Jac(D

Copyright © by The University of Michigan 1975 All rights reserved

ISBN 0-4 72-66000-4 Library of Congress Catalog Card No. 75-14899 Published in the United States of America by The University of Michigan Press and simultaneously in Don Mills, Canada, by Longman Canada Limited Manufactured in the United States of America

PREFACE This book is a slightly expanded version of the series of four Ziwet Lectures which I gave in November I 974 at The University of Michigan. Ann Arbor. The aim of the lectures and of this volume is to introtluce people in the mathematical community at large-professors in other fie lds and graduate students beyond the basic courses- to what I find one of the most beautiful and what objectively spe.iking is at least o ne of the oldest topics in algebraic geometry : cun·es and their Jacobians. Because of time constraints. I had to avoid digressions on any foundat10nal topics and to rely on th e standard definitions and intuitions of mathem.iticians in gerie ral. This is not always si mple in Jfgebraic geometry since its foundational systems have tended to be more abstract and apparently more idiosyncratic than in other fields such as differential or analytic goemetry, and have therefore not become widely l.nown to non-specialists. ~l y idea was to ge t around this problem by imitating history: i.e .. by introducing .ill the characters simultaneously in their co mplex analytic and algebraic forms. This did mean that I had to omit tliscussion o f the characteristic p and .irithmetic sides. Ho1\ever it also meant that I could immedi.itely compare the strictly analytic constructions (such as T eichmuller Space) with the varieties which we were principally discussing. When I first stdrted doing research in algebraic geometry. I thought the subject attractive for two reasons: firstly . because it dealt with such down-to-earth antl really concrete objects as projective curves urn.I surfaces: secondly. because it was a small. quiet ~leld where d dozen people did not leap on each new ideath~ minute it became current. As it turned out. the field seems to have acquired the reputation of being esoteric, exclusive and very abstract with adherents who are secretly plotting to take over all the rest of mathematics! In one respect this last point is accurate: algebraic geometry is a subject which relates frequently with a very large number of other fields - analytic and differential geometry, topology , J...-theory, commutative algebra, algebraic groups and number theory. for instance- and bo th gives and rece ives theorems. techniques and examples with .ill of them. And certainly Grothendieck's work contributed to the field some very abstract and very powerful ideas which are quite hard to digest. But this subject. like nil subjects. has a dual aspect in that all these abstract ideas would callapsc of their own weight were it not for the underpinning supplied by concrete classical geometry. For me ic has been a real ad\enture to perceive the interactions of dll these aspects and to letllll as much .is l could about the theorems both old and new of algebraic geometry.

Dafydd ap Gwilym's The Lark seems to rne like the muse of mathematics: High you soar. Wind 's own power, And on !ugh you sing each song, Bright spell near the wall of stars. A far high-turning journey. ery P1eased I'd l"k If this boo!-. entices u few to go on to learn these "spells." I'll be v thook F.red Gehriog: Pe'.". Dureo '"' .'"' mony people I met " Ano Acbo, . . ' ' to hosp1tal1ty and their w1llmgness to l!sten. A tmal point : in order not their warm text. · we have omitted alm ost all re ferences and attributions m · t h e lectu to interrupt the h 11res t emselves• . Jnd instead written a separate section at the end ocrivinao re fe rences as tor good places to learn vari ous topics. we as suggestions

f~r

CONTENTS

Lecture!:

What Is a Curve and How Explicitly Can We Describe Them? ......... .

Lecture ff:

The Moduli Space of Curves: Definition. Coordinatization, and Some Properties. . . . . . . . . . . . . . . . . . 22

Lecture III:

How Jacobians and Theta Functions Arise ................ . .. . .... 44

Lecture Jr :

The Torelli Theorem and the Schottky Problem .... ... • ... ...... . .. 68

Guide to the Literature and References..... ...... ... . .... . . ... . .............. 95

I

I I

Lecture I:

What Is a Curve and How Explicitly Can We Describe Them?

In these lectures we shall deal entirely with algebraic geometry over the complex . numbers~. leaving aside the fascinating arithmetic and characteristic p side of the subject.

In this first lecture, I

would like to recapitulate some classical algebraic geometry, giving a leisurely tour of the zoo of curves of low genus, pointing out various features and their generalizations, and leading up to my first main point:

the "general" curve of genus g, for g large, is very hard

to describe explicitly. The beginning of the subject is the AMAZING SYNTHESIS, which surely overwhelmed each of us as graduate students and should really not be taken for granted.

Starting in

3 distinct fields of mathematics,

we can consider 3 types of objects: a ) Algebra:

consider field extensions

K

~

cr, where K is

finitely generated and of transcendence degree 1 over b ) ~:

First fix some notations:

we denote by

:uf1

~.

the

projective space of complex (n+l)-tuples (x 0 ,···,Xn), not all zero, mod scalars. coordinates.

x 0 ,·

· · ,Xn are called homogeneous

~ is covered by (n+l)-affine pieces.

Ui = (pts where Xi

Io)

and x 0 = X0 /Xi,···,xn = Xn/Xi

(xi omitted) are the affine coordinates on Ui. algebraic curves c c "JPn:

Consider

loci defined by a finite set of

C URVES AND THEIR JACOBIANS

· f a (x , · · · , Xn ) -- o, and such that for homogeneous equations 0 every x E x, c is " locally defined by n-1 equations with

3 f a 1 , · · · , fa n-1 plus g,

indepen d en t d:J... fferentials", i.e.,

o, such that for all a ,

with g(x ) F

n-1

-l

some polynomials ha, i

i =l

and rk( o f a /ox . (x)) i

=

n-1.

J

consider compact Riemann s urfaces~ .

c ) Analysis: ~

The result is that there are canonical bijections between the set of

[A

isomorphism classes of objects of either type. in case (b):

word about isomorphi

the simplest and oldest way to describe isomorphism in the

algebraic category is that

c1

CIP11 and

c2

"JP~

c

are isomorphic if

there is a bijective algebraic correspondence between

c 1 and c 2 ,

D c c 1 xc 2 defined by bihomogeneous equations

i.e., there is a curve

ga(X 0 ,···,xn 1 ; Y0 ,·· · ,Yn2 l = 0 which projects bijectively onto onto C2.J

c 1 and

To go back and forth between objects of type a),b ) ,c), for

instance, we

1)

associate to a curve C the field K of functions f: C ---?

a:

U

(co) given by restricting to c rational

functions p(X , · · • 0

x

' n

)/q(X

o'

...

x )

' n

1

deg p

=

deg q;and the

Riemann surface just given by C with the induced complex structure from lPn.

----•

Perversely, algebraic geometers persist in

talking about curv

an~ analysts about surfaces when they mean essentially the ObJect!

LECTURE I

2)

associate to a Riemann surface X its field of meromorphic functions; and any curve c which is the image of a holomor?'iic embedding of X in JPn.

3)

from the field K, we recover C or X as point set Just as the set of valuation rings R,

a: c R c K.

To X or C or K we can associate a genus g as usual: g

no. of handles of X

g

dim of')i

or

l" [space of holomorphic differentials w on X)

II

[space of rational differentials m = adx, (1

K

a:(x).

itself

a: U (en)

CURVES AND THEIR JACOBJANS

Here we have the famous theory of elliptic curves:

X

a: L, L a lattice which may be taken to be

=

zi: +z

• w' Im w

>

o.

. any non-singular p 1 ane cubic curve, i.e.,

c

c

c

]p2

defined by f(x,y,z) = o, f homogeneous of degree 3, with some partia . 1 non-zero at each root; in affine coordinates, x,y,

c is given as the zeroes of a

cubic polynomial f(x,y)

=

o.

a: (x,y'f\x)) , where f is a polynomial of degree 3

K

with distinct roots. The connecti ons between th ese are given as follows:

given X,

form the Weierstrass p-function : 1~

(z )

1

2 -

[(

z

z-~ ) 2

\J

a

?

and map ll'g-l

Also, if

3

~ i

~

g , wri te

= dti' ti a local coordinate at xi and then expand

0

$?-.

(we may assume

~i

r O if 3 ~ i ~ y . )

Then Petri ' s fir st step i s to

write down a basis for the vector space o f k-fol d h olomorphic differential forms on C for every k:

8 ....

these are differe n tial f o r ms

18

C URVES AND THEIR JACOBIANS

a{x ) {dx)k with no poles. dimension {2k-l){g-l).

For k .:::_ 2, they form a vector space of The table below summarizes his results.

Look

at it carefully - each column displays a basis for k-fold differentials, 1 ~ k ~

5.

Within each column however, we group the differentials in

rows according to the multiplicity of their zeroes on Uldef x 3 +·· ·+xg. k

k

Thus the first row is always

cii3

at one of the

all other monomials in the

whereas

,···,cpg as each of these has no zero

zero at least to 1st order at each point of {)f .

cii . J.

's will be

The second column

arises like this: a) one checks that every quadratic differential which is 0 on is of the form cpl'. ( ) + r,. 2.(

v1

),

b) hence if c) Omitting these cpicpj' the remaining 3g- 3 monomials form a basis as in di ca ted. The third column arises like this: a) one checks that the triple differentials cp~(

) ttp 1 r,i 2 ( ) ~~( )

are of codimension 1 in the vector space of triple differentials

w, with double zeroes on 01 "fundamental class on c" ·.

th

formed out of

Res

3

d· · e con ition for such an w to be

alone is that

( *) all zeroes y of qi 1 except x , · · ·

This is a reflection of the

>X

g

y

(w/ r;i r;i ) 1 2

0

•

LECTURE I

19

b ) Writing ni as indicated, this has a double zero on (fl every difference c) Hence

ls,

n.-n. J. J

n.-n . J. J

and

satisfies(* ) .

can be rewritten as indicated and this leaves

e.xactly 5g-5 r emaining triple differentials as a basis. The remaining columns are quite mechanical:

the 2 ways of rewriting

differentials reduce us to the attached list, and, by counting, leave e

us with exactly the right number to be a basis! Let us write out the 2 sets of identities by which these reductions are made.

They will be : g

I a iJk(cµ1,(ll2) Cllx:

+ \il.Jcµlcµ 2

k= 3

)

.

g

n.J. -n J

.s

(here the a and in

is

k~3a~jk((lll,(ll2)Cllx: linear, the a

F

3 $_ i,J ~ g, i

~-l

j ).

1

is

quadratic, the

\i's

are scalars,

But what this means in terms of equations

is precisely that 2 sets of homogeneous equations:

ials

g

xixj -

k~3aijk(xl,x2)~ g

l a~jk(xl ,x2 )~

k.=3

\)' 2x ijxl 2

\) " x x 2 ij l 2

of degrees 2 and 3 generate the ideal of C!

C URV ES AND THEIR JACOBI \NS

20

Petri now goes on to prove 3 bea•1tiful results

I)

These equations are related by syzygies: a)

f .. .LJ

b)

~fij

f

gij + gjk g

Ji'

gik

l a . Lfk.t Xjfik + t=3 l.J tlk

I

pijkg jk

a . k.tf .t L=3 .L J

tlj

where 3 ~ i,j,k ~ g, i,j,k distinct, and the pijk's are scalars symmetric in i, j and k.

II ) The re are 2 possibilities:

either pijk

= a ijk =

0 whenever i,j,k

are distinct, and then C is very special ~ i t is a triple covering of lPl or if g

= 6

it may also be a non-singular plane quintic; or

else ~ost of the p's and a 's are non-zero ( prec isely, one can write [ 3 ,···,g} wi t h

= r 1uI 2

pijk ~

ideal of

o,

s o that for all j E ~· k E I 2 , there exists an i

a ijk r

o ),

a nd then the fij's alone gen e r a te the

c.

III ) Given any set off . . 's, g . . 's as above related by the syzygies in l.J l.J (I),

where all Ai

I o,

and at least one pijk

I O,

there exists a

curve C of genus g whose canonical imag e in n$1-l is defined by these equations. In my mind,(III ) is the most remarkable:

this means that we have

a complete set of identities on the coefficients a ,a ' , \!, \! characterizing those that give canonical curves. to

' , \!" ,

A,µ, p

It would be marvelous

use this formidable and precise machine for applications ,

-

- ..... -

-

lfi Cf~ "'. (-) Cf.

['""'""

lo ( - )

r·~,w· J

- . . -J

, cpJ I

l . 'l.

2.

Cf~

,.!. t-C-) ( 1-x,y) carries CA to

c 1 _A; and the map (x,y )

carries

One must cook up an expression in substitutions and no

mor~.

\

i nvari a nt und er these

It is custo mary t o use:

j

(It is readily checked that this J is invariant under these 6 substitutions and since 6

=

max(deg. of numerator and denominator ) , no other

A's give

the same j.)

3.ll We then get a bijection between the isomorphism classes of genus l

ned curves c and the complex numbers~ by taking C to j(A) if c ~CA. Analytically, if c-= ~/L, the j-invariant of c can be calculated from L in the following way:

define*

*The stream of funny constants can best be explained as making a certain Fourier expansion have integral, not just rational, coefficients. This makes the theory work well under "reduction modulo p".

24

CURV ES A ND THEIR JACOB!ANS

Then it can be shown that:

In particular, if L

j ( w) = j ( a:/

zi:+z · W, then

elliptic modular function.

Its

:z+:z·w)

is the famous

most important property is its invariance

under SL( 2 ,Z>:) , which can be explained from a moduli point of view as

{3(: :)

E

w

SL(2,~)

aw 1 +b

-2 -- cw 1 +d

(This is trivial to check . )

\

l

J

isornorp~ism

c/~ +~.

wl

=

But

~]

}

a:;~+~·w2 ~ l

a E

a:

such

a(?UZll..l\) =

tt Hence:

w2



O, then ,6*L is a tensor product of

powers of this bundle and the line bundles

5"."0'.-; l. x c where si: C---? Xis the extension of si.

Now, in fact, by using the

theory of algebraic surfaces, one gets a very good bound:

where q s

genus

*s

c

dimension of biggest

. b a e 1 ian variety which appears in the Jacobian -1 of every curve rr (s) of the family.

cl_(S*~/c) bound.

seems harder to bound:

However, following Grauert

showing first that almost all fibres

I

don't know a nice

one can show that

the cotangent bundle

o! x (o f

small explicit

one exists by

rank 2)

rrl(s) of X" over C and then applying

--

is ample on general results

•

43

LE CTURE II

on ample vector bundles.

A good explicit bound here would be very

interesting. To carry out the second step, one appli e s Kodaira-SpencerGrothendieck deformation theory to calculate the vector space of infinitesimal deformations of

TT: X ----? c-S and of

More precisely, one looks at deformations of

X ~C

rr:

X such

s: c-S

--?

x.

that the map

extends to this deformation and all singular fibres remain

concentrated in

TT- 1 (s

).

It turns out that:

· space of ~ ( infinitesimal deformations of TT; X ~C-S and

(

Space of ~ infinitesimal deformations of s: C-S --? X

To show these vector spaces are

{o ),

that ~/c is an ample line bundle

Arakelov's deepest contribution on

x.

Then the first space is

and the second space is degree. curve

(o )

one shows~ and this is

(o)

by Kodaira's Vanishing Theorem,

because the line bundle has negative

Amazingly, Arakelov's proof here involves studying the D c

X such

D(\TT-1( s)

that the Weierstrass points of TT

-1

(s)

and identifying via differentials the line bundle on X of which D is a section •

Lecture III:

How Jacobians a nd Theta Funct ions Arise

would like to begin by introducing Jacobians in the way that

I

they actually were discovered historically.

Unfortunately, my

knowledge of 19th-century literature is very scant so this should not be taken too literally.

You know the story beg an with Abel

and Jacobi investigating general algebraic integrals I

where

f

=

sf(x )dx

was a multi-valued alg ebraic function of

x,

i.e., the

solution to g(x, f(x )) _ So we can write I

o,

g

polynomial in

2

variables.

as

where Y is a path in plane curve g(x,y}

o;

or we may

reformula~e

this as the study of integrals

a~ r(a)

S o(x,y)

P(x,y )-dx

ao

P,Q polynomials a,a 0 E plane curve C: g(x,y)

of rational differentials w on plane curves

c.

The main result

is that such integrals always admit an addition theorem: there is an integer g such th a1,···,ag+l are any points of

0

i.e.,

at if a 0 is a base point, and

c,

then

one can determine up to

LECTURE Ill

45

permutation bl' ••• ,bg E C rationally in terms of the a's such that a1 ag+l bl bg + .. • + w + + w w, mod periods of a0 a

s

s

s

...

Sw.

0

For instance, if C

=

nJ-,

w = dx/ x, then g

Iterating, this implies that £or all c 1 , ... ,cg EC

1 and:

a 1 ,···,ag,b 1 ,···,bg E c, there are

depending up to permutation rationally on the a's and

b's such that

.I s~

( mod pe riods ) .

l.= l a

0

Now this looks like a group law!

Only a very slight strengthening will

lead us to a reformulation in which this most classical of all theorems will suddenly sound very modern. algebraic group G:

We introduce the concept of an

succinctly, this is a "group object in the

category of varieties," i.e., it is simultaneously a variety and a group where the group law

m: GXG---? G and the inverse i: G --?G

are morphisms of varieties.

Such a G is, of course, automatically a

complex analytic Lie group too, hence i t has a Lie algebra Lie(G),

and an exponential map

exp:

Lie(G) ~G.

Now I wish to rephrase

*

E. g. 1 one can find polynomials gi(x, y; a) in x, y and the coordinates of the a's such

that the bi's are the set of all b

E

c such that gi(b; a)

= 0.

CURVES AND THEIR JACOBlANS

46

Abel ' s theorem as asserting that if C is a curve, and w

is anY rational

differential on c, then the multi-valued function

s a

a !----;.

lU

a

0

can be factored into a composition of c-(poles of w)

¢

~

J «'.

3 functions: exp

. J Lie

L

~~~~

~

where: i )

ii ) iii )

J

is a commutative algebraic group,

L is a linear map from Lie J to

~

p is a morphism of varieties; and,

in fact, i f

then if we use addition on J to extend

p

g

dim J,

to

¢(g ): [ (C-poles w) x· · · x(c-poles w)/permutations ] ---? J

SS then p(g ) is birational, i.e., is bijective on a Zariski-open set. In our example

c then J =

:n>1-(o,CD)

= 1P1,

w =

dx/ x,

which is an algebraic group where the group law is

multiplication, and

p

is the identity.

The point is that J is the

object that realizes the rule by which 2 g-tuples (a 1 , · · · , a ) (b ••• b) g , l' , g are "added" to form a third (c 1 , ···,cg), and so that the integral

LECTURE Ill

47

l g

l r(xi)

becomes a homomorphism from J

to~­

A slightly less fancy way

i =l

p:

to put it is that there is a invariant differential

n

c-(poles w) ---?Janda translation-

on J such that

1'*n =

w ,

hence (mod periods ) . a

0

Among thew's, the most important are those of 1st kind, i.e., without poles, and if we integrate all of them at once, we are led to the most important J of all:

the Jacobian, which we call Jae.

From

property (iii ) , we find that Jae must be a compact commutative algebraic group, i.e., a complex torus, and we want that

p:

C ~Jae,

should set up a bijection: translationdifferentials]=] [ invar iant l-forms] ~[rational ' w on C w/ o poles on Jae

iv)

n

Thus dim Jae

dim R1 (c) genus g of c.

To construct Jae explicitly, there are 2 simple ways: v)

Analytically:

write

L a lattice.

Define:

Jae = V/L, V complex vector space,

CURVES AND THEIR JACOBIANS

48

v = dual of L =iset of

.t( w)

.t Ev obtained as periods, i.e.,

Sw

for some 1-cycle Y on c .

y

a E C

Fixing a base point a 0 E C, define for all

v/L

p(a) = image in

of any

L

E V defined by

a

.t(w)

Sw, a0

where we fix a path from a 0 to a.

Note that since Jae is a group,

v* ;;- (translation-invariant) _ (cotangent sp. to Jae at a) _ Rl(c). any a. E Jae

1-forms on Jae

vi)

Algebraically: sgc

=

following Weil's original idea, introduce

c x ·•· x c/s

g

and construct by the Riemann-Roch

theorem, a "group-chunk" structure on sgc, i.e., a partial group law: m: ui c sgc

Zariski-open.

He then showed that any such algebraic group-chunk prolonged automatically into an algebraic group J with (some Zariski-open

u4 ).

LECTURE III

49

An important point is that p is an integrated f orm of the canonical map _q-1 C ---?Y discussed at length above t: vii)

t

is the Gauss map of

p,

i.e., for all x E

c, dp(T

x,c

) is

a 1-dimensional subspace of T~( ) , and by translation ,., x ,Jae this is isomorphic to Lie(Jac).

If

~-l = [space of

1-diiJ:. subsp. of Lie(Jac)J, then dp: C (Proof:

~~-l

is just t.

this is really just a rephrasing of (iv).)

The Jacobian has always been the corner-stone in the analysis of algebraic curves and compact Riemann surfaces.

Its power lies in the

fact that it abelianizes the curve and is a reification of H 1 , e.g., viii)

Via

p:

C ----? Jae, every abelian covering TT:

the "pull-back" of a unique covering p: G1

c 1 ____, C is

~

Jae

Weil's construction in vi) above was the basis of his epoch-making proof of the Riemann Hypothesis for curves over finite fields, which really put characteristic p

algebraic geometry on its feet.

There are very close connections between the geometry of the curve C (e.g., whether or not C is hyperelliptic) and Jae.

We want

to describe these next in order to tie in Jae with the special cases studied in Lecture I, and in order to "see" Jae very concretely in low genus.

The main tool we want to use is:

CURVES AND TH E IR JACOBIANS

50

] i

Abel's Theorem:

Given

x 1 ,·· .,xk, Y1,···,Yk E c, then

rational function f on C with ( f) = fzeroes of #oles def' I:xi -I:yi

When this holds, we say For instance, when C

=

= Eyi'

Exi JP 1 ,

or

I:xi,I:yi a re linearly equivalent.

any 2 points a ' b are linearly equivalent

via the function f(x )

=

~ x-b

For every k, we consider the map:

~ c x ••• x c

denotes power of

c,

· · the kth symmetric c k divided by permutations, i.e.,

then this map factors via

Define

1 $. k $. g-1

(p(k)

surjective if k

2.. g).

The fibres of this map* are called the

linear systems on C of degree k, and by Abel's theorem they are the equivalence classes under linear equivalence and can be constructed as follows:

*A technical aside:

(k)-1

p

.

the complete ideal of functions on skc vanishing on

{a) is generated by the functions on Jae vanishing at . · a - this is needed to make rigorous some of the points made below.

LECTURE Ill

SI

k

a)

UT.

Pick one point

\ L

i =l

b)

Let

x. E skc. .1.

L(U{) =iv. sp. of fens. f on

C

with (f) + (){ '

o, i.e.,}

poles only at xi' order bounded by mult. of xi

UC .

in c)

Let

d)

Then

lotl

= {set of divisors Ilyi

IUII

= p(k ) -l(p(k ) (U{)) .

16~ I ;;;e)

= (f

) + Exi, f E

L(C~ ),

f -F

o} c

skc.

Note that it follows

projective space of 1-dim.!. subspaces of L (Ur) .

We also want to use the Riemann~Roch theorem that tells us that dim IL{ I = k - g + i where i = ~ dim of v. sp. R 1 (-LO of differentials}

Lw E R 1 (c ) ,

with zeroes on

U[ .

Now let's look at low genus cases: Jae lg

ii:

(a)

p:

= (o) C

~Jae is an isomorphism, i.e.,

C = Jae.

p1 of degree 3. -1

i}The

-1

3 cycles (rr1 (x)} and (rr2 (x)} . ~ - 1 form 2 linear systems E 1 ,E 2 c S 3c, with El = E 2 = lP.

2 families of degree

Then: case b 2 :

w3 =~ ( s 3c

. h E 1 ,E 2 blown d own t o wit

2 points e 1 ,e2)

and

e 1 ,e 2 = ordinary double points of

w3 .

In case b 1 , the 2 rulings "come together"; in fact,

s3c

contains only one non-trivial linear system E, and

CURVES AND THEIR JACOBIANS

56

case b 1 :

w3 e

~ ( s 3c with E blown down to e ) and higher double point of

=

w3 .

In the hyperelliptic case a, it turns out that there is a whole curve of linear systems Ex on a point

x E c:

c

s 3c depending

in fact, take the degree 2 linear

system, and just add x to each of its members. w3

case a:

~ {s3c

Thus:

with the surface UEx blown

)

down to a curve Y isomorphic to C and

=

y

Enough

examples:

double curve of w3 .

the moral is that the wk's and their singularities

display like an illustrated book the vagaries of the which they arise.

curve C from

The general result is the following:

Theorem (Riemann-Kempf):

Let

a E Wk c Jae, let L = p(k)-l(a) c SkC

},

and suppose L ~ :JP •

Then

Wk

has a singularity at a of multiplicity

( g-~+J,), ~ and the tangent cone to Wk inside T

a,Jac

(= tangent sp. to Jae

at a) is equal to:

LJn,o(k) (T

le) U{,s·-c

Ul.EL

Here

Dp(k)

is the differential of

p(k)

sequence:

(*)

0 ---') T,. L ~T ""'

"-

(A,S·-c

and i t gives rise to an exact

LECTURE Ill

57

In fact, this sequence actually "displays" the Riemann-Roch formula in

a beautiful way:

R 1 (c)

using the fact that 2'

translation-invariant differentials on Jae

;;;- T* (=cotangent sp. to Jae at a), for all a, a,Jac w E R 1 (c) corresponds to [w) E T* a ,Jae'

it is not hard to check that if then: 0

in

* ,skc

T(,7

w is zero on

U{

•

Therefore Coker Dp(k)

dual of R 1 ( - l,~, the space of differentials zero on

Ul. •

Therefore counting the dimensions of the vector spaces in dim L

(*):

dim SkC - dim Jae + dim Coker Dp(k )

which is the Riemann-Roch theorem!

What comes next is going to be

harder t _o follow, but we can go much further: let (wi} be a basis of R 1 (-UI) and let ( f j} be a basis of L(U{). Then a general member of L is given by J,

+ lP

if and only if

~

1

E nL/ L.

ii) Modulo the choice of distinguished generators for the associated finite group Gn, we get, up to scalars, a distinguished basis of Sn' hence a normalization of

under projective transformations. by

* )1

~

1 E -L/L acts on

n

In this normalization, translations

t (~g/L) by a simple set of explicit ngxng-matrices. n

set of triples (a,x,0, a E JR,

x Ev,~

EV,

with group law:

(a,x,O•(a',x',~') = (a-+a'+ 1 :

c 1 ~ Jac 1

given by

Then the coverings of Prym via

c2

the canonical map.

i n question are pull-backs of abelian coverings

p_.

Now Jac 1 is very nearly the product of Jae and Prym:

in fact

there are homomorphisms Jae x Prym such that a•a,

~·a

a ---;, Jac 1

~

are multiplication by 2 , and

finite abelian groups isomorphic to (~2~)2 g-l The genus of c 1 will be 2g-l, hence

ker a, ker a (g =genus of

are

c).

CURVES AND THEIR JACOB!ANS

92

dim Jac 1

2g-l

dim Jae

g

dim Prym

g-1.

The beautiful and surprising fact is that the new abelian variety Prym carries a canonical principal polarization too.

e

c Jae, (!ill c Jac 1 and

:=!

In fact, if

c Prym are the 3 theta-divisors,

::;\

is

characterized by either of the properties: a- 1 (e1 ) - 28 + 2 :::

s- 1 ce+1=:) - 281 (where - means the fundamental classes of the divisors are cohornologous; or equivalently that suitable translates of the divisors are linearly equivalent ) .

So far, these facts tie Jae, Jac 1 and Prym into a tight

but quite elementary configuration of abelian varieties, but one that does not impose any restriction on Jae itself.

Thus if

-() , -{) 1 and ~

are the theta functions of these three abelian varieties, one can calculate

-B 1

from

-$ and ; and vice-versa, but -{;) and ~ can be

arbitrary theta-functions of g and g-1 variables respectively.

But

now the underlying configuration of curves comes in and tells us:

(*) where

n

E Jae is the one non-trivial point of order 2 such that

(i.e., the original double cover

c 1/ c

"corresponds" to n).

a(n)

= 0

This follows

LECTURE IV

in fact directly from the interpretations ®1 = W c Jae 2g l"

Now

93

e

=

w

g-l c Jae and

$ and ~ cannot be arbitrary any longer:

(*) turns out to be equivalent to asserting that the squares of the theta-nulls of Prym: ~

2

a,.," E !...,g-1 2 ""

[a](o),

are proportiCf1al to certain monomials in the theta-nulls of Jae:

(after one make the correct simultaneous coordinatization of Prym and Jae ). embed

A third way to interpret(* ) is via the Kummer variety:

Jac/ .±1

and

Prym/ .±1

+:

in projective spaces.

Jac/ .±1

2g-l

c

> 1P

>-#

Prym/ .±1 C

'f :

g-1

F.

J

1P

p

Via suitable normalized coordinate systems as in Lecture III, there is a canonical way to identify JPP

with a linear subspace of JPj.

Then

(*) says:

(**)

'f(O)

=

t(!l ) . 2

Since Im t, Im 'f have such large codimension, one certainly expects that for most g and (g-1)-dimensional principally polarized abelian vari~ties X,

Y, 'f(Y)

and

t(X)

would be disjoint.

CURVES AND THEIR JACOBIANS

· 94

The case when g

4 is the first one where

t(~g)

l ag

and in

this case: dim

a4 =

10

dim t(~4 ) = 9. Schottky was able to show that the above identities on implied one identity on ~

alone, of degree

that this identity holds only on

t(~4 ) !

--3

and

~

8 , and Igusa has asserted

However, when g

>

4, no

efficient method of eliminating ~ from the above identities is known and the ultimate problem of characterizing t(~g) by simple identities in the theta-nulls remains open. I am confident that Schottky's approach has not been exhausted, however, and a full theta-function theoretic analysis of the dihedral (or even higher non-abelian coverings of

c)

remains to be carried out. I hope these lectures have perhaps convinced the patient reader that nature's secrets in this corner of existence are fascinating and subtle and worthy of his time!

\

Guide to the Literature and References Lecture I: The best beginning book on the theory of curves is: R. Walker, Algebraic Curves, (Dover reprint 1962) . With a little more background, the best modern book is: J.-P. Serre, Grou es algebriques et corps de classes Hermann, 1959 . A more analytical treatment can be found in: R. c. Gunning, Lectures on Riemann Surfaces, {Math. Notes, Numbers 2 , 6, 1 2 , Princeton Univ. Press) . C. L. Siegel, Topics in Complex Function Theory (3 volumes, Wiley-Interscience, 1969-1973) . There is a huge classical literature, of which the following are best known to me: K. Hensel and G. Landsberg, Theorie der Algebraischen Funktionen einer Variablen, ( Chelsea reprint, 1 965). F. Severi, Vorlesungen fiber Algebraischen Geometry (Johnson reprint, 1968).

J. L. Coolidge, A treatise on algebraic plane curves {oxford, 1931 • The most famous book which drew together c learly the topological, analytical and algebraic strands is: H. Weyl, Die Idee der Riemannschen Fl~che (Teubner, 3 rd ed., 1955) . For elliptic curves specifically , one can look for instance in : Ch. 7 of L. Ahlfors, Complex Analysis ( 2nd ed., McGraw-Hill, 1 966 ): a good quick introduction to the analysis.

I

q I

CURVES AND THEIR JACOBIANS

96

A. Hurwitz, R. courant, Allgemeine Fu~ktionentheorie und Elliptische Funktionen (springer, 1964).

s.

Lang, Elliptic Functions, (Addison-Wesley

1973)·

ctures at Haverford, to be published J. Tate, Notes from Phillips Le

The Gelfond-Schneider theorem quoted in Lecture I can be found in:

s.

Lang, Introduction to transcendental numbers (Addison-Wesley, 1966) ' p. 22 .

For higher Weierstrass points, see:

B. Olsen, On higher Weierstrass points (Annals of Math., pp. 357-364).

95 (1972 ) ,

J. Hubbard, Sur les sections analyti e de la courbe universelle de Teichrnaller, to appear in Memoirs of AMS ) . For the theory of theta-characteristics, some references are:

D. Mumford, Theta characteristics of an algebraic curve (Annales Ecole Norm. Sup., i (1971), p. 181). Ch . 13, Vol. 2 of H. Weber, Lehrbuch der algebra, (Chelsea reprint) ~ for the case of g = 3. The details of the hyperelliptic vs. non-hyperelliptic story can be found in most of the books cited above, e.g., Walker, Hensel-Landsberg or Severi.

Concerning the explicit description of the inequalities on

generation of Fuchsian groups, see:

R. Fricke, F. Klein, Vorlesungen aber die Theori"e d er Automorphen Funktionen (Johnson reprint 1965), esp. v 0 1 • 1, Part 2, Ch. 2.

GUIDE TO TllE LITERAT UR!o 97

N. Purzitsky, 2 generator discrete free products (Math z · t I ' el. S • 126 , P • 20 9 an d other papers in Math. Zeit., __ Ill. J.)

I

L. Keen, On Fricke Moduli, in Annals of Math. Studies 66 , 1971. Kajdan's ideas on the metrics of varieties

n/ r are contained in :

D. Kajdan, Arithmetic varieties and their fields of quasi definition, (Actes du Cong. Int., Nice, Vol. 2). Proofs of the facts concerning representation of a g eneral curve c of genus g as a plane curve of lowest possible degree or as a covering of

IP1 of lowest possible degree can be found in: S. Kleiman, D. Laksov, Another proof of the existence of special divisors, (Acta Math., 1 32 ( 1974)) . Petri's work can be f o und in : K. Petri, Uber die invariante Darstellung algebraischer Funktionen einer Ver~nderlichen, (Math. Ann. 88 ( 1922)) .

B. Saint-Donat, On Petri's Analysis of the Linear System of Quadrics through a Canonical curve, (Math. Ann., 206 ( 1973)) .

K. Petri, Uber Specialkurven I, {Math. Ann., 93 ( 1924)) .

Lecture II: I

on

~

gave some introductory talks on moduli problems in general and

l,O

, i.e., the elliptic curve case, in particular, in:

D, Mumford, K. Suominen, Introduction to the Theor of Moduli, in "Algebraic Geometry, Oslo, 1 970", Wolters-Noordhoff, 1971 ) . For g

1, cf. also references given above f o r elliptic curves.

genus 2 case is in: J,-I. Igusa, Arithmetic variety of moduli for genus two, (Annals of Math. 72 (1960)) .

The

CURVES AND T H EIR JACOBIANS

98

!Dlg

The precise definition of

. ctive and proof that it is a quasi-proJe

variety are in: · the moduli of abelian W. Baily, on the theor of e-functions varieties and the moduli· o f curves, Annals of Math .,

75 ( 1962 )). n. Mumford, Geometric Invariant Theory, (Springer 1 965 ) . In particular, the f irst reference coordinatizes !Dlg essent i ally by

theta-nulls; the second coordinatizes !Dlg b oth b Y a v ariant of theta-null! using cross-ratios of points of finite order on the Jacobian , a nd also by invariants of the Chow form as in the text of the Lectur e .

The

final method of using cross-ratios of higher Weierstrass points has been suggested b y Lipman Bers.

The method as described i n the tex t

has never been published, but details can be fill ed in as follows : (1 ) one must first assign multiplicities to the higher Weierstrass points so that the divisor with C

l!{k of all of them varies algebraically

cf. Hubbard (op. cit. ) , (2) one proves tha t

with multiplicity > g 2 in

rn k;

no x E

c

occurs

( 3) using this one d educes t hat

~k (Ulk )

is stable in this sense of "Geometric Invariant Theory ", Ch, 3 ; (4) check that if a quadric contains and since

tk(~k),

then i t contains

ik(c) is an intersection of quadrics,

~k(~k)

~k(c),

determines c

up to isomorphism; ( 5 ) apply the results of "Geometric Invariant Theory", Ch. 3.

GUIDE TO THE LIT ERATURE

99

Concerning Teichmtiller space, the best reference is the survey article of Bers: L. Bers, Uniformization( Moduli and Kleinian groups, (Bull . Lona . Math. Soc.~ 1972)) . For the compactification

~

of

!Dlg, see:

P. Deligne, D. Mumford, The irreducibility of the space o f curves of given genus, (Publi. IHES, 36 (1969 )) . L. Bers, S aces of degenerating Riemann Surfaces (in Annals of Nath. Studies 79 1974 , as well as forthcoming articles by F. Knudsen and myself. for the "positive curvature" assertions on

!Dlg

The references

are as follows:

E. Arbarello, Weierstrass points and moduli of curves, (thesis; to appear in Comp. Math . ) . D. Mumfor d, Abelian quotients of the Teichmtlller modular group, (J. d'Anal. Math., 18 (1967)) . H. Rauch, The singularities o f the modulus space , ( Bull. AMS,

68 ( 1962)) . B. Segre, Sui moduli delle curve algebriche,\Annali di Mat.

( 1930 )) . Concerning the Teichmtlller metric and Petersson-Weil metrics on

l ig,n•

see: L. Ahlfors, Curvature properties o f Teichmtiller's space , (J. d'Analyze, 2 (1961) ) . L. Ahlfors, Some remarks on Teichmtiller's space of Riemann surfaces (Annals of Math. , 74 (1961)). H. Masur, On a class of geodesics in Teichmtiller space (to appear) H. Royden, Automor hisms and isometries (in Annals of Math. Studies

space

H. Royden, Metrics on Teichmtiller space (in Springer Lecture Notes,

400).

!!

CURVES AND THEIR JACOBIANS

JOO

For the Ahlfors-Pick lemma, cf.

M. Cornalba, P. Griffiths, Some transcendental aspects of algebraic geometry, §6 (in Proc. Summer Institute on Alg. Geom. 1974, AMS). The rigidity theorem of A-P-M-G has usually been considered separately in 2 parts: family

one on the finiteness of the set of sections of a fixed

TT: x ---? C-S; the other on the finiteness of the set of

families TT.

Both have much deeper, still unsolved number-theoretic

analogs ~viz. given a number field K, then one conjectures that l) given a curve D defined over K of genus g ~ 2 , D has only a finite set of K-rational points and 2 ) given a finite set S of primes of K

r•

and g ~ 2 , there are only finitely many curves D defined over K with "good reduction outside S" and of genus g.

The first ie called v

Mordell's conjecture and the second is called Safarevitch's conjecture (cf. his talk at the Stockholm International Congress of 1962) . one replaces K by the field of rational functions on a curve ~'

c

If over

these conjectures are equivalent to the Rigidity Theorem of the

lecture.

The Mordell part was proven first, independently by Manin

and Grauert: H. Grauert, Mordells Vermutung ~ber Punkte auf algebraischen Kurven und Funktionenk8rper (Publ. IHES 25 (1965 )) . Y. Manin, Rational

oints on algebraic curves over function

ffii~e~l~d~s~,-i~I~z~v~e~s~t~i~J~·a;;'-iA~k~a~d~.~N~a~u~k~,~2~7~~1~9~3~),..E~~~~

P. Samuel, La con·ecture (sem. Bourbaki,

corps de fonctions,

GUIDE TO THE LITERATURE

I 01

The Safarevitch part was proven by Arakelov usi'ng earlier partial results of Par~in: s. Ju. Arakelov, Families of algebraic curves with fixed degeneracies, (Izvest. Akad. Nauk 35 (1971)) . A.N. Parsin, Algebraic curves over function fields , ( Izvest . Akad. Nauk, 32 (1968)) .

Lecture III: The Jacobian is introduced in the standard books referred to in our notes to Lecture I:

esp. Serre (op. cit.) shows that all rational

differentials w on curves C are pull-backs of translation-invariant differentials

n on

algebraic groups J via some ratio nal map

p:

c

- > J.

Moreover Gunning (op. cit., esp. Math. Notes 1 2 , subtitled "Jacobi Varieties") treats in detail many of the topics of this and the next Lecture.

For Weil's original ·algebraic construction of the Jacobian

and its application to the Riemann hypothesis, see: A. Weil, Varietes abeliennes et courbes algebriques, (Hermann,

1948). The result that I called the theorem of Riemann and Kempf on the singularities of Wk was proved for k = g-1 by Riemann.

Kempf's

results appear in: G. Kempf, On the geometry of a theorem of Riemann (Annals of Math., 98 (1973 )). G. Kempf, Schubert methods with an application to algebraic curves (stichting Math. Centrom, Amsterdam, 1971).

CURVES AND THEIR JACOBIANS 102

The elegant proof of Riemann-Roch

using the differential of

(k) iS

P

worked out in detail is: Riemann Roch theorem of algebrai_c:. A. Mattuck, A. Mayer, The Norm. Pisa, .!I (1963)) . ~' (Annali Sc. Here are references for the theory of theta functions; and more theory on abelian varieties: . generally, f unc t ion assical theory of theta functions (In AMS Proc. of W. Baily, Cl _ ) Syrop. in pure math., vol. 9 . · F. Conforto, Abelsche Funktionen un d algebraische Geometrie (Springer-Verlag, 1956) . J. Fay, Theta functions on Riemann Surfaces (springer Lecture Notes 352). J.-I. Igusa, Theta functions (Springer-Verlag,

1972) ·

A. Krazer, Lehrbuch der Thetafunktionen (Teubner,

1903) .

D. Mumford, on the equations defining abelian varieties (Inv. Math.,~ and 2 (1966-67)). D. Mumford, Abelian varieties (Tata studies in Math.5", Oxford, 2nd ed.

1974 H. Rauch, H. Farkas, Theta functions with ap licatione to Riemann Surfaces, (Williams and Wilkins, 1974 . My treatment in the lecture follows more or less my book Abelian Varieties;

e.g., the 2 embedding

of this book.

theorems above are proven in Ch. l

The group-theoretic aspects discussed in the lecture are

in my book, § 13, as well as in my paper and Igusa 's book.

The prime

form Ee and its applications is due to Riemann and is discussed at length with many more applications in Fay.

GUIDE TO THE LITERATURE

103

Lecture IV: Besides the references give above on abelian varieties and moduli problems, one can find further material in: J.-I. Igusa, On the graded ring of theta-constants (Am. J. Math ., 86 (1964) and 88 (1966)). D. Mumford, The structure of the moduli spaces of curve s and abelian varieties (in Actes du Cong. Int., Nice, 1971 ) . G. Shimura,_Moduli of abelian varieties and number theory, (in AMS Proc. of Symp. in Pure Math., vol. 9) . Because of their arithmetic and representation-theoretic i mportance, there is a huge literature on various types of modular forms on ~ g

and on other bounded symmetric domains.

I am not competent even to

begin to list references on these topics, but I would like to emphasize here as in the Lecture one big gap :

the lack of a moduli-theoretic

interpretation of the Eisenstein series. Concerning Torelli and Schottky, good general references are the books of Gunning (op. cit., notes to Lecture I, part III ) , Fay (op. cit. ) and Rauch-Farkas (op. cit.).

In more detail, for Approach I, see:

A. Andreotti, On a theorem of Torelli (Am. J. Math.,

80 (1958)).

H. Martens, A new proof of Torelli's theorem (Annals of Math.,

78 ( 1963 ) ). T. Matsusaka, On a theorem of Torelli, (Am. J. Math., 80

(1958).

A. Weil, Zurn Beweis des Torellischen Satzes, (Nachr. Akad. Wiss. G6ttingen (1957)) .

CURVES AND THEIR JACOBIANS

104

'}

Fay's formula is in his Springer Lecture Notes (op. cit. ) , p. 34 (formula (45); the interpretation via trisecants follows fr om the Proposition on p. 335 of my paper: D. Mumford, Prym Varieties I (in: Academic Press, 1974).

Contributions to Analysis,

Approach II can be found in: W. Wirtinger, Lie's Translati onsmannig faltigkeiten und Abelsche Integrale, (Monatshefte f ilr Math. und Phy sik, 46 (1938)) S. Lie, Werke, Bund 2, Abt. 2 , Teil 2 , p.

481.

Approach III is in: A. Andreotti, A. Mayer, On period relations for Abelian integrals on algebraic curves (Ann. Seu. Norm. Sup. Pisa (1967)). My discussion of Approach IV foll ows my paper " Prym Varieties I " mentioned above.

A more analytic approach is in Fay (op. cit .) or

Rauch-Farkas (op. cit.) as well as in t h e many papers of Rauch and Farkas cited in their book.

The original works of Schottky and Jung

are: F. Schottky, Uber die rnoduln der thetafunktionen (Acta Math. 2

(1903)).

_J_

F. Schottky, H. Jung, Neue S~tze ilber Symmetral funktionen und die Abel'schen Funktionen (2 parts, Sitzung sber. Berlin Akad. Wiss., l (1909)).

.