test and 9T(t) ( Z - 6(t))
velopment
By fixing
3.8:
, ~, 2 IOT(t)(Z - ~(t))I
(54) implies that
is a w e l l - d e f i n e d
analytic
The constant term in the Taylor de-
is
lim @ T ( t ) ( Z - 6(t)) t+0 which gives
.b
= @(z - ½
V ) + e
Zg+io(z
+ ½
I~
v )
a
(56).
Thus, although the Riemann divisor class
A(t) 6 Jg(C t)
corre-
sponding to (St) is not single valued as t goes once around the origin, A(t)+6(t)
£8 a w e l l - d e f i n e d
point in Jg(Ct) , and the bundle of half-
I
~l "'" ~
order differentials
L
on C t for any h a l f - p e r i o d
[~] =
E1
is likewise w e l l - d e f i n e d
if and only if
6g+l : ½.
6g+~ g eg e g + ~
It will now be
shown that ~[lim(A(t)+6(t))]
= A + ½(a+b)
6 t÷0
g
where A is the Riemann divisor class and
~
of degree g,
in (55); here
½(a+b)
with integration
6 J (C)
~g(~)
= ~
£ JI(C)
for C and,
+~(~-
for any divisors
~ ) £ Jg(C)
is given by
ra+b c +½j2~
with ~ the map for any
taken in C cut along its homology basis.
c % C
57
Proposition 3.10.
Let
f(t) = ~ - 6(t)- A(t) & J0(Ct)
positive divisor of degree g with support in
C - U a - U b.
with ~
a
Then
a+b A E J0(C) ~(lim f(t)) : A ---7-t÷0 and the condition
lim @t(f(t)-6(t))
becomes
= 0
t+0
@(e-½1~v) = exp e(e + ½
where
Wb_ a
Vk(X)
v)
k=l
e = A --7--a+b A £ ~g Proof.
~
gq
Let
jt
V q ,~c,
and A is given by (13)-(14). *
(k.qlt) ~ ~g+l
and
(kq0) E @g
Riemann constants for Ct and C O with basepoint the expansions
Wb_ a
k
be the vectors of q ~ C - U a - Ub; then
(53)-(54) give
j0 +½Tj,g+l(t)
: ~
mb-a(X)
j - ½
vj + 0(t)
Ag+l
:
for
j - ½
vj + O(t) = ½
f
v. + O(t) q 3
j ~ g, and
(57) k qg+l,t + ½Tg+l g+l (t) -- zi + ~2~1
Thus
lim f(t) t÷O
is the class in
~0
Vk(X) k:l
k
of the point
C
Wb_ a + O(t).
Z : (Z,Zg+l) E ~g+l,
where
t Z--I
VJgq
Lemma 2 . 7
f a+b ~ I vj : J2q
and See also
k0-
(33).
a+b
2
A ( ~g
58
Zg+l =
The proposition
~0
g+l
(*)
3.11.
(56) of Prop.
for a Jacobi
3.9.
inversion
Let a and b be two distinct
g and suppose
not containing
a or b.
~ 0,
there
not containing
~
is a positive
Then for any
is a unique
problem
for
~
e E J0(C)
g+l ~ ai, i
=
divisor
r ( ~*
positive
points
and
divisor
on a Riemann of degree
e E ~g ~
with
of degree
a or b such that
(IN vl' .... I)% Vg'IA W b - a ) = ( e l
For generic of
is the setting
C of genus
@(~-a-b-A+e) g+l
from applying
~b-a
of the third kind:
Proposition surface
Vk(X)
eb-a - ~i -
now comes
The group differentials
gq
~
..... eg 'In r) E ~g+l
is the divisor
of zeroes
m°dul° r°"
of the section
given by
s(x) --
r@(x+A+a-A-e)
- s ( a , b , x , A )@(x+A+b-)%-e)
8 ( x + A + a - A ) - s(a~b~x,}{ ) @ ( x + A + b - A ) with poles
at
x G~
,
where
~--' A k is a section
of the bundle
b-a
{ J0
with
a simple
E(%~iJ zero and pole at
b and a, respectively. Proof.
If
dim H 0 ( ~ - a - e ) meromorphic
0 ( ~ -a-b-A+e) = dim H 0 ( ~ -b-e)
functions
e E J0' with poles Sl(b)s2(a)
~ O.
Sl(X)
at ~
Then
~ 0, : I,
: 0
and
so we can find multiplicative
and s2(x) , sections
and zeroes s(x)
dim H 0 ( ~ - a - b + e )
of the flat bundle
at a and b respectively,
= Sl(b)s2(x)
+ rs2(a)sl(x)
and
is a section
59
of
e E Jo
with poles at ~
of zeroes of s(x) provides r ~ ~* ,
e E J0"
solution to (*),
r : s(b)/s(a),
and the divisor
a solution to the inversion p r o b l e m (~) for
This s o l u t i o n ~ is unique since if ~A were a second A 8 - ~ would be the divisor of a m e r o m o r p h i c func-
tion f(x) on C w i t h and
and
f(b)/f(a)
dim H 0 ( ~ -a-b+e)
~ i.
= I
so that
f(x) - f(a) E H 0 ( ~ -a-b)
The formula for s(x) can be proved by
c h e c k i n g that s(x), a section of e s a t i s f y i n g n o m i n a t o r v a n i s h i n g for
x 6 ~
s(b)/s(a)
= r, has de-
by r e p e a t e d use of (34) of Lemma 2.7.
The n u m e r a t o r of s(x) can also be found by the c l a s s i c a l Jacobi Inversion T h e o r e m on the curves C t for
t ~ 0:
if
~
(t) - ~
in J(C t) of the point
$ = ( e l , . . . , e g , l n r) £ ~g+l,
then
iff
-~
~
x +A(t) + ~(t) - ~
is a point on the variety
is the class x e ~ (t) of Prop.
3.9.
Since lim ~ ( x + A(t) + 6(t) - ~ t+0 by Prop.
a+b }~ - e a J0(C) - ~) = x + A t--T-•
3.10, and
by (53) and (57), we conclude from Prop.
= lim ~ ( t ) t÷0
=
3.9 that
a +divc[8(x+A+a-~-e)
+ eZg+ls(x+A+b -~-e)]
where
In the e x c e p t i o n a l case w h e n say
e(A-a-b-A+e)
= 0r
dim H O ( ~ -a-b+e)
the s o l u t i o n ~
_> I,
that is to
to (A) must contain the points
a and b unless further oonditions are imposed on
r 6
¢*.
For instance,
60
suppose tinct
a and b are two points
on a torus
C and
ai,a 2 E C
are dis-
from a,b; then
h(x)
=
ro(x+a-al-a2-e)o(x-a)
- ko(x+b-al-a2-e)o(x-b)
ix exp(-~e] dz)
o(x-al)o(x-a 2)
for
k = o(a-al)o(a-a2)/o(b-al)O(b-a2)exp
of
e e J0
h(b)/h(a)
with poles = r E {*
at
if
ai,a 2
is a solution
: al+a 2
and an exceptional
then
(i)
~
- ~
h(x)
= divch
= I
and with
o(b+a-al-a2-e)
= bl+b 2
he(b-a)
h(a)h(b)
~ 0.
to the inversion e 6 {
E 6*, ~ 0
is a section and
On the other hand, problem
satisfying
for
if
r E {*,
o(b+a-al-a2-e)
= 0,
where
°(X-bl)°(X+bl-al-a2-e)
exp(-ne
Ix
dz) ,
i £
{*
o(x-al)o(x-a 2) is a meromorphic
section
h(b) r = -h(a)
(ii)
of
e £ Jo
such that
o(a_al)o(a_a2 ) fa+b la exp(m+ n v) v o(b-al)O(b-a 2) Jal+a 2 b
=
with n given by (46) and
r
v-
e =
~i +
will be a solution
~
only if r satisfies
condition
parameter varies
famly
over C.
$
a,b
of solutions
Iml
E ~,
m ~ Z.
~ ~*,
Thus,
there
*
to (~) in the exceptional
case if and
(ii), in which
is a one
provided
case there
by the zeroes
of (i) as b I
!V.
Cyclic Unramified
One of the high points of classical Schottky's
discovery
of period matrices
of Riemann surfaces
to the Prym varieties Although
theta-function
that his modular form o n e 4
pressed in terms of nullwerthe
curves.
all worked on different
of genus
to the p r o b l e m made it difficult
emerge.
In this chapter,
Relations
covering of degree Let
the group ÷ C,
%: C ÷ C G = {~n,
and write
~*: J(C) + J(C), subgroup
p ~ 2
g = pg-p+l
D (n) = ~n(D)
results
[ 0~ 0 . . . i]
with
are presented
of order
under a
C ~÷
surface
automorphism
C
is
a
cyclic
C of genus
of C generating
of cover transformations
of degree
of
The map
0, has as kernel
of order p in J(C)~
a cyclic and the
is equivalent 8 p< -k l_<, _
C + C.
Since any pth-integer
under Sp(2g,~)
to one of the form
we can assume that a canonical
P homology basis
ad-
p ~ 2.
Suppose that
.
ap-
can be identified with the characteristic
of the discrete bundle
period c h a r a c t e r i s t i c
[29]
for a unified theory to
of order p of the group of points
homomorphism
4
their diverse
for any divisor D on C.
lifting divisors
generator of this subgroup
of genus
[34] and Schottky-Jung
of a compact Riemann
n = 0,...,p-l}
associated
given as an unramified
be a conformal
the locus
on curves of special moduli
automorphism
0 and 0 functions
Riemann surface of genus
g ~ i.
their various
of theta relations
between
coverings
of the same theory,
proaches
mitting a fixed-point-free
describing
3 theta functions
[27], Wirtinger
aspects
theory was
of genus 4 [28] could be ex-
for double unramified
Riemann
general discussion
Coverings
A0,B0,AI,BI,...,Az_I,Bg_I
that the covering is defined,
of HI(C,~)
is chosen so
say, by the characteristic
62
e =
0 0 . i_ 0 .
03 0 .
With such a basis for HI(C,Z)
P larly simple choice of a basis for HI(C,Z)
there is a particu-
such that the corresponding
lattice subgroup in ~g defining J(C) will give rise to a 0-divisor invariant under the action of G on J(C). ^
Just pick a basis of HI(C,Z):
^
io'~o'~l,~l ..... ~g-l'~g-1....'A~_l'B~_1 such that
~Ao = A0'
zB0 = PB0'
p-I n=0 Ak+n(g-l)
p-i ^ ~ n=0 Bk+n(g-l)
= ~-l(Ak) '
= ~
and
~n(~),
Ak+n(g-l)
=
sponding
normalized
(58)
~(~n)"Uo : u 0
Bk+n(g-l)
: ~n(Bk)
differentials
for
l < k _< g-l.
u ,...,u^ 0 g-i
n , ( 9 ) Un(g_l)+k =Uk ,
and
The corre-
on C then satisfy . . . . 1 < k
and the Riemann matrix ~ for C has the symmetric
form
M'
T. A
($9)
where
T
M (n) =
=
(I
u i)
and
}0 =
cn(~ )
J The normalized differentials (60)
v0 = u0'
(I^
uj)
for
i J i,j J g-l.
B0 on C are given by
vi = Ul• +%*u.m + "" . + (¢P-l)*u i
l!iJg-i
= ~
lJjJg-i
with period matrix _i TOO
^ p TOO
~0j
0j
Tij : Tij +Ti,j+g-I + "'" +~i,j+(p-1)(g-l)
i J i,j ! g-l.
63
From
(60)
~(x,y)
and
(30),
the d i f f e r e n t i a l s
corresponding
to the choice
of the
second
of h o m o l o g y
k i n d m(x,y)
bases
and
for C and
satisfy (61)
m(x,y)
Finally,
= ~(x,y) + $ ( x , y
there
) + ... + ~ ( x , y (p-I))
V
x,y
e C.
is a d e c o m p o s i t i o n p-i 1 H0(a I) -~ ~ H0(n~ ® ~ ) C n=O ~--
(62)
where
ns is the s h e a f of s e c t i o n s
i0...0]
the c h a r a c t e r i s t i c
such
linearly
u. + e ]
Elements
independent
n¢;
will
~*: J(C)
÷ J(C)
again be d e n o t e d
line
for each
defined
n ~ 0, t h e r e
and a basis
by
are
is given by
1) (~p_1) *u. ]
can be lifted
by
bundle
of H 0 ( n ~ ® ~ ) are the P r y m
differentials
2~in 27tin( P (¢)*u. + ... + e ~ P ]
The m a p p i n g {g, w h i c h
•
n ... 0
of the flat
P on C w i t h m u l t i p l i e r s
differentials g-i
!
~*: w i t h
i _< j < g-l.
to a map f r o m {g to
the above
choice
of h o m o l -
ogy on C and C,
* z = ~ *(z 0 ,Zl, .... Zg_l) : ( p z 0 , z l , . . . , Z g _ l , . . . , Z l , . . . for
z
c ~g;
in terms
of p e r i o d
characteristics,
,Zg_l) E ~g
(59)-(60)
give:
(63) ~*
B0 6
p~0
B
B
^ T
for all gives
0 , B0 e ~
rise to a group A
for
and
z ~ ~g
A
and
~,~
~ ~g-l.
of a u t o m o r p h i s m s A
0 ~ n J p-l;
Likewise
the a c t i o n
of ~g by A
in c h a r a c t e r i s t i c
of G on J(C)
(58): ~
notation:
64
0
(64)
1
60 61
~p-n
BP
130
Bn + l
...
Bn
~
A "12
for
0,60
Riemann
¢ [
and
k 6k ¢ Rg-l,
i _< k _< p.
If we define
the
form for } by
(65)
H({,~)
=
then H is a positive on the lattice
1
i,3 ~:I
(Re ~)-i z'w'l 3"" l 3 '
definite
in {g defined
form on {~ and by the columns
and (64), H has the invariance H(~,~)
the function
(66)
8({)
This equation,
together
divisor
class
point under the action
Proposition
4.1.*
V
II
o
k=O
Im H
E Cg
has integral
of (2~iI,~)
[33];
values
from (59)
( {g,
o ~ n J p-l.
from ~ has the symmetries
{ E ¢g
A e J~_I(C)
and
0 ~ n ~ p-l.
i.i (i0),
implies
corresponding
that the
to 0 is a fixed
of G on J^ (C). g-I
Let I be the half-period of order
~)[X](~*z)
(67)
{,~
with Theorem
L0 ® ~"(L ~ -01 )
to the bundle
V
0 constructed
= ~(¢n(~))
z'w
property
: H(¢n({),%n(~))
Similarly,
Riemann
^ T
:
2 on C.
c,
in J(C)
corresponding
Then for all
a non-zero
z e ~g,
constant,
[i °0] (z)
so that
~*(0)
= (811]) N ~*J(C).
homology
basis
for C,
In terms
X = ~*(~-l~i,0,...,0) P
* This theorem was observed [i,I, p. 33].
by Mumford
of the above
choice
of
e J(C).
and also proved by Accola
65
Proof. with
Let
f : ~-A
[ a positive
where
=
be a (generic)
divisor of degree
is of degree
is a half-period
if
g-i
p(g-l)
on C; then
=
and
I =
-(Jo(C)
f + (2wik,0,...,0) P (z)
= ~*K C = K~ = 2A.
E (O),
E (O);
w'f-k
8[X](~*z)
@
bundle on J(C)
since it is infinite at most on a
in J(C)
point of (@),
since by (9): 2~*A = w~(25)
Similarly
non-singular
corresponding
is a holomorphic
to the singular
set of
therefore
section of a flat line
@
g-3
dimensional
set
i 001 for
P
k : l,...,p-l. for a family
Since i is a half-period, ~
of curves
Ct covering
unit disc which are being pinched t ÷ 0.
By (59) and Cor.
a sum of
p+l
blocks
O~(t)[k](w'~z)/
that
[l] =
0
PTO0 B 0
along the cycle
along the diagonal;
001 (z)
• " "
C t over the
AoBoA01BO I
as
so the condition
has no poles implies,
to
that
as
t ~ 0,
T(t)
~0 0 ... i] where B 0 0 ...
(pZO)/p-i O
a family of curves
3.2, the matrix ~(t) for C t degenerates
0
[
it has to remain constant
2e0,280
E Z
are such that
is h o l o m o r p h i c
(z 0 )
V
z 0 E 5-
Now a
OTO0
formula from elliptic
functions
[19, p. 402,
#122]
gives
=
~Q for which the above expression
is a constant.
So
I ° o°' :1
[k] = n-i 2
and
O [ X ] ( ~ * z ) /!
(z)
is a constant which
is non-zero
^ T
66
since the multiplicity of
O
of 0[I] at ~*z is the sum of the multiplicities
at z by virtue
of (62):
i(~) ; dim HI(c,~) =
dim HI(c, ~ @ kc) = k=O
where
ik(~) , the multiplicity
number of linearly
0 [0 k "'" ...
of
i]
-p differentials
independent
ik(~) k=O
at
z = ~-A,
is the
with multipliers
ks which
vanish at ~. We will retain
the notation
E(x,y)
for the prime-form
on
C ×C
C xC.
lifted to
Corollary
4.2.
For fixed
x E C,
E(x,y)
is a multiplieative
E(x,y) holomorphic
function
and is a section p. 16,
6(y)
on C with
of the bundle
= ~(y-x)
Corollary
p-i
4.3.
E J(C) For
7*6*(@) @ and
x E C,
~£i
~[I](
f+ks ~ (@)
v +f))
for
~,(~)-i ® I
dy = p y=x 0
and c as in Prop. (f)
4.4.
~*f)vi(x) = p ~ u
Then
Y
4.1,
~zi
from the symmetries
Let f be a non-singular
k = l,...,p-l.
as on
(f)v.(x).2_
l
~i
with,
= y-x E J(C).
0
(67) and the fact that,
~Y
Corollary
y = x , ,x ,, ,... ,x(P-l)
at
:
Use
,(
~(y) f E (@)
~0[X](~,f)ui(x ) = Pc U I
Proof.
zeroes
of 0 and X:
--- (~*f)ui(x). ~zi
point on (0) with
x,y E C,
O[I]([ y u - ~*f)
c E(x,y)
"x
SZ7
P~__I [0 .-. 00]
p E(x,y)
P
Proof.
Write
f = ~-&
with
~ a positive
divisor of degree
g-l;
67
then
for fixed x,
x +[
where
dive0( C
~ = w*f + A + X
@[l](
u-
~*f)
~(x,y)
e(
v-
f)
no zeroes
or poles;
E(x,y)
v-
f) = w*~ + w*x
= w*(f+A)
and
= w*~.
is t h e r e f o r e
div^@[l]( C
By C o r o l l a r y
a well-defined
u - ~*f) = 4.2,
function
on C with
"X
stant
by l e t t i n g
is found to be
~
Proposition
4.5.*
k = 0,1,...,p-l.
Then
E(x,y) P ^ E(x,y)
y ÷ x
e E {g
with
e +ks
p-i V-~ @
w'e)
= >
~[l](~*e)
@(e + ke)
°o]s: /r
0
(
of
y 6 C
a vector
v - e)
(with space
0[I](
fy
E(x,y)
where ,
0(
over the
0
v - e)
E(x,y) E(x,y)
811](
y ,y ,...
This
v-
~ 0
for
e)
appears
e)
v-
...
@[i
o0] (e)
0 J k J p-l,
are p l i n e a r l y
forming
functions
a basis on C.
is a f u n c t i o n
@(
,y(p-l)
(
independent
functions
for the
functions
But by Cot.
on ~ as
4.2,
U - ~*e)
the ~k(x,y)
,,
for
con-
"X
x fixed),
x
E(x,y)
~ (0)
this
~ x,y e C,
O[k](Ix'U-
Since
4.3,
(f).
L_~k:0
Proof.
Cot.
00]
@
Let
and using
u-
z'e)
:
Z 9k (x,y)O k=0
are m e r o m o r p h i c we see that
in [29]
for
on C; so we can write
each
p = 2
functions ~k(x,y)
-
(
on
C x C.
can have
see Prop.
v - e)
Replacing
poles
4.19.
only
y by
at the
68
zeroes
of
• .. 0~ ([Y V - e)
@
oj
the index of speciality placing
i P
y with
of
divc@
(e)
The Prym Variety.
P = {n0~
(
for all
x,y
The Prym variety
of J(C)
+nlR'+
@k(x,y)
is constant
v-
e)
in y since
By re-
is zero.
' ~... ,x(P-l) , we conclude that
x,x
O[X](~*e)/O
the subvariety
so that
~x
~k (x,y) =
E C.
P for the covering
C ~ C
is
given by
"'" + n p - i A (p-I)
I~
e J0(C) '
n.£ m g,
n i = 0}.
Equivalently,
(68)
P= { ~ - ~ '
i A ~J0(~)} = {p@ eJ0(~)l £ + ~' +... + ~(p-1)= 0}.
Proposition there
4.6.
is an isogeny*
versal
The group P is isomorphic i: J(C)
x p + j(C)
Proof. for
~
The projection
~ J0(C),
has kernel
under ~ and P has dimension P is the subgroup 0 1
of J(~)
w*J(C)
i
The unicover
÷ P,
defined
by (63)-(64);
by so
q(~ ) = ~
- ~'
J(C)/w*J(C)
= P
From (64) and
(68),
given by all points with characteristics
~
0 B
and
form H given by (65).
~-g = (p-l)(g-l).
... ePl
(69)
of the universal
Riemann
o: J(C)
J(C)/w*J(C)
of degree p2g-l.
cover P of P is the orthoeomplement
of w*J in @g under the G-invariant
to
J(C),
Sp
P ~ i
k a
and
P ~ i
6 k E ~g-i
,
T
so that by (63), istics
~*J ~ P
of the form
~*
[0 °
is isomorphic
el *
"isogeny"
here means
to the group of ~-character'
P~i and P~i ~ ~'
a group of
eg-IJ T
group epimorphism
with finite
kernel.
69
order p2(g-l);
thus if the isogeny
i(A,B)
= ~*A + B
kernel
~* ×(z*J ~ P)
the degree
for
A E J
and
i: J ( C ) × P ÷ J(C) B & P,
then
is a group of order
of the isogeny.
For
is defined
kernel
and
i = ker ~, = p2g-i
p2(g-l)dim
Z 6 ~
by
W E P,
the symmetry
of H implies 0 :H(Z,
cnw) =H(Z,W) + H ( ~ Z , ~ W )
+ ... + H ( ~ P - I z , ~ P - I w )
:pH(Z,W)
0 so that P and ~
are orthogonal
under H.
Conversely,
if
H(Z,W)
= 0
for some W e {g and all Z 6 ~J, then H(Z, %nw) = 0; taking p~l p-i 0 Z = ~nw, the positivity of H implies ~. ~nw =0 and W 6 i ~ = ~. 0 0 P Proposition
4.7.
For
n *(T00 T01 "CO~-i] , ,..., p p p .
be a point of J(C)
'
for any
x 6 C.
n = 0,1,...,p-l,
let
6n =
such that
Then if z ~ ( ~ ) is the projection
%n(x)- x E P+ 6
to C of a divisor
on C,
{ A eJ(d)I
z:~,(A): 0
in
J(C)}
p-i [J ( P + 6 n) n:0
:
and
Proof.
We first
{7~ e j(~) [ ~
To see this, write ~(k)
= 0
~
implies
k=O in the group of points
show that if
A (k) = 0
= A +B
^ 1 e m : ~ *f2~im ~|---~--'u'''''01' P in J(C)}
with
that
A( k )
=
A E ~*J = pA = 0
p-i ~ m,n=0 and
then
(P + 6
B 6 P;
so that
+ Sm). n then
A is
contained
0 of order p in ~ J .
From the description
of
n
70
~*J m P
given in the proof of Prop.
4.6,
A - 6 n - Sm ( ~*J n P
some integers ~,(}~)
= 0
m and n so that ]~ E P + 6 n + em, as asserted. p-i for ~ [ j(~), ~ ]~(k) : 0 in J(~) so that k=0
+¢ m
~ P+~n
for some integers
(2~im,0,0,...~0) p Finally,
~ 0
of degree
g-I
~i: J(C)
fined by for
~
~i(~) 6 J(C).
tion which
= ~e ~ - ~e(~*A)
÷ J(C)
: ~,(~)
if and only if
and = ~(}~)
a2: J(C) and
~,(I{)
~{ ~ -w*A
:
P+ 6 • n with
: ~ e ~ - pA.
÷ P
a2(~)
Then ~i and ~2 satisfy
can be lifted
f = ~
Now if
by (ii) and
0 = ~,f
Let
0 < m < p,
in J(C), so actually we must have
f £ div @[k] ~ (P + 6n )
positive
m,n~ but if
for
be the projections
de-
= ( p - l ) ~ - A'-...- ~(p-l)
~*~i +~2
to ~g in the following
= pldj(~),
manner:
for
a relaz E {g,
let z = ~l(Z)
= (z0,zl,...,Zg_l)
C {g
with
zi = P91 Zi+n(g-l)' n:0
and set
: o2(2) = ( o , - ~ ° , - s 1 , . . . , - s where, s
If
p-z) e ¢ • (¢g-z)p : ¢&
by (64)
k
^ = (-pzi+k(g_l)
P C {g
phism
+
sk
Zi+n(g_l)) E ~g-1, l
is the universal
0: {g-g = ({g-l) p - 1 %
cover of P
defined
P C J(C),
0
there
= 0 E ~g-1.
is an isomor-
by
i
Then (7l)
V
z E cg, p{ : ~*~i(~)
+ ~2(~)
= ~*z + ~ = ~*z + }(s)
E cg.
71
Proposition P0' an isogeny
4.8.
i: P0 ~ P
82: J(~) ÷ P0
such that
ioo 2 = o 2
(q) is the theta-divisor Proof.
on
J(C) "
on P0"
We will make an explicit construction of P0 as a princi-
(~-g) × (~-g)
with ~ a symmetric
matrix with negative definite real part; the isogeny i
will then be induced from the isomorphism
% given by (70), and the
(72) that
(7~)'
@~(~)/o~(~l(~))n~(¢-l~(~)) T
is a meromorphic identities from [.
and a projection
and
pally polarized Abelian variety {g-g/(2~iI,~)
assertion
polarized Abelian variety
of degree p (p-2)(g-l)
p(@) ~ o~ I(0 ) + 8 ~I( q)
(72)
where
There is a principally
£ 6
function on J(C) will then be a consequence
connecting the 0, 0 and ~-functions,
So let
s = (s l,...,s p-I)
cg
9: {g-g % P E {g-g
of the
the latter constructed
be the isomorphism defined by sending
into
~(s) = (0,sl+..+sP-l,(l-p)sl+s2+..+sP-l,...,sl+s2+..+(l-p)sP-i for
s k ( {g-l;
(73) and
then ~ and ~ satisfy pX'Y
~t(~)
: p~
( {g ;
x,Y ~ ~-g
: %(X).~(Y)
for any
s ( P.
by the matrix ~ can, restricted to
The quadratic
form Q on {~ defined
P C {g, be considered a quadratic
form Q on {g-g by pulling back by ~; if ~ is the matrix defining ~, the matrix representing Re ~ < 0
since
Q on {g-g is given by
Re ~ < 0
on P.
In terms of the matrices
(59)~ *
~ is linear equivalence
p~ = ~t~ ~
of divisors
on J.
with M (n) in
72
If F is the lattice generated in ~g by the columns of ~, the lattice generated by the columns of ~ in ~g-g is mapped by # onto the sublattice
L 0 = {(l-p)y + y' +y(p-l)
{p-i ~. niY (i) 0
I Y E F,
n i 6 ~,
I Y ( F}
~ n i = 0} .
of the lattice
L = F ~ P =
On the other hand , the
lattice generated by the columns of (2~iI) in {g-g is mapped by } onto the lattice A given by the intersection ated by the columns of (2~iI) in ~g.
of P with the lattice gener-
Thus, under ~,
P0 = ~g-g/(2~iI,H)
is isomorphic to P/L0+A , and % composed with the identity mapping on ~g gives an isogeny
i: P0 ÷ P = P/L+A
order of the group L/L 0.
Furthermore,
of degree p(p-2)(g-l),
since the map 02 lifted to ~g
sends the lattice defining J(C) in {g onto the sublattice ~2: J(C) ÷ P
can be lifted to a map
the
°2: J(C) + P0
L0+A ,
such that
io~ 2 = 02 . If
m E 7g,
the orthogonality
of ~*J and P under the quadratic
form Q gives Q(m) = i
,]~ ~ijm.m. = pQ(n+ 6 ~ ( ~ + ~_ =i i ] p) + p)
g where
Q(z) =
and
i~j:=i ~ijzizJ
n E g g,
6 E (Z/pZ) g,
n E Z ~-g,
A
6 (Z/pZ) g-g
are given by pl~'~ol(m) = ~ * ( n + ~)
~2(m) = 9(£ + ~)
(74) ~'6 + from (71).
There is a
n, 6, ~, 6 satisfying
i- i
¢~ 6 (pg) ~
correspondence
(74), and hence
between
m E Z~
and such
73
""
I
I
l-
'f"
,,,
I
|
", ]
~ ~,~ i ~, ~ ~.~-~
~ (,/mt[~ (~-' by (71),
(73) and (74).
So if ~ is the theta function on P0 defined
by ~,
6}({)
where
~ ( (~/p£)g
for any ~,c = 0,
~,~ ~ ~g.
=
and
~P~ 0
~, ~(Z/pz)g-g
7*6 + ~
(Z)np~ 0 (s)
( (pz)g;
Replacing ~ by
~+~
more generally,
with
p~
~ (~/p~)g
by (71),
and
this identity can be inverted to give:
P ~E :o ^
for any
~,~ 6 ~g.
= ~ = 0
The exact statement
in (76) and observing that
(72)' follows by taking ~!I0] th is a p order thetaC
function for J ( C ) w i t h Corollary 4.9. on C or
characteristics If
div~n(o2(x-a-e))
e E P = ~
and
P [~]-
a £ C,
is of degree
see p. 2. either
n(~2(x-a-e))
(p-l)(g-l)
E 0
= p(p-l)(g-l)
and satisfies (77) Furthermore
pe = ¢ + ~*(A +~,a)
- p ( a + ~) E J0(C).
~,(¢) is the divisor of a differential
on C of order ~ (2.
74
Proof. Prop.
Setting
4.8 gives
divcO(ol(x-a))
p~
0 = ~*(~-~,a-A)
: w*~
and
+ %;
which
gives
by Prop.
(77).
: ~,(~)
~.(~)
Double
=
e),
independent (i I^ w.) 2wl -~. ! ] metries
KC
Theorem,
= pA + ~,I
~
= (~
and
pw,l = 0
When
P0 ~ P
w i)
holomorphic
for
C + C
is an unramified
is principally
i J i,j J g-l;
polarized
here
{0 0 ... 0}T ½ 0 0 '
Prym differentials
is the identity
matrix.
2g-2
basically
by Corollary
is the following
E
linearly
so that
The simple relations
sym-
between
due to the fact that the n-divisor 4.9,
is small enough
with the 0[I] divisor on C
theme
with
g-i
on C normalized
(g-l) × (g-l)
double
Wl,...,Wg_l
are
make it easy to give very explicit
on C, of degree compared
p2A : ~..~(%) - (p2-p)A
by (9).
3 with s the half period
involved
begin with,
+ pA - p~,(a)
Cov erin@s.
the @, @ and N functions,
tively
,
: ~*(~,a + A) +
w.,A : w,(l + w'A)
the Prym variety
matrix
H0(K c @
: ~
so that
= p ~
+ p~,(a)
p(p-l)
Unramified
covering, period
dive'x-a-e)
4.1,
0 : ~,(pe)
and so
Since
and
but by Riemann's
e = ~-a-A
pe + p a + p A
: ~
useful
fact,
see Prop.
to be effec4.14.
To
due to Wirtinger
[34, p. 90] and Mumford: Proposition
4.10.
For any divisor
i ( ~ ) is even if and only if i(~ +x'-x) by Prop.
4.7, mult^O[l]
is even for
~
on C with
w , ( ~ ) = K C,
is odd for all
z E P + 60 : P
x E
Thus,
and odd for
Z
( P+~I" Proof. i ( ~ +x')
We'll
= k±l
show that
provided
A
i ( ~ +x) = k ( ~ satisfies
if and only if
A + ~'+x+x'
= K~.
Now this
75
last condition l+dim
gives
H l ( ~ + x + x ') = l + d i m
H0(}{ ') : l + d i m
H0(f{)
: dim HI(}{)
HZ(A by the Riemann-Roeh
theorem.
Since HI(~
obviously implies
dim HI(2~ ) : dim Hl(x+J{ )
is one-dimensional
Hl(}%+x)
and
= HI(j% )
HI(j%+x)
= H I ( ~ +x+x'),
8}~ 0 " ' "
the fact that
implies that e { t h e r
Hl(~{+x ') = Hl(}~+x+x ')
of P giving rise to ~
with i ( ~ ) odd is
~1 ( z ) =
0
fact that
Hl(}%+x ') = H I ( ~ )
or
which proves the assertion.
for
~,
P + 61 = P +
The coset
0]
0
0g
P by ( 6 4 ) ( 6 6 ) ( 6 9 )
0
and the
~
I + 61 =
Proposition
0
4.11.
For all
(78)
s ~ {g-l,
in the construction
the universal
4.1 and { is the isomorphism
of the Prym variety.
If
z 6 {g
n (79)
cover of P0:
1 2 : cq (s)
@[1]({(s))
where c is the constant of Prop.
and
(s)
= c = (-1) 47.6
and
P] (0) (80)
c =
q2~ 0
,
c
n
2
(0)
c
02T[0½ ~] (0)
for all half-integer characteristics *
HI(j9 +x')
dim Hl(}{+x ') = dim H!(}~+x+x ')
HI(}{)/HI()~+x+x')
since
HI( A +x+x')o.
~
and likewise with x and x' interchanged,
and
+x)
)
~
a, B, Y, 6 and p.
These formulas have been proved by Farkas-Rauch
in [i0].
s
(70) used E{g-1
76
Proof.
(79)
To e s t a b l i s h of Prop.
and the s e c o n d
t e r m of
(80) come f r o m
(67)
(78) and the first t e r m of (80), we w i l l use
4.8 w h i c h
for
p : 2
and
(78).
(75) and
(76)
become
(75)' T
aT
and
~
for all
a0,b 0 E ~
law of t h e t a
and
functions
~ ( z / ~ ] ~ -~
a,b,c,d e Rg-l. (3) to
(67) and
Now a p p l y the t r a n s f o r m a t i o n
(76)';
t h e n for
2~ e
~g-l,
• ~ ~ (~1~z} v'
SO
n2~
[:](0)
= c02T
(0).
Using
(4),
(80)
and
(76)', we get
(78):
a~ ~ (m~.] ~-'
'~;] [I~ ~ ~ (z/~z) r Corollary fixed
x 6 C~
lently, with
if
i(}%)
4.12. then
divan(½ = i
If ~(e)
s;
div^n(
c
?w j×
-e)
= ~+x'-x-~*A
, w - e) = ~
-- -'~ {s')
}-I
= ~ ~ C
6 P
~ C, t h e n
and 61 the h a l f p e r i o d
with
for
e E P0
i(~)
= i.
and Equiva-
¢(e) + 61 = ~ - ~'~A E P + 61
° ... 0°]
0
77
!
Proof. Prop.
If
4.11,
e 6 P0
°o] f;
0 ...
0
is such that (
u -%(e))
n(½
w-e)
~ 0
on C since
Y
011](%(½
Therefore -~*A
w-e))
(
and
i(~)
= ~ n (½
u - ~(e))
= ~(e) + 61 : }{ - ~ * A ~
then by
at
y = x ,
t
d i v ~ O ½ 0 ... 0
the d i v i s o r
~ 0,
by
= I
w-e)
: ~ + x
(77), w h i c h
(see
~ 0.
with
means
(88) below).
i(~ ) : i
that
~
and
is in fact
The two s t a t e m e n t s !
in the c o r o l l a r y
are e q u i v a l e n t
since
61 + % ( ½ [ ~ w) ~x
!
and
w
= ½
From this Riemann's
,w
+
w
corollary
Theorem
Corollary
½
~
= x' - x E P +
x,y fi [.
and Prop.
4.10 we h a v e
an a n a l o g u e
of
i.i for the q - f u n c t i o n .
4.13
[34, p.
(q)0
=
94].
Let
P 0 - (q) = {f ~ P0
I q(f)
~ 0}
!
(n)l and for
= {f ( P0
[ n(½
w-f)
~ 0
on C}
2 _< n _< g, fXl+."
+X'n_ 2 -f)
(~)n = {f £ P0
- 0
Xl,...,Xn_ 2 ~ C
| ~ ( ½ ~" x l + W- -+ x n _ 2 ~+ + v fx I .. x n n(½| w JXl+.. + x n
-
f)
~ 0
V
but
x I ..... x n e [}.
Then
(n)
: {f E P0
[ ~(f) + n 6 1
degree Also
(~) : U m~l
at least m at
(q)2m
~-i and
f E (n)2m
a positive
: ~ - ~*A, on C with
PO
since
~,(~
: I I (n)2m+l~ m~-0
) : KC and
and
divisor
of
i ( ~ ) : n}.
(n) has m u l t i p l i c i t y
78
YI+''+Ym_I_ (n)2m = {f£ Po | n(IXl+..w +Xm-i
f) ~ 0
V x I ..... Xm_l,Yl,...,ym_ I 6
YI+..+Ym w - f) ~ o n (| l+..+Xm
but
Vx
l, "'''Xm'Yl'' "''Ym 6 8}.
This corollary appears in the work of Wirtinger extensively the case
g : 4:
then a 3-dimensional
generically the Jacobian of a non-singular varies over P0'
formal involution
s +
genus 4 a Wirtinger
Prym variety P is
quartic curve, and as s o
divP0~(s-s 0) N divP0n(s+s 0)
family of Riemann surfaces
[34] who studied
defines a 3-dimensional
C
of genus 7 with Prym variety P, consO - s +½s O , and quotient Riemann surface Cs0 of
sextic curve constructed
from the quartic curve.
A discussion of this relation between the genus 3 and genus 4 Riemann surfaces can be found in [8] and in the thesis of Recillas Proposition
4.14 (Riemann).
If
e E {g-i
[26].
and c is the constant
of Prop. 4.1, 0[I](
u - ¢(e))
(81) n(e)n(
w - e)
E(x,y)
e~(x,y)
and 8[~](
u -~(e)
+ ~l )
(82)
E(x,y') cE(x,y')
n(½
Proof. e + ½ Iy' w. 'y div~6[l](
w-e)n(½ w +e) IX<' Iy'
(82) follows from (81) by replacing y with y' and e by To prove (81): if (for generic e)
f;-
u -~(e))
= ~
div~n(S y w -e) x
then by (77),
-x-~*A = ~(e) : ~+x'-x-~*A
= ~
and
79
so that
}~ is actually
~(x,y)
~[X]({ u - ~(e)) ~x q(
E(x,y) morphic stant
s;
the divisor
has no zeroes
since
or poles
i(]{)
: 0.
Thus
on C and is a meromor-
w - e)
function
by (72)'
I (e ) by Prop. is [q Corollary
~ +x
4.15.
and Corollary
4.2;
letting
y ÷ x, this
and
2e,26,2p
£ gg-l:
con-
4.11.
For all n
(½
x,y
E C
w) 2 E(x,y)
(83)
:
e
(½
v) o
(½
c
v)
and q2~
(
w )
E(x,y)
(84)
=
Proof. e
=
½
fxy
w
+
(83), 6 ~ {~]
(80) and (3) give in (81) and use
Taking a Taylor Corollary
expansion
4.16.
i:1<x from (61) and Cor. and the identity
For (83),
set
y = x,
we get
(67).
of (83) near
of the second kind on
+[00
Use (83),
(84).
Let [~] be an even half period,
Then if $ is the differential
Proof.
c
C × C,
I::3
(26) and the relation
2.6(i).
and set
This corollary
6~(x,~(x))
= S ( x ) - ~(x)
can also be seen from
(79)
80
~(x,x') : -
2~2 ~21n61~] 6
i,j=O
coming from (29)
~{i~j
and t h e
fact
that
(0)u.(x)uj(x') l
OL½
(
6
u)
Vx~
-
on ~ by
0
Prop. 4.10.
~o~o~~ G
(x) = 1
~et[:I~eono~~er~o~ ~(0)wi(x) )s •
and
H
~n~e~
(x) ............ (0)vi(x). 0 ~z.
I
l
Then
(85)
G
(x)
= c2H
(x)H 0
The "Schottky-relations"
(x)
V x ~ C.
80) and (85), together with the Riemann
theta-formula [17, p. 137] or [19, p. 308], can be used to give equations, respectively, of hypersurfaces in ~ g
containing the locus of
period matrices of Riemann surfaces of genus g, and of hypersurfaces in Pg_l({) containing the image of C under the canonical mapping [13, p. 241], generalizing the Riemann-Weber irrational form of the equation of a non-hyperelliptic singular quartic in ~2(~)
genus 3 Riemann surface as a non-
[7, p. 387].
Both types of equation were
first given in genus 4 by Schottky [28, p. 263 and p. 262 respectively] and later indicated for arbitrary genus in [291, p. 296].
Proposition 4 1 8 function @~ [° ½ Y
0
~o~ an odd cha~octo~iotic [ ~ ] on the Siegel h if pl ne
Prop. 4.11, on the locus ~-2g-i C ~ 2 g - i C of genus
2g-1
admitting
let f ~
be the
i voni hi g
of period matrices of curves
a fixed-point-free
involution.
Then u n d e r
/x
the isomorphism (51) lifted to ~ 2 g - l ' stricted t o "~ 2 g - 1 i s
the
quadratic
by
the differential df(~) re-
differential
81
~I°0~Ix ~I°~~]x- ~-"l~c ~.~ vanishing identically on C by (85). Proof.
If
e = {8½ y@ ~] ~ & J(C)
with [y] odd~ (79) gives
T
eH
(x)H
(x)
0) (~*v(x))i(~*v(x) )j i,j=0 ~.~{. l 3
and 2g-2 a2~[e] IG[Y]2(x) : (0)(%w(x))i(%w(x)) j • -~" [6J i,j=0 ~'~.a~. ~-
3
From the symmetry (66) of 0, the universal covers ~ and ~
in ¢2g-i
are o r t h o g o n a l u n d e r t h e q u a d r a t i c form Q d e f i n e d f o r any h a l f p e r i o d e E ~*J
292 .32@[e](0)~i~ j , Q(~,~) = i,j=0 ~...~.. l 3
by
the q u a d r a t i c d i f f e r e n t i a l (51) l i f t e d
0Si
to'~2g_l
is,
~,~ 6 ~g.
Therefore,
c o r r e s p o n d i n g to df under the isomorphism by t h e h e a t e q u a t i o n
(52),
~j~-2 ) Sf (ui(x)uj(x) +ui(x')uj(x')) : 4 2g-2 ,~] -~2~[e]lui(x)uj(x g-2 ~ij i =0 3~i~2 j ,~o
: eH
In similar manner, if
(x)H
I~]
(x)
e = 1½ ~6 j]
- ~-G
[~]~
with
(x).
I~ ]
even, Prop. 4.5
and (26) give Z21n @Eel
(0)ui(x)uj(x) = ~(x,x') + ½ Q
(x) +Q
(x) ,
i,j---=0 Z~.~. l 3 so that the quadratic differential of Cor. 4.16 2@
I°~:l ~(0) (4~(x,x') B
+ Q
[~:](x) + Q I~l(x) - 2R II B ~x~'
which vanishes identically on C, is the differential
dgl~2g_l
where
(£)
(£)
(x)
(x)
H
H H
(~
(x) h(~
+£
"0 9 £+x
II~ ~o7
9(,£'x)~(£'x)~
£
)
U( £ ' x ) x ( £ ' x ) x +
pu~
~
~e
Ez~Tz~ T : E ~ . ..
(0) ~ [9] g~
(~
) ,£
U( ~ ~
,£~x)g~(£~x)g ~
) z{
U( t
0£ 9
+ s
~q% ¢~q$ puT7
= a
pu~
£'h
a~ s e T % T $ u ~ P T
"do~£ u T
~o7
6"h
CaI pu~
9
+ A
(0)£
~(I8) :
Z
~(L8) ~Aoad oZ
"do~£ u T
"~oo
as~q% mo~£
~x ÷ z
(O)P
x):](&~x)z '
£ 9 a u T peAIOAU T sieT¢u~e77Tp
"(£8) ~ u < s n u~
Z
~X~ ~'~
"(08)-(6L)
9
= ~
~uxsn
'(I8)
UT
0£ 9
~UTZ~% mo~7 sa~oo
: a
(98)
(x>[.+~].}+ +
x
(x)f[]~
~I
:
X
pu~
"7oo~£
"0 9 £ ' x
[ (x)[~.~]',, X
IIe ~o 7
(.~,x)~+ (L£)
9
pu~
{
("
:x
~'
8
(4'x)
I
(M
{ ,J)t° oJ+
,o++ [+,] .,,o [+o] ,i "(6L)
) £
•(~unf'-£:ff:~oqos)
[j
8[
(98)
U (-&~x) g2
6I'~
UOT%TSOdO~£
,~(9L) £q I-~8,J..., uo AIT'eDT%uBPT sB+4sTUeA v
pu~
{,ol[,; ,-'+'o'fl +~ ~,,}[; +*+o°~%,:,, ,,
'-'"
",I]%
88
83
As a consequence of Props. 4.14 and 4.19, one has: Corollary 4.20.
Let s(x) be the non-vanishing multiplioative
^ ,)-i on C; then holomorphic differential E(x,x H0(Z*Kc@ V*(n)-2), where Vx = ½%-loa2(x'-x).
For all
~z i
0
V: C --~P0
e(x') = e(x) 6
is the difference mapping
x E C,
(O)vi(x) =
i
-- e
. ½~l-%TO0
s(x),
2p
6 Z g-I
2g-2 ~. a@[l](6 I- $(s))ui(x) .0... x'~£i x' n(½ w- s)~(½ w +s)
(88)
;x
~[~]~~
:
f
~(x)
Po
e
ce~00
X
oI::]~i~v)
O)d--~in @ ~
V s
o
~](Ix v) }
G[Y1(x)d.d_l n :
!
[L Y] (x) !
....
:
-4e
-½TOO~ 2( x )
and
(89)
~[~]~'Sx'w~
:
,
: -Te
~(x)
~I~]~';Xw~ °
for all even [~] and odd [~]. Proof. (87)
and
Let
y ÷ x
Y
y ÷ x'
in (84),
in
making
#-
eative
section
of
(83)
use
y + x of
in (82),
the
.fact
that,
y ÷ x as
-i
~r"L 0 o n
C ,
E(x,y ) : E(x,y)exp{-½T80-
v0 }
V x,Y E ~.
in
(86)-
a multipli-
84
In all these
formulas
the path of integration
from x to x
within C dissected along the homology base defining
t
is kept
} and the sections
and e. If K(P 0) is the quotient K(P 0) C.~ P2g_l_l({) order ~-funetions
of P0 by the involution
by means of the 2 g-I linearly
of characteristics
i~
will now be shown that the composite
s ÷ -s,
then
independent
on P 0 [9, pp.
second-
212-220];
^ V C -~P0---~K(P0)
mapping
it
>~ 2 g-l-i
actually projects ical imbedding Corollary differentials Proof.
C ÷ ~g-I --~P 2g-l_l
to a mapping
for non-hyperelliptie
C.
4.21
8
(Mumford).
For
(x) and H
H ½
The linear series over P0'
~*IKcl
= K C.
Now use
ated by the squares
is, by Cor.
~
+ ~'
on C with
4.13,
variety
i ^ ( ~ ) = 1 and C [23~ p. 297] which
A ~ the linear series
of the odd and even theta-functions
It should be noted this corollary
w+e)
JX
(89) and a theorem of Mumford
says that on any Abelian
the 4 g-I
~ X T
w-e)n(½ 1
either all of C or all divisors z~(~)
odd,
and V*I2~Ip 0 are the same since
T
div~i½| ~
even and
is the canon-
(x) span H0(C,~I). C
X
as e varies
which
for the purposes
on
A .
of the remark on p. 16 that
does not require the precise
~(x) which itself was constructed
12@I is gener-
statement
from the prime
form:
(89) involving all that is
needed is divan2
(½ xx w) : div~H[~
(x)
and
divan 2
Ix.
(½ x w) : div~H ~ !
which follows
from Corollary
4.12 since ~2[e](½
w)~ for e a half !
period,
vanishes
half period
at the zeroes of a d i f f e r e n t i a l
satisfying
~*(e')
: ~(e) + 81 .
Hie'](x),
e
an odd
(x)
V.
This chapter C admitting
plest
class of surfaces
functions points,
relations
Relations double
cal homology
fixed points,
C.
C ÷ C/Aut
includes double
become,
automorphism
at
double
C ~ C
of a compact
at
a canonical As, + ~ ( A
coverings.
Riemann If
surface
C
%: C ÷ C
QI,...,Q2n,
is
a canoni-
of HI(C,g)
such that
basis
the
be a ramified
QI,...,Q2n { C.
with fixed points
theta-
cases,
AI,BI,...,Ag,Bg,Ag+I,Bg+I,...,Ag+n_I,Bg+n_I,AI,,BI, can be chosen
on the number
with only two branch
in limiting
Let
~ = 2g+n-i points
group Aut
Depending
for unramified
O and O-functions.
sur-
the sim-
both the hyperelliptic
coverings
relations
of genus
basis
with
on Riemann
mapping
g with 2n branch
the conformal
involution
(80) and (85)
between
covering
of genus
e-q
of theta-functions
automorphism
the theory
the
Coverings
C with non-trivial
and the ramified
where
Schottky
a eonformal
projection
of fixed points,
Double
is a discussion
faces
and ramified
Ramified
AI,BI,...,Ag,Bg
of HI(C,~)
) : B , +~(B
is
and
) = 0,
A i + @ ( A i) = B i + @ ( B i) = 0,
,...,A g ,,B g ,
I i~ ig g+l ii i g+n-l. =3
If the corresponding
normalized
holomorphic
Ul,.--,Ug,Ug+l,.--,Ug+n_l,Ul, then for (90) where
I J ~ J g ue(x)
and
= -us,(x')
x' = ~(x)
differentials
are
,...,Ug ,
g+l J i J g+n-l, and
u.(x)l = -ui(x')
is the conjugate
point of
x ~ C.
V
x ~
The normalized
86
holomorphic
differentials
1 5 ~ 5 g,
while
w~ = u~ + u are
g+n-i
,
on C are then given by
i _< ~ _< g
and
linearly independent
The canonical
w.: = u.z
normalized
bilinear differential
v
= u
- u,
for
g+l _ < i _< g+n-i
Prym differentials
on C.
and prime form for C have the
symmetries ~(x,y) and
= ~(x',y')
~(x,y) +~(x,y')
and
E2(x,y)
: E2(x',y')
V
x,y E
is the bilinear differential
~(x,y)
on C.
The
Riemann matrix for C has the form
l.<~jIB.<~
%'
A
T
(9l)
where
=
T is the Riemann matrix
"I
(92)
is a symmetric part
for C and
(g+n-l) x (g+n-l)
matrix with negative-definite
real
(see p. 95). The field of meromorphic
to the functions simple
a 6 C,
- for instance,
e 6 ~g,
on C is obtained by adjoining
on C the square root ¢(x) of a function
zeroes or poles at
elsewhere
functions
QI,...,Q2n ¢2(x)
{il,...,in}
and double
= O(x_a_e_p)2 O(x_a_e)2
u {jl,...,jn}
on C with
zeroes and poles
n E(x,Qik) TTE(Xl 'QJk)
a partition
with
of {l,...,2n}
fQJk The map
7*: J0(C) ÷ J0(C)
lifting divisor classes
87
of degree
0 is an injection:
e J0(C) implies
and
f ~ 0
on C since
f,g or
otherwise
meromorphic g ~ 0
on C, which means ~
of period matrix
mapping
by:
~'~z = ~*(Zl,...,Zg)
"~* { ~ ]
Similarly tion
T
=
notation,
^
¢(~)
for any divisor •
~: J0(C)
^
for
E ~g
÷ J0(C)
c~,6 e 1Rg.
arising
to an automorphism
'''" '~ g ' ) .
.
becomes
~ on C can by (90) be lifted
: (Zl'''''Zg+l'''''Zl'
vanish
for C, we can lift 7" to a
r^. 0 ~ #~ [# 4~ _"~ . E ~g 0 .~
the automorphism
with
of ~ * ~ under
= (Zl,...,Zg,O,...,O,-Zl,...,-Zg)
in characteristic
(93)
= div~(f+g~)
g must actually
+ ~(~)
With the above choice
which,
~* ~
on C, the invarianee
div~ ~ is not of form
~*: ~g + ~g
if
from the involuof ~g sending
to
^
•
^
: - ( z I . . . . . Zg,,Zg+l . . . . Zg+n_l,Z I ..... Zg) e ~g
that is,
(94)
@
v c
~
The theta function
(95) which
v B
for
£
0 constructed
J~_I(C)
by Riemann's
for the divisor
D = A -~*A
6 J (8) n
the line bundle
Theorem
class
V z 6 ~g i.i (I0), that
A corresponding
be the divisor
L0 ® ~*L01
p,v e ~n-l.
from ~ (91) has the symmetry
0(z) = O(~(z)) implies,
~,8 E ~g,
class
to 8.
of degree
A : ~(A)
in
If we let n corresponding
on C, then D is fixed under # since
A is,
and
(96)
2D = 2~ - 2~*a = K e - ~'Kc * = QI + "'" + Q2n 6 J2n(C)
since the pullback
by d~ of any holomorphic
differential
to
on C must
88
vanish at
QI,...,Q2 n e
Proposition
C.
5.1.
For
z E ~g
and
Xl,...,x n 6
-D) n O(z)@(z+~,(Xl+...+Xn-D))
C,
O(~*z+xI+...+x
(97)
where C(Xl,...,x n) is a holomorphic × ~ x... × ~ z 6 ~g.
section of a line bundle on
which is symmetric in
Thus for
= C(Xl,...,Xn)
Xl,...,x n
and independent
of
~,e ~ ~g:
~:~00 (z+½
I X l + " + x [+x ~x n n D w , ~
f~"+Xn W
, -z+½
)
(98)
= C(Xl,...,Xn).
Proof.
Replacing
prove (97), suppose
z by
/Xl+..+x n v z - ½JD
z = ~-A E (@)
in (97) gives
(98).
with ~ positive of degree
To g-i
on C; then by definition of D, w"Z+Xl+ Similarly, degree
if
g-I
...
+x n -D :
~*{+x
I
+. . .+x n -[ £ (8). with { positive of
z + ~(Xl+'''+x.~ n-D) = A-~ 6 (0) on C,
-~(~Z+Xl+'''+Xn-D)
= ~*(-z)-x~-.
so that by the symmetry
.
(95) of 0,
,
.-Xn+D' =
7"
~+Xl+...+Xn -~ E (0)
O(~*Z+Xl+'''+Xn-D)
= 0
again.
Hence the left hand side of (97) is a well-defined meromorphio tion on J(C) with poles at most on the (@) N (0) + ~,(D-Xl-...-Xn), finite for all Xl,...,x n £ C and for generic
dimensional
The section c is non-zero for generic
since for positive
~ of degree
~-A+~*(Xl+'''+Xn-D)
g-i
otherwise
if
ic(~)
= i
on C with
~ (@)
•
l~(~ ~+Xl+'''+Xn ) = i:
subvariety
and so must be a constant C(Xl,...,x n)
Xl,...,x n E C.
Xl,...,Xn,
g-2
func-
and
and
ic(~) =i
89
i~(w*~+Xl+...+x n) = d > i, independent
Prym differentials
+ w , ( X l + . . . + x n) there are only
genus
C
g+n-i
~i({), for any
g > 0,
on C, contradicting
independent
p ~ ~i(~),
i ~ j
the fact that
and
of genus
on C.
n-i,
C(Xl,...,x n)
is the
which vanishes
(see p. 13).
Similarly
ex-
for
one has
Corollary
5.2.
For
X,Xl,...,Xn_ 1 6 ~
and
z E {g~
+XI+-
X
X
(99)
linearly
Prym differentials
O(Xl+...+Xn-W*p-A)
for some
d-I > 0
at the generic divisor
C is h y p e r e l l i p t i c
theta function
x i = x~
vanishing
linearly
=
= A + w*p
hypereliiptic actly when
of degree g+n-i
In the case D = A -~*A
there would be at least
y
=
X+Xl+''+Xn-i
e(y',xl,...,Xn_
1)
v+z
and
C(Xl, • ..,x n) = 0
i ~ j.
For
~*J0(C)
rXl+..+Xn + {]D u } C J0(C)
! x i = xj
m > 0,
O has multiplicity
for m disjoint pairs
Proof.
For g e n e r i c
x i = x~3
if and only if
m on the subvariety
for any
Xl,...,x n E C
(i,j) with
i ~ j
i,j e l,...,n.
and z by
z = ~-x-A
£ J(C)
v- z ) ~ C ,
with
~ :divc@(
~ × ... x ~
Xl,...,Xn_l,
z-
v.
= ~*~ + X l + . . . + x n l - X - A £ J o ( C )
Xl,...,Xn_ I 6 C.
seen that O(Xl,...,Xn_l,y) generic
and
such that
(99) by replacing x n by y
so that the left hand side of (99) has zeroes
subvariety of
i,j ~ I .... ,n,
(97) gives
~z+x'+xl+...+Xn_l-D
for generic
for some
at
Since the locus
of codimension has simple
i
zeroes
y : Xl,...,Xn_ I £ C(Xl,...,x n) : O
is a
and since we have just at
' ' I y = Xl,...,Xn_
the symmetry of c implies that the variety
for c = 0
98
x i = x[3,
is the union of the divisors assertion
concerning
The section divisor
in
@ now follows
c in (97)-(99)
the class
i ~ j,
in
~ x ... x ~.
The
from (97).
depends
D 6 J (C); n
of course
the divisor
on the choice
class
D itself
of
is given
by (i01) and (103) of the following Proposition of {1,2,...,2n}
5.3.
For any partition
with m a non-negative
period of the form
D -
{00 ~0 1 ~0 E v
integer,
there is a unique half-
such that
. . .+Qin_2m
For a partition
with
m > 0,
0] + ~*J0 (C) "~ so that,
{00
J0 (~)
{il,...,in_2m}U {Jl'"'']n+2m }
[00 U ~] ( SY u +~*e)
- 0
0 has multiplicity by Riemann's
for all
m on the subvariety
Theorem
e E ~g
and
i.i,
x,y E C
whenever
0 ^c°rresp°nds to a "singular partition" with m > i. If "T is a non-singular odd half-period coming from a partition with
yU
+ ~*e -
x
V
e 6 ~g,
if {00 U ~
i00 01 = c(y',x v 0
'''" 'Qi
is a non-singular
v + e)@(
v- e)
n-2
Q. +.. 31 +Q JR+2 v d = a + kI J Qil + . .+Qin_ 2+4a
where
)@(
m = i,
even half-period
for any
a E C.
corresponding
Finally,
to a P ar-
v
tition with
m : O, Q. +..+Q. f ]I 3n
(iOl)
n = ~-~*A
= Q.
+..
iI
and for all
*
e 6 ~g
and
+~*(k|
"+Qi n
v
JQil+..+Qin
2~,2B ~ ~g,
This has been proved also in [i, II, p. 22].
)+ 100~ 01 E ~n(C),
91
(i02)
: c[~] : c(Qil,...,Qin)eXp
½~_ t
QJl +" "+QJn by Prop. 5.1, where
= ~
and
v
~ : (~ij),
for
IQil+" -+Qin g+l -< i,j -< g+n-l. Proof.
For notational convenience,
let
QI
Qil + . . . .
+
Qin_2 m
m
and
QJ = QJl + "'" + Qj
; for any positive divisor n+2m
X = Z xi I
of
degree m on C, set Qj £ Jm(C) %(Qj - Qi ) = x + kj v QI+4X where the integration is taken within C out along its homology basis. Then the divisor class (i03)
: ~'(~*(QI
Q J ) ) - Q I + n E Jo(C)
is invariant under ¢ and, from (96), satisfies 4~
: ~*(QI- QJ)- 4QI +4D : -2QI- 2Qj + 4D : 0.
~-~
E
Jo(C)
and,
by (97) , @(~'e+Y-X- ]% ) (iO4)
fQl +4X e(e-kJ
for all
Qjv
e 6 {g
fQI+4Y
= c(Qil, ....Qin_2m,Yl,Xl,...,Ym,Xm)
)O(e+k] Qjv - < , . ~") and positive divisors
Now the characteristics of
{~ ~ -~]
Y : Z Yi of degree m on C. I must remain constant for a
family of surfaces Ct obtained by pinching C along a loop homotop to
92
zero,
enclosing
applying
the formulas
the divisors have,
and not separating (47)-(48)
the points
of §3 to (104) with
X and Y near some of the points
for all
A
Qi:
Qi' we
e 6 {g,
OT(e- {~I)@T(e-
{~})@s(IYw-{~] ) !
t
: lim ct(Qil,...,Qin_2m,Yl,Xl,...,Ym,Xm) t+8 O'r(e)OT(e-
where
2g )
T and s are the period
Riemann
surface
of genus
I s] ~ T = 0
Thus
matrices
n-i
in J 0 ( C ) a n d
with Weierstrass
of genus
{il,...,in_2m}
u {jl,...,Jn+2m }
and ( 1 0 4 ) ; is for
n-i
Formulas
corresponding
(I00)
and (102)
hyperelliptic
according
to the
now eome from (i01),
o f 0 on t h e
5.2 o r d i r e c t l y
QI,...,Q2n.
to the partition
of {1,2,...,2n}
and t h e m u l t i p l i c i t y
c o m p u t e d f r o m Cor.
points
I~ 1 is the half-integer
characteristic
rule on p. 13.
for C and the hyperelliptic
subvariety
from (104),
(103)
~ 0 ~
letting
Yk ÷ Xk
k = l,...,m. In contrast
to the unramified
on C does not become
case in §4, the prime-form
a multiplicative
lifted to a multiplioative
section,
~*L -I on C. However, the pullback 0 plicative inverse differential on x,y : QI,...,Q2n
and double
retain
the notation
ential
of
x,y
ential
on
C × C
and simple ential
E2(x,y)
E C, so that
zeroes
lifted to at
dz(x)dz(y) (z(x)-z(Y)) 2
-½ order differential
on C when
in x and y, of the induced [(d~)*X
zeroes
E(x,y)
(d~)*]E2(x,y)
~ ×C
with
at
y : x
bundle
is a multi-
2n simple poles
at
and
y = x'.
for this multiplicative
inverse
differ-
is a bilinear
differ-
@2(y-x)/E2(x,y) C ×C,
QI,...,Q2 n
with double poles -
analogous
in the hyperelliptic
ease
at
We will
y = x,x'
to the bilinear ~
differ-
z }C : ~i(~).
93
Proposition
E Jo(~) be a non-singular even ~0Jv 0 r half-period corresponding to a partition [il,...,i ~ U {jl,...,j ~ Let
of {l,...,2n} as in Proposition B (105)
(
u-w'e)
2
5.3.
@(
Then
~
e E {g
v-e + ~)
8(
=
O~ Z u](~*e)E(x,y) ~ uj
and
+
O(e-6)E(x,y)
x,y E ~,
v - e - ~) .....
~o(x)
O(e+~)E(x,y)
Q. +..+Q. where
f 31 In $ : ¼| v ~ JQil+..+Qin
~*(2~) E J0(C)
In E(x,Q- )" o(x) = ' ~ { ~ " "']k ~ i E(x,Qik)
and
is a section of
with simple zeroes at the QJk and simple poles at the
Qi k.
(The sign of the square root in (105) is chosen to be positive
when
y = x.) Proof.
since
First of all, the right hand side of (105) makes sense
E2(x,y)~(x)a(y)
has
at the Qi k ' so that for
fixed
x £ C,
double
zeroes
E ( xI, y ) ~ o~( x ) a well-defined
at
y = x,x
section
of
the
and double
poles
i ^~ ^ o(y) ) E(x,y)
(respectively
n
!
bundle
on C with
is,
di-
n !
visor
~ Q J k l - x - x'
@( ~i (x'y) :
(respectively
~Qikl -x-x
v - e + ~)
@(e-~)E(x,y)
).
For fixed x,
@([Yv-e-~),x a ( ~ "~--7-~. V~<x;
and
~2(x,y) :
are multiplicative half-order differentials except for simple poles at
y = x
of
77) y E C,
holomorphic
with residues i, and simple poles
!
at
y = x
with residues of opposite sign since
definition of 6, ~*(2e) ~ K^ C
%(~i +92 )2
o(x') = -o(x).
By
is therefore a meromorphic section of
with all double zeroes and a double pole of residue i
* For e a half-period, this appears in [29 II, p. 746]; in the hyperelliptic case see (17) in §i.
94
O[B]2(Ix~U-w*e) at
y = x;
period
as such, it is given by
B 6 Jo(C)
where the half
satisfies
div^e(ix" C
:
~ e + ~ + ~
@[612(0)~2(x,y ),
v-e
+ ~)
+
1
~
E J~- l(~)
div^ C E(x,y) ~ o ( x )
so that, by (i01), 6 = -z~'~e-A +~*(e-$+x+A)
The Prym Variety. P = { ~ - ~' (106)
I A
~
+ El Q J k - x - x '
In analogy with J(C)}
i( ~ , ~
) -- ~ * ~ + ~ ,
phic to the group
02: J(C) + P
i: J(C)
}[ E J(C)
~*J ~ P
by
=
~ ~0 0
~4, the Prym variety
~,B 6 Rg
and
× P ÷ J(C) and
~
p,v E ~n-i
6 P,
with kernel(i)
A',
isomor-
in J(C) lifted to J(C).
dl: J(C) ÷ J(C)
02(}{ ) = ~ -
.
of degree 4 g defined by
of 4 g half-periods
Let us define the projections and
Eeikl
is the subgroup of all points
6~6¢
and there is an isogeny
-- D - ~ * ~ -
by
so that
Then from (94) and (106), it is readily seen that
dl(~)
= ~.~(/~)
~*°i + o2 = 21dj(~). ker °l = P
and
ker G 2 = ~ (y +z*J(C)), where F is the group of 4 n-I half-periods y~F of the form I? ~ 0] E J ( C ) w i t h 2~ and 2v £ gn-l; and this implies [0
that
]~ 0 T
J(C)/z*J(C)
projections
is, under d2, isogenous
to P with kernel
~i and °2 will be lifted to ~g by setting,
z = dl(£)
= (z I .... ,Zg)
~ cg,
for
F.
The
~ ~ ~g,
z~ = £ ~ - £ ~ ,
and = a2(~ ) = (Sl,...,Sg,2Sg+l ,...,2sg+n_l,sl,...,sg)
6 Cg
where, by (94) and (106), ^
SC~
= z C~ + z (~ , ,
i -< ~ --< g
and
2s.i = 2z i'
g+l -< i -e g+n-i
.
95
are the coordinates P C J(~).
of a point in the universal cover
2~ = ~*oi(~) +o2(~)
= ~*z +~ = v*z + ¢(s) e ~g
where ~ is the isomorphism from ~g+n-I onto
(108)
of
Thus
(107)
sending
P ~ ~g
s = (Sl,...,Sg+n_l)
£ ~g+n-i
¢(s I, . ,Sg, . . . ,Sg+n_l) . . . . =.(Sl,
If the isomorphism
~: ~g+n-I $ ~
~(Sl,...,Sg,...,Sg+n_l)
P C ~g
defined by
to
,Sg,2Sg+~,...,2s± g+n_l,Sl,...,Sg) E ~g. is given by
= (Sl,...,Sg,Sg+l,...,Sg+n_l,Sl,...,Sg)
e ~g
then the Riemann matrix ~ restricted to P and pulled back by ~ becomes on Cg+n-i twice the symmetric matrix ~ given by (92) which has negative definite real part since ~ does; also that
V
~,6 E IRg
and
~,v ~ R n-l, ¢
Consequently,
if
¢~t(~) = 2~
for any
~ ~ P,
so
(108) gives
=
E
P0 = {g+n-i/(2~il'Z)
P.
is the
g+n-i
dimensional
principally polarized Abelian variety formed from the Riemann matrix 11, ¢ induces an isogeny by the
group
of
¢: P0 ÷ P
half-periods
of
the
of degree 2n-I with kernel given form
1000}
with
2K ,~ ( Z / 2 Z ) n - 1 .
K 1I
Proceeding
exactly
Proposition for
~ £ Cg,
set
as
5.5.
in
Prop.
4.8,
Then for all
(i09)
09
a,b,c,d ~ ~ g
26c(Z/2Z) g which inverts to:
has:
Let q be the Riemann theta-function
z = ~e(~) ~ Cg
(107)
one
and and
for P0 and
s = ¢-l(d2(z)) ~ Cg+n-i ~,v C ~n-i
as in
96
(ii0)
2g@2T
(z)n2~ d 2¢~(~/2Z) g
In particular,
O2(~)/0 (~l(~))n T morphic function on J0(~). Analogous
(¢-I~2(2))
is a well-defined
mero-
to Cor. 4.9 is:
Corollary 5.6. n( I x w - e) - 0 a
If
for all
e 6 {g+n-i x E C
and
a E ~,
x w-
or
then either e) = ~
is of degree
a
2 (g+n-l) satisfying (iii)
¢(e) : ~ + a' - a - (QI +...+Q2n ) - ~*A 6 J0(C),
and ~,~ is the divisor of zeroes of a differential of the third kind on
C with at most simple poles at Proof.
Let
= ½¢(e) £ {g;
divcO(dl(x-a))
~+ ~*~
: ~
and
div~O(x-a-~)
~ J0(C)
= 2}{ ~ J2~(C)
and
~ : }~ - a - A
oi($)
where
= 0,
~ J0(C).
by Prop. 5.5, so (96) gives
25 = $ + ~"~(~,a + A) - 2 a - 2A = ~ + a' - a Since
= }{
then (i0) implies
0 : ~ -~,a-A But
QI,...,Q2n.
(QI +...+Q2n ) - ~*A.
~ , ~ - ( Q I + . . . + Q 2 n ) = ~.,~*A = 2A = K C
which gives
the last assertion. When there are only two branch points, the differentials
of the
third kind arising in Cor. 5.6 are given explicitly on p. i01. Proposition of Prop.
5.1,
(112)
n2z
5.7.
For
x l,...,x n E C,
2p £ ~g
and c the section
[~ ~] I~ I+~" +xn Xn)O2T[~I ([xl+''+xn ( ) = C(Xl,... , ~ v ).
97
if the half_period {00 M v O~ 0)~ e°rresp°nds to the partition {il,...,in_2m} u {jl,...,Jn+2m } q2~i p BI l
xy'w ) vE 0 j On ( ~ x I~
n2~ for
°0]
(Sy w
as in Prop. 5.3, then
if
m >
:]
-
) = c(y"x'Qi I'
m = I
and d as in (i00); and
m : 0
and eJ~,/ , ~ g i v e n by ( 1 0 2 ) ,
r
"'Qin_2 )¢2T
(a2d v)
(ZZ3)
for
rut
so t h a t Q.
P
(½[
q2~ Proof.
(0) = c
For any
gether i m p l y
+..+Q.
e E Cg
92~
and
v
)
.
aQil+''+Qin
2p ~ Z g,
(ii0), (98) and (3) to-
that
le e Lz.t~-ZI ~
which gives ( 1 1 2 ) . repeat
the
above
For [00 B :] a non-singular odd or even half-period~ argument
with
(100)
or
(105)
and
(102)
instead
of
(98); and when I~ ~v 001 is a singular half-period, use (Ii0) together with the fact that by Prop. 5.3.
O
(
u-~"e)
- 0
on
~ x ~
for all
e E ~g
98
From (113), (109) and (4) we have Corollary 5.8.
q([Yw-s)q(S)
For
s 6 cg+n-i
and
~ x,y 6 C ,
- E(x,y) 2E(x,y)
~X
(114) 8
[oo o
where the summation is extended over all non-singular even half-periods of the form y = x,
~0 ~ 0~ l U o oj ~
with corresponding ~,~ as in Prop. 5.4.
(i14) and (i09) give q2(s) = ~, e[~]O~ 0 ~ 0] % ( s ) + ~ * ~ )
p
6
,
~-iz*
~-period which is even or odd with ~ { T : w, 0 and ½
w
~ s (~g+n-l.
Lo] LO ojo
Now for any half-period
½
When
(letting
(
by setting
is a half s _ ~ - i ( ~ , ~:}_)~T
z ÷ x), the above corollary, together
with (102) and (105), gives relations analogous to the Schottkyrelations (83), (80) and (85) respectively in the unramified ease; for the form of the relations in the simplest ease of 2 branch points, see (117)-(120) below. relation for the odd
In addition, there is the following special 0- ~
functions, due to Schottky-Jung [29 I,
p. 292]. Proposition 5.9. a :
~ T 6 J0(C),
let
Prym period with
For any non-singular odd half-period a =
I:°I
~(9) = ~*~.
0 ~6 P0
Then
E2(x,y)q[9]( (i15)
be the corresponding odd half
V x,y ( C ,
w )
G[~](×)
2
--
E2(x,y)@[~](lY v) X
_
_
H[~](x)
G[9](Y) +
H[~](y)
99
where
H[~]
Proof. rie
g i~
=
~@[~](0)vj ~-~i
By Prop.
bilinear
5.5,
holomorphic
and
G[~] -
g+n-i ~I
=
E-2(x,y)n[9](
differential
~n[~]( .. ~sj" 0)w]
w)@[~](
on
C x C
Ixy v)
which
is a symmet-
can
be written
as
~l(x,y) + ~2(x,y)%(x) where
¢ is defined
entials
on
on p. 86, and the ~k(x,y)
C x C.
the condition
+ ~3(x,y)¢(y)
Replacing
x,y
by
that ~ is odd implies
H[e](y), ential
differential
so that
~(x)
since
~(y)H[~](x) n[~](
Letting
which,
y ÷ x,
÷ C
v ) =
(x,y)
G[~](x)H[~](x)
s;'
w )@[~](
by (19),
Ramified
= i.
= 0. is a
zeroes
of
for some meromorphic Likewise
~3(x,y)
differ-
: ~2(Y,X)
=
so w )¢[~](
2n[~](
of E and
that ~2(x,y)¢(x)
at the double
= ~(x)H[~](y)
i(½div H[~])
the symmetry
differ-
~l(x,y) +~4(x,y)¢(x)¢(y)
in y vanishing
~2(x,y)
are meromorphic
x',y',
By fixing x and sending y to y', it follows holomorphie
+ ~4(x,y)%(x)¢(y)
s;
v)
implies
Coverinss
H[~](x)¢(y)~(y)
+ H[~](y)¢(x)~(x)
= 2H[~](x)¢(x)~(x)
= E2(x,y)
.
and therefore
(
H[~](x)G[~](y)
+ H[~](y)G[~](x)
)
(115).
with Two Branch
has only two branch
points
Points.
When the double
a and b, the g-dimensional
covering Prym
¢ variety
P0 = P
is principally
the 8, @ and q functions fact that the H-divisor By (i01) of Prop.
polarized
and the relations
have a particularly on C is a translate
5.3, the divisor
fb D : &- ~*A = a + ~ * ( 4 | v)
class
simple
form,
between
due to the
of the S-divisor
on C.
D in this case is given by Ib
= b - T*(¼
v) "a
= a + ~*~ = D'
i00
so that
~ x 6
W
:
W
:
W
:
½
The section o(x) of Prop. O(x)
=
ffTx,b) ~ ~ ,
v
and
W X ~
=
v
-½
v
:
v+½
v.
D
a
5.4 for the partition
while the section
c(x)
{a}
u
{b}
of Prop.
= c(x')
is 5.1 has
no zeroes or poles on C. Proposition characteristics
6.10.
For all
s ( {g,
x,y 6 C,
and h a l f - i n t e g e r
y,
x÷O
= "4.,1qJ
Proof. s +½
w
To establish
in Cor.
the left hand side of (116), replace
5.8 and let
x ÷ a,
s by
making use of the fact that by
~D (106) and (97): E(x,y) (118)
lim
~(~
½ ----
x-~a
E(x,y)
o ( [ Y u - ~*e) _
~ o(y)
-a
0(e+
O(~{e) '
O( I
~)
: c(y___)_)
y v - e - 6)
Vy~.
c(a)
a
The right-hand the left-hand S
side of (116) side of (116).
= {Y} + ½ 1 ~ w
6 {g
comes by replacing Finally
(117)
in (116) and using
From (116) and (lll) we see that if divan(
w-s) X
:
~ ~ C
for
s E Po'
then
s with
[y+x s - ½J w ~x+D
in
is obtained by setting
(98) of Prop. x E C i(~)
5.1.
is fixed and = 0
if
and
only
if
i01
a,b ~ ~; equivalently, ~(s) : ~ - b - a - ~ * A
if div~n(½jYr w - s ) y'
by (Iii) with
Cor. 5.6, the g-dimensional !
n(½
: ~
i(~ ) = 0
~ C, iff
then
D(s) ~ 0.
By
linear series on C, generated by
!
I
f;
w-s)n(½ w +s) for s E ~g, cuts out the divisors of zeroes x of differentials of the third kind on C with at most simple poles at a and b, and which are holomorphic
for
s 6 (~):
these are given ex-
!
plieitly by substituting
s +½
f
X
w
for s in (114) and letting
y ÷
X ~
:
X
#n(½
w +s)n(½
w-s) :
O(~*~+#(s))Wb_a(X)
ic(a)E(x' ,x) The differential c(a)-in2(s)
is holomorphic
iff
s 6 (n)
g ^ + ~ ~@(#*~+#(s))v m=l ~ z since
0(~*[ + %(s)) =
by Cot. 5.8 (see also (88)).
Proposition
5.11.
y GE~](x)n[~]( I w ) x
For any odd half-period ~ and
2 ~2 I [Yv) @[~]2( v) E(a,b) - c (a) (x'7) e[~]( 4 E2(x,y) -x @[~](2[)E(x,a)E(x,b)
+ H[~](x) [e [~]( Iy v + 2 0 x while for any even half-period
nE6](0)n[6](
w)
~ x,y 6 C,
-
4
[6](
v-
2~) ~-T~YJJ
6,
E2(x'Y) E2(x,y)
I~ + eE6](o)
o(y) + e[~]( o(x)
2@[6](2[)816](
o(y) v + 2~) o(x) + e[~](Iy v
v )
- 2~) ~(y)jj
In particular, G[~](x)n[~]( (I19) H[~] (x)O[~](
w)
:
G[~]2(x) =o2(x),
ID
v)
H[~](x)
o2(a ) @[~] ( v) E(a,b) - -@[~](2~)E(x,a)E(x,b) 2
(x).
102
q[S](0)q[B](IDW)
n[612(0)
(120)
2
= c2(x),
o[6](o)o[6](
= c
v )
o[6](o)o[6](½
(a)
= c2(b).
v ) a
Proof.
Set
s-½
w
(102) and (105) and let by (21). and
Since
G[m](x)/H[m](x)
that
d In g(x)
by Prop.
5.12.
a+b+~*(½divcH[m])
depending
a and
have already
V x,y
v)
E C,
@[a](
D
=
is a constant
of degree
for the case of two branch points becomes
e[a](
-x
y ÷ x
2.11.
For any odd half-period
q[~](Yw)
- -
= ½mb_a(X)
5.11 has at most simple poles,
of the divisor
5.9, which
5.8, apply
can be proved by letting
for all odd m; these special divisors
in Prop.
Corollary
where o
in (I14) of Cot.
making use of (118) and Cor.
on C is positive
2 E2(x,y)
[8}
observing
(119)-(120)
the index of speciality
appeared
{m} or
z ÷ x,
The equations
y ÷ a,
=
JD
+
on ~ (and C) and satisfying ....
b
x
f
(121)
02 _ c2(a ) 9 [ a ] ( a V )
2
0[~](½
and
~
= c2(x)
x
[e]( aV)O[~](
nell(
v) *a
for all
x E C.
Proof.
Since
G[~](x) H[a](x)
a
is, by (119),
0[~](
a
I x v) D
meromorphic
function
it must be a constant (115),
on C with poles only at the zeroes of o
independent
gives the first equation
pansion of (119) near
x = a.
of x which,
above;
(121)
substituted
9[a](
v),
into
comes from a Taylor ex-
i03
This
corollary
also
follows
from
Prop.
~) valid,
by
(45),
that
a
and
then
by
b
Cor.
for a l l
~ a are 2.11,
differential
@[e](½
e[e](
v + ½ X
has,
in b o t h
In case
x and y,
e = B
e +½
12x
~ 0
v ) "a+b
g-2
is a s i n g u l a r
of
q[B]2(
Ix w)/c4(x)E(x,a)E(x,b)
identity
(q),
E[xpl
v ~ (0)
for
C and
e.
Now
the
v - ½
zeroes
symmetric
v )E(x,y) -2
and
simple
q[6](0)
and the
suppose
some half-period
2 s:
half-period,
{B} 7
the
and h a l f - p e r i o d s
on
v )BEe]( double
is an e v e n point
e C
that
2
and
E(~,~)
x,y,a,b
such
5.11
holomorphic
on
zeroes
= 0
e;
from
C x C at a and b.
(120),
differential
on C is h o l o m o r p h i c
with
zeroes
at
D a and b and
g-2
double
an odd h a l f - p e r i o d , on
C and
period
of p o i n t s
y is
4g-lg
are the
simple
maining
b are
the
when
of d e g e n e r a c y Example C with that
which i.
If
slit
(121)
(119)
that
hand,
when
zeroes.
O[y](½
I I @[y](½1 xv 2y=0EJ 0 a
For
v ) = 0
) ;
to
say,
the
q[~](
w) ~ 0 on C
fixed
a 6 C,
for
some h a l f -
½(4 g - 2 g) of t h e s e
while
the
re-
(83).
see h o w
illustrate
is
differential
for y an odd h a l f - p e r i o d , by,
e = ~
is i n f i n i t e ,
a holomorphic
double
a and b a p p r o a c h
examples
ramification
the
interest
the p o i n t s
following
such
b = a zeroes
in
g-2
= deg d i v C
double
It is of some ates
b £ C
zeroes
a
is by
at a and b and w i t h
number
On the o t h e r
constant
G[~]2(x)/H[~](x)
vanishing the
the
zeroes.
the each
double other
covering along
essentially
two
C degener-
some
path
possible
in C;
types
can arise: ~
is a f a m i l y
points
atb t in the
of r a m i f i e d
double
a t and b t a p p r o a c h i n g limit
becomes
a cycle
coverings
a point homologous
p
E C to,
Ct o f so say,
104
the cycle A I in C, then the limiting is an unramified ~
double
surface
cover of C defined
00 "'" ~] as in §4, with an ordinary
C0 has genus
2g-i
and
by the characteristic
double
point where
the surface
!
crosses
Such a family as in §3, p. p,p
C C0
~
to a point
to
such a way that,
near p,
surface
x = y : t : 0 Ul(X't)
= I
Al(t)+Al,(t) ferentials region,
a
Al(t) +Al,(t) ~
in
is the
x 2 _ y2 = t,
and b t are the points
I
ii
50, starting with C0 and
and pinching
non-singular
on C O .
will be constructed
cycle homologous
point
^
itself at p and p , the image of p under the involution
x : -~
at X = ~,
and
x = y
on the surface Ul'(x't)
for
= 2~i
and p (resp. (resp.
t ~ 0,
x
p ') is the
: -y).
Since
the normalized
dif-
Al(t)+Al,(t)
u (x,t) with symmetry
expansions
(90) on Ct have,
away from the pinched
of the form
Ul(X,t)
= ~p_p,(X) + Ul(X) +0(t),
u (x,t)
= u (x)+ O(t),
Ul,(x,t)
= ~p_p,(X) + 0(~)
(122)
where Ul(X) , u~(x)
u ,<x,t)
= -u~,(x) + O(t),
and u ,(x) are a canonical
basis
of
i < ~ ~ g
HO(~0 ,~^1
) with
C0 respect
to the involution
as in (58),
~p_p,(X)
= - U l ( X ) - ~p_p,(X') ^
is the normalized poles
of residue
morphic
differential +i,-i
differentials
zero with t.
p,p
!
, and the expressions
on a t outside
Since
+
B w2,...,wg
at
of the third kind on C O with
the pinched
uI =
w
O(t)
simple are holo-
region which tend to
for the Prym d i f f e r e n t i a l
B
on C0' the Prym matrix ~(t)
7
of Ct has, by (122),
an expansion
105
for some constant o!, where (H 6) is the Prym period matrix for C0 and 0(t) is a matrix satisfying
lim O(t) = 0. t+0
(mI . .',mg)~ Z g 2m2 =
t
exp { ~ t
~g-I ml~ Z
so that
Thus ~
(Wl,...,Wg) E ~g,
i
1
g + ~miwi+ml(mlCl+Wl 2
g I~ O(t)] + 2~mi w.) + 2 , m
lim n2~(t)(Wl,...,Wg) = q2 (w2,...,Wg) t÷0
and, from Prop. 5.7
and (84), n2~(t) (0) lim ct(at) = lim
t÷O
t÷O
n2~(0)
[bt 02T(½/; v) -t
=
[00l 02mLoJ(O) ½
= c
where ct(x) is the section of Prop. 5.1 and c is the constant of Prop. 4.1.
Thus (120) implies Schottky's relations (80); the relations
(85), on the other hand, are implied by (119) and Cor. 5.12: an odd half period of the form
[e] = [00 ~j
for
if ~ is
26,2E [ (~/2Z) g-l,
then
lim G[m](x,t) = lim t÷0
t+0
Eg ~ t [ a ] ( i
O)w.(x,t) = g ~q[J . (O)wj(x) = G[~] (x)
~sj
]
~sj
provided x is kept away from the pinched region in C0; so by (121),
P-1
2
G~6~ (x) = lira G[~]2(x,t) = lim h~J t÷O t÷O
=
x bt 2 H[~]2(x)@[~]2(I v - ½1 v) g~,t at at
@[a](
P
v)O[~](
v)
at
bt
= c2H
(x)H
½
106
Example
2.
Let
unit t - d i s c w h o s e distinct
branch
two b r a n c h sists
points
points
for
to a p o i n t
of C j o i n e d
x2 _ g2 = t
and
differentials
a t and b t
at p.
p 6 C
u (x,t)
double
2g o v e r the
coverings
t # 0,
while,
p 6 C
and the
of C w i t h
at
t = 0,
the
f i b e r C0 con-
~
by p i n c h i n g
37, in s u c h a w a y that
~
x = y = t = 0.
a
is the
n e a r p, a t and b t are the p o i n t s
is the point
on Ct h a v e
of genus
We c o n s t r u c t
to zero as in §3, p.
surface
x : ~,
Ct are r a m i f i e d
coalesce
cycle homologous
and
be a f a m i l y of c u r v e s
fibers
of two c o p i e s
analytic
~
x= -/T
The n o r m a l i z e d
expansions
= v (x) + % t v ~ ( p ) ~ ( x , p )
+ o(t) i _< ~ < g
u ,(x,t) = ¼ t v
for
x 6 C
malized
outside
holomorphic
(p)~(x,p)
the p i n c h e d
there.
where
on C o u t s i d e
The Riemann matrix
T({'I
x = p,
the p i n c h e d
matrix
is the d i f f e r e n t i a l
-
for C and
of
are h o l o m o r p h i c i lim [o(t) t÷0
region with
-
are the nor-
= 0
g i v e n by
-
-
t
1 lim ~o(t t+O
o[~)
= O.
Using these
we h a v e
2.6
a second-order
5.13.
(iii); theta
For any and,
for
function
x,p
{ C,
e E {g, on ~g.
set let
Zx(p) ~(z)
d = ~in
E(x,p)
: O(z+x-p+e)O(z-e),
Then
, 5] = ~
Vl,...,Vg
and o(t)
for Ct is thus
-
Proposition as in Cor.
at
where
on C, w(x,p)
"~
T is the R i e m a n n
expansions,
region,
differentials
the s e c o n d k i n d on C w i t h p o l e differentials
+ o(t)
~ ¢= I
{o)
107
is a cochain in the sheaf of q u a d r a t i c d i f f e r e n t i a l s for all x and p 6 C and independent of Proof.
When
x £ C
in p, h o l o m o r p h i c
e E ~g.
is away from the point p, the e x p a n s i o n s
above give
we(y,t) =
,t) + u s , ( y , t ) =
v e +½t(ve(P)Zx(p) + d e ) +o(t)
at for some constants of i n t e g r a t i o n d e to be determined.
On the o t h e r
hand, x
bt
x
; v , I v: I at so that (97) of Prop.
at
v-
½tv'(p)
+ o(t)
P
5.1 for two branch points gives,
V
e 6 {g:
c=.|
The coefficient of t is i n d e p e n d e n t of e and thus is h o l o m o r p h i c
V x,p
6 C;
taking
e +½
zero the c o e f f i c i e n t of
conclude that
implies that
v = f & ((~) i x-p
in the Laurent series at
g ~ (d i + ½ v ~ ( p ) ) ~ ( f ) i dzi di
= 1
1
-~vi(P).
non-singular,
= 0
V f
which,
and e q u a t i n g x = p,
we
by Cot. 4.21
to
VI.
This unitary
chapter
symmetric
Riemann
boundary
F 0 , F I ~ . . . , F n _ I. Riemann
surface
anticonformal symmetric set
(resp.
z , = zso ~
(without
C of
~ with
R U DR
local
~' e I'
for
of genus
s
set
in case
neighborhood
of this
covering
the c a n o n i c a l
cocycle
dz B (k fl) = (d--~-)[ H I (C , 0 C)
having
and w i t h
symmetry
k fl(x) = k
bundle
canonical
is a c o m p a c t
for
choose
suoh that U , = ~(U s
)
( I0~
U s = ~ ( U s)
a local
coordinate
p a r t on
can be d e s c r i b e d > 0
x ( U
for
basis
(resp.
Ap+ k = F k
for
Ai,,BI,,...,Ap,,Bp,)
the r e l a t i o n s
in H I ( C , Z ) :
k = l,...,n-l, are c y c l e s
and
is
U~ N R. by a
x E DR
~ Ufl, so t h a t v v on C.
on C:
AI,BI,...,Ap,Bp,Ap+I~Bp+I,.--,Ap+n_I,Bp+n_I,AI,,BI,,''',Ap,,B
such that
a
Us C R
for a n y d i f f e r e n t i a l
homology
an
~ in an i n d e x
I'),
set ~
admitting
We w i l l
U s with
k sB(x)
, ,($(x))
of the s a m e b u n d l e
Let us fix a s y m m e t r i c
DR.
imaginary
with
curves
g = 2p+n-i
z : Us + {
DR and w i t h p o s i t i v e
and ~*v are s e c t i o n s
on a
a positively
analytic
on an o p e n
I);
In t e r m s
p with
¢ I (resp.
coordinate
(resp.
boundary
r e a l on
functions
[3, p. 107]
fixed point
is the l o c a l c o o r d i n a t e
to be a s y m m e t r i c
of genus
of C by n e i g h b o r h o o d s
as follows:
will have
to the s t u d y of
and k e r n e l
of n d i s j o i n t
boundary)
involution
for some u n i q u e
z : Us ÷ {
surface
DR c o n s i s t i n g
open covering
~(R))
of e - f u n c t i o n s
surface.
The d o u b l e
I 0 I 0 U I'
Surfaces
differentials
Let R be an o p e n R i e m a n n oriented
Riemann
is an a p p l i c a t i o n
functions,
finite bordered
Bordered
p,
AI,BI,...,Ap,B p
in R (resp.
#(R))
satisfying
109
¢(A i) : A.,,
¢(B i) = -B.,,
1
%(A i) = Ai,
If
i! i !p
C
1
¢(B i) : -Bi,
p+l ~i 5 p+n-i
Ul,...~Up~Uo+!,...~Up+n_l~Ul,,...~Up,
are the corresponding normalized differ-
p--i
#
entials on C, then (123)
¢*u i : -u.,, i
i J i J
and
p
¢*ui : -Ui'
p+l i i J p+n-l,
and the period matrix for C has the symmetric form
where
real
a and
(n-I)
c x
are
p x p
(n-l)
matrices,
matrix.
b is
a
p ×n-1
matrix
and
d is
a
The normalized differentials of the
second and third kind on C have the symmetries (124)
for
e(x,y) : e(~,9)
x~y~a,b E C
and
and
= ¢(x)
the conjugate point of
(123), we also conclude that, for "ml(x)''mp(X)
mb_a(X) = ~ _ ~ ( x )
b0 E F0
and
mp+l(X)..mp+n+l(X)
x ~ C.
From
mp(X) 1
~ cg
x ~ C~ ml(x) • •
!
nl(x)..np(X)
0
..
0
-nl(x). -np(X)JT
(12S)
x+~ {ml(Xl"~p(x)
i
2b~ = nl(x ) .~p(X)
o
. .
o
np+l(X)..np+n_l(X)
-ml(X)''-mP(X) 1 nl(X)
np(X)]T
where the m, n, m, n are the general harmonic measures on C~ and mp+l(X),...,mp+n_l(X) 0 and 1 for
x ~ R.
are well-defined functions bounded between
e ~g
Ii0
The mapping J0(C):
if
~ gives
D = ~ -~ _
then
%(D)
=
rise to an antiholomorphic
E J0(C)
with ~ , ~
positive
_
~
in J0 so that,
involution divisors
~ ~
- ~
is the class
by (123),
~ lifts
of the point
on
on C
~
(]~ u) = (]A %~u)
to the antiholomorphic
involution
on
{g given by ~(Zl,...,Zp,Zp+l,...,Zp+n_l,Zl,,...,Zp,)
=
-(~i ..... '~p''~p+l' .... ~p+n-l'~l ..... Zp)" In terms of T-characteristics
(126)
~
for all
of a point
v ~ T
v
B
e,6,Y,~
E [P
and
Proposition
6.1.
The theta-function
period
matrix
for all
~,6,y,6
A • Jg_l(C)
has the symmetry
establish
(z)
e [P
=
and
satisfies
Proof.
equation
becomes
E
P,v ~ [n-l. for J(C)
formed
from the
T has the symmetry
(9 6 Pv
(127)
in ~g, this
-P
~,v ~ ~n-l.
A = ~(A)
E2(x,y)
form
divisor
and the prime-form
x,y
it suffices
But the quadratic
V z ~ ¢g
The Riemann
e Jg-l'
= E(x,y------~ 2 V
By (i) and (126)
(127).
v -~6 (
(9
class on
• C. to prove Q(~)
=
@(z) : @(~(z)) g ,~=~16J~jk{k for
j
e ~g
has the symmetry
Q(~)
= Q(%(~))
from the symmetry
of T;
therefore O((~(z)) = E
exp{½n~nt+
n.(~(z)} = E
n~g g : ~-~, exp{½mTm t +m-z} meg g
exp {½~)(m)~%(m)t +O(m).~(z)}
m¢~Z g = O(z)
CxC
V
z e ¢g .
to
iii
To show that A is a symmetric e 6 ~g
such that
divc@(X-a-%(e)) %(e)
odd-half
divc@(x-a-e)
= ~
= A -a-A
and
period
point
since the middle
A = A
in Jg-l"
then,
6.2.
J0 is a disjoint points
5~
E
meromorphic
-[-A =
divisor
i
along
on
C × C
y : x.
all unitary
if
Jf(x) J : i
functions
functions
D on
are
f on C with
C - ~R.
S = {s e J0(C)
~p ,
~,86
= i
function
f on C un£tary
principle
The subvariety
with
~
o2[f](y-x)
given by meromorphic
J0
and
if f is any non-singular
union of 2 n-I real g-dimensional
I~½~-~]
(i0),
H[f](x)H[f](y)
function
for some positive
Proposition
Theorem
and with the value
by the reflection
divcf = D - D
a E C
give:
term is a well-defined
up to a constant,
choose
then by (127),
Finally,
02[~(f)](y-x)
Let us call a meromorphic x 6 ~R;
~ C;
: H[~(~)](~)~[~(f)](~)
with no zeroes or poles
for
= A
so that by Riemann's
(19) and (127)
(~m~,y) II
in Jg_l(C),
J ~(s)
torii
w & ~n-!
S
= s}
of
given by all
and
~J
T : (Zl,...,~n_l) function _ ~k
E (Z/2Z) n-l.
If
D -D
is the divisor
f on C and b is any point on F0, then
1 I darg 2~ rk
f (modulo
Proof.
s =
If
2),
I~ ~' y]
D - (deg D)b @ S
with
k : l,...~n-l.
for {g
6
of a unitary
for
e,6,y,66
[p
and
, [n-i ~ ,w& ,
[ ~ J ~6 r
then by (126), if
s - ~(s)
~+y 6 Z p,
where
S
8-6 6 Z p
: ~@+y [8-6 and
2B' 0
2~' 6 Z n-l.
is the set of all points
[~ ½Zv -~J ~i = 0 or I;
for
e~8 E ~P~
y+a] 6-8~ r = 0
S =
if and only
I J S , Ze(~/2Z) n-I
in J0 with characteristics
u ~ ~n-1
S O is a real Abelian
Thus
in J0(C)
and
~ = (ZI ..... Zn-i )
group of real dimension
with 2p+n-i
= g
112
atron
e
°
real torus of dimension and
D- D
g.
Now suppose
o
0]~
o
s = D-(deg
D)b
the divisor of a unitary function f on C.
with
b £ F8
By (125) we can
write
~(s)-s
= ([D u) :
0 -
E
Cg
T with
m,n 6 Z p
conclude that
and
~ e ~n-i
s ~ S
where
a p p l y i n g Abel's
t h e o r e m (8), we then
~Ik = + 2~ IF d arg f (modulo 2). k
A divisor D on C is said to be
symmetric if D is fixed under 0;
symmetric divisors are thus of the form are divisors on C with torii S
D2 c
DR.
D I + 51+ D 2
An alternate d e s c r i p t i o n of the
is then given by
Proposition
6.3.
For any fixed d i v i s o r
entirely in F0, S v consists of all points tive symmetric divisor on C containing, k = l,...,n-1, Proof.
and
n-i-
n-i ~ ~k i
~
- ~
with
(128)
e S0,..., 0
~
and
b k ~ Fk
and
modulo
e J0
with
~
a posi-
2, ~k points on Fk,
~ = 0,
let Z 0 be the
Fk,
k = l,...,n-l;
then
u ~ S0,...,I,..., 0 (k)
p ~ C.
We will prove that
~0 = SO
~0 is an A b e l i a n subgroup of S O c o n t a i n i n g some open
n e i g h b o r h o o d of real d i m e n s i o n g. g x g
- ~
a positive symmetric divisor con-
0
b 0 ~ F0,
by showing that
the
of degree g contained
since by (123) and (125),
[ J2b0
for any
~
points on F 0.
taining an even n u m b e r of points on each SO
J
To prove the assertion first for
set of all points
~0 c
where D I and D 2
matrix
Now for generic
X l , . . . , X g ~ DR,
(ui(xj)) has n o n - v a n i s h i n g d e t e r m i n a n t since other-
wise there would be a h o l o m o r p h i c d i f f e r e n t i a l v a n i s h i n g i d e n t i c a l l y
113
on ~R, corresponding
to the hyperplane
of ~R under the canonical The Jacobian
imbedding
of the map
in ~g_l(~)
6
points
tains
of real dimension
Jacobi divcf
AI-
~
and
Inversion = X +~
Theorem,
- (AI
divc(f(x) + f(x)) symmetric
Proposition constants
6.4.
k = l,...,n-i Proof.
For
is given by (13);
and
~ = 0,
=
-
2
2~i 2
where,
as in Prop.
Then
is a positive
equivalent fact,
b ~ F 8,
and
zeroes
to a positive
together with
the vector
so that modulo
for 2,
for arbi-
of Riemann
s 6 S , @(x-b-s) 1 + ~k
zeroes
on
on F 0. k b. of 3 of ~:
k b = &- (g-l)b
1 g IA ~k i~ 2~i k--~l = ~j ' k k~j ' -
2 ~i
fl
u.
'
Uk '
]
k' Tjj
on C with
the proposition
from (123) and the symmetry
3~
g.
the jth component
~b - ~ j , j , -2~i j' . . . 2. . .
(129)
proves
on C or has,
n-i ~ ~k i
i j j j g,
A3
and this
For any point
identically
function
we also see that
k b = A- (g-l)b ~ SI,I,...,i'
vanishes
in Z0 and, using the
From this last argument,
divisor of the same degree,
~.
On the other hand,
~ ) = A3 - ~ e Z0
g is (linearly)
for
the set Z0 con-
of degree
where
e J0
(A i - ~ ) + ( A2-
divisor of degree
trary
Fk,
g; thus
of S O .
and the assertion
either
are two points
for X positive
of degree
(128)
g.
let f be a meromorphic
+ A2 )
and Z0 is a subgroup
symmetric
~
and consequently,
= A3 + ~ - ( A i + ~2 )
divisor
a symmetric
A2-
E~g_l(~).
(~R) g + x I + ... + X g - ~
has rank g at such generic
suppose
the image
x £ C + [Ul(X),...,Ug(X)]
(Xl,...,Xg)
an open neighborhood
containing
i ~. 2~i ~¢j
1.2, the paths
u~
+
2~i
uj
}
k' u]
of integration
£:i
A k are considered
part
114
of the positively oriented boundary ~l(C,b).
Thus
from (126)
kb-~(kb)
since
suppose
that
setting
~
of C dissected
= - ll0 10 ... i0 i] 0 T 6 {g
kb £ SP
for some
s = ~ - b - A E Sp
with
= 7~i + ~ i + ~ with
~ a
along generators and
kb 6 SI
of
.... i
~ £ (~/2g) n-I
by Prop.
6.1.
Now
b ~ F0
= divc@(x-b-s)
~ C;
~R
and
~
by Prop.
6.1,
(126) implies
that
~gb
This means,
°
u
by (128),
Fk; and since
that
deg ~
~
on F 0.
Corollary
(G) n S
6.5.
sisting of points g-I
Proof. s = ~ -b-b
having
(mod
s E 0 n Sp
for some positive
is a symmetric
positive
the divisor of a meromorphic
*
Vl'''''lJn-1
T
2)
i + Pk ~
function
divc(f(x) c+f(x))
= b + {-~
then by Prop.
symmetric
divisor
by Prop.
b £~
divisor
6.3,
If ,
con-
s ~ S .
b 6 ~
,
b + ~i- ~
f for some positive : 0
variety
symmetric
points on Fk if
~; if
on
must have
sym~metric divisor ~ . divisor
points
real-dimensional
b 6 F 0,
and then
{ which,
*
~ a positive
i + pk
and
''"
(modulo
g-i
with
2)
*
~ n-i (mod 2),
is the
s = %-A ~ J0
If A
must have
= g = 2p+n-i
n-I ~ Pk (mod 2) points i
of degree
I;
(deg~)b
is
divisor
~I'
in J0 for some positive
6.4, must have
(modulo
2)
i + Pk
points on Fk. Let
Sp C {g
the half-period 6
~
is real for all functions
be the universal
i 00 ½p0 0 0} r E ~g; with
~,B E N p
s E S .
Obviously (O) n S empty - see p. 126.
$ ~
Sp C J0
passing through
then S P is given by all points and
The spaces
on C with the minimal
cover of
v E S
, and by (126-7)
parametrize
@[s](0)
the generic unitary
(g+l) number of zeroes:
for all
p ~ 0;
but
(O) n
SO
can be
115
Pro2osition variety V a
6.6.
For any fixed
(S n 0) u V a u W a c S
= U {b-a-a+~-A be~R
let Z a be the sub-
a e C - DR,
where
I ~ positive
symmetric
of degree
g and
i(~) > 0}
and Wa :
U
{~-a-a+@sing~
S I ~ positive
Then for
s & S~ - S# ~
(@),
(13o)
f(x)
~ @(x-a-s)
=
E(x~a) E(x,Z)
O(x-a-s)
symmetric of degree
n-i exp ½ E ~k laUp+k , I
2 on C}.
I~I
: 1
f 2~ jFkd arg f = I + Pk
q
is a unitary
function
on C vanishing
(modulo 2) and with at most g (resp. Every unitary satisfying
s e ~
-
function
i(D)
~p~
= 0
g-l) poles if
s 6 V a (resp. W a)-
f on C with a divisor D of exactly has the form (130) for some
a e D
g+l
poles
and
z a.
Proof.
If
s 6 S
- S
f] (0), P
function on C vanishing by Props.
at a, with
@(x-i-s) @(x-a-s)
on
is a meromorphic
at a, with change in argument
6.2 and 6.4, and with constant
n-i i_a exp ½ ~i uk Up+k
E(x,a) E(x,~)
absolute
along £k given
value
IE(b,a) [ = IE(b,A)[ for
3R since
b 6 £0
and
a
n-i
lo(b-~-s)l = lO(b-a-~(s))l : lO(b-a-s)lexp Re E ~ 1 by Prop.
6.1.
Thus
C with at most degree
e g+l
the form
b+~
this case a ~ ~,
and
g+l
f(x)
as defined by (130)
poles;
~+a = ~+a E J
cases.
b e 3R g
s : b-a-a+~-A
~b function on
and f(x) will have a divisor of poles of
in two possible for some
is a unitary
a
['%+k
First,
divc@(x-a-s)
and ~ positive
of degree
could be of g-l;
in
is a special divisor of degree g since e Va
where
~ : ~+a
may be taken positive
116
by
Prop.
6.2.
for s o m e g-i
c ~ C
poles
implying s & Va (130) tion
W a)
has
~ 0
= a
some
U b~R
metric g+l
vb a
zeroes
in R w i l l
actly
g+l
in Prop. unitary by the
6.16
functions torus
Now poles
S
suppose
is a u n i t a r y
by
(130) = f(o),
that
aZl
-
Sp n
of the
there
are
holomorphic
general
Ea
Va u
case,
the
is e m p t y
when
form
(S ~ 0). unitary
for
g+l
since
f(x)
or poles
of the
has
complement It has
domain,
a 6 R
zeroes
sym-
of f in S
been
functions
R is a p l a n a r
on R w i t h
has
=
~ a positive
of zeroes
always
in R; and w h e n
then
real-dimensional
components
variety
S0,..., 0 n
g-i
the n u m b e r
= 0,
Za •
I s = ~+a+a-b-A, In the
at a,
on each
f is a u n i t a r y i(D)
function
given
S
func-
O(x-a-s)
f(x)
of
function
if a u n i t a r y i(D)
g-l,
a choice
unitary
Theorem, thus
of d e g r e e such
satisfying
~ S;
at most
with
it is
and
that
exshown
the
are p a r a m e t r i z e d
0"
satisfying
If f is
g-2}.
one
126]
that
poles
is at most
real-dimensional
zeroes
Finally,
V b = {-s 6 S a
constant
[3, p.
poles.
s e
Va
divisor
to a s p e c i a l
= c+~+q
f has
Conversely,
and by R i e m a n n ' s
variety
then
rise
g+l
and
of d e g r e e
g-2;
~ Wa .
s = D-a-~-A
= 1
including
g-i in
Is I
where
be
proved
C for
The
divisor
of the
on
g-l)
a ~ D
divc0(X-a-s)
is a s p e c i a l
gives
D of
is t h a t
of d e g r e e
e Jg-i
g (resp.
a divisor
Remarks. V
= ~+a
for any
O(D-a-x-A) for
~ positive
obviously
at m o s t
= 0
possibility
s = c+c-a-a+(n+a-A)
(resp.
i(D-a)
n+a
that
f(x)
other
and
and
with
(130)
The
by
: 0; with
(130)
for a r e p l a c e d Ill ~ I;
function
then
fl(x)
a divisor
for by
observe
on C w i t h
a divisor
- 1 ' : f(x) i - if(x)
of
g+l
poles
s = D-a-a-A
and
a 6 D,
~ 6 D1
and
s by
that sI ~ a+[ Ic+ u, ; and
Vc u
for
D of III
~ i,
D1 = D 6 Jg+l" then
fl is g i v e n
c+~ s I = s - ]a+ u, We
g+l
since
where
V c and W c are
r
translates
of V a and W a by
sI ~
(O)
since
i(D-c-~)
> 0
117
for
c 6 C
iff
Ill = If(c) I : i.
r a m i f i c a t i o n points
Corollary of
g+l
df I = 0
6.7.
and locus
i(D)
zeroes of the d i f f e r e n t i a l
= 8,
dln
f: C ÷ ~i({)
@(x+x+A-D)
dln
By Prop.
f =
dln
then the s y m m e t r i c d i v i s o r of 4g
- are given by divc@(2x+A-D),
over the unit circle in ~i({)
Proof.
given by
f - that is, the r a m i f i c a t i o n points
order theta function on C by (2).
the locus
Ifll = 1
If f is a unitary function on C with a d i v i s o r D
poles s a t i s f y i n g
of the c o v e r i n g
All functions fl have the same
The curves
If(x) l = 1
a fourth on C lying
are the components of SR t o g e t h e r w i t h
= 0. 6.6, f has the form (130), so (38)' implies
O(x-a-s) O(x-a-s)
E(x,a) E(x,~)
0(s)@(2x-a-a-s) E([,a) @(x-a-s) @(x-i-s) E(x, a)E(x,Z)
w h i c h gives the first a s s e r t i o n since
s = D-a-a-A.
On the other hand,
the addition t h e o r e m (45) gives
f ( x ) - f(x)
O(x+x-a-a-s)O(s)E(x,x)E(a,a) = e 0(x-a-s)O(x-[-s)E(x,a)E(x,a)
and thus the zeroes of the h a r m o n i c the locus
If(x) l = i,
are
or equivalently,
isfying
div C w
= divcw
if
f ( x ) - f(x),
describing
divcO(X+X+~-D)E(x,x).
We say that a m e r o m o r p h i c v = ¢*v
function
n-i exp ½ ~ B k l a U p + k k:l
d i f f e r e n t i a l v on C is v(x) = lw(x)
and for a suitable
symmetr£c
if
for a d i f f e r e n t i a l w satconstant
In terms of the symmetric b o u n d a r y coordinates
I d e p e n d i n g on w.
given on p. 108, such a
d i f f e r e n t i a l v is then real on ~R, and the sign of v at a point of ~R (not a zero of v) is w e l l - d e f i n e d positive on ~R.
since the canonical cocycle
A s y m m e t r i c d i f f e r e n t i a l will be called d ~ £ n £ t e
it does not change sign along each contour
F0,FI,...,Fn_ I
if all its zeroes or poles on ~R occur w i t h even order.
*
(k ~) is
This can be empty - see Prop.
6.16.
if
that is,
118
Proposition
6.8.
of J0 is a d i s j o i n t
The s u b v a r i e t y
union
g i v en by the p o i n t s v = (Vl,...,Vn_l) t = D - A 6 J0
of the
2 n-I real E J0'
½v -6
E (~/2~) n-I
with
D +D
T = {t e J0(C)
Each torus
on C, h o l o m o r p h i c if t 6 (@), vk (-i) a l o n g £k' k : l,...,n-l.
Proof. p,v
,
Let
~n-i
6
; then by
if and only if T =
~
(126),
~-¥ 6 zP,
~ Tv v~(g/2g)n_l
acteristios
f~ ~, ~] T E
t =
Where
T
T
t + ~(t) B+~ e Z p
is the
for
a,6,y,6
6 ~P
and : 0
?v' e ~n-l.
set of all
points
differ-
on F 0 and real with
'I
and
and
of all points
symmetric
a-y 0 , B+6 2v 6+B
I
=
Tv
~ e ~n-i
T v consists
non-negative
= -t}
torii
~,B 6 ~P,
of a d e f i n i t e
ential sign
g-dimensional
with
the d i v i s o r
I ~(t)
in J0(C)
Thus
in J0 with
char-
v ½v~ - ~
,
v 6 ~n-l,
a translate
by the h a l f - p e r i o d
~U
0 0~ of the g r o u p T O of r e a l d i m e n s i o n g. Now by the J a c o b i ½v 0 ] T I n v e r s i o n T h e o r e m , any t 6 J0 can be w r i t t e n as t = D - A for D of
L0
degree means
g-i that
definite
and, by Prop. D +D
on
and
the two s y m m e t r i c multiple
+
mj
all zeroes
the a p p r o p r i a t e
t 2 = D 2- AeT
2
b (F0,
along
if
t E T;
differential
o c c u r to even o r d e r on
~R.
suppose
corresponding exp
m6Z p
{2
on C, In o r d e r
tl= D 1 - A E T l
~ D I + D2I _ -D 2 _ DD2+D 2 = * *
* T
E "DI DI
function
is p o s i t i v e for x 6 F 0 by (124) and r e a l w i t h ek Fk g i v e n by (-i) w h e r e , for any b k 6 F k, .bk w b
+ DI+DI-D2-D2
of
to t I and t 2 w i l l be a
and '
E k : ~ a r g exp
this
D 1 and D 2 on C; then the r a t i o
function
j This
= 0 ~ J0
sign a r r a n g e m e n t s ,
differentials
for
+ D-A
of a s y m m e t r i c
for d i v i s o r s
of the s y m m e t r i c
(uj-uj,)
D-A
is the d i v i s o r
~R s i n c e
to d e t e r m i n e
6.1,
P bk ~ mj (u.-u.,) i 3 ]
sign
119
bk
bk
D2 ID u k 1
I {Im = ~-
P
bk
~ mj I m 1~jk} -- ,~k 2 - ~kI (modulo 2 )
by (7) and the symmetries
(123-4).
Thus two symmetric definite differ-
entials arise from points in distinct torii Tv if and only if they have a different sign arrangement along 3R.
Now there are points in all
torii, except possibly T0,0,..., 0, giving rise to holomorphie definite differentials
since for any
in Tv making
(@) ~ T w
are no holomorphic
v # 0,
there is always an odd half-period
non-empty.
But by Cauchy's Theorem, there
symmetric differentials
non-negative
everywhere on
~R; thus T0,..., 0 must be the torus giving rise to the differentials non-negative on ~R and always meromorphic. Let
Tv C {g
be the universal cover of
the half-period
½v 0
and
~'~ e
O(t) is real for all
t E Tv.
Corollary 6.9.
If
J0
passing through
then Tv is given by all points
6 {g;
I~ ~ ~ -6~} { T Ep ~g ½ with ~
TvC
~ & ~n-l,
and by Prop. 6.1,
O(x-a-t)O(x-[+t) E(x,a)E(x,a)
t E Tv,
is a symmetric vk
differential along Fk,
on C, holomorphic if
k = 0,1,...,n-I
linear differential whenever
x £ Fk
Proof.
for any
and real with sign (-i)
(with the convention
O(y-x-t)O(y-x+t) E2(x,y)
and
Since
t E (@)
y ~ F~,
t + ¢(t) =
~0 = 0).
is real with sign (-i)
~ 0
6 ~g,
(2) and Prop. 6.1 imply
@(b-a-t)
O(b-~-¢(t)+2~iw)
E(b,a)E(b,~)
E(b,a)
E(b,a)
and
a E C
~k+~
0 ~ k,~ ~ n-l.
@(b-a-t)O(b-~+t)
b £ F0
The bi-
-
2 IO(b-a-t) ~
near b; from continuity in a then, the
_> 0
120
symmetric
definite
@(x-a-t)O(x-[+t)
differential
> 0 -
E(x,a)E(x,~) which
gives the first assertion by Prop.
comes
from setting
a = y 6 F i,
@(b-y-t)O(b-y+t)
8.8.
for
x e F0 ,
The second assertion
since we have just seen that the sign
v~
of
is (-i)
for
b 6 F 0.
E(b,y)E(b,y) This corollary, and
y 6 F~,
together with
for any n o n - s i n g u l a r
(-i)
with
over
the partial
g ~ i,j:l
+
¢ 0.
and
signs along
derivatives
Hf(x)Hf(y)
that ~ x E Fk
~ 0
f ~ Tv ~ (O)~
@(t)
y £ Fi
with prescribed
~k+mi
point
Vk+V ~ g [~(x'y)
t E Tv
ferentials
(39), implies
k,£ : 0,...,n-l,
(-i)
for any
(25) and
and
~2
] in @(t)ui(x)uj(y)j _> 0 ~zi~z j
Integrations
x 6 Fk
of these b i l i n e a r dif-
give holomorphie
~R, as well as various
of O.
From Prop.
differentials
inequalities
6.4 and Cor.
for
6.9, one also
coneludes Corollary
6.10.
For each
4p symmetric half_periods If
e 6 S
n Tv
either vanishes
and
of the form
b e F 0,
identically
and is real on F 0 and real (resp.
{~
½~ 6}
the h a l f - o r d e r
on C or has (resp.
S ,
{] T v
consists
26 and 2e 6 (~/2~Z) p.
differential
O[e](x-b) E(x,b)
i + Zk (modulo 2) zeroes
imaginary)
of
on Fk for
on Fk
vk = 0
i).
The transition of h a l f - o r d e r
on
functions
differentials
Proposition described
~,v e (Z/2~) n-l,
p.
6.11. 108,
the
defining the eorresponding
bundles
L
e
can be found from the following
In terms of the symmetric bundle
of
half-order
open cover
differentials
{U s} of C L
e
can
be
121
given by a cocycle gaB(x)
(gaB) £ H I ( c , ~ )
= ga,B,(x)
2 gaB(x)
with
= kaB(x)
and
if and only if e is one of the 2 g half periods
in T 0 . Proof. eaB(x)
Any cocycle
= ea,6,(~)
of the form
(ea6) E HI(c,~ *)
if and only if (caB)
2 eaB:
will satisfy
corresponds
I
and
to a line bundle
e =
6 T O ~ S since -e = e : ¢(e) e J0(C) ~2 0 6 2 and the characteristic homomorphism of (eaB) over the cycle Bp+j, N.
j = l,...,n-1,
N.
3 ~ e. • k=l ik-llk
is
3 2 e. , . , = ~ e. . ik-llk k=l lk-llk
= 1
for a chain
• ,. of'neighborhoods in R joining some boundary n e i g h b o r h o o ~ Um 0 "''UiN. ] • and UiN" for F 0 and Fj, respectively. Therefore, by a standard Ux 0 3 construction of a cocyele from the characteristic homomorphism of a line bundle
- see [13, p. 186] - it will
L 0 can be described gab 2 : kaB cycle
and
by a cocycle
(gaB) with
2 gab = ka6
since
ka, B,(x) = kaB(x),
since
¢ L0 = L0
assume
that
since
kaB(x)
> 0
then be finished defines @(x-b) - E(x,b)
a trivial ~ 0
a positive
6.1.
whenever
for
(caB)
x e ~R
CaB(X)
is a section differential
ga'6 '(~)
=
with
2
; then
aB = i
in HI(c,~ *)
With no loss of generality
we can
a,6 are in the index set I 0 (see p. 108) with the positive cab =
(~a6) e HI(c,{*). of L 0 which
orientation;
Ii saB
But for
we will
a , B ~ 1 0 u I' a E I, B e I0~ I b 6 F 0,
is real on ZR since
on ~R by Prop.
0(x-b) is the section E(x,b-------~on Ua,
only that
L 0 is given by the co-
gaB (x) is a trivial cocycle
if it can be shown that cocycle
to prove
(gaB) 6 HI(c,0 *)
So suppose
and set
and
by Prop.
sa6 = i
of the form
: ga'B ,(x).
gaB(x)
suffice
its square
is
6.8; this means that if ga(x)
g (x) = ga~(x)gB(x)
for
x 6 Ua ~
US
ga,(x) and g(x)
- s (x) = ±i a
for
x 6 Ua,
where
e
= i a
if U
(a 6 I 0) a
122
is a boundary neighborhood. ~ ~(x)
: ~ B(x) = --~(x)
~B
therefore
(131)
if
x e U s ~ U B,
6.12.
The prime
E(x,y)
: E(x,y)
x,y E C,
b £ F 0.
form on and
If
(resp. negative) Proof. C ×C.
of
for
p 6 R (resp.
By Prop. Therefore
6.1,
(~ 6) is
real C~-section
of
since
by Prop.
factor
exp Re
-
loop Bj (resp. Aj). ly positive
~ E
p E R,
functions
respect to the point p - see For all
a section of
IKcI @ 2 Re t,
a 6 R (resp.
R).
Ik ~I, iE(p,p)
of
function
i
~,~[i ® L0
by Prop.
6.11,
it
if
tran-
@ ( p - p) picks up the
i) as p describes
the
is never zero and is strictare positive p 6 R
coordinate
choice of homology
is called the capacity
and
a bundle with positive
iE(p,p)
with a symmetric boundary
6.13.
C xC
(resp.
iE(p,p)Idz0(p) I : 2 Im z0(p) + ... > O
Corollary
a section
6.11 and the fact that
since the transition
For a suitable
on
@ ~,(@),
+ 2
For
p E C,
lim E(x,y)/E(x,y) = i. x,y+b~F 0 = -iE(p,p) defines by (131) a
iE(p,p)
L~I @ % , ~ i
functions
for
is strictly positive
which
E(~,~)/E(x,y),
y : p,
sition
= E(b,x)
R).
given by
and
has the symmetries
(E(~,~)/E(x,y)) 2 is the constant
be the constant
x : p
C xC
IKcI -I @ 6*(0)
must actually
i/iE(p,p)
B e I u I0,
and the cocycle
defined by the cocycle
funotion
P0 6 F0
and
: p - p E J0(C)
in x and y, is a w e l l - d e f i n e d
Taking
~ ~ I
E(b,x)
6(p)
IKcI is the real line bundle
is a real C~-section
on
for
if
trivial on C.
Corollary
for all
Consequently,
and
is near a point
z 0.
basis on a planar domain R,
(or transfinite
diameter)
of R with
(133). t 6 ±0'
strictly
@(t)
positive
> 0
@(t-a+a) i@(t) E(a,~) negative) for
and
(resp.
is
123
Proof.
By Prop.
6.1,
O(t) E R
~ t E T0
is never
zero for any
t £ TO.
function
of the moduli
and so must remain
along a loop e n c l o s i n g
and by Prop.
6.8,
O(t)
But the sign of @(0) is a continuous
~R as in §3.
constant
From Cor.
as R is pinched
3.2 and the symmetry
of T, the limiting value of 0(0) is the positive
quantity
~-~ ~ ~ ~ 1 ~ t t - t ~nan t z ~mam exp ~{nlan I + n2an 2 + mdm t} : I L e I L e ~zP n&Z p m ~ n-±
nI
m ~ n-I where d is the real period matrix of a planar domain, period matrix of a compact Riemann surface of genus assumed
generic - that is,
@(t-a+a)
then follows
i@(t)E(a,[) Cor.
6.12,
@a(0)
~ 0.
concerning
from the property of iE(a,a)
and is also a direct consequence
t : ~0 ~6 Z 0]J ( T0 0 -6 ~
is a positive
p which may be
The assertion
e(x-a-t)O(x-a+t) O(t-a+a) = i Res 02 iO(t)E(a,a) x=a (t)E(x,a)E(x,a)
When
and a is the
differential
with
of Cor. i
~
2~@2(t)
~ £ l~n-i
given in
6.9: @(x-a-t)O(x-a+t)
~R
and
6 ~ ~Rp '
on R defining a Riemannian
O(t-a+a) iS(t)E(a,~)
metric with
Gauss curvature 4@2(t)E2(a,[) @2 (t-a+a)
by (41),
22 --In Sa~a
(126) and Cor.
O(t-a+~)
483(t) O(t-2a+2[) =
E(a,a)
-
< 0 @4(t-a+a)
This metric generalizes the Poincar6 P metric in the unit disc D since if C ÷ Pl({) is a conformal homeomorphic with
p(R)
6.13.
: D, /
1 iE(a,~)
V
a£
R.
2~ae(a,a)
When
= ~dp(a)dp(~)/:~ i fp
t £ T0
where ae(X,y),
is a h a l f - p e r i o d
p(a)
>0.
E(x,a)E(x,a)
Idp(a) l
=
1 - Ip(a) l2
e, this metric comes
the Szego r e p r o d u c i n g kernel
from
for sections
124
i
of Le, is given by sections of L
e
on
•
%j(x) %j(y)
R u DR
for a complete
orthonormalized
set of holomorphie
by the conditions
S~R *jSk k] Proposition Oe(~,y ) =
6.14.
For any (even) half-period
i @ [ e ] ( y - x) 2~i @[e](0)E(y,~)"
except for a pole along Oe(X,Y)
Then Oe(X,y)
y = x,
: - Ce(Y,X)
is holomorphic
let
in x and y
and satisfies : - Oe(X,Y)
For any section % of L e holomorphic
#(X) = I ~ R O e ( 9 , x ) ~ ( y ) =
so that Oe(X,y)
e E TO,
on
V x,y E C.
R u DR,
Vx
I~Roe(X~Y)~(Y)
R
is the Szego reproducing kernel for the space of holo~
morphic sections
of L e on
R ~ ~R
with the norm
II~II = (S
I~I2)½ DR
Proof.
First observe that a e actually exists since
by Prop. 6.8; from the symmetry properties Oe(~,y
)
1
o[¢(e)](~
- x)
(127) and (131):
1
-
2wi @[~(e)](0)E(y,x)
(@) sing {] T O = @
O[e](x
-9)
:
- o
2~i @[e](O)E(y,x)
(x,9).
e
By Prop. 6.11, this means that in terms of the symmetric open covering {Us}
= - °e( x,y-)B,
°e(X'Y)$
tion o on the open set of L e on Us,
U
,,
where
Oe(X' y )B,
× U ,; consequently,
Oe(X,y)B,~
(y)
,
is the
see-
if % (y) is any section
is a section of
IKcI in y and of L e
in x with the property that
- Oe(X,y)6, if
y 6 U s n DR
~ (y) : Oe(X,y)B,
,% (y) : Oe(X,y)~,~%~(y)
for some boundary disc U s.
i ~ @[e](y - x) EDR °e(X'Y)~(Y) = 2--~i eDR O [ e ] ( 0 ~ ( ~ ) ~ ( y )
Therefore,
if
x ~ R:
@[e](y-x)~(y) : y:x Res O[e](0)E(x,y)
:~(x)
125
In the case of a p l a n a r domain
(p = 0), there is a global uni-
valent function Z on R w i t h dZ(x) a n o w h e r e v a n i s h i n g d i f f e r e n t i a l h a v i n g a w e l l - d e f i n e d square root on R cut along segments joining FI,...,Fn_ I to F0; then
/ ~ i IdZ(x) 1½ = exp F Arg
dZ(x) IdZ(x) l
is a multi-
valued function on R which picks up a factor of (-i) as x traverses any loop Fk, ~ E (Z/2~) n-I real with
i j k j n-l. and
i +~k
e = ~ 6 T0 ~ S' tuJ T zeroes
fore by continuity, on
R x R
Fk.
e =
(mod 2) on Fk by Props.
Oe(~,y)//dZ(~)dZ(y)
t°t
0 T,
o0(~,y)//dZ(~)dZ(y)
the classical
b £ F0, is
6.4 and 6.8.
There-
is a m u l t i p l i c a t i v e ~k
function
as x goes around the loop
Szego kernel
is w e l l - d e f i n e d on
kernel for a space of
if
°e(b'Y)/IdZ(b)dZ(y)1½
w h i c h picks up the factor (-I)
So w h e n
e # 0
On the other hand,
R × R
functions on R, while
and a r e p r o d u c i n g
~ (k,y)//dZ(~)dZ(y) e
is a r e p r o d u c i n g kernel for sections of
(6) - that is, functions with m u l t i p l i e r s
(-i)
e ~ J0 Bk
for
as given by
along Fk,
k = l,...,n-l. Now in the case w h e n R is the unit disc D, the inner product on the h o l o m o r p h i c h a l f - o r d e r d i f f e r e n t i a l s grating two analytic functions over normal derivative of the Green's basepoint
0 6 D.
can also be o b t a i n e d by inte-
~D with measure given by the inner
function G(x,0)
at
x 6 ~D
To d e s c r i b e this situation in the general case, re-
call that on a finite surface R, if
(132)
ab_a(X)
for the
= a~ ~(~)
g ~
= ~0b_a(X) -
-
j ,k:l
-i u . ( x ) ( R e T) 3
Re
Ia
uk
jk
is the unique d i f f e r e n t i a l of the third k i n d on C w i t h simple poles of residue -i and +i at a and b r e s p e c t i v e l y and with purely i m a g i n a r y periods over all cycles on C, then the Green's function
G(x,y)
= ½
~_ = ½ Y-Y ~
m(p,q)
+ ½
~ (Re j,k=l
Re uj
Re u k
126
is a harmonic
function
G(x,y)
in x and y with the symmetries
= G(y,x)
and with a local expansion
(133)
= - G(x,y) at
i G(x,y) = in ~ + i n
= G(x,y)
y : x:
g iE(x,x) + ½ E ( R e 1
in terms of the harmonic
~ x,y E C,
measures
T)jkmj(x)mk(x) + O ( I x - Y I)
of (125).
The bilinear differential
is the Bergman kernel of C with the reproducing II B ( x , y ) A
V(y)
for any differential
property:
V(x)
=
V(x) holomorphio
on R u DR.
a 6 R,
(x) =
R Proposition
6.15.
For any fixed
let
~_ a-a
dG(x,a) + i *dG(x,a) poles of residue
be the differential
of the third kind on C with
-i,+i at a,a and purely imaginary
closed paths in C.
If
~ is the divisor of zeroes
periods
along all
of ~_
in R, then
a-a
i(~ ) = 0
and
e : ~ -a-A
is a point of T O ..
satisfying ° ~0
(134)
~ In@(e+a-a) ~zj
for mj the harmonic
~ inO(e) Szj
measures
= m.(a) 3
of (125).
j = l,...,g
For any
be the Cauchy kernel
(37) formed from the divisor
meromorphic
of
function
K(x,~)
is, for any
~ C R;
let Aa(x,y) then the
6 C :
Aa(X,y)
O(x-y+e)
O(a-a+e)
E(y,a)
E(x,a)
~_a - a (x)
O(a-y+e)
G(x-a+e)
E(x,y)
E(a,a)
a holomorphic
function
of
=
Y e R u ~R,
f(x)
x,y
x,y 6 C,
~ R : ~i ~
f(y)K(x,y)~_a-a (y)
x E R
~ x 6 R
such that
127
for all h o l o m o r p h i c functions
f on
R u ~R.
Thus K(x,y)
d u c i n g k e r n e l for the H i l b e r t space H2(R) of functions
i llfll = lim (- ~-~ +
with finite n o r m
[
If(x)l2 , dG(x,a)) ½ G(x,a)=e
The Green's function
the m a x i m u m principle,
f a n a l y t i c on R
J
s+O
Proof.
is the repro-
G(x,a)
0 < - *dG(x,a)
> 0
= in_
for all (x)
x E R
for
so by
x ~ 3R;
by
a-a
Prop.
6.8,
e -- ~ - a - A
i(~{) = 0
and
is t h e r e f o r e a point of T O
e ~ (0)
@(x-a-e) 0(x-~+e)E(a,~)
since
(8) {] T O = ¢.
, Prop.
2.10
(38),
,.o.
Since
~0
with
~_a - a (x) =
(125) and (132) imply that
@(e) @(a-~+e)E(x, a)E(x, ~) ~x
EC,
c--J
which gives
(134).
and hence K(x,y) morphic for
e~R
0(p-a+e)
is h o l o m o r p h i c
x,y E R
h o l o m o r p h i c on
i 2~
Now
~
x,y £ R
never vanishes
for
and K(x,y)~[_a(y)
except for a simple pole at
R u 3R,
y = x.
P e R,
is holo-
So if f is
the residue t h e o r e m gives
f(y)K(x,y)~_ (y) a-a
:
: @(p-a-e)
= _
i 2--~ E3R
- R e s f(y) Aa(x,y) y=x
The r e p r o d u c i n g p r o p e r t y of K(x,y)
f(y)K(x,y)C_ (y) a-a
~_ (y) a-a - f(x) ~_a - a (x)
for the Hilbert
a c o n s e q u e n c e of the general Poisson r e p r e s e n t a t i o n
~ x ~ R.
space H2(R)
is t h e n
formula for func-
tions in the Hardy class HI(R).
Planar Domains. of genus
g = n-i
lytic curves
For the r e m a i n d e r of this chapter, we assume that C is the double of a planar domain b o r d e r e d by n ana-
r0,...,rn_ I.
128
Proposition g zeroes
6.16.
in R for any
For all a E R.
phic on R with the minimal @(x-~-s) a O(x-a-s)
E(x,a) E(x,~)
Proof. positive
If
s 6 S0'
Every unitary
a 6 R
s 6 SO ~ ( 0 ) , g-i
function
and
t h e n by Cor.
6.5,
for all
since @(s) is real on S0 by Prop.
6.13.
Now if
-s = D+a+a-b-A
where,
g-i < n-l;
since
g-d
b 6 F0 b k ( Fk
points and set
ek = 6
where
is a differential
> 0
6.4, D is positive with an odd number Fk (and possibly
and
a ( R,
F0); again this is
Thus as a varies over the in-
has a fixed number of, say,
d points
6.4.
If 6 is a local coordinate
in
of a near b, and e k a ~u
for ak near bk, then the condition = u ( ~ )e
g ~ I
=
where u ( ~ ) is the non-singular
ak u k gxg
From (35) of Lemma 2.7,
2 ~2 in @(x-b-s) g -1 E u ( ~ )kj u(b) : 6 lim E(x,b k) SxSb 1 J x÷b k
f : bk-b-s
O(0)
To compute d, let a ~ R approach a point g g divc@(X-a-s) = ~ a k and divcO(X-b-s) : ~ b k where 1 1
du(b)
(ui(bj)).
@(s) > 0
6.1 and
b C Fk
-
in R.
a local coordinate
matrix
and
by Prop.
deg D = g-2 < n-2.
by Prop.
implies that
on FI,...,Fn_ I
(@) ~ S0 : %
for some
[ is
= 0
terior of R, divcO(x-a-s) and
thus
where
O(b-a-s)
of points on each contour except impossible
on C, holomor-
s = [-A
with an odd number of points
since
by Cor.
has
IeI = i
an impossibility s £ S0
and @(x-a-s)
(g+l) number of zeroes has the form
for s £ ^ , SO
of degree
O(s) > 0
(~] 6 (O){h S0,..,I,.., 0 with a symmetric
Hf(b) :
@
-
-
H f ( b k)
g
and
Hf(x)
-- ~ i
~.(f)ui(x) 1
divisor of zeroes on C.
However,
bk Hf(b)Hf(b k) = H f ( b ) H ~ ( f ) ( b k) = H f ( b ) H f ( b k ) e x p { - ½ T k k -
~
Uk + s k}
(k)
: - E-l(b,bk)~-l(b,bk)O(s)O(s
+ 2b- 2b k-
i00
.. 0] 0 0 ) e- ~ k ~ Y
129
by (20), (25), (127) and (131).
S i n c e s+2b-2bk -
"
'
"
(k) if
s 6 S0'
we c o n c l u d e that
all zeroes of @(x-a-s)
~k/6 < 0
lie in R for
for
a 6 R
k = l,...,n-l;
and so
near b, and by c o n t i n u i t y
d = deg div_@(x-a-s) = g for all a £ R. Finally suppose that f is a R unitary function on C such that divcf : ~ +a-}{-a with a & R and N
=
~ a. 1 j
c o n t a i n e d in R; then the h a r m o n i c measures
mi([ j) > 0
m. (a) > 0 i
and
of (125) satisfy
g
1 A+a T] uk k=l ik ~+~
:
N ~ m i ( a j) = M i > i ~:i
mi(a)
+
--- i
V x 6 R,
for
M. 6 Z. 1
n-I But
l~mi(x)
+ m0(x)
w h e r e m0, the h a r m o n i c measure
of R with respect to F0, satisfies a similar condition N
mo(a)
+
~_lmO(aj)
= MO _> 1,
M0 6 Z.
Therefore
J n < -
n-i n-i ~ M. = ~ i=0
l
{mi(a)
i=0
+
N Z m i ( a j)} : N + 1 j :i
and any unitary function h o l o m o r p h i c on R must h a v e at least zeroes.
Furthermore,
if the function has exactly n zeroes, equality
must hold in the above inequalities, s :
I
A u - ka E ~g ga
@(s) - s
=
u
and this means that
must be in S0 since~ by (129):
+ k b - @ ( k b) =
+~
for any
n = g+l
0
T
= 0
in
0 T
b 6 F 0.
Using this result,
a solution can be given to an extremal p r o b l e m
for b o u n d e d a n a l y t i c functions as f o r m u l a t e d in [3, p. 123]:
See [2, p. 7]; the i n e q u a l i t y n _< N+I of course holds for arbitmary b o r d e r e d surfaces by the argument principle.
130
Proposition let ~
6.17.
For two distinct
fixed points
be the family of all differentials
cept for simple poles at a and b with the family of functions where
IFI j i.
F vanishing
~ analytic
a and b in R, on
Res ~(x) = i, x=b
R m ~R
ex-
and denote by
at a and analytic
on
R ~ DR
Then
IEb) l ~ ~
DR b+~
with equality
attained if and only if, for
~(x)
= @2(x-a-s)
s = ½ ~
e S0'
E(x,[)E(b,~)E(a,b) 5
02(b-a-s)
E(x,b)E(x,b)E(x,a)E(b,a) ta+b
F(x)
e(x-~-s)
: e @(x-a-s)
E(x,a)
I~1
E(x,a)'
:
1
and
e(½1 u IF(b) l = ~+~
ra+[ ~S(b,a)
e(½]
Proof.
We will find the extremal
enee; the explicit properties.
Now by Cauehy's
with equality of absolute
if and only if
value i; thus
function on C with by Prop.
6.16.
differential with
construction
Theorem,
F and ~, assuming their exist-
IF(b) I = 2 ~
F~ = Sll~ I
IFI : i
on ~R and so extends
II F~ I ~ 2 ~ ~ I~I DR R
on DR for some constant
~
where
~
C R
--IF(x)m(x) : ~(x) e1 to a symmetric
and
dog ~
zeroes on
R u ~R
and that
is a positive
differential
on C
we conclude deg ~
: g.
that ~ actually has no By Prop.
6.16 then,
@(x-a-s)E(x,a) : g
with O(x-a-s)E(x,a)
and by Cor.
6.9,
~ g
+ a + div ~ ; since ~ on R has poles only at a and b, R u~R
deg div C 9 : 2g-2,
F(x)
E1
on DR and F extends to a unitary
divcF = a + ~ - a -
and since
"
u )
will show they satisfy the required
On the other hand,
div ~ = A R u ZR
)
lel : i,
S = ~-a-A
E SO
131
~(x)
O(x-b-t)e(x-b+t)
= r
with
r 6 ~+,
t = ~-b-A
6
TO"
E(x,b)E(x,b)
Since
S O ~ T O = J(C)
and
.[+ff
b+F~
s+t
ib+b s = ½ a~Z
we find
where
= s
~ ~ ~t ~
and
~
b-a
: ½[ u J a+g
~(x) = eI ~
m(x)
b ÷ a,
Corollary
6.18
J(C),
&
air @(x-a-s)2E(x,a) e E(x,a)E(x,b)E(x,b)
-
0(½1 u ~+b
e 0(x-a-s)2E(x,a) - - = Res ¢I r x=b E(x,a)E(x,b)E(x,b)
Letting
+ ½~ u Ja+b
)2E(~,b)
= E(b,a)E(b,b)
the above proof gives: (Sehwarz'
Lemma).
For
a ~ R
fixed,
let ~ be any dz(x)
differential
analytic on
is h o l o m o r p h i c
except at a where
for some local coordinate
and let F be a function vanishing
R U 3R
analytic
on
in the same local coordinate
z.
Then
(z(x)-z(a))
z in a n e i g h b o r h o o d
R u 3R
at a with Taylor development
~(x) -
where
F(x)
of
IFI j i,
2
x : a;
and
= F'(a)(z(x)-z(a))
+ ...
1 ~ IF'(a) I ! 2-~ ~R I~I, with equal-
ity if and only if O(x-[) F(x)
:
~
E(x,a) - - ,
e(x-a) E(x,~)
l~l
: 1
02(x-a) ~(x)
=
@(a-i) ,
and
IF'(a) l =
92(0)E2(x,a)
Observe that the extremal ferential
of Cor.
6.13;
for g0 the Szego-kernel [Ii, p. 22].
i@(0)E(a,[)
derivative
also, the extremal of Prop.
~F'(a)
is the positive
function F(x) is
¢
dif-
~0 (x,a)
6.14, a fact observed by Garabedian
in
For a relation with the span of R, see [36, pp. 97-107].
132
As
an e x a m p l e ,
let R be the
annulus
1 <
IPl
< 1
and
R the
annu-
r
ius
1 <
under
IPl
the
< r
for
p 6 ¢.
anti-eonformal
Then
identifying
= 1
and
r
IPl
= r
= __i, C = R u DR u ~(R) is P s u r f a c e of genus 1 w i t h n o r m a l i z e d d i f f e r e n t i a l s u(p) = dp = P i/r and p e r i o d m a t r i x ~ = [ u = -2 In r < 0 w i t h r e s p e c t to
a compact -u(¢(p))
involution
IPl
~(p)
Jr
the
canonical
B = {r-p,
and
a & C,
for f i x e d
tained
in R i f f
fundamental -in
r21al
Re
z
< Re
-are
=
6
e(
f>
-z)
a 6 R,
z <
~(R)
r
in
u-z)
r l a I.
The
for
a
s =
6
class
if
A = ~i - in r,
in -r___XXa= z
in J0"
~
of the
J!l~///~/~//i
~'o(C)
by
S O and
Jo;
divisor
and
is con-
in a h a l f
defined
circles
0 -< 0 < 2~}
if and o n l y
dive@(
-In
for
Riemann
= 0
z is a p o i n t
of two - in
A = {rei0~
The
parallelogram
is the u n i o n and
basis
O < p -< r - l } .
for
Thus
homology
variety
S
-l~r
S 1 defined~ E R
SO
-Mr
respectively
o by
Re z = 0
fixed,
divcO(
u -s)
=
(v £ [9),
while
dive@(
u - s)
union
of t w o
=
T
-ae
E R
circles
T O and T I d e f i n e d
by
a ~ C,
a r g divc@(
= arg
(resp.
TO).
2~i and ~(z) and
so,
is real
6
u-t)
have
: -~(-z)
for any
:
s =
The W e i e r s t r a s s
-2 in r
= %(z)
h(z)
for
~
a 6 R,
of n, a n d
continuous
since
h'(z)
The v a r i e t y
Im z = 0
functions
z e ~; the
~(~i
for z real. has
zeroes
Im z = ~
for the
the p e r i o d
-
T is the
~ + arg
~ (z)
elliptic
r +z)
and
a (resp.
the s y m m e t r i e s
~ ( i n la}2" + zi + I n
and
S I.
a)
in J0; for
lattice
= ~
(~)
n given
any
t E TI
generated
= ~ (-z) in
for
by
and
(46)
is r e a l
+ i n l a l"2 ( ~
_ ~___d___)
function
+In
Since
in T O at
r + z)
-2 in r h(z)dz
= 0
z = - i n l a I and
21nr
by d e f i n i t i o n - i n rla I , we
133
conclude that there is exactly one zero of h(z) (-In r [ a [ , - i n
la]) C R,
producing kernel
~ [m(x,~(y)) V x,y 6 C,
and this is the point in T O giving the re-
for H2(R)
on the other hand,
in the interval
in Prop.
6.15.
The Bergman kernel
function,
is given by :
- (Re T)-lu(x)u(%(y))]
~
1 ( ~(in
and the Szego kernel function
xy) - ~ +
1 ) dxdp 2 in r xy
for the h a l f - p e r i o d
{O]o T
is
~0(~(y),x)
O(]x u )
1
=
+m
1
2
+m
r-n (xy)n
E
O(in xylO
.
(01 dC~xd9
.~.2
'k
(n+a)r-
(-l)n'a
~9
--¢o =
V x,y
6 C;
xy : i.
-
-
2rt
o0(%(y),x)
+~
r -n
2 +~ _(n+½)2 ~ r (-x~) n+½
is zero when
Observe that as
! d/6~xd9 27
--
r ÷ +~,
for the unit disc D,
xy = -r
o0(~(y),x)
Ix[ < i,
(x~)½
and is infinite when becomes
x £ {;
the Szego-kernel
and Cor.
6.18 implies
1 - x9
that if IF' (a) I <
]F] j i
2~
in D and
°O(~(a)'a)
Ida I
-
F(a)
= 0
i
1 -lal 2'
for some
a ( D,
the classical
then
Schwarz
lemma.
134
Notation T
period matrix,
I and 3
Siegel half-plane,
i
point with characteristics
I~I, i L
~ r
first order theta-function
with characteristies
V1
normalized holomorphic
differentials,
J(C) : Jo(C)
Jaeobi-Picard
3
~ -A ' ~b-a J (C) n HO(D)
variety,
differentials
components
of the divisor class group, sections
of the third kind,
canonical
point in J2g-2'
A
Riemann's
divisor class,
kb
Riemann constants
Hf(x)
differential
L
bundles of half-order
E(x,y)
prime form , 16
S(x)
projective
~(x,y)
bilinear
7
with basepoint
b, 8
formed from gradient
differential
of (0) at f, i0
differentials,
connection,
II
19 of the second kind,
sheaf of germs of holomorphic
i(D)
index of speciality
divcf(X)
divisor of zeroes- poles of f on C
differentials of a divisor
on p. 5.
20
functions
sheaf of holomorphic
convention
5
6
1 ~c
See also the notational
4
of the line bundle with divisor
KC
e
3
normalized
holomorphic class D, 5
~
I~I, i
on C
(class)
D
135
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