Theta functions on Riemann surfaces (Lecture notes in mathematics)

Lecture Notes in Mathematics Edited by A. Dold, Heidelberg and B. Eckmann, ZLirich Series: Dept. of Mathematics, Univ. o...

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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


02[el(z)

is a second-order

[2] [19, p. 39]. theta-function

TLBJ


[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.

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. 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,

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


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_ 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


with

(Za(Pa) , ~

(Pb,t) ~ W ~ U b × 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) : LaO

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

~[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[.+~].}+ +

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. 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 ( 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 fp0.

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

Theta functions on Riemann surfaces (Lecture notes in mathematics)

Theta Functions On Riemann Surfaces

Theta functions on Riemann surfaces