Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups

Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann 483 Robert D. M. Accola ETHICS ETH-HB *00100000135731* ...

Author: Robert D. M Accola

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Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann

483 Robert D. M. Accola

ETHICS ETH-HB *00100000135731*

II II U~IUII II II UlIII II MII III Riemann Surfaces, Theta Functions, and Abelian Automorphisms Groups m

Springer-Verlag Berlin. Heidelberg. NewYork 1975

Author Prof. Robert D. M. Accola Department of Mathematics Brown University Providence, R.I. 02912 USA

Library of Congress Cataloging in Publication Data

Accola, Robert D M 1929Riemann sufaoes, theta functions, and abelian aut omorphism groups. (Lecture notes in mathematics ; 483) Bibliography: p. Includes index. i. Riemann surfaces. 2. Functions, Theta. 5. Aut omorphisms. 4. Abe lian groups. I. Title. II. Series: Lecture notes in mathematics (Berlin) ; 483. QA3.L28 no. 483 [QA333] 510'.8 [515'.223] 75-25928

AMS Subject Classifications (1970): 14 H40, 30A46

ISBN 3-540-07398-1 ISBN 0-387-07398-1

Springer-Verlag Berlin 9 Heidelberg 9 New York Springer-Verlag New York 9 Heidelberg 9 Berlin

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 w 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. 9 by Springer-Verlag Berlin 9 Heidelberg 1975 Printed in Germany Offsetdruck: Julius Beltz, Hemsbach/Bergstr.

Contents

Part I I Introduction

I

2 Remarks on general coverings

3 Resum~ of the Riemamnvamishlmg theorem RAmified normal coverings

7 8

5 Abeliancovers

12

6 Main ~esults

19

Part II i Introduction

32

2 Completely ramified abelian covers

@0

3 Two-sheeted covers

50

@ Other applications

56

5 Closing remarks

63

Part III i Introduction

66

2 Castelnuovo's method and P0-hyperellipticity

70

3 Extensions

7@

The p - 2 conjecture for p = 5

79

5 Elliptic-hyperelliptic surfaces of genus five

81

6 Elliptic-hyperelliptic surfaces of genus three

88

7 Cyclic groups of order three for genus two

9@

8 Some local characterizations

95

9 Closing remarks

98

References

100

Index

102

PA2~ X x)

i.

Introduction.

Torelli's

type of a Riemann surface

theorem states

is determined

class of) one of its period matrices. some property

If a Riemann surface has

by some property

a property which

period matrix at hand. characterizations problem,

by {the equivalence

not shared by all Riemann surfaces

should be characterized hopefully

that the conformal

then this fact

of the period matrix,

is independent

of the particular

The main tool for effecting

is Riemann's

solution

often called Riemann's

such

to the .~cobi inversion

vanishing

theorem.

Riemann's

theorem relates vanishing

properties

of the theta function

the. Jacobian of a surface

to certain

linear series on the surface.

Since special properties existence

of special

will be reflected, properties

on a Riemann

linear series,

via Riemann's

for

surface often imply the

these special properties

theorem,

in special

vanishing

of the theta function.

l)The research for this paper has been carried on during the last several years during which the author received support from several sources, i) Research partially sponsored by the Air Force Office of Scientific Research, Office of Aerospsr~ Research, United States Air Force, under AFOSR Grant No. AF-AFOSR-II99-67. ii) National Science Foundation Grant GP-7651. iii) Institute for Advanced Study Grant-In-Aid.

The special property that a surface might possess considered in this paper is the existence of an abelian group of automorphisms.

This subject has a long history,

The vanishing proper-

ties of hyperelliptic theta functions have been known since the last century

[/~J

Recently Farkas

[@] discovered special

vanishing properties for theta functions associated with Riemann surfaces which admit fixed point free automorphisms of period two.

The author has discovered other vanishing properties for

some surfaces admitting abelian automorphism groups of low order. The purpose of this paper is to present a general theory which will include most of the known results. Part I of this paper will concern the general theorem on vanishing properties of theta functions for surfaces admitting an arbitrary abelian group of automorphismso

Part II will be

concerned with applications

to particular situations where the

order of the group is small.

The case where the order of the

group is two will be considered in some detail. 2) Part III ws

deal with the problem of the extent to which

special vanishing properties characterize surfaces admitting

2) The problem of vanishing properties of theta functions for surfaces admitting automorphisms of period two dates from the nineteenth century. The unramified case was treated by Riemann ~|, Nachtrage p. 108] Schottky-Jung [~q] and more recently by Farkas [9],~0] and Farkas-Rauch ~ ] . For the hyperelliptic case Krazer ~5] has a complete ~reatment. The elliptic-hyperelliptic case was treated by Roth ~ ] o Recently the general ramified case been treated extensively by Fay [l~]. The above remarks are by no means a complete bibliography. For further references to the work of the nineteenth century the reader is referred to the article by Krazer-Wirtinger ~6] and Krazer ~ . The papers of Farkas and Farkas-Rauch contain further references to more modern work.

abelian difficult

groups of automorphisms.

p r o b l e m f o r w h i c h no g e n e r a l

The s e t t i n g Wl, o f g e n u s

analytic. arbitrary normal.

for

Pl'

automorphisms, naturally

T h i s seems t o b e a more

G.

these

P l z 2,

results admitting

setting

however, where

so t h a t

of

abelian G,

W/G

group of (=W0), i s

map, b ,

some r e s u l t s

b : W1 * W0

exists.

Riemann surface,

the quotient

consider

the cover

presently

a closed

a finite

The s p a c e o f o r b i t s

a Riemann surface We w i l l ,

is

theory

is

i n a more

n e e d n o t be

2.

Remarks

arbitrary

on General

n-sheeted

of genera

Pl

meromorphic

Coverings.

ramified

and

P0

functions

M1

abelian group,

covering

respectively. on

W1

of the field of meromorphic subfield of

Let

os index

and let functions

n.

*

*

*

Now let

be the maximal

MA

i:

A

over

(omitted).

Now, canonical

uI

be the lifts, W0o

Then

via

M0

is a

5)

of

M0

in

M I.

group of

and

in the case where

WI,

homology bases

Zl, and let

in

W1

differentials

and

u0

J(W0),

P0 = 0

M 1 = MA

will

5.

from

dual

W1

J(W0)

and

z 0 = b(Zl). W0

Fix

and choose bases

to these homology bases.

and

are defined:

S) If M is a field, M group of non-zero elements If

on

be the field of

to the (dual of t h ~ Galois

A proof

fix a point

the analytic

4)

M0

M1

abelian extension

is isomorphic

follow in Section

J(WI)

Let

surfaces

M0.

Proof:

maps

of closed Riemann

A, as follows:

A = (f e MII fn e M0}/M 0.

Lemma

be an

We now define an important

Definition:

MA

b : W1 * W0

W0

for Thus

into their Jacohians,

4)

will stand for the m u l t i p l i c a t i v e of M.

will be taken to be the one element

abelian group and the theta function will be the function which takes the value one on J(W0).

b,

uI

,)(w~)

W1

u% J

W0

The maps way.

u1

A map

of

W1

as

x0

under

and a

u0

5)

is

follows: b

with

branch

Thus

linearity

to arbitrary

J(W0)

to divisors

is easily

on

is

points

according

divisors

counted degree on

of degree

n.

W0.

zero,

extension

of

M0

a

inverse

is

~

in

to

those

i~ape of

to multiextended

as follows:

then

usual

W0

Now we d e f i n e

seen to be a homomorphism.

abelian

the

the

on

x 0 e W0, ~x 0

always has

W0

in

from divisors

§ J(WI) , again denoted by

is a divisor

unramified

ax 0

extended

now d e f i n e d

for

plicity.

from

are

(w 0 )

by

a map

if

DO

au0(D 0) = ul(aD0). Let

MUA

Ml;

thus

be the maximal

M 0 c MUA c M A c M I.

Lemma Galois Proof:

2:

The kernel

group of

MUA

(omitted).

follow in Section

of

~ : J(W0)

over A proof

5 of Part

§ J(W1)

is isomorphic

to the

M 0. in the case when I.

The case

MUA = M 1

MA = M 1

will

will be

5) The symbol a will be used consistently to denote homomorphism from a b e l ~ n groups associated with W 0 into the corresponding abelian groups associated with W I. The particular group will be clear from the context.

covered in Part II. With the homology bases and the dual bases of analytic differentials chosen, let

(~iE;B0)pOx2p0

and

be the corresponding period matrices where priate identity e[Xl](U;B1)

matrix.

Finally let

E

(~iE;B1)plx2pl

is the appro-

e[•

O)

and

be the corresponding first order theta functions

with arbitrary characteristics.

Lemma 3:

F o r any c h a r a c t e r i s t i c

function

E(u)

function Proof:

on

so t h a t

J(W0) , i s

The p r o o f

of transformation

proof

of this I,

an n t h o r d e r

theory.

1emma a r e q u i t e

the proof

there

is

an e x p o n e n t i a l

E ( u ) e [ X 1 ] ( ~ u ; B 1 ) , as a m u l t i p l i c a t i v e

i s an i m m e d i a t e

parts

in Part

•

theta

adaptation Since

different

is deferred

function.

until

of the simplest

the technics

used in the

from those used elsewhere Part

II.

5.

Resum6 of the Riemann Vanishing

results

in this paper depend on Riemann's

inversion problem.

We summarize

Let

W

a canonical

the existence

p

be chosen, W

of a point

K

J(W)

then there is a integral

Riemann's

p, p ~ I, let

and

theorem asserts

so that if we choose any

divisor

D

on

W

of degree

so that

e(e) f 0, then

D

(mod J(W))o

is unique.

If

e(e) = 0, then the above

equation can be solved with an integral Moreover, at

of the

let a dual basis of analy-

J(W).

in

u(D) + K m e

If

to the Jacobi

let a base point be chosen,

into

of the

6)

homology basis be chosen,

be the map of

e e J(W),

solution

be a closed Riemann surface of genus

tic differenitals u

The proofs

here those portions

theory that will be needed later.

let

Theorem.

e

in this latter case,

equals

Riemann-Roch

i(D),

the degree of whenever

D

Finally,

if

D

i(D)

equals

of

is

p - I.)

--- -2K

Moreover,

0(u(D))

of

divisor,

p - i. e(u)

(By the

-D

indesince

+ K) = 0

divisor of degree at most

is a canonical

u(D)

D.

of

the number of linearly

functions which are multiples

is an integral D

the order of vanishing

the index of speciality

theorem,

pendent meromorphic

divisor of degree

p - i.

then

(rood J(W)).

6) The material in this section is a complete and more modern treatment

covered in Krazer ~5]. see Lewittes [18].

For

4.

Ramified

Normal

Coverings.

b:

be an n-sheeted

(possibly)

group of automorphisms, x01,x02,

Let

W I § W0

ramified

G, of

W1

normal

occurs.

Let

X0

For each

vj, so that above

there are

multiplicity

vj, j = 1,2,

for this cover

(i)

2Pl

where

the total

...

of

W0

the Let

over which

the

be the divisor

x01 + x02 + **. + X0s. x0j

where

need not be abelian.

... ,X0s , s ~ 0, be the points

ramification

covering

,s.

x0j n/vj

there is an integer, branch points

each of

The Riemann-Hurwitz

formula

is

2 = n(2Po

2) + r

ramification,

r, is given by

s

(2)

If

r - n ~ (1 - v i l )

b

is unramified

(r = 0) then most of the following

discus-

sion is unnecessary. Let

Xlj

as a divisor

be the inverse of degree

Xll + x12 + .*- + Xls.

n/vj.

image of Let

X1

x0j

under

be the divisor

Then

~Xoj " VjXlj

and

~Uo(Xoj

z O) ~ Ul(~xoj

b

~Zo)"

considered on

W1

So

(3)

s

There are vj

give

~ ul(aXoj ) vj

2P o

points on

u0(x0j)

denote it by

for

ul(az0).

Let

formula "(3)

(4)

j = 1,2 ....

J(W1)

which when multiplied by ,s.

Similarly,

Fix one of these and n-lul(s

)

denote

which when multiplied by

n

gives

let

vj-lul(az0 ) = (n/vj)n'lul(s by

~j

vjcj ~ 0.

Then dividing

and rearranging terms yields

Ul(Xlj) -= a ( v j ' l u 0 ( x 0 j ) )

where

cj

+ v j ' l u l ( a z 0 ) * cj

depends on the choice of

n-lul(~Z0 )

~ j ' l u o ( X o j ).

and

Now we determine WI, in terms of D0

J(W0)

vj-lu0(x0j).

a certain point of

ul(az0)

K I, the vector of Riemann constants on

aK 0 and other quantities already defined.

be an integral canonical divisor on

W 0.

Let

Then

S

a_D0 +

a divisor of degree W1.

[ (v~ j=l J

l)xlj

n(2p 0 - 2) + r ( = 2Pl - 2), i s c a n o n i c a l on

Now -2K 0 ~ u0(D 0

(2P 0 - 2)z0)

So

-2aK 0 ~ Ul(~D0) - (2P0

Since

-2K 1 -: ul(aD0)

2)ul(az0).

S

we have

§

j=~l(Vj

1)Ul(Xlj)

10 S

-2X 1 ~ -2~K 0 + (2P0 - 2 ) U l ( ~ Z 0 )

+ j=~l(Vj

l)Ul(Xij)

"

Substitute formula (4) into this last equation.

-2K 1 5 -2aK 0 + (2P0 - 2 ) u l ( a z 0 )

+

S

l) la(vj'luo(Xoj))

+ vj-lul(aZO)+

cj]

j=~i(vj S

Noting that

~ 0

vjcj

and

2P0 - 2 +

[ (vj j=l

l)vj "I = n-l(2Pl - 2),

and dividing this last equation by two gives

(5)

K1 = aK 0 - ( P l - 1 ) n ' l u ] ( a z 0

where

e0

and

eI

are points

of

) - ae0

- el

J(W 0)

and

J(W 1)

respectively

so t h a t s

(6)

2e 0 -

~ (vj j=l

(7)

2e 1 = "j =~lCj 9

Thus

2ne 1

l)vj-lu0(x0j)

7)

S

If

~ 0.

the cover

k : W1 * ~0

is unramified,

we h a v e s i m p l y

the equation

7) For f u t u r e r e f e r e n c e s we d e f i n e e O so t h a t ne 0 = ~ ( n ( v j 1)/2~vj-luo(Xoj). If n is even this will a l w a y s be t r u e . I f n i s odd so a r e a l l vj se t h a t i f i s odd we d e f i n e

e0 =

s[ ( ( ~ j j=l

1)/2)vj _lu 0 ~x0j).

n

11

(8)

K1

---. a I ( 0

where

2e I ~ O,

e0

taken

is

-

(Po-1)ul(gzD)

since

to be

zero

PO

-

1 =

in this

e 1

(Pl-l)n "I case.

in

this

case.

~2

8) 5.

Abelian

Covers.

In this

(possibly)

ramified

an a b e l i a n

group of automorphisms of

have fixed

points.

the set

abelian

section

Let

cover;

R

G

isomorphic

The g r o u p suppose from if

A s

W0.

is,

however,

the

f § Xs

If

f

identity

is

and

have a function s

if

f

field

A

functions

is

R.

kernel

to

R.

easily

H1

over

Thus,

s e e n to be then

f/g

Thus t h e map map i s o n t o

extension

under

os a

w h e r e we G, we s h a l l

Xf. M0

c a n h a v e as a v e c t o r

to each character

then let

lifted

This completes

invariant

character

For

G.

In the above situation is

R

isomorphism.

That this

X.

is

group,

under

HO.

G

group of

same c h a r a c t e r , in

s

G; i . e . ,

a function

X e R, t h e c y c l i c

corresponding

function,

•

into

for

to the

is

is

may

os

a finite

invariant

a n d so l i e s

M1 = HA.

extension

fn

the

whose d i v i s o r

s

an a r b i t r a r y

is

yield

is

isomorphic

and

f

for each

corresponds

Thus t h e space basis

g

an i s o m o r p h i s m o f

o f Lemma 1 when

G

i s no c a n o n i c a l

where

character

g i v e n by t h e f i x e d

say that

of

whose e l e m e n t s

the multiplicative

canonically W1

where

of characters

Since

there

loT = •

s e e n by e x a m i n i n g ,

proof

s

although

Then the d i v i s o r

T e G, t h e n

gives

H0

G

i s m e r o m o r p h i c on

a character.

is

to

set

b : W1 ~ W0

W0 = Wl/G

W1

into

complex numbers of modulus one. is

i.e.,

be t h e

os h o m o m o r p h i s m s o f

assume

f X

of

R.

If

= n -1 ~ x ( T ' I ) f o T . TeG

8) The a u t h o r w i s h e s t o e x p r e s s h i s t h a n k s t o P r o f e s s o r H. S. N a r a s i m h a n f o r many v a l u a b l e d i s c u s s i o n s concerning the material of this paper, especially this section.

13

Then

f =

[ f and the f's which are not xeR X X independent since they correspond to different The next

of

this

part

paper

classical

of

the

section

discussion

statements

elliptic

this

of

of

the

theta functions.

discussion

zero

linearly

characters.

generalizes

to

half-periods

vanlshZng

are

the

that

context

procedes

properties

of t h e

If the cover is u n r a m i f i e d

the

hyper-

then this

is unnecessary. I

Lemma 4: there X0

Let

is and

(9)

a

be

a divisor an

(=)

D0a

~-tupIe

= ado=

a function

of

on

in

W0

M1

so

that

a n e M0.

Then

not containing any point of i (al,a2 .... ,a s ) so that

integers

s [ ~,.•

9

Moreover

S

(I0)

nDoa +

~ (n/vj)ajXoj j=l

-0

on

WO.

!

If

a

!

has

same character

the

as

same properti]es

a,

and

!

(al,a2,

t

I

as

a, a !

...

,as)

corresponds is

the

to

s-tuple

the for

!

a

then

for

j = 1,2,

....

s

i

aj

Proof: given

Since form.

(a] an

this

function

left

hand

side

is is

on of

)

invariant

a function

W0, Co)

and

formula

(]0).

under lifted the

G,

it

via

b

from

of

a0

must

with

the

given

divisor

For

a

must

be of

the

W0 ( c a l l be

the

14

properties

it follows

character;

that is,

that

a/a'

a/a'

corresponds

itself

is in

to the identity

M0

Thus

!

aj

ajm

0 (mod ~j)

Lemma

4 shows

as a point

in

for all

j.

q.e.d.

that the s-tuple

(al,a2,

Z 1 x Z 2 x ... x Zv

. . . . as)

viewed

(call this abelian

group

S

V)

depends

Let

Xj

only on the character,

be the smallest

and consider

the map

X, to which

non-negative

R § V

it need not be onto. cyclic

extension

homomorphism

Lemma 5: Proof.

a given

of the points There

exists

group

.

X

(mod ~j)

a

There

case shows

which generates

that

an unramified

the homomorphism

are, however,

properties

of this

which we will need later.

For

Let

an

the zero s-tuple,

need not be one to one.

aj

Xs).

The hyperelliptic

Since

gives

of

corresponds.

given by

X § (XI,X 2 .....

This map is a homomorphism.

residue

a

be

of

j, the

there stabilizer

Xlj.

Call

a character Thus

is

there

X

a X e R of

the point which

one

q.

so (and

Xj ~ 1 .

therefore

Then

is faithful

is a function,

that

all)

T 2 = id.

on the cyclic

a, corresponding

to

We can find a local parameter,

z,

so that

~oT - ~

where

~ - Exp(2~i/vj}.

at

so that

q

z(q) = 0

and

T

has the representation

15

Tz = wz.

(ii)

Now

Thus in this parameter

a(mz)

=

wa(z)

a(z) =

~ ~.z k near k=t z Formula (ii) implies that

Consequently

Lemma 6:

q

for some coefficients

~j = 0

aj ~ i (mod vj)

For any X e R

unless

and so

•

~k

j ~ 1 (mod vj). = I.

q.e.d.

we have

S

(12)

~. (n/vj)Xj --0 (mod n) j=l

Proos

By Lemma 4, formula (i0) S

n deg D0a + j=l ~ (n/vj)aj - 0

since

deg x0j = i.

Since

Xj - aj (mod vj) the result follows. q.e.d.

Now consider

S = (SI,B2 ....

for each

Assume the

j.

,Bs) 9 V

where

0 -< 8j < vj

Bj's are chosen so that

S

(13)

~l(n/vj)Bj =- r/2 (mod n).

j=

where

r

(14)

Bxj ~ Bj + Xj (mod vj)

where formula

is the ramification.

0 ~ Bxj < vj.

For

X e R

For

X e R

define

define

tX

Bxj

by

by t h e f o l l o w i n g

~6 S

(IS)

~ (n/vj)Bxjj=l

By Lemma 6

7.

Lemma

(16)

Given S e V

Z t

=

j.

satisfying formula (14) we have

0

~ !(n/vj)Bxj

= j~(n/vj)~RSXj

R Fix

Since

Xj = 1

numbers

0,1,2, . . . . vj

through

R.

for some 1

•

(n/~j)

The same is true for

R~Sxj = (n/vj)CCvj

Thus

txn.

this is possible.

xeR X

Proof.

(r/2)

~ ~(nluj)Bxj R j

By formula (15)

2

Xj

runs through the

times as

runs

B•176 Consequently

- vj)/2) = nCvj - l)/Z

- ~(nluj)(n(~j j

~((r/2) - t•

- 1)/2) = (nr)/2

= (nr)/2.

R

The r e s u l t

X

q.e.d.

follows.

The next part os this section examines more closely the divisor in formula (i0). By Lemma 4, formula (10) S

u 0(nD 0

§ j=1 ~.(n/v -i)a.x J 0.) J -: 0

Thus there is a (1/n)-period in ca, (since it seems to depend on

J(W0) , provisionally called a) so that

17

(17)

If

u0(D0a)

and

a

s[ a~v~ j=lJ j

+

a'

_lu0(x0j) -

s a

correspond

to the same character

then

!

D0a - D 0a ' + ~((aj Thus

9a ~ ea,.

character map

Consequently,

to which

X * ex

Applying

a

from

s

to formula

formula

)x0j ~ 0

depends

o n l y on t h e

s o we d e n o t e

it

e . X

The

is clearly a homomorphism.

(17) and letting

d = deg D0a

gives

- dz0) + ~aja(vj'lu0(x0j) ) ~ a e a .

(4) gives

ul(aD0a~

~aj[ul(Xlj)

But

ea

corresponds,

R * J(W0)

au0(D0a

Applying

aj)/vj

aD0a + ~ a j x l j

~ 0.

and so

d + ~(aj/vj)

Xj ~ a j

(mod v j )

Lemma 8.

(18)

- dUl(s

+

- vj-lul(az 0) - cj] ~ ~ea

Consequently

= 0.

we o b t a i n

nd § ~ a j ( n / v j )

Since

vjcj

~ 0

the

following

= 0,

and lemma~

S

~e x ~ - ~ I X j C j

J We conclude

this section by giving a proof of Lemma 2

in the case where

b

: W1 § W0

is an unramified

In this case Lemma 8 shows that the map phism from

R

into

ker a.

If

e• = e•

• § ex then

abelian cover. is a homomor-

18

u0(DoQ)

E u0(D0a,}

character. that

Thus

t h e map i s

d e g DO = 0 , principal corresponds

then

and so X " X'

onto,

a/a'

corresponds

a n d t h e map i s

suppose

h e ~ 0.

a_u0{D 0) ~ Ul(~D0}

and invariant

under

t o some c h a r a c t e r

G.

If

X, a n d

to the

identity

one to one. If

~ 0.

To show

u0(D 0} ~ e Thus

aD O = (a} e = e•

~D 0 then

where is a

19

6.

Main Results.

The statement and proof of the main result

of this paper are technically complicated.

However, the idea

of the proof is exactly the same as in the unramified case. 9) In the

statement

summarized,

but

At t h i s notation. sion

let

us

are

is

related and

then

some o f

introduce reminded

homology bases

function

row v e c t o r s ,

proof

the ~revious

notation

is

all.

The r e a d e r

results

theta

not

point

canonical

all

of the

to

that

on

W0

these

is

abuse

throughout and

bases.

z = ~ i ~ + B~ there

a convenient

W1 If

where

g

an e x p o n e n t i a l

this

are

discus-

fixed

O(u;B) and

of

h

is are

function

and a real

E(u)

so

that

O(u + r ; B )

In this context write

= E ( u ) O[Igl ( u ; B ) .

0[T](u)

for the usual

0[~l(u;B). L

=l

This notation will be extremely convenient and will lead to no confusion provided the canonical homology bases remain fixed.

Theorem 1:

Fix

s

BI,B 2 . . . .

integers

0 ~ 8j < ~j

and so that

$ =~ (n/vj)Sj

r/2

,B s

for

n.

For each

X

9)

See Accola [q].

in

R

j = 1,2 . . . . .

s

is an integer which is a multiple

j t of

so that

define

B•

by

20

(19)

Bxj 5 Bj + Xj

and

0 ~ BXj < v j .

(20)

j=~l(n/.j)Sxj

(mod vj)

Then d e f i n e

by the equation

tX

s

Fix

= r/2

go e J(Wo).

- t•

X e R consider the equation

For each

s

(21)

go + eo " j=; 1 8 x j v i l u 0 ( x o j

Define the non-negative

) + z x -- u 0 ( ~ 2 1 5 + K0~

as follows. If formula (21) X admits no solution with an integral divisor a of degree X P0 - 1 + t X let N X = 0. If formula (21) admits solutions with an integral divisor

integer

aX

N

of degree

P0

1 + tx, let

N X = i(o x) + t x

where

i(av)

Then at u

=

is

the

index of

8 4= 1X[ 8.c.3 3 - e

s x-

(U;Bl)

vanishes to order •

NX

~go"

Proof.

Let

O[~8.c. - ~ Jj J If NX > 0

N

be the order

eli (U;Bl) N = 0 X for m

at

for all

a_g0. X

of

We show first that

characters

N >x!RN X.

there is nothing to prove.

(z) ,X (2) , .-. ,X tm). X f

ence introduce the notation Nx(k),r

of vanishing

N (k),e (k),Sj(k),t(k) Now fix

X [k)" "

Assume

~

and apply

For convenifor a_ to

21

formula (21). on

Denote by

~(k)

an appropriate integral divisor

W0.

ago § ae0 _ ~sj(k)a_(vj "lu0(x0j))

In this equation replace aK 0

by formula (5).

§

_ae(k) ---au0(~ (k) ) + aK 0.

a(vj-lu0(x0j))

by formula (4) and

We then obtain

ag o + ae 0 - ~sj(k)[ulf~Xlj) - vj'lul(az0)-

=- ul(a--~(k))"

(P0

1 + t(k))ul(az0

+ K1 + (Pl " l ) n - l u l ( a z 0

Since

cjl + ae (k)

)

) § a--e0 § e l "

~SJ (k)~''13 § (P0 " 1 + t (k))

= n'l(p I

l)

,

~Bj(k)cj ---XSjcj§ Xxj(k)cj and

~xj(k)cj - - ae (k)

(22)

ag o + ~.Bjcj

where t h e d i v i s o r ~Bj(k)(n/vj)

we obtain

e I --- X B j ( k ) u l ( X l j )

~8j(k)xlj

+ a_a (k)

+ n ( p 0 - 1 + t (k))

Suppose f i r s t X e R with

that

NX 9 O.

N(1) I= i ( ~ (1))

m = 1, t h a t

has degree Pl

1.

is,

there

i s o n l y one

By t h e Riemann-Roch t h e o r e m

§ t (I))

i s t h e number o f l i n e a r l y

multiples of

-o (I)

lifts via

to a multiple of

b

or

+ ul(a__~(k)) + K1

on

W 0.

Any multiple of

independent

-o (I)

on

W0

-(aa (I) + ~8j(1)Xlj )

on

W I.

22

Since

the degree

of

(a_a (1)

+ [8j (1)x lj )

Riemann-Roch theorem implies divisor

is

at

~g0 + [ S j c j If

~

- eI

N (1)

j(k)ul(Xlj)

divisor

is

at

(22)

8(u;B1)

least

of

H (I) H (k)

,m

let

H (k)

the integral at

is,

k = 2,3 ....

N ~ N( 1 ) . ,m

that

+ ul(a_a(1) ).

be a f u n c t i o n

on

W1

whose

as f n l l o w s .

(aa(1) _ + ~Sj (I) Xlj).

be a non-zero constant function. is invariant under

G, H (k)

which will now be shown to be For each X (k).

that

+ UlCa_a(k) ) ~ ~ 8 j ( 1 ) U l ( X l j )

...

1

vanishes

N(1);

shows f o r

(H (k)) = (a_o(k) + ~8j (k)xlj)

Let

Pl

the index of this

and so

to order

m > 1 formula

k = 2,3,

For

least

that

is

k

let

a (k)

Since the divisor

corresponds to a character

X(k)/X (I). be a function which corresponds to

We may write the divisor of

a (k)

as follows.

(a(k))_- a_(D0(k)) + ~• where

DO (k)

may i n c l u d e

,jXlj = aXOj.

points

By f o r m u l a

Uo(D0(k))

+

of

XOO

This

is possible

(17)

•

(k) "J -lu0 (x0j)

--- e (k)

Substituting this into formula (21) we have

since

23

g0 + e0 + ~(Xj (k) " Bj (k))~j'lu 0 (x0j) + u0(Do ok)) 5 u0(o(k)) + K 0

for

k = 1,2 ....

function on

W0

,m.

Thus for each

k = 2,$, ... ,m there is a

whose divisor is

~.(Xj (1) - Xj (k) . Bj (1) + Bj(k))vj-lx0j ] +DO

since

(1)

(Xj (I)

I f we l i f t

this

- o

(1)

- DO

(k) + ~(k3

Xj (k))

(Bj (13

Bj (k)) ~ 0

function,

via

to

b,

W1

the

(rod vj). lifted

function

has the divisor

~.(Xj (1)

- Xj (k)

,3

+ a_(D0(1)

which is

the divisor

corresponds

to the

to the character Let .~(k)

L (k)

on

W0.

theorem. to

W1

p

Let via

independent

- a(1)

of

§ o(k))

character,

Thus

H ( k ) a ( 1 ) / a (k)

and so

H (k)

corresponds

x ( k ) / x (1) k = 1,2, Then

be t h e

Thus

functions

,m

be t h e s p a c e o f m u l t i p l e s =

a L (k)

lifts is

Let

of the functions the span of

in

the

H(k)aL(k) functions

H ( k ) a L (k)

is

of

by t h e R i e m a n n - R o c h

which are multiples

by m u l t i p l y i n g

Then e v e r y f u n c t i o n

e , e

N (k) = dim L (k)

-(a_~ (k) + ~ B j ( k ) x l j ) . obtained

_ D0(k)

HCk)a(1)/a (k).

identity

aL (k) b.

f~jCk))xlj

- Bj C1) §

N (k)

L (k)

linearly

of

be t h e f a m i l y in

in

aL (k)

by

a multiple

of

of functions H( k ) .

24

-(as (1)_

+ XSj (1)xlj).

H(2)a_L (2)

9

Thus the families m

.... H(m)aL (m)

multiples of

represent k!l N

-(a__o(I) + ;Bj(1)Xlj)

to different characters. that

0(U;Bl)

H(1)al (I)_, , (k)

since the

As in the case

vanishes at

linearly independent H(k)'s

correspond

m = i, we now conclude

a_g0 + [8~cj

eI

to order at least

m[IN (k) ; that is, N ~k-~IN(k) k--

m

The p r o o f the

above

that

argument.

vanishes

to

to prove

so assume

there

an integral

is

N ~ ~ N tk)"" is essentially k-l Suppose then that e[[Bjcj

order

N

at

u ~ ago o

N > O.

If

By t h e

divisor,

~1'

N = 0

the

reverse

- e l ] ( U ; B 1) there

is

Riemann vanishing of

degree

Pl

- 1

nothing

theorem on

W1

that

(23)

ag o + XBjcj

e I -= Ul(~ I) + K I.

By the solution to the Jacobi inversion problem there is an integral divisor of degree

(24)

P0

on

W0

so that

gO + eo - ~Sj'j'lu0(x0j ) --- u(~0) + K0"

Applying

s

a- to formula

+ a-eo

Now apply formulas

(24) yields

XSja-f~j'luo(Xoj)) (4) and (S).

= a-Uo(r

of

§ a-Koo

so

25

~g0 + a-e0 " ~Sj[Ul(Xlj)

Ul(~0)

" P0Ul(~Z0)

Now eliminate

~g0

" vj-lul(~Z0)

+ K1 * (P1 " l)n'lul(~z0 ) + ~e0 + el" between this last equation and formula (23)

noting that in the coefficient of

ul(az 0)

~ S j v j " I + PO - ( P l " l ) n ' l

where

t (0)

is

- cj]

defined

we have

= I - t (0)

by

~8j~j'ln

= r/2

- t(O)n.

We then obtain

Ul(~ l) ~ ~SjUl(Xlj)

+ Ul(a~ 0) + (t (0)

1)Ul(~Z0).

Let

(25)

D 1 - ~BjXlj § e~ 0 + (t (0) - l)az0.

Then

D1

is

a divisor

of degree

~ 1 ' and i n v a r i a n t

under

each

f

X e R

let

X multiple of

-D 1

X

Then "

Let

= ~ x(T'I)f TeG

f o T.

-D 1

N - ~ N~. xeR

linearly

equivalent

be a m u l t i p l e Then

f

of is

-D 1.

also

to For

a

X

since each function

the space of multiples of N 1 = dim L•

G.

P l - 1,

f o T

is.

which correspond to

Let •

LX

be

and let

26

If

f(#O)

is

in

L•

then

divisor

the

of

f

can be written

S

(f) = aa 0 + ~ f.x

j-1 J

-

lj

integral divisor on (for each

DI

where

0 s f. < Vo

W 0.

Since

f

corresponds

x$

It follows that

fj = 6Xj.

Thus the degree of

(26)

n(p 0

a0 If

Zfj Xlj

1 + tx).

is

and so

= a(a~

- aO).

identity character we see that that the map

f0 § f(f0 ~ ~)

of mulitples of + iX"

-o 0

on

is of

DI

is another function in

(fl/f)

ao 0

Thus

is an integral divisor of degree fl

N1 X = i(ao)

we have

(14)).

r/2 - txn

LX

( f l ) = ~a~ + ~BxjXIi -

and s o

X

is an

(mod ~j).

(See formula

(f) = a--aO § ~BXj~ lj

where

to

a0

(mod vj)

fj ~ Xj + Bj

or

degree

and

)

j)

fj - S j s

W 0.

)

W0

Since

1 § t•

on

then

D1

fl/f

e 0 ~ a~

P0

corresponds on

W 0.

to the

It follows

is an isomorphism from the space onto

L x.

Consequently

27 For

f

in

LX

we combine formulas

(f) ffi aa 0 + ~S•

(24)and (25) to obtain

a_(60 + (t o . l)z0)

~BjXlj.

By formula (17) we have

e x

or

u0(%) + ~Bxj.j'lu0(x0j

-

u0(%)

- XBj,j'lu0(x0))

u(~0 ) + ~Sjvj'lu0(x0j) - u0(o 0) + XS•

Substituting this into formula

(24) yields

go § e0 " ~S•

We conclude that if

- ex

) + e• ~ u(~ 0) § K 0.

N •1 9 0

then

•

satisfies the conditions

required in the statement of the theorem for formula have a solution. list

•215

Consequently ... ,•

Consequently

if

N •1 > 0

then

•

(21) to is in the

given at the beginning of the proof. q.e.d.

N - X N~ ~k!l N(k). •

Since a

t•

X t ~ 0 either all t are zero or else there is • • • which is positive. Considering these two possibilities

yields several corollaries.

Corollary exponential

(27)

1.

Suppose function

H(u)O[~Bjcj

ty = 0 of

u,E(u),

- ell(a._u;B1)

for

all

• e R.

and a c o n s t a n t

Then t h e r e ~B ~ 0

is

an

so t h a t

28

= ~8-FFo[eo xeR Moreover,

as an

of formula

n th

that for

order theta-function

(27) has a (1/2)-integer

is independent of

Proof:

- ZSxjvj'luo(xoj ) . ex](U;B0).

8 = (B1,B2,

By Lemma 3 t h e r e

is

J(W0).

theory

characteristic [~8jcj - e 1]

after

is

an

the statement

of this

n th

each side

... ,Bs).

order

function

n th

By t h e same g e n e r a l i z a t i o n

mentioned

J(W0)

theta characteristic which

an e x p o n e n t i a l

E ( u ) e [ X B j c j - e l ] ( a u ; B 1)

for

order

E(u) theta

function

of transformation o f Lemma 3, t h e t h e t a -

function

is

rational

since

is a rational theta-characteristic.

We now consider the theta-characteristic on the right hand side of formula (27).

of the product

Since theta-character-

istics add when theta functions are multiplied,

the character-

istic in question is

(2s)

X

so

Eeo

xeR

As i n t h e p r o o f

~

8•

+ eX].

o f 1emma 7

~BxJVj'lu0(x0j) " ~nvj'l((~j2 - vj)/Z)~j'lu0(x0j) a sum i n d e p e n d e n t

of

B.

By f o r m u l a

ne 0 = Z ( n / 2 ) ( ~ j

(6)

and n o t e

- 1)vj'lUo(Xoj).

(8)

29

Thus

xeR

Since

Ee o - ~ B •

+ e•

~

X § ex

is a homomorphism

= ~e

X

,

and

-~X is an element of XeR order two we see that the characteristic in formula (28) is a (ll2)-integer characteristic.

Now fix degree

P0

solution.

go e J(W0). 1

for all

Since •

t•

m

0

such that formula

vanishes to order

N x (= i(~x))

has

(21) admits a

Consequently,

+ cx;B0)

whenever formula (21) has a

the product on the right hand side of

of formula (27) has order

N = ~N X

Since

has order

8[~Bjcj - el](U;B I)

at

go" N

left hand side of (27) has order at least function on

X, o•

Thus

8(g 0 + e 0 - ~8xjvj'lu0(x0j)

solution.

for all

J(W0).

at

~g0 9 J(W1), the

N

considered as a

Thus the quotient of the left hand side

by the product on the right hand side is an entire function on cP0

Because the theta-characteristlcs

involved are rational,

a suitable power of the quotient is an entire function automorphic with respect to the periods defining quotient is a constant,

~8"

That

~B ~ 0

J(W0).

Thus the

follows from the

theorem since the two sides of (27) are non-zero for the same values of

go' and there are clearly values of

the product non-zero,

gO

which make q.e.d.

30

Corollary

2:

If

is an unramifs

b__ : W1 § W0

abelian cover,

then

I'i(u)O[el](a-u;B1) = ~ee-~ker aO[e](U;Bo) 2e I ~ 0.

where

This is now immediate since t h e map

Proof:

isomorphism os

Corollary 5: BZZBjCj for any

R

onto the kernel os

Suppose there is a

in

X

iS an q.e.d.

a.

~ 0. Then X vanishes to order at least xeR~max (0,tx)

el](U;B1) u

X § r

a_J(W0).

X 9 R

so that

For the general point on

t

aJ(W0)

this

lower bound is achieved. Proof:

(21)

For each

go + e0

X 9 R

reconsider formula (21)

~BXjvj'lu0(x0 j) + e x ~ u0(~ X) + K0

where we wish t o solve this equation with an integral divisor a•

of degree

P0 " 1 + t x.

regardless of what

gO

t >1 X X To p r o v e

go e J(W0)

the

N x = i(~ x) + t x for any

tX 9 1

this is always possible

Consequently, for any

t >1 (iCa x) X

last

so that

is.

If

+ ix) ~xeR X max (0,ix).

assertionwe

show t h a t

N X = max (0,iX)

for all

we need show that there are

X, the following:

(i)

u = ~g0'

if

t

X

~ 0

there

are

X.

Since

g0's

which satisfy,

the formula (21)

31 admits

no solution

But f o r

each

satisfying least

X

these

one in

have degree

aX, a n d it

is

has

J(WO). P0

index

easily

lie

For if

zerop

the

tx 9 0

that

If

t•

general

the proof

the

on a s e t

t x -< 0

1 or less. Since

if

seen

requirements

P0 " 1 + t X 9 P0" ~P0

(ii)

then 9 0

then

set

of

gO's

-- 0. not

of co-dimension

at

~X

to

is

then

Integral

of the

i(Ox)

required

aX

has

divisor

corollary

is

degree

of degree

complete. q.e.d.

Remarks:

A necessary

Corollary

5 be applicable

This

follows

In case periods other

from formula

P0 = 0

e(u;B1)

4)

This

not

sufficient

to a ramified

condition cover

is

that that

r ~ 2n.

(20).

Corollary

is non-zero,

(1/2n)-periods

(See note

but

0(u;B1) generalizes

I says

that

and Corollary vanishes

on c e r t a i n 5 says

to various

the hyperelliptic

(1/2n)-

that

on

orders.

situation.

PART I I

1.

Introduction.

results

of Part

where

n - 2

other

In Part

cases

will

where the

because

it

will

admitting

some p r e l i m i n a r y

class Part

section. all

the

can be quickly

and five.

these

cases

are

the

derived

that

vanishing

included

properties the

those

we m u s t d i s p o s e

2 we show t h a t

after

chosen

group.

some o f w h i c h i s

derived

to

three

cases

vanishing

case

two,

abelian

In section

the

in addition

sometime characterize

the particular

material,

of covers, I,

function

the particular

examining

introductory

theory, III

apply

The m o s t s i m p l e

have genus

general

theta

we w i l l

i n some d e t a i l

b e shown i n P a r t

of the

Before

surfaces the

paper

cases.

be considered

illustrating

surfaces

of this

I to particular

Besides

properties

II

of

in this

for

a large

o f T h e o r e m 1,

case

P0 = 0

has been

investigated. The n o t a t i o n The f i r s t up i n t h i s

the map

of Part

I will

o f t h e two p r e l i m i n a r y

introduction

concerns

~ : J(W O) § J(WI).

Yl'

"'"

'Y2P0'

W1

respectively.

be continued.

and

y~,

....

considerations

explicit

matrix

to be taken

formulations

for

Suppose canonical homology bases, Y2Pl

The i n t e r s e c t i o n

have been chosen matrices

on

(7 i x 7 j )

W0

and

2p 0 • 2p 0

33

= J0

2Pl x 2Pl

(y~

and

x yi )

= Jl

are

,.Io E

where

is the appropriate identity matrix.

We now define a map, a, from singular one-chains on singular one-chains on

W I.

If

parametric disc of

and

k

ay

is the

n

same way as

W0

copies of y.

If

b

y

y

W0

into

is a small arc defined in a

is unramified over this disc, then

in

b-l(y), each copy oriented the

is branched over

y

then the definition

m

is suitably modified.

Now extend

a

to arbitrary one-chains by

linearity. If on

HI(Wi)

is the first homology group (over the integers)

Wi, i = 0, I, then we can consider

HI(W0)

into

(aij)

so that

HI(W1).

a

Thus there is a

also as a map from

2P0 x 2Pl

integer matrix

2P I

~Yi

=

Now define a map singular one-chain on i = 0,1

j=l aiju j

i = 1,2, ... ,2P0.

a : C p0 § Cpl W0

and

as follows.

du i = (dUil,dUi2,

If

7

is a

... ,dUip i)

,

is a basis of analytic differentials dual to the given

canonical homology bases,

let

~ fduo = fdu l,

34

Since

is a mapping of homology t h e map

a

a : cPo . cPl

m

takes

periods into periods and so reduces to the original a, : J(Wo) * J(W1)

by reducing modulo periods.

and l e t

f ~Oi = ]duo

let

(~iF Bj) pj

"

and

j ffi 0,1

2pj

X

~j

For

~lj = [dUl" ~j'

Yi Since

a

Iduo = .IdUl Yi

~ijyj 3

or

2P 1 a_~oi =

j~=laljaij

we see that (1)

a_~o = ~ l a

where in this context

~

a

2Pl x 2Po

is a

Pl x PO

complex matrix and

integer matrix.

Since the intersection matrix of the nJ 0

(2)

it s

aYi's

that

nJ 0 - ( a y i x a y j )

. ajla.

is seen to be

is

35

Let

where each

aj

is a

P0 x Pl

matrix.

Then formula (2)

leads

to the equations

(~)

e2~ 1 - al~ 2 = 0

t

~4~1

3~2

~ 4 ~3

~4

nE 0

.

With these considerations we will indicate hew Lemma 3 of Part I is proven.

Lemma 1 :

Let

(i)

describe the map

(2)

a_~0 = ~I~

a : J(W0) § J(WI)

where

nJ 0 = ~J1 ~.

Then I

"~2

36

where

~(u)

is

a~ n t h o r d e r

characteristic

Proof:

If

[;]

n - 1

transformation first

order

(2)

formally,

the

theory

result

theory

for

is

first

transformations. forgetting ~

result

for

is

first

function

for

J(W0)

with

w h e r e 1)

the

and a s s u m e t h a t then

theta

a

simply

order

order

a restatement theta

I f we c o n s i d e r

for

and

simply

functions

square

matrices

a restatement

of the

theta

functions

under

under

formulas

t h e moment t h e p r e s e n t are

of the

(1)

and

context,

(i.e.,

P0 = P l ) '

transformation n th order

trans-

f o r m a t i o n s 2) . The s l i g h t this

degree

lemma f o l l o w s

go t h r o u g h

when

w o r k when

~

are

from the

P0 ~ P l ;

and

satisfied.

of generalization

a

are

observation

that

is,

that

all

rectangular

We o m i t t h e

needed

for

the

the proof

classical

the matrix

of

proofs

computations

and formulas

(1)

and

(2)

details. q.e.d.

Another varying curves

1)

If

the

conformal

on t h e v a r y i n g

A

up o f t h e 2)

classical

is

a square

diagonal

See Krazer

technique structure

be used

o f a Riemann s u r f a c e

that

matrix,

SpA

will

be t h e

column vector

A

taken

in the

same o r d e r .

166.

of

so that

are

of

"squeezed

is

surfaces

elements

[ISJ p .

which will

to points"

and the

made

37

limit "surface" We will

formulate

formulations

In the a

is the union of two or more punctured

of this procedure

annuIus

radially

conformal

the procedure

l~(z)l § 1

on

equivalent

as

Izl § I.

extend the definition 0 ~ t < 1

a conformal

A'

let

equivalent

definition

~t

outside

of

structure but as

to

{0 < M < [z I < ~}.

~

and let ~t"

to another

S(t)

defined by

U t.

t § I, S(t)

"approaches"

case we will say the unit circle

Necessarily

in the unit circle. be the annulus with

Then

A(t)

annulus.

(0 -< t < I)

Now extend the

sphere by letting

S(t)

A'

A(t)

be the Riemann Then

be

A = (0 -l < ]z[ < p}

by reflection

to the Riemann

A, and let

U(= U(z)dz--) H-f which defines a

differential

defined by

is conformally

let

so that in this new structure

Ut : t~

structure

< 1}

In the annulus

of

Other

may be found in U~] and [17].

symmetric Beltrami

is conformally

For

a little more precisely.

A' = { p - 1 < [z]

structure

surfaces.

~

be zero

sphere in the conformal

is conformally

two p u n c t u r e d is "squeezed

a sphere,

spheres.

to a point"

In this as

t § I. Now let CI,C2,

W

be a Riemann surface

... ,C r

be

divides

W

disjoint

annuli,

analytic

Suppose

p. curves

there

circle

A.. J ~ = 0

0 ~ t < 1

tu

each of which

are pairwise

We obtain in

a global Beltrami differential r W - ~J A. and ~ equal in each A.

j=0~ differential

Let

Aj, j = I, 2, ... )r so that A. is conformally -i 3 {pj < Izl < 0j} and Cj corresponds to the unit

to

setting

simple closed

into two components.

equivalent in

r

of genus

the corresponding

a conformal

Riemann surface

by

to a

J

like the one in the previous defines

U

paragraph.

structure W(t).

As

on

For each

t)

W) and we denote

t ~ I

the curves

38

Cj

are squeezed to points and

union of

W(t) § W(1) where

W(1)

r + 1 punctured s u r f a c e s .

Suppose that

A01 ~ . . .

,A0P0, All , . . .

,AlPl, . . .

Arl, ... ,ArPr, B01 , ... ,B0P0, ... ,Brl , ... ,Brp r canonical homology basis of BjI~ ... ,Bjpj

W

so that

is a

Ajl , ... ,Ajpj,

is a canonicalr basis modulo dividing cycles for

the jth component of (du0(t)

is a

, dUl(t)

W - iViAi, j - 0,i,...

, ... ,dUr(t ) )

differentials of

W(t)

,r.

Let

be the basis of analytic

duel to the given canonical homology

basis where

/duj 1 (t) ~ duj(t) =

]I duj2(t)

(t

\•Ujpj Let the corresponding period matrix for If

k ~ j

duj(t) = 0

and

(~iE,B(t).)p x 2p

f | duj (t) § 0 4

Aks t ~ i.

be

then

f

as

W(t)

Bks

Thus as

t § 1

B(t) § diag (Bpo,Bpx , .... Bpr )

where

(~iE,Bpj) pj • 2pj

component of W(1).

is the period matrix for the jthr

In the

k th

component of

W(O)

iglAi

39 the cons

structure of

Consequently, if

WCO)

~A. iffil I

y

and

W(t)

is unchanged as

is a path lying in the j + k

then

duj(t)§

0

8 IO gl "'" g~l (u;B(t)) 0 hl h theta function for istic,

then as 8

where

F~i]

as

is

varies.

component of t * i.

Finally,

a f i r s t order

is a P i - t h e t a character-

t § 1

I~

0 hl

8Ihll (u; Bpj )

component of

W(t)where

k th

t

W(1).

I

. h

CO;B(t)) -~ -~-e j-O

Ill

(O;Bp

is a f i r s t order theta function for the jth

40

2.

Completely Ramified Abelian Covers.

arbitrary abelian cover. M0

and

M0

in

b : W1 § W0

MUA

M I.

Thus

be the maximum unramified abelian M 0 c MUA c MI.

admits a corresponding factorization

Then

where

b

MUA ~ M 1

Suppose

x01,x02,

the cover

W0

... ,X0s, s ~ 1.

W - {x01 , ... ,X0s }

into

3)

If

be the element of

WUA = W0,

In any case

is completely ramified.

is a completely ramified abelian

Then the covering

W 1 § W0

is determined

of the fundamental group of

G

G, the Galois group of the cover. 3) which

~

x0j , j = 1,2, ... ,s. 4)

We will describe

WUA ~ W 1

has ramification occurring at

~

"circles"

W I § WUA

~ : W1 § W0

by a representation

aj

If

will be called completely ramified.

Suppose that cover.

~ : W 1 § W0

W 1 § WUA § W 0.

then, of course, the original cover was unramified. the covering

be an

If the corresponding function fields are

MI, M 0 c M1, let

extension of

Let

u

Let

assigns to a path which Then we assert that

a little more carefully.

Let

G

Y0

is

b~ a

point of W 0 - {x01Px02, ... ,X0s} which serves as the base point for the fundamental group. Then if we number the n points of b-l(y0 ), say {yljY2, ... ,yn }, every closed path y based at Y0 i-nduces a permutation, ~(y)~ of ~'l(y0). The image of U is a regular transitive subgroup of Snt the symmetric group on n objects, and this subgroup is isomorphic to G. The cover transformations also induce a regular transitive representation of G in Sn. Since G is abelian these two subgroups of S n are identical. We identify this subgroup with G for the purposes of this discussion. 4) Since G is abelian all conjugates of aj there is no ambiguity in the choice os aj.

equal

aj

and so

41

generated subgroup

by the elements of

G

generated

then the cover

Since

The following

Lemma

2:

Let

Then there following

admits

the cover

If

H

H

a proper

WI/H § N 0

the desired

lemma gives

ramified

For let

by these elements.

we have reached

completely

... ,a s,

N1 § W0 (= W1/G)

W 1 § WI/H § N 0. unramified,

al,a2,

be the

is not

G

factorization

is seen to be

contradiction.

a topological

description

of

covers.

b : W1 * W0

is a disc

be a completely

A0 c W0

properties.

(let

ramified

A 1 = b'l(A0))

All the ramification

of

abelian with

b

cover.

the

occurs

over

D

A0

and

W1

homeomorphic Proof:

If

The c o v e r

A1

-

to

PO = 0

group of

curves

group of

sects

Again let

50 .

which corresponds corresponds

to

where

a~j

~ O.

on

it

N

topy class

7~

Xoj under

Let

suffices on

to

W0

described

by a r e p r e s e n t a t i o n

~

...

Let

N0

aj,

N -

is

to p r o v e so assume

Xo1 , . . .

W0

each of which

~.

N0 - {xOl ,

on

fundamental

components

is nothing

is

containing

n

under

there

~ : W1 § N0

simple closed

into

N0 - A0

the fundamental be any d i s c

divides

,XOs.

,XOs} Let

into

71,

and s u p p o s e d t h a t

G.

...

whose h o m o t o p y c l a s s e s

PO ~ 1.

,72p 0

7~,

A0

be

generate

no curve,

from

the

inter-

j = 1,

...

,s,

under

~.

Suppose the element which

~

the number

of

G

is

~.a~j.

to show~at

be t h e e l e m e n t s

To p r o v e by m o d i f y i n g

N

the

lemma by i n d u c t i o n

7~

c a n he d e c r e a s e d

within

its

by one.

homoFor if

42 N = 0

a disc

satisfy

A0

corresponding

the conclusion

To show t h a t where

a~j

tours If

N

+ 0.

is a little

contours

of

W0 - X0

to a simple

7s

on

W0

a YE

XOO

around

,Yg_ I, Ys

reduced

by one.

Xoj

7~+ I,

A0

...

has

which

of

curve

a

y~'s

W0

will

along

two b o u n d a r y

of the non-dividing

T

so that the path closed

of the

b y one c u t

surface

from a point

Now pick

71 , .-.

cut

two s i d e s

circle

we can find a path

to

can be decreased

to the

choice

1emma.

The r e s u l t i n g

corresponding T

of the

to that

includes

"1

7 E'

cycle

y~.

no other

xOk ,

is homotopic

on

is clearly homotopic

complementary

,72p O.

con-

to one of the two

7s

7E'.

y~

to the curves

For this new

A0

N

is

q.e.d. We now t u r n abelian

covers,

to a characterization among a r b i t r a r y

of completely

abelian

covers,

ramified

in terms

o f t h e map

a : J(Wo) § J ( W l ) .

Lemma 5: is

Suppose

completely

b : W1 + W0

ramified

if

is

and only

an a r b i t r a r y if

abelian

cover,

a : J(WO) § J ( W l )

is

b_ an

isomorphism.

Proof:

We show t h a t

extension

if

and only

Suppose degree

f~(=

and so

admits

if

ker ~ + (0).

ker ~ + (0).

If go ) ~

(fl) is

in

(eM1)

The c o n v e r s e

= ~D 0 M0,

then

obtained

is

(fl)

is nD 0

an u n r a m i f i e d by reversing

unramified

a divisor

so that

Consequently

generates is

a non-trivial

Then t h e r e

zero which is not principal

principalo so

M0

DO

abelian

on

W0

of

~ u 0 ( D 0) = u I (~D 0) invariant

under

is principa extension the

of

G

I on

is and W0

M0 .

above argument. q.eod.

43 At t h i s M1

p o i n t we can c o m p l e t e t h e p r o o f o f Lemma 2, P a r t

i s an a b e l i a n e x t e n s i o n

W1 * W2 * W0 maps

~ij

then

~21

know t h i s

is clear

: J(Wi) § J(Wj)

M0.

Considering the covers

from t h e d e f i n i t i o n then

lemma t o be t r u e

of

a

~21 ~ ~02 ffi ~10"

i s an i s o m o r p h i s m , and so

Section 5), fact,

it

of

I when

that If

i f we have

W2 ffi WUA

k e r ~10 = k e r ~ 2 "

for unramified abelian

the p r o o f i n t h e g e n e r a l a b e l i a n

covers

S i n c e we (Part

case follows.

t h e above a r g u m e n t s show t h e lemma t r u e

for arbitrary

I,

(In covers.)

We now use t h e i n f o r m a t i o n o f Lemma 2 t o o b t a i n i n f o r m a t i o n about the p o i n t s

Lemma 4:

Let

cj

~ : W1 * W0

and s u p p o s e we c h o o s e (Part

in

A0

J(W1)

(Part

I , S e c t i o n 4~

be a completely

as in Lemma 2o

ramified

abelian

Let

vj" l u 0 (x 0 j)

A0o

Let

a r e fixed and c o n t a i n e d w i t h i n

A I.

cover

I , S e c t i o n 4) e q u a l

Xoj (l/vj)

where the p a t h o f i n t e g r a t i o n

J du o z0

lies

within

n'lul(~Z0)

I , S e c t i o n 4) e q u a l

(Part

~z 0 (l/n)

where t h e p a t h s

Then for all

e•

0

for all

X 9 R

•176

we have

I dul nz I

~XjCj

= 0

and consequently,

44

Proof:

Let

A~

Pl = nPO + q

A 0 , B~, '

"'"

(du o

Po

'

the 1 AI'

let

B0 "'"

A1 A~, ' Po ~

be the lifts of Let

)[1' "~

AZ ' Po'

"'"

B0

An

n ~AI'

''"

to the

An ' Po'

"'"

n

1 Blp

components of

be a canonical basis of

t

"" ~

dUls

of differentials.

vector

where

it is clear that

2P0

aI = a4 = [O,E,E,

e e J(Wo)

E

W1 A O.

is the

x 2pl

Then

v

where

0

is the

identity matrix.

v

then

Po x q

We now show that for any

g g

"""

g\/,

h h

,,~

h

ae

has period

/

j, cj(zJ(Wl) )

characteristic of the form

~

1 a o

i o 0

\

... o\/" /

O/

zero

Consequently if

/g\

has period characteristic ~ h )

(s)

and

has the form

.,. ,E'I Pn x p^

W0

characteristic

(6)

A 1.

is a q-vector and

dUq

With respect to these canonical homology bases on

matrix and

Bn ' Po

'

~ ~ .o. ,dUln)

where

"'"

~i'

P0'

~ ~ du I = (dUq,dUll,dUl2,

W1

W0

Bn is a canonical homology P0 Denote the dual basis by du I where o.o

is a P0

If

'Po

')~q']~l' "'" '~q

W I.

g 1.

analytic differentials dual to this bases)

"~

)[I' "'" ,~q0 Al1 , basis for

is the genus o f

is a canonical homology basis for

A~ '

q

' Po

Po-Vector o f "'"

where

has a period

45 By formula (4) of Part I

cj E Ul(Xlj)

C7)

Now squeeze

~A 0

and the n-components of

simultaneously. cj

a(vj'lu0(x0j)) - vj'lul(~z0).

to

n + 1

points

During the process the period characteristic of

is constant since it is rational and a continuous function

of the varying surface.

In the limit !

diag

where for

~A 1

Bq

is

W0 - A0

du o ~ 0

in

the

a 0.

becomes

!

!

(Bq, BO, BO, . . .

B matrix

in the

B1

limit.

for

A1

and

Moreover,

Thus i n t h e l i m i t

,Bo)

B0

is

the

du0~ § 0

in

the right

B matrix 41

hand side

and of

%

formula Thus

(7) b e c o m e s a p o i n t

cj

has the indicated

Now c o n s i d e r Since the

left

hand side

(5) and t h e r i g h t

formula

(6) b o t h s i d e s

(i)

Continue

(eq,0,0,

...

,0) ~ .

s aE x ~ ffil X j C j . ~ j characteristic as i n F o r any

•

has period

Since

a

characteristic

is

as i n

an i s o m o r p h i s m ,

the

the hypotheses

o f Lemma 4.

Suppose further

that

i s a B s V, B ffi (B1, B2, . . . ,Bs) so t h a t s ~ nv.'lsj = 0 (mod n) and (ii) ~8jcj = 0 j=l J

Then t h e r e Proof:

I.

hand side zero.

form

characteristic.

has period

are

of the

is complete.

Lemma 5: there

J(Wl)

period

Lemma 8, P a r t

formula

proof

in

is

a

X e R

Suppose first

so t h a t that

Xj = Bj

PO = O.

for

all

By f o r m u l a

in

J(Wl)

j. (4) o f P a r t

I

46

~BjUl(Xlj)

~ ~Bjvj-luI(~zo)

or

(XSjvj'1)~z0 ~

ZBjxIj

Let

a

be a function on

its divisor. definition

Then •

a

= Bj

W1

o.

with the above invariant divisor as

corresponds to some character

for all

X

and by

j.

In the general completely ramified case the period characteristic of points.

cj

is uneffected by squeezing

The same is true of the s-tuple

~A 0

and

(XI, ... ,•

depends on the topological nature of the covering Thus the proof for

PO = 0

~A 1

to

since it

b : A 1 § A O.

suffices for the general case. q.e.d.

It can be shown that hypothesis

Lemma 6:

Continue

-I) (nvj

s

that

~

o f Lemma 4.

the hypotheses

= 0 (mod n ) .

(i) is necessary.

Then

2e I - 0

Suppose further if

and only

if

there

j=l exists

a

Proof: the

fact

X

so that

This

follows

that

Lemma 7:

of the

eI

for

immediately

.~ c j =

Continue

(1/2n)-period

Xj = 1

all

j.

from the previous

two lemmas a n d

-2elo

the hypotheses

of formula

o f Lemma 4.

(5) P a r t

I has period

form

gl 0 0 . . . O~ 1 0 0

Then t h e

0" / J

characteristic

47 Proof:

We repeat formula (5) of Part I

(8)

On

K1 ~ aK0 " (Pl

W0

A0

which divide

consider

l)n'lul(~Z0 )

dividing

W0 - A0

into

Jordan

P0

a-e0

curves

el"

6s

~ = 2,3,

...

'Po

components, each of genus one and

each component containing a handle pair of the given canonical homology basis for W1

Now lift each

dl, one for each component.

to points on W1

W 0.

W0

to

Now squeeze

and squeeze the

simultaneously.

6s

nP0

In the limit

W0

n

copies on

8A 0

and the

6~'s

lifts of these curves on becomes

P0 + 1

punctured

surfaces, one of genus zero and the remaining components of genus one.

W1

becomes one surface of genus

zero surface below) and nP0

q

(lying over the genus

surfaces of genus one.

Since the

vector of Riemann constants for a torus has period characteristic (~), in the limit

As b e f o r e (e,O,O,

K1

K0

n-lul(aZo) ...

,0) ~

where

has period characteristic

becomes a p o i n t e

is

in

a q-vector.

becomes

~

i kl kl "'" k~2 ~ 2 k2 k2 " "' @

J(W1) In the

of the limit

form e 0 = O.

48

comes from each set of 2 component of

WI

AI.

P0

tori arising from a

By putting this information into formula (8)

the 1emma is proved.

q.e.d. Now let

u

characteristic

be a point of

J(W0)

with

a fixed

period

C~); that is;

As we squeeze

BA 0

the surface.

(The

to a points 6's

u

is a continuous function of

of the previous proof will no longer be

considered).

e[[Sjcj

-

el](au;B 1)

then approaches

t

!

Ol'[Bjcj

since

B 1 § diag (Bq, B0, B0, ... ,B0). t

[~Bjcj [~Bjcj are

t

e I ](0;Bq)(eE~l[u;B 0 )} n

The q-characteristic

t

- eI ] e 1]

is obtained from the Pl-characteristic

by deleting the last

nP0

columns, a11 of which

zero. We now s u m m a r i z e

Theorem I: cover.

Let

this

discussion

~ : WI ~ W0

in

a theorem.

be a completely ramified abelian

Continue the other hypotheses of Lemma 4.

Then the vanish-

49

ing p r o p e r t i e s of a_J(WO)

eE~Bjcj

el](U;B1)

are precisely the vanishing properties of !

O[[Bjcj'

on a general point of

e I ](U;Bq)

at

u = O.

50

3.

Two-Sheeted Covers 9

possibly

ramified

Let

cover.

~ : W1 § W0

In this

P0-hyperelliptic

Riemann s u r f a c e

P0-hyperelliptic

function

hyperelliptic

case

h~perelliptic.

If

r

situation and

field.

and if

the

M1

If

P0 = 1

is

W1

total

case

degree

s.

s = r Also

Choose

A0

and

as i n t h e

basis

W0

for

W1

is

c a s e . 5) is In]

A1

in accordance

classical

in

A1

case.

(aj

a

a

in the

e11ipticthen

1.

+Xls

is

a divisor

and the ~A0

w i t h Lemma 4 ,

Fix a canonical of the

canonical

as i n t h e c l a s s i c a l

characteristic

_ 1)

then squeezing

part

exactly

If the period

denoted

we a r e

called

-

be c a l l e d

be c a l l e d

ramification

X1 = X l l + . . .

and choose that

lying

will

will

of

2q + 2 = r .

zI = Xll on

and

WI

P0 = 0

P l = 2Po + ( r / 2 )

In this

be a t w o - s h e e t e d

of

theta

and

ah I

...

1 0

cj,

and l e t homology

homology basis hyperelliptic

j = 1,2,

characteristic to 3 points

~

[ e 1]

,2q + 2 is

denoted

simultaneously

shows

that r-

In]

= 11 1 1 2 3

and the

(aj)'s

to those

given

5)

See K r a z e r

j = 1,2, in K r a z e r

...

9

...

qO ,2q + 1

U6"]p.448.

.

0 0

00

o..

~

6)

~ 2

are a n a l o g o u s Since

in the same way

G = Z 2 R = {id,

El~J p. 445.

6) A subscript 2 on a p e r i o d o r t h e t a indicate a (1/2)-integer characteristic.

characteristic

will

X}

51

where the corresponding (s times) Let

are

(~a),

distinct

elements

(0,0,0 . . . . .

~ = 1,2 . . . . .

0)

2q § i

characteristics, 2q+l (~a) = (0).

period

By Lemma 6

We could use Corollary usual vanishing properties

in

V = Z 2 x Z 2 x ... xZ 2

and

stand for an arbitrary sum of (aj), none of which is

for the hyperelliptic

If

W1

results.

is a P0-hyperelliptic

Pl = 2P0 + q' the s

(a0) = (0).

theta function,

to tabulating the results by Theorem

I of Part If, assuming the classical

1:

,i).

3 to Theorem I of Part I to derive the

but we will confine ourselves

Corollary

(1,1,1 . . . .

(Krazer D53 p. 459.)

Riemann surface and

table gives the vanishing properties

that follow from Corollary 3 of Part I. If

o

equals

o e[n + ~a](u;B1)

then

(o ~ q)

on

q

-

i

or

q

-

2

q

-

3

or

q

-

4

q

-

5

or

q

-

6

vanishes

a_J(W0) , in general,

to order

We now use the table to derive the highest order vanishing properties

of a P0-hyperelliptic

hyperelliptic odd or even~

theta function.

case we distinguish between

Pl

As in the (and

q)

being

52

Corollary 2: Pl

If

is odd then

is a P0-hyperelliptic

e[n](u;Bl)

general point of least

WI

a_J(W0) o

(Pl + 1)/2 - P0

Corollar~ 5:

If

W1

Riemann surface where

vanishes to order Thus

on the

e[n](u;B I) 4p0

(q + 1)/2

vanishes to order at

half-periods

is a P0-hyperelliptic

at a

of

a_J(W0).

Riemann surface where

Pl

is ~ven, then e[n](u;B1) and the 2q + 1 functions 1 e[n + ~a](u;B1) vanish at a general point of a_J(W0) to order

q/2~

Thus these

Pl/2 " P0

2q + 2

at the

4p0

functions vanish to order at least half-periods

of

a_J(W0).

We now consider the elliptic-hyperelliptic for in

Pl ~ 5.

Putting in the four half-periods

e[n](a_u;B1)

Corollary 4:

of

aJ(W0)

for

Let

characteristics

W1

be an e11iptic-hyperelliptic

surface of odd

Then there are four bali-integer

Ink] , k = 1,2,5,4

Sin k](u;B 1)

vanishes

so that

theta

nI + n2 + n5 = n4

to order precisely

(Pl

I)/2

at

u = 0.

Proof:

The word " p r e c i s e l y "

C o r o l l a r y 5 t o Theorem i , at

~g0

to order

~N X

integral divisor on

~t X t >0 •

needs e x p l a n a t i o n .

Part I,

where

W 0.

all integral divisors. order precisely

u

gives the following.

genus, five or more.

and

case of Corollary 2~

when

- el](U;B1)

N• = t X + i(Ox)

But if Thus

e[~Bjcj

P0 = 1

e[~Sjcj

In t h e p r o o f o f

then

and

~X

i(o•

el](U;B I)

vanishes is an

= 0

for

vanishes to

PO = i.

q.e.d.

53

Note that Corollary 4 applied to the case

Pl = 5

gives

the correct count on the codimension of the e11iptic-hyperelliptic locus in Teichm~11er space for genus 5, the codimension being 4. The theta characteristics

of the last corollary are of the

form

3

2

where

I:,~

Pl "

r

2

2

is an arbitrary half-integer

characteristic

for

2

genus one,

If

Ft'll 2 r

of

Pl = 3 EILI ~ (u'B)

l_112 e ' / z

~'/z_l

teristic

is

vanishes

for all

au

odd.

at

a r e nc s p e c i a l U = 0

However,

the fact

u ~ J(Wo) (O0 ~ ~)

t h e quarter-periods and

Corollary 5:

the t h e t a

properties charac-

(au;B1) [_1/2 a and t h e p e r i o d c h a r a c t e r i s t i c of

0

~ ~1/211/2 '~:~ : ' c'~ (O;B1)

[~,~ .

Letting

whose period characteristics

~/21/, ,/y

Let

that

shows t h a t

f o r any c h a r a c t e r i s t i c

~/, 1/,1

since

vanishing

' 1

i s o f t h e form

vanishes

there

W1

fl

and

fz

are

gives the following corollary

be an elliptic-hyperelliptic

Riemann

surface of genus three.

Then there are two quarter-periods,

and

fl ~ • s

f2 ,7)

so that

Fo. the . i t

\112 0

0J

11/2 3/4 3/4'x~ /2 1/4 1/4/"

f

i)

and

we

~,112 1/2 1/2//

be

ii)

or

fl

2f I ~ 2f 2 ~ 0,

(1,2 I,,

\112 1/4 114j

havo taboo

and

54

iii)

[(2fl),(f

1 + fz)]

= 18)

Now we a p p l y C o r o l l a r y P0-hyperelliptic only consider so that

the first

r z 2.

q (aj)ts.

Let

(27),

that

O ( f l ; B 1) = e ( f Z ; B 1) = 0. \

I, to the

Since

x01

= (0) 1 q * 1

Select

one i s a l w a y s

Xoj k ' k = 1, . . . . q.

iv)

1 o f Theorem 1, P a r t

case where sums o f

and

points

and t h e o t h e r s

(~a) = (~a r

)

k=l~Jk-1)

we n e e d of

have indexes

In a p p l y i n g

"

X0

formula

S

Part

s-tuple

in

positions, s-tuple

I, V

observe

4e 0 = ~ u 0 ( x 0 j ) .

w i t h ones i n t h e f i r s t

and l e t

B

modified

w i t h ones i n t h e o t h e r

by

Let

and

jl,J2,

(call

X

positions.

S

B

be t h a t

.o.

it

BX)

Finally

,jq be t h e

let

S

o IBjUo(Xoj) xjUo(Xoj). I =

Formula ( 2 7 ) , Part I , then reads

(9)

e [ n + ~al(aU;Bl)

=

zBe[ ( 1 / 4 ) o ] (u;B0) o f - ( l / 4 ) a ]

8) and

For half-integer (a')

= (|h 1' 1

Exp{wi ~ ( g . h . '

j=lJJ

p-period

"~ 'gP'~ . . . ,hp 2

/

(u;B0)

characteristics the

* g j ' h 3 ) } = _* 1

integer If

]o',a[

[c',a[

characteristics a r e s a i d t o be s y z y g e t i c . We draw a t t e n t i o n to this fact about (fl we will use it in the characterizations paper.

= 1

(a) = /~hl ' ' " ' 1

g~ 'hpJ2

is d e f i n e d

t o be

t h e two p e r i o d

See K r a z e r [/SJ C h a p t e r v i i . + f2 ) and ( 2 f l ) since

given in Part III of this

55 E(U)

i s t h e c o n s t a n t one by Lemma 1, P a r t IX. Xoj

(1/4)~

i s t o be

f

d e f i n e d by o n e - f o u r t h

of the values of

I duo

where t h e p a t h os

x01 integration to points

is restricted in t h i s

to

AO~ f o r i f we s h r i n k

8A 0

and

8A 1

c a s e we o b t a i n ~ i n t h e l i m i t O[n' + ~ a ' ] ( O ; B q ) { O F g ] ( u ; B O' )2 "

s

(u;Bo')2

The d e p e n d e n c e o f t h e c o n s t a n t s i n g p r o b l e m which we b e l i e v e

is open.

dependence is complicated since the denoted

~8

9

on

B

The l a s t ~B's

seems an i n t e r e s t f o r m u l a shows t h e

in t h e l i m i t ~ h e r e

~B' ~ are t h e n o n - v a n i s h i n g h y p e r e l l i p t i c

thetanulls.9)

9) For a n o t h e r d e r i v a t i o n o f f o r m u l a (9) and a d e e p e r i n s i g h t t h e n a t u r e o f t h e ~Bts see Fay [133 C h a p t e r S.

into

56 4.

Other Applications.

of Corollary

3, P a r t

In this I,

section

to certain

we c o n s i d e r

applications

g r o u p s os g e n u s two~ t h r e e t

and five. If

W1

Z2 x Z 2

let

admits

a group of automorphisms,

G = {~1,~2,~3,~4 }

*~ = *~ = *~ = ' l ' thus have the

Let

following

where

W1/<*j~ - Wj covers

G, i s o m o r p h i c

~1 " i d , and l e t

~2 ~5 = ~4 W0 = WI/S.

to and

We

of surfaces

W1 W2

h'3

W4

Wo where each line

Pi

segment corresponds

i s the genus Of

(10)

Wi~ i = 0 ~ 1 , 2 , 3 , 4

Pl

admitting

for the quotients.

( P l ; P2' P3' P4; PO) a

Z2 x Z2

We w i l l

with

consider

ii)

(5; O, I, 2; O)

iii)

{5; 1, 1, 3; 0 ) ,

E l i p. 478.

denote a s u r f a c e o f

the three

(3; 1, 1, 1; O)

See Accola

If

10)

the corresponding

i)

10)

cover.

then

Pi + 2P0 " P2 + P3 + P4

Let the symbol genus

to a two-sheeted

cases

genera

57

A fourth i)

case,

iv), w i l i

(31 1~ 1, 1; 0).

formula

r = 12

and

be

on a s u r f a c e

In this

case

n = 4.

s = 6.

Order

the p o i n t s

X 0 (X 0 - x01 + x02 + ... + x06 )

For t h e

Z3

of genus

By the R i e m a n n - H u r w i t z of

so that:

above

xOl,Xo2,Xo3,X04

W2 ~ W0

is b r a n c h e d ;

above

x01,x02,x05,x06

W3 * W0

is b r a n c h e d ;

above

Xo3,Xo4,Xo5,X06

W4 ~ W0

is b r a n c h e d .

cover

two.

W 1 § W 0 vj = 2, for

j = 1,2,

...

,6, so

V = Z 2 x Z 2 x Z 2 x Z 2 x Z 2 x Z2. The e l e m e n t s

of

V

X •

X

X

Admissible

B e V

to

corresponding

R

are

(I)

(0, O, O, O, O, O)

(2)

(I, I,

i,

(3)

(I~

O, O, i , I)

(4)

satisfy

I,

I , O, O)

(09 O, I , I , 1, 1). 6 ~ 2Bj j=l

-= 6 (mod 4)

*) B 9 V will be c a l l e d a d m i s s i b l e of f o r m u l a (15), S e c t i o n S, Part I.

or

if it s a t i s f i e s

~Bj

is odd.

the c o n d i t i o n

58

Let

S (j)

be that element o f

V

position and zeros elsewhere. the i, j

and

k

positions.

with a single one in the jth

Let

8 (i'j'k)

Modifying

have ones only in

B (1)

by the

xWs

gives

(i, O, 0, O, O, O)

(0, i, i, i, 0, 0)

(0, i , O, O, I , I) ( i , 0, i , i , i , i ) .

Thus

xeR~max (0,t X) = I

for

B Cl)

there

are

B's

in all

of four

32 a d m i s s i b l e

e a c h when m o d i f i c a t i o n s

sideration. the

The

other

B(J)ts

8 arising

last two

B's

Part I,

B(1'3'S)

it is seen that

2cj

2e I ~ 2e I ~ 0.

by Lemma 4, Part II. e(u;B I)

and

into

are

24 o f t h e

~ max xeR

is a

Also

Thus

sets conBts,

For these

(0,tx) = Oo

and

into

admissible

B( 2 ' 4 ' 6 ) .

Now

eight

taken

By Corollary

i,

8[c 2 + c 4 + c 5 - el](U;Bl)

implies the vanishing at

i/4

period.

For any 6 ~scj ~ 0

(c I - e 1 ) + (c 2 - e 1 ) ~

c I - e I - -(c 2 - el).

is an even function of

corollary.

X's

B Cj}.

u = 0.

By Lemma 6, Part II, e I j

for

8[c I + c 3 + c S - el](U;Bl)

a r e non-zero a t

which divide

by the

account

from

and for any other

Since

u, the vanishing of

c2 - eI .

je

at

We now have the following

cI - eI

59

Corollary 6:

Suppose a surface os genus three, W1,

(3; 1, 1, I; 0) o

are distinct;

iii] ii~

Then there are three quarter periods

ii)

(3; 0, I, 2; 0). Analysis

hyperelliptic

= 0(f3,Sl)

f2' s

= 0.

Again in this case

as in the previous

vanishing property of

the cover

eI

since

r = 12,

and

O(u;Bl).

properties

W1 § NO

n = 4,

case will lead only to the one

The elliptic-hyperelliptic

Consequently,

s

2f I ~ 2f 2 ~ 2f3; and

e(fl,B I) = O(f2,B1)

s = 6.

admits a

do not show up in analyzing

is a half-period

in this case.

the following corollary merely records previous

results.

Suppose a surface of genus three admits a

Corollary 7:

(3; 0, 1, 2; 0).

Then

e(U;Bl)

vanishes

to order two at one half-

period and to order one at two quarter periods

as in Corollary

Since a surface of genus three which admits a necessarily

admits a

(3; i, i, I; 0)

S.

Z2 x Z2 x Z2

and is hyperelliptic 11)

we

obtain the following

Corollary

8:

automorphisms

Suppose a surface of genus three admits a group of isomorphic to

function vanishes

Z 2 x Z 2 x Z2.

to order two at one half-period

order one at three quarter-periods iii)

(S; i, 1, 3; 0)~

and so

ii)

s = 8.

The corresponding

as in Corollary 6o

By the Riemann-Hurwitz

Order the points os

See Accola [I] po 478.

and vanishes

X0

formula

so that

r = 16

theta to

60

above

x01,

o..

,x04

W2 § W0

is branched

above

x05,

...

,x08

W5 § W0

is branched.

W4 § W0

is branched

j - lj2~

...

in

V

,8

B's

8 (0)

Since

V

eI

is

seven

to

B (11)

P

xZ 2

(8 t i m e s ) .

(1, I ,

X[3)

(0, 0, O, O, 1, 1, 1, 1)

X (4)

(1, 1, 1, 1, 1, 1, 1, 1)

satisfy

zero element

a half-period

6 (12)

~

of

V

in this

i

case

vanishing

that

and let

only ones in the

which yield

The e l e m e n t s

1, 1, O, O, O, O)

~2Bj ~ 8 (mod 4 ) ;

the

for

are

X (2)

to order P

R

v 39 ~ 2

X0.

(0, O, O, O, O, O, O, O)

with

B's

hals

B (0)

V = Z2 x . . .

of

X( I )

be t h e

element of

are

and

corresponding

Admissible Let

over each point

it

of the

is,

~Bj

B(i'j) and

is

j

positions 9 there

function

the

Corollar~ (S;

B (15)

P

B (45)

P

B (46)

J

B (47)

9

Since we

following.

9:

Suppose

lm 1~ 5; 0 ) .

distinct

at

two~ n a m e l y

B(0) + B{11) + B (12) + B (15) - B {0) + B (45) + B (46) + B (47) obtain

even.

be t h e

seen that

theta

is

half

W1

Then

periods

is

a surface

e(u;B1) s

of genus five

vanishes

i = 0,I~2~

...

to order ~6

where

admitting

two a t

seven

a

61

0 = fO § f l

iv)

Suppose

W1

+ f2 § s

- fO + f 4 + fS + f6"

of

is a Riemann surface

genus two admitting

cyclic group os order three.

By the Riemann-Hurwitz

see that the genus of

is zero,

vj ffi 3. images of

The requirement R

for a suitable satisfy the

jth

W1/Z 3

in

V

(= Z 3 x Z 3 x Z 3 x ZS)

and

(o, o, o, o)

x (2)

(1, i, 2, z)

x (3]

(2, 2, 1, 1)

~Bj ~ 4 ~ 1 (mod 5 ) .

Let

os

X0.

B( j )

and z e r o s e l s e w h e r e .

formula we

s ffi 4. means

Each that the

are as s

x (1)

ordering of the points

position

r = 8

4 ~ (n/v~)X~ - 0 (mod 3) J J j1=

a

Admissible

be t h e B (1)

B's

B w i t h one in

m o d i f i e d by

R

gives

(1, 0, 0, 0)

(2, 1, 2, 2)

(0, 2, 1, i ) .

Thus f o r B's

give

B(J) xeR~max (0, t x ) = 1, j = 1 , 2 , 3 , 4 . ~ max (0, 0 xeR tx) = "

t h e n m o d i f y i n g by

R

gives

(I,

For example i s 2, i , 0 ) ,

Other permissible B

=

(1, 2

(2, 0, 0, 2)

'

1

and

'

0)

62

(0, 1, 2, 1 ) . However, 6(cj

el)

Corollary

Thus

8(u;B1)

vanishes

a 0

10:

sixth periods iii)

cj - 9 1

( c I - e 1 ) + (c 2 - e2) ~ c I + c 2 + ~cj ~ 0. and

5(cj

Suppose

W1

el)

= -3e I

fl'

f2

so

that

i)

Z5.

j = 1,2,5,4.

Also j.

o f genus two a d m i t t i n g a Then t h e r e a r e two one-

< f l > ~ ___' i i )

e(f I, B I) = e(f 2, BI) - 0.

for

independently of

is a surface

group o f automorphisms i s o m o r p h i c t o

and

at

3s 1 -

3s 2

63

5.

C l o s i n g Remarks. o f vanishing

number 3pl - s

which

of surfaces

In t h e f i v e properties

admitting

Z 2 x Z 2 x Z2

and

direct

vanishing

corollaries

consider

W 1 § W 5 (= W 1 / ~ 3 ~ ) In general,

cover

W 1 § W1/G.

ing properties example,

For large

since

properties

revealed

W1

Pl

admissible

zeros

9 to non-

groups

I.

isomorphic

1 of Part

To see how this

becomes

there

Theorem

W 1 ~ W 0.

more complicated Theorem

are,

fewer

1 directly

to the

in fact, more vanishsubgroups.

As an

are no vanishing

1 directly

P0 = 0, simply because

I seems

complicated

to a

Z 2 x Z2

there

are no

W1 § W2

and

of most use for cyclic

groups

9, (5; i, 1, 3; 0). W1 * W3

one zero in common

so there

groups~

An

can be seen by considering

The two elliptic-hyperelliptic each

of order two at half-periods.

to

are not~

at the cover

to check that there

by applying

of the locus

at the cover

will have more cyclic

is even and

to more

Corollary covers

G

G

=

B's.

Theorem approach

ip Part

show up by applying

the reader may wish

where

G

space

extensions

than by looking

as the group

(s - 3)

The elliptic-hyperelliptic

from looking

rather

properties

7.

5)

of Corollary

These

of Theorem

Corollary come

(Spl

five admitting

Z 2 x Z 2 x Z 2 • Z2.

properties

vanishing

of genus

to

4 the

group.

the type of result

surfaces

of Section

in Teichm~11er

the particular

hyperelliptic

might happen

is equal

is the codimension

One can extend

however,

corollaries

give rise to a set of four However,

are seven which

these

satisfy

two sets have the conditions

on

64

of Corollary

9o 12)

quarter-periods However,

only

corollaries

Also in Corollary

correspond

t o one o f t h e

in exceptional

does a direct

a complete

picture.

There

intersection

is needed

the

applying

Theorem 1 to cyclic

properties

4p - 5

canonical

homology

<~i,~

admitting basis

where

characteristics Corollary

a general of sets of

can arise

and

0 0 .

2 of T h e o r e m

~2

1, Part

two

procedure of zeros

in a m o r e

I,

give

for

obtained

general that

by

W0

context.

is, a s u r f a c e

Z 2 x Z2.

on

Suppose

that a

so that

are h a l f p e r i o d s

2

two.

G.

an u n r a m i f i e d

can be c h o s e n ~I

of order

these

(4p - 5; 2p - 1, 2p - i, 2p - I; p);

of genus

ker ~ =

like

three

o f T h e o r e m 1, P a r t

subgroups

The same k i n d of p r o b l e m a

subcovers

circumstances

application

deciding

Consider

6 a n y two o f t h e

1 0 ...

with period

6

2~

I yields

E(u)O[e 1] (au;B 1) = 2 -

s There

of the

are

type

2

(u;B0)e[~l]2(U;Bo)8[~2]2(u;B0)e[~

Up . 2 ( = ( 4 p ' 2 0 0 eI 0 eI '

...

" 2P'2)/2)

Cp_2~

.c:2 P

so that

1 + co2] ( u ; B 0 ) ~ 2

odd p e r i o d

if

go

characteristics

h a s one o f t h e s e

2

12) A proof of this last assertion will be omitted. an e x a m i n a t i o n o f t h e c o v e r s more d e t a i l e d than that this paper.

I t d e p e n d s on presented in

65

period characters then

e[el](U;B1)

vanishes to order four at

u ffi ~g0" Consequently, higher

unramified

order vanishing

the subgroups of order a further

problem is

of higher

order

Finally,

properties two.

zeros arising

properties

except in isolated

instances.

Wirtinger varieties

from

Z2 x Z2 x Z2

of the seven sets

from t h e s e v e n s u b g r o u p s o f o r d e r

case further

seem i n a c c e s i b l e

four yield

to zeros arising

Thus f o r an u n r a m i f i e d

the intersection

unramified

groups of order

in addition

one s h o u l d m e n t i o n t h a t

t h e two s h e e t e d w h i c h , so f a r ,

non-cyclic

[~$Jdiscovered o~ z e r o s on

four. in J(W1)

by t h e m e t h o d s o f Theorem 1, P a r t

I,

PART III"

i.

Introduction.

In the third part of this paper we shall show

how some of the vanishing properties

of theta functions derived

in Part II characterize the existence of automorphism groups.

As

remarked in Part I, Riemann's solution to the Jacobi inversion problem allows one to infer from the existence of certain vanishing properties

of the theta function the existence of certain linear

series of degree

p - i.

The problem then is to show how the

existence of such linear series leads to the existence of a u t o morphisms.

With one exception we shall show only the existence of

automorphisms

of period two (involutions).

This work can be viewed

as a generalization of the classical characterizations

of hyper-

elliptic Riemann surfaces of genus three and four and the modern work of H. H. Martens

[20] for arbitrary genus.

ize elliptic-hyperelliptic further P0-hyperelliptic

surfaces

for all genera except four and

cases for higher genus.

will be a theorem essentially due to Castelnuovo the P0-hyperelliptic

We will character-

characterizations

The main tool [6].

However,

for the lowest genera are

reduced to the case of higher genus by extensions

of a method due

to Farkas [93.

Supported by National Science Foundation Grant GP-21191(A2)

67

For genus hyperelliptic elementary

three and five it is possible information

ahelian

two-groups

Also for h y p e r c l l i p t i c is derived confirming non-involution

ponding

two.

surfaces

Finally,

five a c h a r a c t e r i z a t i o n The case where

for genus

automorphism

five two local charac-

groups.

In the final section

indicated.

If

D

is a divisor

is a finite group of automorphisms

called the quotient

genus

finite order the quotient WI/~T >.

Frequently

of

If

G.

If

~enus of we

and notational

will

so that

W1

the genus of T

theta function".

WI/G

Statements

homology basis

"the

homology basis

is a closed Riemann

of

the quotient genus

surface

is chosen

of

a canonical first order

will be designated

concerning

and

will be

refer to "the theta function"

zero characteristic

not depend on the canonical

canonical

surface

is an automorphism

T will mean

unless

is canonical,

to have been chosen and the corresponding

theta function with

WI

2D

is a Riemann

In order that this make sense we will assume basis

conventions

All divisors will be integral

G

If

space corres-

some open questions.

will be called half-canonical.

homology

a

is that of a cyclic group of order

D

W I.

of certain

eight and sixteen.

for loci of Teichmueller

which are used in this paper.

of

four,

of genus

We now record some known results

otherwise

the existence

the "p - 2 conjecture".

are obtained

to two other

we discuss

of orders

is characterized

three for genus terizations

to characterize

to use the elliptic-

by "the

theta function"

will

chosen.

of genus

Pl ~ 1

and a

then to each of the

2P 1 2

1

non-zero

period-characteristics

1

smooth two-sheeted

corresponds

one of the

2P 1 2

coverings

of

W1

of genus

2Pl - I.

68

If

(o)

tation

is

characteristic

of the first homotopy

defines If

a period

the two-sheeted

[s I]

and

[E 2]

[E 1] + [~2 ] = (o)

then

(homology)

) (pc W 1 and function

odd then the function on to special vanishing

and

Wc

(T)

to

for which

Wo, WT, and

W

so that

a single

If both characteristics 2Pl

2

of Farkas

are

and gives rise

for the theta function of

W o.

[9].

WI, corresponding

then

to the group

on which a fixed-point-free

operates whose quotient are the quotients

cyclic groups of order two.

(o).

of theta functions

is of order

<(o),(T)>,

of automorphisms

The methods

is

modulo

of Farkas

W1

and

the three

extend to this

and we will make use of this in characterizing

P0-hyper-

surfaces.

For hyperelliptic

surfaces we need the following

is a covering

hyperelliptic Weierstrass

so is

points

involution. elliptic

W . o

cover of

four-group

W 1 § W0

corresponding

are distinct period-characteristics

of period-characteristics

elliptic

whose kernel

u : W 1 § J(WI) ) defines

observation

there is a four-sheeted

situation

on

properties

This is the fundamental (o)

Wo

a represen-

Z2

are two theta-characteristics

8[r

If

defines

group onto

smooth covering

then the quotient

valued meromorphic

(o)

determined

W 0.

by

D

For hyperelliptic

are the fixed points

Also if

surface

of closed Riemann surfaces

D

are Weierstrass

then if

surfaces

is

the

divisor on a hyper-

for the linear series

points.

Finally,

for surfaces

of genus four or more a surface cannot be hyperelliptic elliptic-hyperelliptic

W1

If

for the hyperelliptic

is a half-canonical

then the fixed points

facts.

at the same time.

and

69

For surfaces hyperelliptic two-groups

o f g e n u s f i v e we w i ~ n e e d

involutions ( [ 1 ] p.

commute and so g e n e r a t e

480).

last

fact

involutions

Finally with quotient

we n o t e t h a t genus

P0

5 of this

for surfaces

is,

automorphismsD and i t alizes

we i n c l u d e

P l , an i n v o l u t i o n

is unique if

therefore, will

the hyperelliptic

Since

paper.

of genus

Pl > 4Po + 1.

Such an i n v o l u t i o n

e1iiptic-

involutions

f o r genus f i v e .

d o e s n o t seem t o be i n t h e l i t e r a t u r e

a proof in the appendix to section

that

elementary abelian

Also e11iptic-hyperelliptic

commute w i t h f i x e d - p o i n t - f r e e this

the fact

central

be c a l l e d

situation~

([1] p. 479 or [5] p. 521)

in the full

strongly PO = O.

group of

branched.

This gener-

?0 2.

Castelnuovo's

method

l e m ma is an a d a p t a t i o n p.

and P 0 - h [ p e r e l l i p t i c i t y .

The f o l l o w i n g

of a t h e o r e m of C a s t e l n u o v o

([6])

or [7]

294). 1)

Lemma

i:

Let

a Riemann

Proof:

C

be a s i m p l e h a l f - c a n o n i c a l

p-I

surface

Use

a curve

gr

W

of genus

p.

the l i n e a r

series

in

of d e g r e e

]Pr(C)

of f i x e d p o i n t s

of

gr

If

n

series

so that the p o i n t s in g e n e r a l there

containing Let Zl,

in

is a h y p e r p l a n e

2{r - i) p o i n t s

...

points

,z Zl,

...

passes ...

W

onto

is the n u m b e r

a linear

and so

H0;

theorem n > 2r + 1 -' r2 s eries g 2n w h i c h is part

p - 1 > r 2.

Fix a h y p e r p l a n e

section,

Zl,Z2,

that is, t h ru any

r - 1

containing

no f u r t h e r

z's.

...

H0

,Zn, are

of the p o i n t s

Thus

thru any

(that is, a quadric)

z's.

thru the r e m a i n i n g

,z

By the p r e c e d i n g

m a p of

be a set of z's so that i) any q u a d r i c

,z

on

p >_. 3r.

p - 1 - n

one can find two h y p e r p l a n e s

no f u r t h e r Zl,

C

of the h y p e r p l a n e

position

where

series

By C l i f f o r d ' s

cut out on

of the c a n o n i c a l

then

to give a b i r a t i o n a l

p-l"

The q u a d r i c s

r > 2

linear

impose

argument

n - 9

independent

z's and ii)

conditions

thru

the

on the q u a d r i c s .

2r - 1 < 9 < n. w

Now consider

the n o n - f i x e d

points

of the linear s e r i e s r1

[gr22n Thus

z I - z 2 - ... r < rI < r2 - ~

z)[. or

It is a

g

n

and c o n t a i n s

gr n.

p - 1 > r 2 > r + ~ > 3r - i.

i) The a u t h o r thanks A l a n L a n d m a n for v a l u a b l e c o n c e r n i n g the m a t e r i a l in this section.

discussions

q.e.d.

71

In Corollaries

2 and 3 of Part II of this paper, the highest

order vanishing properties of P0-hyperelliptic recorded.

If

Pl

is odd and

theta function vanishes at (Pl + 1)/2 - P0" at

If

Pl

(2Pl + 2 - 4P0)4 p0

W1

4p0

theta-functions are

is P0-hyperelliptic

then the

half-periods to order

is even then the theta function vanishes half-periods to order

use Lemma 1 to show that if

Pl

is large and

Pl/2 - P0" P0

We now

small one of

these vanishing properties suffices to insure q0-hypere!lipticity where

q0 ~ P0"

Theorem i: PI"

Suppose

Suppose

W1

W1

is a closed Riemann surface of odd genus

admits a complete half-canonical

(Pl-l)/2-P0 g

where Pl-i q0 ~ P0"

tic where Proof:

Pl > 6P0 + 5.

Then

W1

is q0-hyperellip-

If the linear series is simple then by Lemma 1

Pl --> 3[(Pl " 1)/2 - p0 ] or Pl ~ 6P0 + 3. (Pl-l)/2-P0 Thus

g

Pl_l

b : W1 § W0 g

of

(Pl-1)/2-Po n/t

on

W I.

series.

n If

(Pl O~

t on

is composite and there is a covering

sheets and a complete simple linear series W0

which lifts via

b

to the original series

is the number of non-fixed points of the original linear t > 3

1)/2

then

- PO < n / t < (Pl

l)/t

< (Pl

1)/3

72

Pl ~ 6Po + 1 (Pl-1)/2-P0 a contradiction.

Thus

t = 2.

If

g

n/2

on

W0

is

special then by Clifford's theorem

(Pl

1)/2-

[(Pl " i) - 2P0] ~ n/2 - (Pl

I) + 2P0 _> 0

or Pl ~ 4P0 + 1

again a contradiction.

Thus

g

(PI'I)/Z-p0

n12

is not special and

so by the Riemann-Roch theorem

qo

= n/2

-

[

(Pl

1)/2

q.e.d.

- po ] < PO

The following immediate corollary will be useful. Corollary 1:

If it is known in Theorem 1 that

q0

=

P0

then

(P1-1)/2-P0 g

Pl.1

is without fixed points and every divisor in the

linear series is invariant under the involution of quotient is

W1

whose

W 0.

As in the hyperelliptic case, the results for even genus are not as neat.

This is because the linear series derived from the

highest order vanishing property always has at least one fixed point.

Theorem 2: Suppose

g

Suppose

W1

is a Riemann s u r f a c e of even genus

Pl"

admits a complete half-canonical

(Pl- 2) / 2-P0

t i c where

W1

Pl_l

where

q0 ~ P0"

Pl ~ 6P0 + 8.

Then

W1

is q 0 - h y p e r e l l i p -

73

Proof:

If the linear

series

is simple

Pl ~ 3 [ ( P l

then by Lemma 1

2)/2 - po 3

or

Pl ~ 5P0 + 6

a contradiction.

As in Theorem

1, there

is a map of two sheets

n/2

W0

which

is complete

2)/2 + P0 > n/2 - [(Pl

" 2)/2

- p0 3 = q0

(pl- 2)/2-P0 b : W1 § W0

and a

non-special.

Thus

(Pl - 1)/2

g

- (Pl

on

and

or

q0 < P0 + 1/2

Notice sheeted

that in T h e o r e m l and 2

covering

involution

of

WI

b : WI § W0 whose

group of automorphisms

of

Pl > 4q0 + I

is strongly

quotient W 1.

q.e.d.

is

W0

so the two-

branched; is central

that isp the in the full

74

3.

Extensions.

for genera

To obtain characterizations

lower than in the last section we employ Theorem

I and its Corollary Let

G

G = {fl,f2,

be a finite subgroup ...

,fn }.

Suppose

G; that is, the map

has genus

and

the h a l f - p e r i o d series

an.

of dimension

note that the vanishing sj

at

1

and degree

the dimension

extensions

Corollary

2:

2)

Suppose

of Theorems

W1

admits

a complete,

Then

n ~

j:l

s. J

at

half-canonical

To state this a little differently,

if

sj _> I. series

on

J(W I) R1

If all W2

is

to order

of d i m e n s i o n s 3. _> 1

then

(n - i) + ~rj. the

1 and 2 possible. 2)

two distinct half-canonical

where Pl-i

This

G.

2, Part I

to a linear series on 1

be

to the

has kernel

has a zero of order

i.

W2

at

E f.~G 8[fJ](U;Bl) J

admits

W2

Let

corresponding

§ J(W2)

n,

~, 0 vanishes

(n - i) to what might be expected makes

following

rI

~

of the half-canonical

The addition of

g

Pl

order is

s.3 -> 0.

of the theta function on

fj + n corresponds

r 39 = sj.

W1

By Corollary

e[el](.,B2)

([sj)

to order

~ : J(WI)

=

Thus

whose

for some half-period

n(p I - i) + I.

2e I ~ 0

J(W I)

cover of

E(u)OEel](s

where

of

f39 + ~

the smooth abelian n-sheeted

W2

I, Part

2.

each point of finite order

group

of P 0 - h y p e r e l l i p t i c i t y

technique was used in E4] p. 17.

series

?5

a)

rl = (Pl - 1)/2 - P0

if

Pl

is odd and

b)

r I = (Pl - 2)/2 - P0

if

Pl

is even and

Then

W1

Proof:

is q0-hyperelliptic The previous

the characteristics are

ql

and

q2

for some

Pl ~ 6P0 + 1 Pl ~ 6P0 + 4.

q0 ~ P0'

discussion can be applied with corresponding

let

n = 2, for if

to the two half-canonical

G = {0,n I + ~2 }.

W2

series

then admits a half-

r2 canonical

g

2Pl_ 2 (2Pl - 2 = P2 ~ I)

in case

a)

r2 = Pl

2P0

in case

b)

r2 = Pl

1 - 2P0.

Thus in case a) P2 > 6(2P0 tic where

r2 = (P2

I) + 5. ql ~ 2P0

again

P2 > 6(2P0)

where

ql ~ 2P0"

quotient is ql"

W1

1)/2 - (2P0

i.

Thus

and

Consequently,

§ 5.

where:

i)

and

by Theorem 1

In case b)

r 2 = (P2

1)/2 - 2P0

Again by Theorem i, W 2 W2

W 2 is ql-hyperellip-

is ql-hyperelliptic

admits two involutions,

and the second whose quotient

one whose

is a surface of genus

Since the latter involution commutes with the former,

erate a four-group

on

W2

whose quotient

W0

and

has genus

they genq0

satisfying

2qo Since

W1

1 ~ ql

or

qo ~ PO"

is a two-sheeted cover of

W0, the theorem is proven. q.e.d.

Corollar 7 S:

Suppose

W1

admits four distinct half-canonical

series

r1 g

Pl_l

J(WI)

so that the corresponding where

four half-periods

sum to zero in

76

a)

r I = (Pl

1)/2 - P0

if

Pl

is odd and

b)

rl = (Pl - 2)/2 - P0

if

P0

is even and

Then

W1

Proof:

is q0-hyperelliptic Let the half-periods

nl,n2,n 3, and

u4

where

for some

Pl ~ 6P0 - 1 Pl ~ 6P0 + 2.

q0 ~ P0'

in the statement of the corollary be

nl + ~2 + ~3 -- n4"

of the discussion preceding Corollary {0'nl + ~2' ~i + n3' nZ + ~3 }

If we let the group

G,

2, he the four-group

and let

n I be the half-period by

which G is translated, we see that the condition 4 [ nj - 0 allows us to apply the conclusions of that discussion. j=l We obtain a smooth four-sheeted

abelian cover

W2

of

W1

on which

r2 there is a half-canonical

series

P2_l

in case

a)

r 2 = 2Pl

2 - 4P0 + 3

in case

b)

r 2 = 2Pl

4

Thus in case a)

r2 = ( P 2 -

hyperelliptic

where

I) and

W2

where

ql-hyperelliptic

Thus W1

W2

+

where

i)):

(P2 " 1 = 4(Pl

and

3.

(4P0

and

3)

Consequently,

ql ~ 4P0 " 3.

r2 = (P2 - 1)/2 - (4P0 is

- 4P0

1)/2-

P2 = 4Pl - 3 > 6(4P0 - 3) + 5.

is

g

In case

by Theorem 1

W2

is ql-

b)

P2 > 6(4P0

i) + S.

Consequently,

ql ~ 4P0 " 1

admits a four-group os automorphisms

and an involution whose quotient

whose quotient

is a surface of genus

ql"

Since the latter involution commutes with the involutions of the four-group,

all together they generate an elementary

of order eight whose quotient

4qo - 3 ~ q l

W0

has genus

or

qo ~ PO"

q0

abelian group

satisfying

77

Since

W1

is a two-sheeted

cover

of

W0

the theorem

is proven. q.e.d.

The last corollary as seems possible. ogous

squeezes

4:

as much out of this method

We omit the proof since

to that of the previous

Corollary

about

Suppose

W1

it is entirely

anal-

corollary.

admits

eight distinct

half-canonical

r1 series

g

Pl_l

a translate

so that the eight

(by a half-period)

corresponding

of a subgroup

half-periods

of

J(WI)

are

of order

eight where a)

r I = (Pl - 1)/2 - P0

if

Pl

is odd and

b)

r I = (Pl - 2)/2

if

Pl

is even and

Then

- P0

W 1 is q0-hyperelliptic For

P0 = 0 Theorems

characterizations

For

vanishes

for hyperelliptic

Also

four-group

P0 = 1

and

Pl = 6

we derive

additional

We can summarize

P0 ~ 2

it is

or

Pl odd, Pl ~ 5

to allow Corollary Z2 • Z2 • Z2

We treat

for

again an 3 to apply. can be

P0 = i, Pl = 3

this case in Section

6 where

information. the results

that P0-hyperellipticity

ties of the theta

Pl ~ 8

4 to apply. However,

4 does not apply.

For

i, Part If, that the theta

an appropriate

Corollary

2 give the known

on a coset of a Z 2 x Z 2 x Z 2 in J(WI).

can be found

found to allow Corollary

q0 ~ P0"

surfaces.

Corollary

appropriately

P0 = i, Pl even,

appropriate

some

- 3

Pl ~ 6p.

1 and 2 and Corollary

easy to see from the table, function

for

Pl ~ 6P0

of the last two sections

is characterized

function

at half-periods

by the vanishing for Riemann

by saying proper-

surfaces

of

78

genus

Pl

even

Pl

we have

Pl ~ 6Po - 3, Pl ~ 5

and f o r

Pl ~ 6PO' Pl ~ 6. It's

branched P0 = I genus

where f o r odd

worth remarking that involutions and

characterized

Pl = 9, P0 = 2.

five is treated

t h e two c a s e s o f n o n - s t r o n g l y by these methods

are

The e 1 1 i p t i c - h y p e r e l l i p t i c

in section

5 of this paper.

Pl = 5, case for

79

4.

The

p - 2

conjecture

for

p = 5.

3 allow us to give the following surfaces

of genus

five.

characterization

p

the h y p e r e l l i p t i c

space has codimension

p - 2.

Corollary

be a Riemann

Let

W5

theta function vanishes and

n3

and vanishes

of sections

surface

is m o t i v a t e d

locus

of genus

by

in Teichmueller

5.

Suppose

to order two at three half-periods

to order one at

2 and

of hyperelliptic

The title of this section

the fact that for genus

5:

The methods

~i + n2 + nS"

the

nl,n 2

Then

W5

is

hyperelliptic. Proof:

hyperelliptic precisely without

El2

is evident.

that a

W5

g 14

the

gI

must contain

admitting

is unique

the surface

divisor containing

W5

gl 3

has a

x + y + z

1 g 4"

two divisors

divisor of

in

let

of a divisor

El 4

containing must contain

containing y

be a half-canonical

El4

be three distinct points

El4

Dx + DY

also contains t

is not hyperelliptic,

two points

El3, no point being a fixed point

canonical,

1 g 3

a

1 g 5

in

the third.

Assuming and let

3

the

can have

fixed points can admit at most one half-canonical

and any canonical

where

has two fixed points

We now show that no

one fixed point by showing

Necessarily

of

g 1 4 'S

If one of the corresponding

since

is the fourth point; gl 4

constitutin E a divisor

gl 4.

Let

Dx

and

and

y

respectively.

z, say

z

is in

no point of

D + Dx

is a second half-canonical

x

of

then

D x.

x + y + z

is canonical. that is

gI

divisor

2s ~ 2t

+ t and so

be

Being If

D

we see that

Thus

4 = g 51

Dy

is a Dx

Dx = x + y + z + t If

gl 5 + s

s = t.

80 Thus,

if our surface

canonical

gl4's

beginning of

W5

is not hyperelliptic,

has a fixed point.

out fixed points. is a two-sheeted

W 2.

and a central

a complete

By Lemma 1 the

g616

Thus

is composite

WIT

admits

is

abelian group of order eight,

call

We wish to show that the genus of

quotient

is

four-group W0

W 0.

a complete

cover of

W 0.

Applying

zero, and

WI7

g68

But

formula

W2

W 2.

W5

is

W5

These generate

an

G, whose quotient we will W0

is zero since

W5

admits a four-group whose

(10) of Part II shows that any

on a surface of genus two must be a (2;1,1,0;0).

has genus

with-

and so

a four-group whose quotient

involution whose quotient

is a two-sheeted

6 g 16

This surface must have genus two and will be

elementary W 0.

at the

abelian cover, WI7 ,

half-canonical

cover of a surface admitting

fixed points.

denoted

By the discussion

of section 3 there is a four-sheeted

of genus 17 admitting

without

none of the half-

is hyperelliptic

after all.

Thus q.e.d.

81 5.

Elliptic-hyperelliptic

of genus

five can admit

it is important

where according

involutions

genera

Let

five

by several

3 the theta function

These

gl4's

W5

give rise

3, Part II, via the Riemann Proof: from

2D 5 5 X 5. then

f5

of the cover

If

f5

vanishing

T

of

symmetric W5

is invariant

at the branch points symmetric

at

under f5

quotient Since

and

D5

precisely

from l i f t i n g

of

four

gl 2 , s

properties

gl 4

from

of Corollary

If

X5

then W5

is

is the divisor

X5

is canonical

whose

divisor

W1

fl

and fourth order via

is

because of

on

2D 5

to the

are simple

Thus

(f5) 2

where

zero at b.

X5

the divfsor

f5

W1

of

and

with respect

the poles

D1

g12

is the lift of a

must be anti-symmetric.

is the lift of

two-

involutions.

are

or anti-symmetric

T.

xiI=b(Xs) )

(i) The number

whose

on

abelian

theorem.

and is the lift of a function

simple poles IDII = gl 2

W5 § W1

to order

cover of surfaces

to the vanishing

is a function

is either

involution f5

which arise

IDsI = gl 4 on WS, where 1 and g 4 is half-canonical.

branch points

between

vanishes

elementary

Then t h e r e

Let W1

involutions,

correspondence

be a t w o - s h e e t e d

on

a surface

and sets of four half-periods

a n d one r e s p e c t i v e l y . gl4's

Since

elliptic-hyperelliptic

b : W5 + W1

half-canonical

five.

elliptic-hyperelliptic

a one-to-one

to Corollary

generated

Lemma 2:

of

of genus

This will allow us to characterize

groups

W1.

several

to establish

elliptic-hyperelliptic

two.

surfaces

D1

fl

is has

where

Thus

4D 1 ~ X I. of

gl2's

satisfying

formula

(i) is sixteen,

the number

82

of quarter-periods since

W1

on the Jacobian

of

W1

(which is

WI

itself

is a torus). !

Now suppose

D1

and

!

D1

lift to

WS

to be

D5

and

D5 ,

!

half-canonical

divisors.

Then

2D 5 ~ 2D 5

and it follows

that

!

2D 1 s 2D 1

on

W I.

Thus

lift to half-canonical on (the Jacobian

of)

That these

the

Dl'S

divisors

gl4's

five.

Suppose

properties Proof:

from

W1

to

involution

~ : J(Wl)

In the proof of Corollary

smooth four-sheeted

abelian

cover

4 and

§ J(W5)

Riemann surface of genus theta-characteristics

and

8[nk](U)

Then there is an elliptic-

T, W 1 = Ws/,

arise from the cover

to the half-periods

W 5.

~i + n2 + n5 = ~4 u = 0.

(i) and

four in number.

there are four even half-integer so that

formula

from Corollaries

be a n o n - h y p e r e l l i p t i c

to order two at

hyperelliptic

correspond

from the fact that the map

W5

Ink], k = 1,2,5,4 vanishes

W5

are the ones derived

was defined by lifting divisors Let

on

WI, and so are precisely

1 of Part II follows

Theorem 3:

which satisfy

and the four vanishing

W5 § W1

as in Corollary

3 we showed that WI7 , of genus

W5

4, Part If.

admits

a

17, which admits

an

!

involution

T

which

is hyperelliptic

or elliptic-hyperelliptic.

!

If

!

G

is the four group on

WIT

whose quotient

is

W5

then

T

!

and

G

generate

on

eight whose quotient

WI7 W1

an elementary is covered

abelian

two-group

in two sheets

by

W 5.

of order Since v

W5

is not hyperelliptic

the genus of

be an e l l i p t i c - h y p e r e l l i p t i c 1

g 4's T'

on

W5

involution

lift to the same complete

is e l l i p t i c - h y p e r e l l i p t i c

Theorem

W1

must be one and

of

WIT. 7

g 16

the conditions

T

must

Since the four on

WI7

of Corollary

2, this part, hold and so each divisor of

7

g 16

and since 1 to is invar-

83

'

iant under

T .

If

D4

is a divisor

1

in a

g 4

on

W5

then it !

lifts

to a divisor

well.

Thus

this

DI6

must he invariant W I. W5

Thus

DI6

on

and the proof

which

is invariant

under

the four

WIT

is invariant

under

T, the involution

gl 4 ,s

arise

is completed

Definition: surface

If

of genus

T

by an appeal

T1

T-families distinct Proof: for ~i

W 5.

T-families

Then

T1

with

T2

have precisely

= W5/<TI>'

we have

and

for a surface

Suppose

and so whose

g 1 2's

to Lemma

as D4

quotient

from

W1

is

to

2.

sets of four half-integer

involution

five then the set of four half-integer

Theorem

involutions

W5

is an elliptic-hyperelliptic

associated

If

G

G

intersect.

characteristics 4:

of

from lifting

We now investigate how the different theta-characteristics

all of

under

T

will be called

are two distinct

of Eenus

five,

~TI,T2>

(=G)

2WI = Ws/'

the followin E diagram

in common.

is a four-group W5 = Ws/<TIT2>' of covers

2Wl

\ / WO

Let

W 0 = Ws/G.

corresponding

W3

involutions

(5;i,i,3;0). and

Three

in common.

are elliptic-hyperelliptic

/

iWl

a ~-famil 7.

the corresponding

one theta-characteristic

T2

theta-

elliptic-hyperelliptic

have no theta-characteristics and

on a

to

G:

Then

84

W5

is hyperelliptic

lifting 1 g 2

and the hyperelliptic

the unique

on

W5

gl 1

on

is half-canonical

Consequently,

this

gl 4

corresponds

to a

gl 4

T2, and therefore, g 1

on

W 0.

TI, T2, and

T3

A

9

in each T-family

This

arises

g14

on

W5

arising

under

G.

statement

4

Thus

it comes

contradiction

completes

the proof,

eight

4 we can now characterize abelian

two-groups

those

implies

five can admit

hyperelliptic

Corollar 7 6: admits

a group

A non-hyperelliptic of automorphisms

to order two at

[~]) corresponding qorollar[ admits

7:

a group

generated

each divisor by the

surfaces

Notice

o f genus

by two,

three,

that Theorem

isomorphic involutions

u = 0

surface to

of genus

Z2 • Z2

five

generated

if and only if

@[~3(u)

for seven theta-characteristics)

to two T-families.

A non-hyperelliptic of automorphisms

by three

Riemann

Riemann isomrophic

elliptic-hyperelliptic

surface to

4

at most five elliptic-

involutions.

by two elliptic-hyperelliptic vanishes

and

involutions

generated

generated

involutions 9

of genus

T1

q.e.d.

and four-elliptic-hyperelliptic that a surface

in

from the unique

of degree eight since of order

elementary

T 2.

to a theta-characteristic

the group

five admitting

under

elliptic-hyperelliptic

corresponding

from

4 is proven.

under

Using Theorem

and

from

in both T-families

are invariant

of Theorem

three

T1

to a theta-characteristic

divisors

all of

admits gl

W5

and invariant

must have divisors

would be invariant T's.

under

W5

the

on

a theta-characteristic

whose

The first

Now suppose

Thus

corresponds

each T-family. Conversely,

1

W 0.

1 g 2

of genus

five

Z2 • Z2 • Z2

involutions

if and only

85

if

0[n](u)

vanishes

characteristics) Corollary admits

8:

to order

[~]) corresponding

if and only if

8[~](u)

theta-characteristics, By Theorem

three T-families

Riemann

isomorphic

vanishes In]

u = 0

for nine

theta-

to three T-families.

A (non-hyperelliptic)

a group of automorphisms

Proof:

two at

surface to

to order

corresponding

of genus

Z 2 • Z2 • ZZ x Z 2

two at

u = 0

for ten

to four T-families.

4 and the inclusion-exclusion

counting

principle

contain

3"4

theta-characteristics,

4.4

-

3"1

+

i'0

=

9

and four T-families

contain

-

i0

6.1

§

4"0

-

I'0

=

theta-characteristics.

q.e.d.

In the last three corollaries space

for genus

groups

loci are seen to be The surfaces

G

elliptic eight.

of conditions complete

8 deserve

further

Z 2 x Z 2 • Z2 • Z 2

admit elementary

Tk) k = 1,2)3,4

all have quotient hyperelliptic.

genus

Thus

abelian

generate

three while

there

admitting

the particular

Consequently,

by four elliptic-hyperelliptic

surfaces If

a

given.

in Teichmueller

the various

intersections.

of Corollary

five admitting

generated

the codimensions

five of the loci of surfaces

is the number

of genus

five

G

comment.

(=G)

necessarily

involutions two groups then

TiT k

TS(= TIT2T3T4)

is a fifth T-family

among

A surface has

since hyper-

of order and

at most

TjTkT s

is again ellipticthe ten theta-

86 characteristics. Corollary

This is, however,

8 since,

consistent

by the inclusion-exclusion

with Theorem

4 and

principle

5.4 - i0-i + i0.0 - 5.0 + Io0 = i0.

These surfaces

are known as Humbert's

in [8], [~@], and ~ 7 ] . abelian differentials reducible

quotient TkT m

surfaces

and have been studied

They have the property

that a basis of

of the first kind can be achieved by five

abelian differentials

five differentials

surfaces

which are, in fact,

are the elliptic for subgroups

for the five choices

of

integrals

elliptic.

lifted from the five

of order eight generated by Tj.

involution,

quotient

genus three.

It follows

quotient

genus of such a group of order eight must be one. are linearly

that they correspond

Tj, TkTs

Such a subgroup has only one

elliptic-hyperelliptic

five differentials

These

Tj, and thus six involutions from formula

independent

to five different

of

(6) [i] that the

follows

characters

of

That the

from the fact G.

(See [2],

pp. 600-601.) We also remark, similar vanishing surfaces,

without proof,

properties

it here.

Proposition:

Let

T

involution

W

be

a

would imply a sixth

proposition

involution.

in the next section

closed Riemann surface of genus five.

be an elliptic-hyperelliptic

fixed-point-free

gl 4

or a hyperelliptic

We will need the following

so we include

Let

of the theta function for Humbert's

for an eleventh half-canonical

elliptic-hyperelliptic Appendix:

that there are no further

involution.

Then

involution and let T

and

S

commute.

$

be a

87

Proof:

Let

order

2n.

Dn

be the dihedral group generated by

We wish to show that

reflections in

Dn

n

is two.

generate the cyclic group of order are conjugate and so

D n. that

Therefore

T T

and

n.

TR 2

and

TR 2

If

or

n = 4

n

T

and

T

and

commute and so Dn(n > 2)

then

Now consider

is a (5;1,1,3;0)

D4/

zero, two, and two.

acting on

WS, ,,

reduce to commuting involutions on

n ~ 4.

would

Let

R( = TS)

invo-

is central in

is

R n/2

we see

n = 4.

three groups of order four on

and so

of

TR2( = R'ITR)

R2

and so

is fixed point free of quotient genus two and

(5;3,3,3;2).

S

is even.

Then

S

were odd all

fixed-point-free involution of quotient genus three.

and

are elliptic-hyperelliptic

Since the only central element of n = Z

n

would be conjugate and thus

have the same quotient genus;consequently,

lutions.

If

T

Ws/

is a

Consequently <S,SR2>

WS/. and

R2

is a

The

<S,SR2>

of quotient genera

But this is impossible by formula (i0) Part II

The proof is complete.

88

6.

Elliptic-hyperelliptic

surfaces

this paper become more circuitous section

is no exception.

of genus three.

as the genus decreases,

However,

any surface

admitting an involution is hyperelliptic, or both.

Consequently, abelian

properties

of the theta function.

two-groups

three is indicated

Corollaries

automorphism

characterization

vanishing

of the theta function

need only consider 5:

of non-abelian

iv)

characterize

three by vanishing

automorphism

remarks,

Let

groups

Section 9.

fl ~ ~f2; ii)

be a Riemann

of genus

fl

f2

at

admitting

the

to order two, we

5 and 6.

surface

and

2f I ~ 2f 2 ~ 0; iii)

of

in terms of the

at one h a l f - p e r i o d

to Corollaries

the theta function vanishes

the surfaces

Since we will assume

of h y p e r e l l i p t i c i t y

converses

W3

II characterize

that there are two quarter-periods i)

three

How these characterizations

groups.

classical

Theorem

of genus

in the closing

5,6,7 and 8 of Part

the indicated

and this

of this section are to show that the conclusions

The tasks

of

elliptic-hyperelliptic,

on surfaces

lead to characterizations

for genus

of genus

we will be able to completely

elementary

might

The methods

three.

Suppose

so that

l(2fl),(f I + f2) l = i; and

fl

and

f2'

Then

W3

is

elliptic-hyperelliptic. Proof:

Let

o ~ 2f I ~ 2f 2 ~ 0; o is a half-period.

smooth two-sheeted

cover of

W3

involution

with

as quotient.

admits S

of

W5

W3

an elliptic-hyperelliptir

so that the quotient

genus of

This will prove the theorem and) (5;3,3,1;i)'s

leading

corresponding

a

and

W5

be the

S

be the

We wish to show that

involution IS)T>

to

Let

T

that commutes

W5

with

is one.

as a bonus,

give information

to a local c h a r a c t e r i z a t i o n

in Section

about 8.

89 By a classical

technique

([15[] p. 280) we may assume, without

loss of generality,

that the canonical homology basis on

been chosen so that

(0 0 2 , (fl + f2 ) = (~ ~ 0) (o) = ~i 00 0)

cfl

f2) :

01 0) 0 2.

Thus t h e

syzygetic

has

and so

2 G = <(o),(fl

group

consists of these three period characteristics

W3

+ f2)>

(0 0 0 . Now ~0 0 0) 2

and

it is easy to see that there is a unique set of four odd thetacharacteristics

obtained by adding to each element of

theta_characteristics; [~ of

i] ~ 1 2 G.

namely

G

the same

[~ 0 ~] [~ 0 ~] [~ 0 ~] , and 0 2' 0 2' i 2

which are obtained by adding

I~ 0 ~] 0 2

to each element

We now write these four theta-characteristics

as

[nl], [n I + a], [n2], [rl2 + ~] 0

where

By a classical theorem

([15] p. 258) there are four other pairs of

odd theta-characteristics 0 by Farkas the two g 2's

[nk]' [~k + o], k = 3,4,5,6. on

W3

corresponding

As shown

to a particular

pair lift to equivalent gO 4 's on W 5 and give rise to a complete i g 4 which is half-canonical. Thus the six pairs [nk], [n k + o] give rise to six half-canonical

series

rise to the special vanishing properties a

W5

kgl4

on

Let

fl

and

a : J(W 3) § J(W5)

by lifting divisors.

Thus

f2

and so give

of the theta function for

which admits a fixed point free involution.

the quarter- periods i g 4 'S on W 5.

W5

We now show that

also lead to half-canonical

be the map defined in Section 2, Part I ~o ~ 0.

Since

2f I ~ 2f 2 ~ o

on

W3,

90

~fj

is a one-half

on

W3

to the vanishing

then the lifts ~(fl) have

period

in

J(Ws).

which we will denote

gl4's

1

7g 4

fl + f2 + ql + ~2

8 g 4"

characteristics However,

+ (o

Theorem

satisfying

involution

with

with any automorphism

~S,T 5 >

or one.

The proof

quotient

genus

genus

zero.

g 4"

We thus periods

0

1) 2

+ Io 1

in

the hypotheses W5

W3

that

on

J(Ws).

give rise to theta-

of Theorem

2, Part

is not hyperelliptic

T5

W5

might be.

whose

III.

since we do

Nevertheless,

be a hyperelliptic

S of

for if W5

T5

existence

by

or elliptic-

is infered

by these

is a four group whose

Therefore,

Then the involution involution

it commutes

at the end of section quotient

of the theorem will be complete

is one.

the hyperelliptic

is hyperelliptic

and if it is elliptic-hyperelliptic

is that of the proposition

Consequently,

since 1

properties.

commutes

the result

W5

-fl'

from the quarter

k = 1,2,7,8

2, Part III we can let

four vanishing T5

k gl 4 's

the possibility

hyperelliptic

on

and

and so

we do not know that

not exclude

fl

correspond

Now

a_fI + _af2 + a~l + a_~ 2 5 0

the four

at

g02'

canonical

arising 1

and

1

characteristics

Consequently,

and

are equivalent

and so give rise to a half-

two more half-canonical

as period

g02

of the theta function

of these divisors

5 ~(-fl)

If

of

suppose T3 W 3.

on Let

genus

5. is zero

if we show this

W 0 = Ws/<S,Ts> W3 D3

induced

by

has T5

is

be the divisor

on

91

W3

of degree

two corresponding

function at the quarter period W 5. on

Then W5

under

D5

corresponds

at a h a l f - p e r i o d T5

and so

D3

fl"

to order two.

that is, fl

contradiction

shows that

hyperelliptic

involutions,

T3

(i0), Part

W5

Then

be a Riemann surface O[qk](U )

vanishes

The two sets of

and the four sheeted

cover

,8

[~k]'S

where

correspond

Theorem While

are elliptic-

<S,T5>

on

W5

must

corollary.

of genus

five admitting

to order two at

u = 0

[qk]'S

k = 1,2,

form a T-family

and

...

,6

W5 + W3 3) for the two-

W 5 + W I.

5 three times

is, in fact,

the different

D3 This

ql + n2 + "'" + n6 ~ 0

of Corollary

6, Part

to obtain a converse,

5 we must know that hypothesis

this

T5

to the two covers

k = 1,2,7,8

We now turn to the converse apply Theorem

to

is invariant

Consequently,

the following

nl + q2 + n7 + ~8 E 0. ...

T 3.

D5

must be a half-period.

[qk ]

k = 3,4,

D3

of the theta-function

Therefore

under

for eight half-periods

and

be the lift of

If, the four group

Corollary

a (5;3,3,1;1).

of the theta

q.e.d.

We thus obtain

Let

D5

and therefore

be a (5;3,3,i;i). 9:

Let

to the vanishing

is invariant

must be half-canonical,

By formula

to the vanishing

iii)

the case we prefer

holds

II.

We could

but to apply in all cases.

for the sake of brevity

proof which follows.

3) That the second set of six theta-characteristics do, i n f a c t , correspond to the second fixed-point-free involution is not proved, although it is extremely plausible. The o n l y p r o o f we know u s e s techniques different from those of this paper and is quite tedious. Therefore, we omit the proof.

92

T h e o r e m 6: there

are

cyclic ii)

three

be a Riemann surface

quarter-periods

groups

,

fl'

and

of genus

f2

and

f3

three.

Suppose

so that

i)

the

are distinct; = 0

for

j = 1,2,3.

Then

W3

a (5;i,i,I;0).

Proof:

Applying

denote

the Riemann

divisors

of degree

k = 1,2,3.

six divisors same

W3

2f I E 2f 2 E 2f3, and iii) 0(fj)

admits

for

Let

linear

quarter

counted

U(Bk)

+ K -- -fk

four

Ak

gl 4

twice

which we will

ical divisors

2Ak,

where

each

call

x. g12

Ak

Thus by the Riemann-Roah

let

Thus

r = 1

or

Ak

and

Bk

would

points.

point and so each It follows

x

Ak

that the

Since

fk

is a

r = i.

contain

points

and the non-common

theorem

are all part of the

it would have the fixed point Bk

2x

we see that the

r = 2. so

the variable

contain

j,k

2Bk, k = 1,2,3

and

Weierstrass

A k + Bk

contradiction.

for all

is not half-canonical

since

be hyperelliptir

~ ?2f k

had a fixed point

the hyperelliptic

Weierstrass

+ K - fk

of degree

period

If the

U(Ak)

2fj

gr 4

theorem,

two satisfying

Since

series

vanishing

points

Also

in of

the point, g 14

would be

A k, B k

would

the three distinct

and so

2x

has index

canon-

2.

is also a hyperelliptic and gl

Bk 4

is half-canonical, must be without

a

fixed

points.

The sphere

El

4

defines

whose total

W3

as a four-sheeted

ramification

is

twelve

cover

of the Riemann

by the Riemann-Hurwitz

93

formula.

Each

2A k

total r a m i f i c a t i o n tinct points possibility twelve,

2B k

depending

ever occurred

so each

on whether

Ak

two or three to the

Ak

twice.

(or

(and

Bk)

the m o n o d r o m y

in the symmetric for

would

gl 4

is thus ramified

two.

being a four group

(3;0,1,2;0)

or

of the canonical

(3;1,1,1;0).

1 g 4

a complete

Corollary

i0:

the hypotheses vanishes

a Riemann surface

of Theorem

to order

three gives

4, and suppose

two at a half-period.

But in the first 2 g 4'

divisor,

doubled.

Since

the fiber

must hold.

and 6 with the classical

for genus

Suppose

2

the second alternative

Combining Theorems5 of h y p e r e l l i p t i c i t y

gl

four-group

There are two possibilities

case each fiber of the cover must be a canonical

defines

there are

But then the covering must be

group on four letters.

since it consists

above six

and above each such point

group of the cover is the normal

Z 2 • Z2, namely,

exceed

is two distinct points.

given by the

of m u l t i p l i c i t y

is two dis-

But if the latter

normal with the group of cover transformations since

Bk)

the total ramification

on the Riemann sphere

two points

contributes

or one point connected

The covering points

and

q.e.d.

characterization

the last two corollaries.

of genus

three satisfies

the theta function also Then

W3

admits

a

(3;0,1,2;0). Corollary

ii:

the hypotheses vanishes Z2

•

Z2

Suppose

a

of Theorem

Riemann

surface

5, and suppose

to order two at a half-period. •

Z2.

of genus

three satisfies

the theta function also Then

W5

admits a

94

7.

Cyclic

converse

groups

to Corollary

T h e o r e m 7:

Let

Suppose there i) {fl> ishes

of order

W2

are

~ ;

at

fl

three

10, Part

be a closed

Then

We now prove

Riemann surface periods

ii) 3f I ~ 3f2; f2"

two.

the

II.

two o n e - s i x t h

and

for genus

fl

and iii) W2

of genus

and

f2

so that

the theta function

admits

a cyclic

two.

van-

group of order

three.

Proof:

By t h e R i e m a n n v a n i s h i n g

divisors

of degree

one s o l v i n g

U(Ak)

theorem, the

let

Ak

and

Bk

be

equations

+ K -- fk

u(B k) + K - -fk

for

k = 1,2.

points W2

Then

3A 1 ~ 3B 1 ~ 3A 2 ~ 5B 2

are distinct

which

by i).

represents

Riemann

sphere.

Hurwitz

formula.

W2

The total Thus

Thus

by ii) and the four i defines a complete g 3 on

3A I

as a three-sheeted ramification

is eight by the Riemann-

all the ramification

B's and so each of the four branch points covering the cover

has multiplicity is a cyclic

the group of cover

three.

occurs

at the A's and

of the three-sheeted

Consequently,

group of order

transformations.

cover of the

the monodromy

of

three and so, therefore,

is

95

8.

Some local characterizations.

In all the characterizations

discussed in this paper for genus five or three,

the number of con-

ditions imposed on the t~eta function turns out to equal the codimension in the appropriate Teichmueller faces admitting the particular

space of the locus of sur-

automorphism groups.

By viewing the

theta functions evaluated at particular points of finite order as functions on Teichmueller

space

(or Torelli space or the Jacobian

sublocus of the Siegel upper half space) give global defining equations stay off the hyperelliptic

these derived conditions

for the subloci except that we must

locus for genus five.

In this section

we discuss cases where some derived vanishing properties give local rather than global characterizations; of surfaces admitting particular

that is) we show that the loci

automorphism groups are irreducible

components

of varieties defined by vanishing properties of the theta

function.

Two cases of genus five are considered.

the analysis

for obtaining a local converse for Corollary 9, Part Ill

and briefly discuss fixed-point-free definitive.

the case of surfaces

involution.

T

k = 1,2,5,4,

One expects that the local equations

be Teichmueller

locus of

T

T

(F)

involutions.

(E - H) k,

Then

(E - H) 1

k

or

and the

are disjoint since a surface of genus five

stand for the locus of surfaces

By the Riemann-Hurwitz

Let

of surfaces admitting

cannot be hyperelliptic and elliptic-hyperelliptic Let

are, in fact,

locus.

space for genus five.

stand for the loci in

more elliptic-hyperelliptic hyperelliptic

admitting at least one

The results of this section seem hardly

global if we again avoid the hyperelliptic Let

We will indicate

at the same time.

admitting a (5;1,3,3;1).

formula it is seen that the codimension of (F)

96

is eight in

T

and that of

group of automorphisms involutions

contains

(E - H) 3

generated

submanifold

a (5;i,3,3;i)

rains points

of

of

nine

we see that

defining assumed and

Moreover,

half-integer

as a function on

[qk ]

such that

equations

of

q2 + q5 + q6

To

Thus for

(E

H) 3

at

W 0.

The

is

+ ~9 = 0) corresponding

W 0.

,8

But a component

lies in an irreducible

component

is four-dimensional

of

(F)

con-

consider

...

there are

,9, are

[qk]'S

will be

to the T-families

in Corollary

a pure

of

is true for all of

(F).

(F)

4) That this naive counting settled in Baily [5]~

W 0.

contains

for

Thus

four dimensional

(F)0

of

(F)

V4

at

W0

passing

thru

by Corollary of

9. 5)

V4

on the theta functions

9 give local defining

Since any component

of

we see that this component

(F)0; that is, the eight conditions

derived

is a

nl + n2 + ~7 + ~8 = 0, n I + n 3 + ~4 + n9 = 0

defines

(F)0

of

W 0 ~ (E - H) 3

8[Dk](0;B ) = 0, k = 1,2 . . . .

Since

(F)

theta-characteristic,

involutions

W0

and

each component

the three e l l i p t i c - h y p e r e l l i p t i c

V4, at

(E - H) 3 c (F) o

(E - H) 3

8[nk](0;B ) = 0, k = 1,2,

for

to satisfy

variety,

Since every

(E - H) 3

For a particular 8[~3(0;B)

T.

4)

by three elliptic-hyperelliptic

It is also known that each component complex

is nine.

equations

points

of dimensions

of

(E

for -

(F)0o

H) 3 ) the result

is, in fact,

correct

is

5) To prove this last statement precisely calls for much detailed analysis of the T-families associated with the three elliptic-hyperelliptic involutions which generate the elementary abelian group of order eight. We omit this analysis.

97 Let

(FPF)

denote

the

fixed-point-free

involution.

fixed-point-free

involution,

Component defining

of

equations

k = 0,1,2, At

W0

(FPF) at

V6

(FPF)

passing

we see that equations locally

6) This methods.

e[ok](0;B)

for all of

define

points

...

(5;1,1,3;0)

of

be given by

a

contains

Moreover,

(E - H) 2.

W0

six-dimensional

,6.

a

every

Let the 8[nk](0;B)

= 0

Since

variety

defined by

the component,

(FPF)0 , of

is six dimensional

= 0, k = 1,2,

...

,6

(FPF)0 , and so similar

all components

last result

admitting

n0 + nl + n2 + n5 = n0 + n4 + n5 + ~6 = 0.

be the purely

through

of surfaces

(E - H) 2 c (FPF).

W 0 r (E - H) 2

8[nk](0;B ) = 0, k = 1,2,

T

Since any

contains

... ,6, where

let

loci in

of

and lies in give

V6,

locally defining

sets of six equations

(FPF). 6)

was proved by D. Mumford

(unpublished)

by other

98

9.

Closing remarks.

presentation.

We mention several

The first problem of genus four. groups

There are many obvious

that seem of immediate

is to characterize

Any information

gaps in the preceding

elliptic-hyperelliptic

on non-hyperelliptic

for genus four would be a significant

to finish the discussion the 2-hyperelliptic reasonable

conjecture

strongly branched terization

of involutions

case remains. exists.

involutions

of groups

be a significant

advance.

It should be remarked elliptic

on surfaces

automorphism

surfaces

automorphism

advance.

Secondly,

of genus five,

For all of the above cases,

Thirdly,

(Pl > 4P0 + i)

of order three for

would admit a charac4.

p ~ 5

Finally,

quite different

this paper. 7)

has shown how to distinguish

elliptic

surfaces by vanishing

Consequently,

properties

A similar characterization

morphism

groups

interesting

is an obvious

automor-

is completely

of elliptic-hyperelliptic

auto-

difficulties.

Finally, we include a few remarks paper might yield information

hyper-

next problem which seems to present

but not insurmountable

at least for low genus.

hyperelliptic

of the theta function

solved.

from those used in

of the theta function.

the problem of characterizing

phism groups by properties

hyper-

of the theta function has

been solved by classical methods Martens

any

would seem to

that the problem of characterizing

groups by properties

Moreover,

no

one would think that the

for the cases not covered in Corollary

characterization

importance.

as to how the methods

about non-abelian

The surfaces

7) We know of no explicit reference; in [K53 and more recently in [11].

automorphism

of this groups,

of genus three admitting

however,

the methods

the

occur

99

two largest automorphism

groups we will denote

the orders of the groups under consideration. these surfaces (3;1,1,1;0).

is hyperelliptic, By consulting

groups for a surface

existence

iately from the vanishing A similar situation

respectively of Theorem

four-groups

properties

automorphism

two non-commuting

seven and fourteen 6 we see that the

can be discovered

immed-

of the theta function.

occurs when considering

these will generate

involutions. a dihedral

group of order eight contains

which the dihedral

group of order six does not,

by vanishing properties

surfaces For a

of genus

non-hyperellip-

group of six or eight.

Since the dihedral

be distinguished

for

of genus three we see that the two automorphism

of seven or fourteen

tic surface,

WI6 8

on them must be a

a table of the possible

By the characterization

three admitting

and

Since neither of

any four-group

groups are the only ones containing four-groups.

W96

two four-groups, these two cases can

of the theta function.

References

[l]

Accola, R.D.M., Riemann surfaces with automorphism groups admitting partitions. P r o c e e d i n ~ of the American Mathematical Society Vol. 21-~qg"5-g[ pp. T77--~2.

[2]

Accola, R.D.M., Two theorems on Riemann surfaces with noncyclic automorphism groups. Proceedings of the American Mathematical ~ V o l . 25--[l-9-T~-p~. ~-g8---~2~

[3]

Accola, R.D.M., Strongly branched coverings of closed Riemann surfaces. Proceedings of the American Mathematical Societ i Yol. 26 (1970) pp.-~l~-/-~227

[4]

Accola, R.D.M., Vanishing properties of theta functions for abelian covers of Riemann surfaces (unramified case). Advances in the t h e ~ of Riemann surfaces Princeton University-Pr-~s (1971)?

[s]

Baily, W.L., Jr., On the automorphism group of a generic curve of genus > 2. Journal of Mathematics of Kyoto University Vol. 1 (195]-/2~--pp?-I01-I08, CS-~r~-6tion-p. 325.

[6]

Castelnuovo, G., Sur multipli du una serie lineare di gruppi di punti. Rendiconti del Circolo Matematico di Palermo Vol. VII (1-~--pp?--8~---flS.

[7]

Coolidge, J.L., A treatise on algebraic plane curves. Dover

[8]

Edge, W.L., Three plane sextics and their automorphisms Canadian Journal of Mathematics Vol. XXI (1969) pp. 126377.

[9]

Farkas, H.M., Automorphisms of compact Riemann surfaces and the vanishing of theta constants. Bulletin of the American Mathematical Society. Vol. 73 ~577--~p__2 1~-232.

[lO]

F a r k a s , H.M., "On t h e S c h o t t k y r e l a t i o n and i t s g e n e r a l i z a t i o n to a r b i t r a r y g e n u s , " Annals of Mathematics, Vol. 92 ( 1 9 7 0 ) , pp. 5 7 - 8 6 .

[11]

F a r k a s , H.M., P e r i o d r e l a t i o n s f o r h y p e r e l l i p t i c Riemann surfaces. I s r a e l J o u r n a l o f M a t h e m a t i c s Vol. 10 (1971) pp. 284-301.

[12]

F a r k a s , Ll.M. and Rauch, t t . E . , "Two k i n d s o f t h e t a c o n s t a n t s and p e r i o d r e l a t i o n s on a Riemann s u r f a c e , " Proceedin~_s o f t h e N a t i o n a l Acade_m_~ o f S c i e n c e s , Vol. 6 2 - ~ - 9 - ~ y ~ - - p p .

101

[13]

Fay, J.D., Theta functions and Riemann surfaces. Lecture Notes in M a E ~ e m a t ~ - 3 - - ~ . - - - - ~ 1 ~ r eV-~T-ffg,-?-q-g~3.

[14]

Humbert, G. Sur un complex r e m a r q u a b l e de c o n i q u e s e t s u r l a s u r f a c e du t r o i s i e m e o r d e r . J o u r n a l Ecole P o l y t e c h n i q u e Vol. 64 (1894) pp. 123-149.

[15]

K r a z e r , A . , Lehrbuch der T h e t a f u n k t i o n e n .

[16]

Krazer, A. and Wirtinger, W., "Abelsche Funktionen und allgemeine Thetafunktionen," En__ng~kl. Math. Wiss. If; BT, pp. 604-875.

[17]

Lebowitz, A. Degeneration of Riemann surfaces. ~eshiva University, 1965.

[18]

Lewittes, J., "Riemann surfaces and the theta function," Acta Mathematica, Vol. Iii (1964), pp. 37-61.

[19]

M a c l a c h l a n , C . , Groups o f a u t o m o r p h i s m s o f compact Riemann surfaces. Thesis--~ Unz"--~'-~v~FsTt-yo f ~irmingfi-ffm~-Ern-~nff?--1-Fb~.

[201

M a r t e n s , ~ . H . , V a r i e t i e s o f s p e c i a l d i v i s o r s on a c u r v e . I I J o u r n a l f u r d i e r e i n e und angewandt M a t h e m a t i k . Vol. 235 l~TY~y-p~. 8 9 - 1 0 0 .

[21]

Riemann, B., Gesammelte M a t h e m a t i s c h e Werke.

[22]

Roth, P . , Uber e l l i p t i s c h - h y p e r e l l i p t i s c h e Funktionen. M o n a t s h e f t e f u r Mathematik und P h y s i k . Vol. 23 (1912)

Chelsea

Thesis,

Dover, 1953.

~.--r~rr~.

[23]

S c h o t t k y , F . , Uber d i e Moduln d e r T h e t a f u n k t i o n e n , M a t h e m a t i c a , Vol. 27 (1905) pp. 235-288.

[24]

S c h o t t k y , F. and J u n g , H., Neue S a t z e ~ber S y m m e t r a l f u n k t i o n e n und d i e A b e l s c h e F u n k t i o n e n d e r Riemannsche T h e o r i e . S i t z u n g s b e r i c h t e P r e u s s . Akademie d e r W i s s e n s c h a f t e n (1909)

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Thomae, J . , B e i t r a g zur Bestimmung von 0 ( 0 , 0 , . . . ,0) d u r c h d i e Klassenmoduln a l g e b r a i s c h e r F u n c t i o n e n . J o u r n a l f u r d i e r e i n e und an~ewandt M a t h e m a t i k . Vol. 107 (187-0~j---~? 0 ~ -

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[27]

Wiman, A . , Uber d i e a l g e b r a i s c h e n Curven yon den G e s c h l e c h t e r n p = 4, 5, and 6 welche e i n d e u t i g e t r a n s f o r m a t i o n e n in sich besitzen. Svenska Vet. Akad. H a n d l i n g a r Bihang t i l l l t a n d l i n g a r 21 ( 1 8 9 5 ) , a f d 1, no. 3, 41 pp.

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W i r t i n g e r , W., U n t e r s u c h u n g e n u b e r T h e t a f u n k t i o n e n , T e u b n e r , L e i p z i g , 189S.

Al_~ebr___aais c u r v e s .

Acta

Dover

vr

B.G.

Index

Abelian covers

12

admissible

57 2

automorphism groups dimension of spaces admitting certain

63, 95

genus 2

50, 52, 81, 86, 87, 88ff 95, 98 61-62, 9@

genus 3

53, 56, 59, 88ff, 98-99

genus 5

53, 59, 79, 81, 95, 98

elliptic-hyperelliptic

hyperelliptic

31, 79, 98

involutions (see Po-hyperelliptic automorphism groups) po-hyperelliptic strongly branched unramified canonical divisor Castelnuovo character

50, 51, 52, 70, 7~ 69 17, 65 7 66, 70 6

coverings abelian completely r-m~fied abelian ramified

12 4Off 8

strongly branched

69

unramified abelian

65

half-canonical linear series

67

103 Humbert' s surfaces

86

involutions (See automorphism groups, Po-hyperelliptic) Jacobian linear series

"p-2 coz~ecture" quotient genus Riemann vanishing theorem

l, 7 66 67, 79 67 l, 7

strongly branchea covering

69

T-family

83

theta function

2

transformation theory

36

Wirtinger

65

Notation

b

analytic map

3, ~, 5

w/o

orbit space

3

A

abelian group

#

J(W I)

Jacobian of W I

4

u O, u I

maps of surfaces into Jacobians

4

M O,M I,M A,M~A meromorphic function fields

~, 5

a

homomorphism

5, 33,

(~ iE,B)

period matrix

6

e[n] (ujB)

first order theta function with theta characteristic In]

~)

multiplicity of branching

~

J XI

'~

Ko,K I

cj

divisor of branch points sad image under b

8

vectors of Riemanm constants

9, i0

jcj o in JCWl~. _

9

n-lul(~Z o)

9

~lul(a._zo)

9 io, ii

eo,e I R Xf,

X

characters of the abelian group, G

12

characters in R

12

f x V

meromorphic function associated with x 12

(Xl,---~ s )

an element of V corresponding to x

a product of cyclic groups

14

105

an admissible element of V

15, 58

an integer associated to •

15, 16

1/n-period in J(W0)

17

homomorphism of HI(W0,Z)

34, 55

A0

a disk on W 0

41

A1

b-lG 0 ) c w o

41

(Sl,S2,---,g s) t• C

x

Po-hyperelliptic

5O

Z2

group of order two

50, 51

In]

generalized hyperelliptic theta characteristic

50

generalized hyperelliptic period characteristic

50, 51

(PljP2,P3,P@jP0)

symbol for a four-group on W 1

56

(o)

period characteristic

68

[c]

theta characteristic

68

gr n

linear series of dimension and order n

(aj)

Also see page 3.

r

70