Diophantine approximation on algebraic varieties

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

M

ICHAEL

N

AKAMAYE

Diophantine approximation on algebraic varieties

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{2 (1999),}

p. 439-502

<http://www.numdam.org/item?id=JTNB_1999__11_2_439_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques

Diophantine

Approximation

on

Algebraic

Varieties

par MICHAEL NAKAMAYE

RÉSUMÉ. Nous donnons un _apergu de

_progrès

récents en théorie

de

_{1’approximation diophantienne.}

Le

_point

de

_départ

étant le

théorème de

_Roth,

nous nous intéressons d’abord à la

_conjecture

de

_{Mordell, puis}

ensuite à des résultats

_analogues

en dimension

supérieure,

résultats dûs à

_{Faltings-Wustholz}

et à

_Faltings.

ABSTRACT. We _present an overview of recent advances in

dio-phantine approximation. Beginning

with Roth’s

theorem,

we

dis-cuss the Mordell conjecture and then _pass on to recent

_higher

dimensional results due to

_{Faltings-Wustholz}

and to

_Faltings

re-spectively.

0. INTRODUCTION

The

_theory

of

_{diophantine approximation}

often

_begins

with a finite

col-lection of

_polynomials

_{f 1 (X), ..., In (X) E}

_{(~~.X l, ... , Xm)}

with rational

co-efficients. There are then two distinct

_types

of

_{questions commonly}

asked.

First,

one can look for rational

_points

(pl/ql, ...

which lie on the

common set of zeroes of the

polynomials,

i.e.

pm/qm)

= 0 for

all i. This line of

_inquiry

leads to the Mordell

_conjecture

_(which

deals with the case of a

_{single polynomial}

in two

_variables)

and

_higher

dimensional

generalizations

due to

_Faltings

and

_{Vojta. Alternatively,}

one can look for

rational

_points

_{(pl/ql, ... ,Pm/qm)}

which are

_{’approximate}

solutions’ or in

other words lie _very close the set of common zeroes of the

f i’s.

This

question

is

_perhaps

older than the first and is

_aptly

called

_diophantine

approxima-tion because one is

looking

not for solutions to

diophantine

equations

but

for

_approximate

solutions. In this

_direction,

the most

_striking

results are

Roth’s

_{theorem, dealing}

with the case of a

single

irreducible

polynomial

in

one

variable,

and the Schmidt

subspace

theorem which allows for several

variables but deals with _very

_specific

_(linear)

_polynomials.

One of the

_simplest

results in

_{diophantine approximation}

is Liouville’s

theorem. For

_this,

we consider a

_single

irreducible

polynomial

f (X )

E

_Q(X~

of

_degree

d &#x3E; 2 with a root a E R. Since

_f

is

_{irreducible, a g Q}

and it makes sense to

_approximate

this zero of

f (X)

by

rational numbers.

Liou-ville’s theorem states that for any rational

p/q

one

_always

has an

_inequality

of the form

where

_c(a)

is an

_explicity

determined constant

_depending

_only

on a. It

turns

_out,

_however,

that the

_exponent

d in the denominator is not best

possible

except

for

_quadratic

_{f (X ).}

In

_fact,

Roth

_proved

that 0.1 can be

improved by replacing

d with 2 + E for _any f &#x3E; 0. The

_disadvantage

to

Roth’s theorem is that the constant

_c(a)

is no

longer explicitly

determined.

Roth’s theorem was

generalized

fifteen _years later

by

Schmidt who deals

with the case of n

_independent

linear forms

Li(X)

E

_{k[Xl,... Xnj}

where

Q

has now been

replaced by

a finite extension

k/Q.

The Schmidt

subspace

theorem

_(see

_§5

for an

explicit

statement)

states that the set of rational

points

(pl/ql, ... , pn/qn)

which are close to the n linear

_subspaces

0 are

_degenerate;

here

_degenerate

means contained in a finite union of

proper linear

subspaces.

In the one variable _case,

_choosing

L(X)

= X - _a, one recovers Roth’s theorem since a

_degenerate

linear

subspace

of

_Q

is

just

a

_point.

Schmidt’s

_subspace

theorem has

_recently

been extended

_by

Faltings

and Wüstholz to deal with the case when the

_Li

are no

_longer

linear.

Returning

to the first

_question

_posed

in

_{diophantine approximation}

we

take the

_subvariety

X C pn defined over

Q

as the common zero locus of our collection of

_polynomials.

As noted

_above,

in this case one is not

’ap-proximating’

a

_{geometric object}

with rational

_points

but rather

_looking

for

actual rational solutions to the

_system

of

_equations

_defining

X. The

con-nection between the

_problem

of

_finding

actual as

_opposed

to

_approximate

solutions to

_polynomial

_equations

was

developed by Vojta

in his

proof

of

the Mordell

_conjecture

_[V2].

The Mordell

_conjecture

states that if X is a

smooth

_projective

curve of _genus at least

_2,

defined over a number field

k,

then X has

_{only finitely}

many rational

points.

Like Roth’s

theorem,

the result is ineffective in that it does not bound the size of the solutions.

Building

upon

Vojta’s

ideas,

Faltings

[Fl, F2]

was able to obtain similar

results for

_higher

dimensional varieties under certain restrictions.

The

_goal

of these notes is to

_give

an introduction to the ideas and

argu-ments

_occurring

in

_{diophantine approximation, beginning}

with the

_simplest

result,

Liouville’s

_theorem,

and

_culminating

with the difficult results of

Falt-ings

and

_Vojta

on the one hand and

_{Faltings-Wilstholz}

on the other. The

rough

structure of these notes is as follows: the first section

begins

with

Liouville’s theorem and

_procedes

to

_analyse

the difficulties encountered when

_trying

to

_sharpen

Liouville’s result to obtain Roth’s theorem. The

second section then deals with these

_{difficulties, concentrating}

on the

geom-etry

behind Roth’s

_theorem,

_specifically

the

_product

theorem and

_Dyson’s

lemma. The third lecture moves from Roth’s theorem to

_{Vojta’s proof}

of the Mordell

_{conjecture, developing}

_{the necessary}

_theory

of

_height

functions and the

_{corresponding}

metrics on line bundles

_along

the _way. We will once more

emphasize

the

_geometric

_aspects

of the

proof

and

_give

a

rough

idea of

the difficult arithmetic

_machinery

involved. Lectures four and five will be devoted to

_higher

dimensional

_problems,

one to

_higher

dimensional Mordell

type

theorems and one to the Schmidt

subspace

theorem

respectively.

The

proofs presented

here will

_not,

in

_general,

be

_complete

but we

_hope

to

em-phasize

all of the

_{important points}

so that the interested reader can then

consult the literature for

_{rigorous proofs}

and

_hopefully

the account here will facilitate this process.

There are several

expositions

of the material covered in these lectures.

For Roth’s theorem and the Schmidt

_subspace

theorem one can consult

Schmidt’s two collections of lecture notes

_{[Sl, S2].}

For

_Dyson’s

_lemma,

in addition to

_{Dyson’s original}

paper

[D],

one can consult

_[Bl,

_EV,

_Nl,

N3].

Vojta

has two

_proofs

of the Mordell

_conjecture,

_{[V2, V3],}

the second of

which is closer to the

_exposition

_given

here. He also has a beautiful _paper

[V4]

giving

a full

_proof

of the

_higher

dimensional results of

_Faltings

[Fl , F2].

For _surveys of the material behind

_{Faltings’ work,}

one can consult

[H]

for a beautiful overview or

[EE]

for a more

_thorough

treatment.

These notes form the basis of a series of five lectures delivered at the Isaac Newton Institute from March 23

_through

March

_27,

1998. It is a

pleasure

to thank the

_organizers

of the

_{workshop, Christophe Soul6,}

Jean-Louis

Colhot-Th6l6ne,

and Jan

_Nekováf,

for

_inviting

me to

_present

this material.

I would also like to thank all of those who attended the lectures and whose

comments have

_helped

to remove _many errors from these notes as well as

to add better and more

_{complete explanations.}

I also thank those who

attended courses I _gave on

_part

of this material at Harvard in the

_spring

of 1995 and at

_Bayreuth

in the winter of 1996: it is

_only

after

_trying

to teach the material several times that I have reached a full

_appreciation

of the

subtleties involved. The

_spirit

of these notes is

_hopefully

the same as that

of the

_{lectures, namely}

informal

_{but aiming}

to

_give

a

_{reasonably complete}

picture

of the difficult

_techniques

involved in these

_questions

of arithmetic

geometry.

1. FROM LIOUVILLE TO ROTH

We

_{begin by recalling}

the

_quick

and

_{elegant proof}

of Liouville’s theorem.

Theorem 1.1

_(Liouville).

_Suppose

a E R is an

_algebraic

irrational

The

_proof

can be

formally

divided into three

_stages

which will _reappear

in all of our

_arguments

but which are

particularly

_transparent

in this case.

We

_begin

with the irreducible

_polynomial

_{f (X )}

E

_Q[X]

for a over

Q.

To

specify f uniquely

one can take

_f

E

_Z[X]

with

_relatively

_prime

coefficients.

The outline of the

_argument

is as follows:

Step

1: 0 for _any

_p/q

E

_Q

since otherwise

_f

would not be

irreducible over

Q.

Step

2: since

_qd

is a common denominator for the terms

in

_1(P/q).

Step

3:

_a~

for an

explicit

constant

_b(a).

Liouville’s theorem

_follows,

with

_c(a)

=

1/2b(a),

by comparing

the bounds

in

_Steps

2 and 3.

_Only

the _upper bound in

_Step

3

_requires

further comment.

Suppose

we take the

_Taylor

series

_expansion

of f (X )

about a. Since

f (a) _

0 the first term is zero:

Thus

This establishes

_Step

_3,

_provided

_{lplq -}

_al

_1,

with

_{b(a) =}

Ed 1

_jail

and _proves Liouville’s theorem

_(taking

_c(a)

= As for

effectivity,

the numbers ai

depend only

on a as

_they

are the coefficients of

the

_Taylor

series

_expansion

of

_f(X)

about a.

Consequently,

b(a)

and

c(a)

depend only

on a.

Suppose

now that one tries to

_improve

the

_exponent

d in Liouville’s

theorem. One idea would be to run

through

the same

_argument

with a

different

_{auxiliarly polynomial}

_{f (X).}

Of course this risks

_{losing effectivity}

because

_{f (X)}

was determined

uniquely by

a.

Nonetheless,

one could

_try.

A

review of the

_argument

reveals that the

_exponent

d in the theorem comes

by taking

the

_quotient

of

_{deg f(X)}

_by

_{multa ( f (X )).}

Thus if we

replace

f (X )

with a different

polynomial

g(X)

which also vanishes at a we will

_get

But for

_Step

2 to

_work,

we need to have

_g(X)

E

_Q[X]

and thus

_g(X)

will be divisible

_by

Thus we

always

will have

degg(X)/multa(g(X))

&#x3E; d and there is no

_improvement.

Thus some new idea is needed in order to

_improve

the

_exponent

d of Liouville’s theorem. At the

_beginning

of the

_century,

Thue had the idea of

running

the same

_argument

with an

_{auxiliary polynomial}

in two variables.

Thus in order to bound how close a

single

rational number can be _{to a,}

Thue considers a

pair

of rational numbers both of which are

close to a. The reason

_why

an extra

_{approximating}

rational

_{point helps}

is that now the

auxiliary polynomial

will be in two variables and this

_gives

additional

_degrees

of freedom

_allowing

for an

_improvement

in the

_important

ratio

which occurs as the

_exponent

in Liouville’s theorem. There are of course

immediate

_{complications}

in the

_argument,

_particularly

with

_Step

1 as with

two variables there will not be a well-defined choice of

_auxiliary

polyno-mial with

_big

_multiplicity

at

_(a,

_a).

But without a canonical choice of

f (X,Y)

there is no

_longer

_any reason

_why

f (pl/q,,P2/q2)

should be

non-zero, without which the

argument

runs

aground.

Thus the

simplest

_step

in

the

_proof

of Liouville’s

_{theorem, Step}

_1,

becomes the most cult when

one considers an

_auxiliary

function in more than one variable.

Of _course, once one is

_willing

to consider

_{f (X, Y)}

there is no reason to

stop

there and Roth was

finally

able to obtain the best result

_{possible by}

considering

an

auxiliary polynomial

with an

_arbitrary

number of variables:

Theorem 1.2

_{(Roth’s Theorem).}

_Suppose

a E R is

_algebraic

and

irra-tional. Then

_for

_any e &#x3E; 0 there are

only finitely

_many solutions to

Note that the formulation of Roth’s theorem is

_slightly

different from that of Liouville’s theorem. It follows from the fact that 1.3 has

_only

_finitely

many solutions that there exists a constant

_c(a)

such that

The

_problem

is that the constant

_c(a)

is not effective and this is

_why

Roth’s theorem is formulated in this alternative

_style.

The

_strategy

for

_proving

Roth’s theorem is identical to that of Liouville’s theorem

_except

that we will use several

_good

_{approximating}

_points.

So we

assume that Roth’s theorem is false and this

_gives

a _sequence of

_good

approximating points

The

_strategy

for the

_proof

will be to choose m of these

good approximating

points,

where m -+ oo as e --~

_0,

and construct an

auxiliary polynomial

in m variables with

large

order

vanishing

at

(a, ... , a).

Steps

2 and 3 of

Liouville’s theorem will then tell us that the

_{approximations}

are too

_good,

giving

a contradiction. In order for

Steps

2 and 3 to

_yield

a

contradiction,

one needs to choose the

_appropriate

notion for the order of

_vanishing

of our

polynomial

f (X 1, ... , X,,,~ )

at

_{(a, ... ,}

_{a) .}

We will see as we _go

_through

with

the

_proof

that the

_following,

_though

at first

_awkward,

is the

_apt

definition:

D efinit ion 1.5. Let

_{0 7~ /(Xi,...,X~)}

E and let

(al, ... , 7 am)

E Consider the

_Taylor

series

_expansion

of

_f

about

(a , ... ,

7 am):

Suppose f

has

_multi-degree

_{(dl , ... ,}

The ind ex of

_f

at

_{(a~ , ... ,}

is defined as follows:

The index of a at a

_point

x E cm is a

weighted multiplicity.

For

_example,

if

di ==...== dm

= d then

In

_general,

each variable is

_weighted

so that it can contribute a maximum

of 1 to the index.

To see the relevance of the notion of

index,

we run

through

the three

step

proof

of Roth’s theorem which we would like to model after Liouville’s

theorem: we construct an

auxiliary polynomial

f (Xl, ... ,

of

multi-degree

(dl, ... ,

with

_large

index at the

_point

_{(a, ... , a).}

Then the

proof

should

_procede

as follows:

Step

1: Show that 0.

Step

2: I

as in Liouville’s theorem.

Step

3:

_If(PI/ql,...

_I

is small since

_{lpilqi - al}

is small and

_f

As it

_stands,

much work is needed in order to make this into a

_proof.

Only Step

2

_requires

no further

justification.

We

begin by explaining

the

upper bound for

,Pm/qm)1

in

Step

3 as this is what motivates

the notion of index. As we did in Liouville’s

theorem,

consider the

Taylor

series

_expansion

of

_f

about

_{(a, ... , a):}

Using

1.4 will

_give

the

_following

_upper bound

for some constant C which

_depends

on the number of terms in 1.6 and the size of the coefficients of P. We _see,

_ignoring

the constant C for the

moment,

that 1.7 contradicts the lower bound of

_Step

2

_{provided provided}

aj=0

whenever

Since we do not know the relative sizes of the _qz, the

only

reasonable _way

to

_guarantee

_inequality

1.8 is

_{by choosing di}

so that

qai

are all

roughly

proportional,

i.e. we need to choose

Moreover,

with this choice

_{of di}

_inequality

1.8 translates into the

_following:

we want

_aj=0

whenever

And here the index has

_{finally reappeared}

as a natural _consequence of the

argument:

indeed,

1.10 says that we want

To

_summarize,

the lower and _upper bounds for

_given

in

_Steps

2 and 3

_respectively

contradict one

another, assuming

we can

suitably

bound the constant C in

_1.7,

_provided

that

_m/(2+e).

At this

_point

there are three technical issues to deal with in order to

Problem 1: Show that

_{f (pl/ql, ... ,}

0. Problem 2: Bound the constant C

_occurring

in 1.7.

Problem 3: Show that one can find

0 ~

_f

with index &#x3E;

m/(2 + e)

at

(a, ... , a).

We will deal with the three

_problems

in reverse order. Problem 3 is

essentially

a

_{counting problem.}

We want to kill

_leading

terms _aj in the

Taylor

series

_expansion

1.6

_provided

J satisfies 1.10. In order to

_guarantee

that one can force all of these a~ to

vanish,

it suffces to show that the number

_{of m-tuples}

_{( jl, ... ,}

_jm)

with

_{0 j~ c~}

_satisfying

1.10 is a small

f raction

of the total number of

monomials;

for if this is true then a

dimen-sion count shows that there are lots of

polynomials,

with coefficients in a

finite extension

_having

index at least

_m/(2

+

_e)

at

_{(a, ... , a)}

and all of its Galois

_{conjugates. Taking}

the norm over

Q

of such a

polynomial

and

_clearing

denominators

_gives

the desired

_auxiliary

_polynomial.

To show

that all of this works in our

_{particular setting}

is a

_special

case of what

Falt-ings

and Wiistholz

_[FW2]

refer to as the ’law of

_large

of numbers’.

_Rougly

speaking,

this _says the

_following:

for each

_i,

the

_"average"

value of

_{ji/ dï}

is

_1/2

so if one

_randomly

chooses a set the

_probability

that 1.10

is violated

_approaches

zero as m

approaches infinity.

For a

_rigorous

state-ment and

_proof

one can consult

[FW2]

Proposition

5.1.

Alternatively,

for

the case in which we are

_interested,

one can make an

_explicit

_computation

as in

_[L1]

_pp. 170-171.

Next we deal with Problem 2. There are two

_separate

issues

_here,

first

counting

the number of terms aj and second

_bounding

the size of the

_jail.

The first of these is

_simple

as the number of aj is

Bounding

the size of the aj is

equivalent

to

_bounding

the size of the

co-efficients

of the

_{auxiliary polynomial}

_{f (X)}

which is

_potentially

a serious manner as

_f

was constructed

_abstractly

to

_satisfy

certain

_vanishing

condi-tions. The fact that the size of the coefficients of

_f

can be controlled is a

consequence of the famous

Siegel

lemma:

Lemma 1.11

_{(Siegel’s Lemma).}

Consider a

_system

of

rra linear

_equations

in n

unknowns,

with m n:

Suppose

az j E Z

for

all

i, j

and that

laijl

I

A

for

all

i, j .

Then there

exists a solution

(xl, ... , xn)

E Zn to the

_system

_of

_equations

with

_ixi

_I

We

_apply

_Siegel’s

lemma

_{by viewing}

the condition

_{ind~a ... a~ ( f )}

&#x3E;

_m/(2+

e)

as a

_system

of linear

_equations

in the

_coefficients

of

_{f ,}

i.e. we view the

coefficients

of f as

variables.

_Siegel’s

lemma cannot be

_{applied directly}

in

this situation because our

_system

of linear

equations

will have coefficients

in the number field K =

Q(a)

since _{we are}

_trying

_to

_impose

_large

index _at

a.

_Choosing

a basis for K as a vector _space over

_Q

allows one to make the

reduction to Lemma 1.11

_(for

_details,

see

[S2]

Lemma

9A).

One then

ob-tains a

_{polynomial g}

E

_{0~[Yi,... ,}

with

_large

index

_along

_{(a, ... ,}

_a)

and its

_conjugates

such that the coefficients have bounded

_size;

_taking

the

norm

of g

then

_gives

the desired

f .

The crucial observation to make about

_Siegel’s

lemma is that the

ex-ponent

m/(n - m)

is small

_provided

that the number of unknowns is a

fixed

_multiple

of the number of

_equations.

At this

_point,

one needs to do

a lot of

_accounting

to check that

_Siegel’s

lemma

_gives

an

_auxiliary

polyno-mial

_{f E Z(Xl, ... ,}

such that the constant C in 1.7 is small

_enough

to obtain a contradiction when

comparing

the lower and _upper bounds on

In

_practice,

to

_get

the numbers to work out one

chooses m

_{big enough}

to

_impose

index

_m/(2

_{+ e/2)}

at

_{(a, ... ,}

_a)

and the

extra

_e/2

allows one to absorb the bound for C in 1.7 and still obtain a

contradiction when

_comparing

the numbers in

_Steps

2 and 3.

2. INTERLUDE: DYSON’S LEMMA

We still have not faced the most difficult of the obstructions to

_proving

Roth’s

_theorem,

_namely

the issue

_arising

at the

_beginning

of the

_argument

in

_Step

1: how can we

_guarantee

that

Without this

_information,

all of our

_computations

to obtain a contradiction

from

_comparing

_upper and lower bounds of

_[

are in

vain. This is

_by

far the most difficult

_part

of Roth’s theorem as the rest

of the

_argument

_essentially

consists in

_counting.

In

_general,

there is of

course no _way to

_guarantee

that 0 because

_f

has

not been

_explicity

constructed: we

_simply

know from dimension

_counting

that such an

_f

exists but of course there _may be several and most of them will vanish at the

_{approximating}

_point

Roth

_originally

dealt with this

_problem

in an

_essentially

arithmetic manner. He

_proved,

in what is now called Roth’s

_lemma,

that

_provided

di

W

_{d2 ...}

»

_d",,

(or,

equivalently, given

1.9,

ql « q2 « ... «

_q."a)

the

_polynomial

_{f (X )}

constructed

_{using Siegel’s}

lemma cannot have

_large

order of

_vanishing

at

(pi/9i)’ "

Taking

the

_appropriate

derivatives of

_f

then

_yields

a

contradiction

_(since

the number of derivatives is

_small,

it does not affect the contradiction arrived at in

_Steps

2 and 3 of the

_argument).

Roth’s

lemma is

_essentially

arithmetic in

_nature,

_using

the facts that

_f

has

_integer

coefficients and that we are interested in its order of

_vanishing

at a rational

point.

For a

proof,

one can consult

[Ll]

_pp. 179-181.

An alternative

_geometric

_argument

to establish

_{non-vanishing}

of

,pm/4m)

was

developed thirty

_years later

_by

Esnault and

Viehweg

[EV]

building

upon

previous

work of

Dyson

[D],

Bombieri

[Bl],

and Viola

_[Vi].

Esnault and

_{Viehweg approach}

the

_problem

from the

following

point

of view.

_Suppose

that whichever

_{f (X)}

we choose with

large

index at

_{(a, ... ,}

_a)

we

_always

find that

f (X )

also has

_large

index at

(pl/qi,...

This _says that certain linear conditions on the _space

of all

_polynomials

of

_degree

_{(dl, ... ,}

fail to be

_independent.

Thus if

one could establish this

_independence

then this would also

_yield

a

contra-diction. The

_advantage

to this

_viewpoint

is that it is

_entirely

_geometric;

the fact that the

_points

in which we are interested are all

algebraic

becomes

unimportant.

To state the

_powerful

result which Esnault and

_Viehweg

were able to

prove, we introduce some new notation. Let

be a

_product

of m

_projective

lines defined over

k,

an

algebraically

closed

field of characteristic zero. We will assume for

_simplicity

that k -

C,

the field of

_complex

numbers

_although

the

_argument

remains valid in the

general

case.

Let 7ri :

P -

Pl

be the

_projection

to the

ith

factor. For

positive integers d 1, ... ,

dm

write d =

( d 1, ... ,

dm )

and

For fixed d let

0 # s

E

H°(P,Op(d)).

The index of s at a closed

_point

,

of P is defined

_{by locally identifying s}

with a

polynomial

and

applying

Definition 1.5. We need to define certain volumes as in

[EV]

Definition 0.2:

Definition 2.1. Let 1m =

_{l~ = (~1, ... ,}

_£m)

E Rm

_{0 ~~}

1 for all

_i}

and let

_Vol(t)

denote the volume of

I

Of course 1 -

Vol(t)

_measures,

asymtotically

in

d,

the

_proportion

of

sections of

_{HO (P, Op (d))}

with index &#x3E; t at a

point

C. Dyson’s

lemma

gives

conditions on a set of

_points

_{~~}

C P and a line bundle

Op (d)

so that

_requiring

_{index ti}

_imposes

almost

_independent

conditions on

global

sections of The

_following

is an

alternative,

slightly

more

transparent

formulation of Esnault and

_{Viehweg’s Dyson}

lemma:

Theorem 2.2.

_{Suppose 0 f=}

s E

H°(P,

and

_{~1, ... , Cm}

C P so

Note that

_if

t 0 then

_Vol(t)

= 0.

Theorem _{2.2 says} that the linear conditions for

_imposing

_{index ti}

- m8

are

independent

inside

H°(P, Op(d)).

The

key

observation about

Theorem 2.2 is that if

di

»

_d2

» ... » then mJ is small so in this case

Theorem _{2.2 says} that the conditions to

_{impose index ti}

at

_(i

are

_nearly

independent

inside This will then allow us to derive the

desired contradiction and prove Roth’s

theorem,

provided

we can _arrange

for

_dl

W

_d2

» ... »

d",,.

But

_by

1.9 this is

_equivalent

to

_choosing

rational

approximations

with denominators

_satisfying

_ql C q2 « ... « _q",, and we can achieve this since we have assumed that there are

_infinitely

_many

_good

rational

_{approximating points.}

It should be noted that there is still some

arguing

left because

_Dyson’s

lemma

_only

shows that there exists

_g(X)

E

C[X1,... Xm]

with

_large

index at

_{(a, ... ,}

_a)

and its

_conjugates

which has small order of

_vanishing

at

_{(pllql,... ,P."6/q",,).}

From here one can

construct

_{f (X )}

E

_{Z[X1,... Xm]}

with

_large

index at

_{(a, ... , a)}

and small index at

_(pi/9i,...

but a

_priori

we have no control on the size of

the coefficients of

f .

Thus more calculations of dimensions and of

Siegel

type

are _necessary

(see

_[EV]

_§10

for

details).

We now turn to the

_proof

of Theorem 2.2. In the statement of Theorem

2.2,

the volumes which occur have been

_{perturbed by}

m8. The reason for

this is that suitable derivatives of the non-zero section 8 E

with index

_{at (;}

still have

_{index &#x3E; ti}

- m8

at

_z.

The

_product

theorem

will tell us that the common zero locus of these sections is contained in a

finite union of _proper

_product

subvarieties. Let P’ C P be one of these

product

subvarieties and let 7r : P - P’ denote the natural

projection.

Taking Ci’

= _one then

shows,

_except

in certain

degenerate

_cases, that

on P’ one can

_produce

a new non-zero section s’ E

H°

_{(P’, Op (d))}

with suitable indices at the

_(I.

This

_procedure

is then iterated to

_produce

a

zero

_cycle

_representing

and the

_part

of the class associated

to

_~’=

measures the cost of

_imposing

index ti

- m6

at

_~t.

These classes have

_disjoint

_supports

and this

_implies

the desired conclusion: the cost of

imposing

index ti

- mJ

at (i

for all i is the sum of the individual

_costs,

i.e.

the conditions are

independent.

The details of the

_proof

of Theorem 2.2 are as follows. We are

_given

a non-zero section s E

H°(P,

_{with ti}

=

ind(¡ (s ).

Given a

_point

₍

E

_P,

an af&#x26;ne _open subset U =

Spec

A _C

P,

and _a local trivialization of

Op

(d)

on

_U,

the sections Q E

H°(P, Op (d))

such that &#x3E; t

_generate

an

ideal c A. This ideal does not

_depend

on the trivialization and as U

varies over an a£ne cover of P this defines an ideal sheaf

_{2£, d t}

c Op .

Let

Since we are

_trying

to show that the various index conditions are

inde-pendent,

as a first

approximation

one should

_expect

the

_supports

of these

ideal sheaves to be

_disjoint.

This is not cult to show because one can

write down

_explicit

monomial

_generators

for and hence determine its

support

(see

[EV]

Lemma

_2.5).

It then follows

_readily

_([EV]

Lemma

_2.8),

as

desired,

that

As discussed

_above,

we would like to be able to

_produce

sections

sat-isfying

the relevant index conditions on _proper

_product

subvarieties. To this

_end,

let

_{Id (s) =}

_IÇj,d,tj

and for P’ C P a

_product

_subvariety

let

_Op’

_(d)

=

Op(d)[P’.

If _one is

unlucky

and P’ _C

Zi

for _some i then

H°

= 0 for all k _&#x3E; 0 so there will be no

hope

of

producing

a new section. On the other

hand,

[EV]

2.9

(ii)

_implies

that

whenever

_{P’ 0}

_Zi

for

_{any i}

then for k

_sufficiently

divisible

There is one more crucial

_ingredient

in the

proof

of Theorem 2.2 as it

allows us to use the sections

guaranteed

by

2.4 to construct an intersection

class

_representing

cl

( Op (d))m.

This is the so-called

product

theorem which establishes a certain

_degeneracy

in the locus where the non-zero section

s E can have

large

index.

Initially inspired by

the

product

theorem of

_Faltings

_[Fl],

the result

_presented

here

_strengthens

Theorem

3.1 of

_[Nl].

Theorem 2.5

_{(Product Theorem).}

Let

_f

E

_{C[Xl, ... , Xm]}

be

_of

multi-degree d 1, ... ,

d,~

and

_for

an m -

1--tupl e

of

non-negative

integers

a =

1

₁

Let W C

_{X ( f )}

be an irreducible

_component.

Then W is contained in a proper

product subvariety.

Proof of Theorem 2.5. We will show that either is a

_point

or

in a _proper

_product

subvariety

and we are done. In the latter _case, choose

a

_general

A E C and

_{replace f}

with

Since

_{a/axl, ... ,}

commute with evaluation at A we have

"""""- --

-B ut

’9011

...

irm-2

2

2

f

(

X 1, ... ,

X m )

vanishes

along

W’

x C for aZ But

5X " ... "!-2 f

(Xi,... X.,,,)

vanishes

_along

W’ x C

for

_{ai :5}

a1 ,gxm 2

assumption.

Hence

_by

2.5.1

Thus, using

the notation in the statement of Theorem

_2.5,

W’

C

and we can conclude

by

induction on m that

W’,

and hence

W ,

must be contained in a _proper

_{product subvariety.}

We now

proceed

to establish that if is not a

_point

then W =

W’ x

_{C, concluding}

the

_proof

of Theorem 2.5. Choose a

general

point

’11 E W

so that

Let denote the

_tangent

space to W at 17. Since we are

assum-ing

that W - C is

_surjective

it

_{follows, choosing q}

still more

gen-eral if _necessary, that we _may assume the induced _map on

_tangent

_spaces

is

_{surjective. Consequently, viewing}

_Tl1(W),

after

translation to the origin, as a subspace

We claim that for some 1 i m - 1

Observe first that if DO. is _any differential

_operator

of order

_{tll(f) - 1}

and if

_8/8X

E

_{T,7 (W)}

then = = 0 since

D°‘ ( f )

vanishes on W and E Thus

Suppose,

contrary

to

_2.5.5,

that for have

By

2.5.4 and 2.5.6 we also have But this

would

_imply

that +

_1,

a contradiction. Thus 2.5.5

holds.

_Applying

the same

_argument

inductively

and

using

2.5.3 shows that

for some

_non-negative

_{il, ... ,}

But

_by

the definition of

_{X ( f ),}

= 0 for _any

0 ~ Qi ~ dj.

Thus

+ 1. Since

_f

has

_degree

_dj

in this is

_{only possible}

if W is of the form W’ x C. This concludes the

_proof

of the

_product

theorem.

Observe that if we

_replace

d with Nd for N

_sufficiently

divisible then we can assume without loss of

generality

that if

Z 0

Zi

for

_{any i}

then there

exists

Indeed, applying

the

_product

theorem to the section s of Theorem 2.2

gives

sections of

H°

_(P,

_{Op (d)}

o which

_generate

off a finite

union of _proper

_product

subvarieties. If P’ is one of these _proper

product

subvarieties,

not contained in

_Zi

for

_{any i,}

then 2.4

_gives

a non-zero section

for some

_positive

_integer

n. Since 2.4

only

needs to be

applied

finitely

_many

times in order to obtain sections on _any

subvariety Z 0 Zi,

we can choose

N

_sufficiently

divisible to

_give

all of the sections sz at once.

We will use 2.6 in order to construct an effective

cycle representing

This is done

_inductively

as follows. With sp as in

_2.6,

_Z(sp)

represents

cl(Op(d)).

Let Z be an irreducible

_component

of

Z(sp).

If

Z C

Zi

for some

_i,

then choose some

0 ~

Sz E

H°(Z,Op(d))

and

_Z(sz)

is

an effective

_{representative}

for

cl (Op (d)) fl Z.

_Suppose

next that

_{Z 0}

_Zi

for

any i. Then

by

2.6 there exists E

H°

and

_Z(sz)

_represents

_{cl (Op (d))}

fl Z.

_Repeating

this

_procedure

_inductively

gives

an effective

_{representative}

for

_Moreover,

since

_by

2.3 the

Zi

are

_{mutually disjoint,}

we can write

where is the

_part

of the intersection

_supported

on

_{Z; .}

Since R is

_effective,

We have

_deg

= m!

die

Thus to conclude the

proof

of

Theorem 2.2 it suffices to prove that for

1 j

M we have

Combining

2.8 with 2.7

_yields

Theorem 2.2.

A

_{rigorous proof}

of 2.8 involves

_blowing

_up the ideal sheaf

but

_intuitively

it has an _easy

explanation.

Suppose Cl, C2

C

p2

are two

distinct,

irreducible curves

meeting

in a

point

P.

Suppose

moreover that

both

₀₁

and

_C2

have

_multiplicity

m at P. Let

_{C2 )}

denote the

_part

of the intersection class

C2

supported

at P. Then one has

But of course

m2

measures

_precisely

the cost of

_{imposing multiplicity}

m

at P. In the situation of

_2.8,

_things

are more

complicated

because the

condition we are

_imposing

will force

_vanishing

not

_only

at

_points

but also

along higher

dimensional subvarieties. The fundamental

_{principle, however,}

is the same: the

_part

of the intersection class we have

produced supported

on

_Zj

measures

_by

definition the cost of

_imposing

_{index t~}

- m6 at

~~ .

The

right

hand side of 2.8 is the cost of

_{ixnposing tj -}

mb

_{at (;}

whence the

inequality.

More

_precisely,

suppose 7r : Y - P is the

blow-up

of P

along

_-EC,

_{-m5 .}

The ideal sheaf has been defined so that it is

_{generated locally}

by global

sections of

H°

_{(P, C~p (d) ) .}

Hence if E denotes the

_exceptional

divisor of _7r, it follows that is

_numerically

effective.

_Lifting

the construction of to Y one verifies that

Since

_x*Op(d)(-E)

is

_numerically

effective is

deter-mined

_by

the

_growth

of for

_large

n.

Pushing

back

down to

_P,

a direct

computation

(see [N3])

then derives 2.8 from 2.9.

We now _pass to

_Vojta’s

version of

_Dyson’s

lemma on

products

of curves

of any genus. This is a

simple

and beautiful theorem which

initially inspired

Vojta’s diophantine proof

of the Mordell

_conjecture

_[V2]

and serves as a

key ingredient

in that

_proof.

We

_{begin by fixing}

some notation. Let

Ci

and

_C2

be smooth

_projective

curves over an

_{algebraically}

closed field of

characteristic zero.

_Suppose

L is a line bundle on

Cl

x

_C2.

Let

F1

=

Pi

x

C2

be a fibre of the first

projection

and

F2

=

C1

x

_P2

a fibre of the second

projection

for

_{arbitrary points}

PI

E

_Cl

and

_P2

E

_C2.

Let

_di

= L -

F2

and

_d2

= L -

Fl

so

_d1

and

d2

measure the

’degrees’

of the line bundle L in

analogy

to the

_multi-degree

of our

_auxiliary

_polynomial

in Roth’s theorem.

isomorphic

to the

_projective

line. If s is a non-zero section of L it makes sense to talk about the index of s at a

_point

x E

₀₁

x

_C2:

Definition 1.5

applies

in this

_situation,

with

_{weights di, d2,}

as it is a local definition. We

let the genus of

Ci

be gl and the genus of

C2

will be denoted 92.

Theorem 2.10

_(Vojta).

_Suppose

there is a non-zero section 8 E

H°(Cl

x

C2, L).

moreover that are

_{points in}

Ci

x

_C2

such that no two are contained in a _proper

product subvariety

and let =

t~.

Write

_{Z(s) =}

_E~=1

aiZ¡

where

_Zi

is irreducible and let

Several remarks are in order to

_explain

the

_strength

of Theorem 2.10.

If,

as will be the case in the Mordell

conjecture,

the _genus of both

01

and

C2

is at least

_2,

then

_{2g1 -}

2 + M &#x3E; 0 and one does not need to take the maximum. In these

circumstances,

inequality

2.11 reads

..- ..

Moreover,

we know that e

d2 }

since _any curve

Zi

which is not a

fibre of either

_projection

satisfies 1 and 1. If we assume

that

_d1

»

_d2,

as we did in Roth’s

theorem,

this will kill the second term

of 2.12

_leaving

us with In the case we dealt with

previously

where both

_Cl

and

_CZ

had _{genus zero,} we have

cl (L)2

=

2d1d2

and this

gives

the 1 on the

_right

hand side of our

_previous

Dyson

lemma. On

higher

genus curves,

however,

there are more line bundles because the

_diagonal

inside C x _{C is}

_{only linearly equivalent}

to fibres of the two

_projections

when the _genus of C is zero. Thus if one can choose a line bundle with

cl (L)2

« then

_Vojta’s

version

_{of Dyson’s}

lemma _says that no section

of L can have

_big

index _{at any}

_point

of C x C. It is

exactly

in this fashion

that

_Dyson’s

lemma will be

_applied

to _prove the Mordell

_conjecture.

The

_proof

of Theorem 2.10 is

_essentially

identical to that of Theorem

_2.2,

the

_only

difference

_being

that

_taking

derivatives is

_slightly

more

_complicated

on a curve of

_larger

_genus:

this,

in

fact,

is what creates the

_{2gi - 2 occurring}

on the

right

hand side of 2.11. To see the

_similarity

between Theorem

2.2 and Theorem 2.10 we consider first the

_special

case of Theorem 2.10

where both

_Cl

and

_CZ

are

_projective

lines. In this _case, the theorems

appears on the left hand side of Theorem 2.2 while it has been moved to

the

_right

hand side in Theorem 2.10.

To see how our

proof

of Theorem 2.2 can be

_adapted

to this new

situ-ation,

suppose we make _sure, when

_constructing

our intersection

_product,

that the sections which we intersect all have rather than

settling

for

_{index ti}

- 28

as we did. In other

words,

after

identifying s

with

a

_polynomial

f (X, Y)

on an affine _open subset and then

_{differentiating}

,f (X,

Y),

we increase the index

by

multiplying

by

some other fixed

polyno-mial. Of course, since the

goal

of the

proof

is to induct on the dimension

of

_product

_{subvarieties,}

the

_polynomial

one

_{multiplies by}

should if

_possible

have zeroes in a _proper

product subvariety,

i.e. one should

_just

add fibres

of one of the two

projections.

Thus the

simplest

solution is to

_replace

the

derivatives

_by

Indeed,

the

_polynomials

in 2.13 still have index at

_{least ti}

at as for all i and

the

_possible

decrease in index incurred

_by

_{differentiating}

has been remedied.

The

_{only problem}

is that the

_degree

of the

_polynomial

has been increased.

To minimize the

_increase,

note that one of the

points,

_say can be taken

to be

_(oo,

_oo)

with

_respect

to the affine _open

_patch

where we differentiate.

Then the derivatives do not affect the index of the

_{projectivization}

of

_f

at

_~1.

_Also,

since each derivative decreases the

_degree

of

_f,

we have some

additional room to twist in order to

_get

a section of In

_particular,

taking C2

=

(0, 0),

_we note that

Xi axi (f )

has

degree

(di, d2)

and index

&#x3E; t2

at

_~2.

So with this modification 2.13

_gives

us

polynomials

of

bi-degree

(d,

+

_{(M -}

_{2)i, d2).}

The number of derivatives which need to be taken in order for the common zeroes to be contained in a _proper

_{product subvariety}

is

_by

definition the number e because is transverse to all non-fibral C and so these

partial

derivatives

always

decrease the order of

vanishing

along

such

_components

of

_Z(s).

Thus the

_conclusion,

if we run

_through

the

argument

used to _prove Theorem

_2.2,

is that

Expanding

2.14

_{yields exactly}

_{2.11 up} to a factor of two on the second term:

for a discussion of how to remove this factor of

2,

see

[EV]

_§10.

We now turn to

_{proving Vojta’s}

version of

_Dyson’s

lemma in full

gener-ality.

In order to take derivatives of a section of L on

₀₁

x

_C2

we recall

the method used

_{by Faltings}

_([F1]

_p. 560 and _p.

₅₆₆₎

in a more

general

derivatives of a section s E x

_C2,

_L)

is first that there are no

_global

vector 2 and second that after

_{identifying s}

locally

with some

polynomial

and

differentiating

this

polynomial,

the derivatives

no

longer patch together

to

give

a section of L. To

remedy

this we

con-sider the

_following

construction.

_Suppose

we are

_given

a dominant _map

pl :

Ci -

Then one can

pull

back the derivation on

Pl

via with

poles along

the ramification divisor R of pl and so obtain:

Definition 2.15. Let s E

H°(Cl

x

C2, L)

and _suppose Z C

_Z(s).

Then

is the

_global

section of +

_L(1riKcl)IZ

obtained

_{locally by differentiating s}

via the

_pull-back

of the derivation

on

_Opi

_by

the

_composition

_{pi - 1r1.}

Higher

order derivatives are defined

analogously

so for a &#x3E; 0

provided

all

_partial

derivatives

_{D/3 ( s)}

with

_Q

a vanish

identically along

Z.

Using

Definition

_2.15,

one can derive Theorem 2.10 in

_exactly

the same

manner in which we

_proved

Theorem 2.2. The

only

difference is that when we take e derivatives of the section s we have to twist

_by

_{e7ri Kc¡.}

Since we

also have to add one fibre for each of the

points ~~

in order to _preserve the

index of the derivative of s at

_(i

this means that the total twist we need in

order take the necessary e derivatives is

and,

since

_{deg(Kcl )}

=

2g1 -

2 this is

exactly

what occurs on the

_right

and

side in 2.11. Thus

_{arguing exactly}

as before one

_finds,

for J =

max2g

-2+M,0}

which is 2.11 _up to a factor of 2 on the second term. We

_already

saw this

factor of _{2 appear} after 2.14 and to avoid it one makes a direct intersection

theoretic construction as in

[Vl]

or

[EV]

_§10.

Two further remarks are _necessary in order to turn this sketch into a

proof. First,

one needs to be careful about

_twisting

the line bundle L

because this can make it difficult to

_compute

deg R~

in

2.8;

indeed the

proof

is

_{cohomological}

in nature and

_requires

that the sections constructed

all be sections of the same line bundle. This

problem

is not

_serious,

_however,

as we are

_{merely twisting by}

fibres of the first

_projection

and so one can

twist all bundles involved without

_{adversely affecting}

the construction. The second

_{possible problem}

is that on

_higher

_{genus curves,} the derivative of

not as a section of a line bundle on

Cl

x

_C2.

This

_too,

_however,

_plays

no

crucial role in the

_proof

of 2.14

_(and

its

_analogue

on

Cl

x

_C2).

_Again

the

proof

is

_{cohomological}

in nature and

_only

_depends

on numerical

effectivity

of a line bundle on a blow _up of

01

x

C2

(see [N3]

for

details)

and for this one does not need

_global

sections but

_just

non-zero sections on _any curve

a c 01 x C2.

To conclude this

_section,

it is worth

_pointing

out that Theorem 2.10 does

not

_hold,

as

stated,

on a

product

of three or more curves of

_{genus &#x3E;}

1: in

other

_words,

the data of the

_degrees

_di

of L and the

_top

intersection number

c1 (L)t°p

are not sufficient to bound the sum of the relevant volumes. The reason for

this, thinking

of the

proof

of Theorem

2.10,

is that the

only

proper

product

subvarieties X of

Cl

x

_C2

are

_points

or fibres of the two

projections.

In both _cases, the

_degrees

_{dl, d2}

_determine,

at least _up to

numerical

_equivalence,

the line bundle In

_higher

dimension this is no

longer

the case and we take

_advantage

of

_precisely

this fact in

producing

a

_higher

dimensional

_example

where the direct

_{generalization}

of Theorem

2.10 fails.

Let C be a smooth

projective

curve of genus g &#x3E; 1 and let Y = C x C.

Let

Fl

C Y be a fibre of the first

projection,

FZ

C Y a fibre of the second

projection,

and A’ =

tl - Fl - F2.

Finally,

let n =

(nl, n2, n3)

be _a

3-tuple

of

_integers.

We consider divisors of the form

Since

_F1

+

_F2

is

_{ample and Vn ~}

_(Fl

+

_{F2 )}

= _{nl +}

n2, some

multiple

of

Vn

is effective

_provided

nl + n2 &#x3E; 0 and

Vri

&#x3E; 0. We have

Hence

_given

_any E &#x3E; 0 there exist _{nl, n2, n3} such that nl » n2 »

0, Yn

is

effective,

and

LetX=CxCxCandfixapointPEC.

Let _{p :} X -~ Y denote the

projection

to the first two factors and let x3 : X ~ C denote the

projection

to the third factor. Define

Choose

_{0 0 s}

E

H° (Y, OY (Vn) )

and let t E

_HO(C,Oc(P))

be the section whose divisor of zeroes is P. Then 1 for _any

_{point C}

E X

Thus if Theorem 2.10 held for G one would find

(up

to an

o(1)

_depending

on the constants

_implied

in nj » n2 W

₀₎

that

contradicting

2.17.

It should be noted that the

_particular

divisor

_Yn

which we chose above

will

_{play a}

central role in

_{Vojta’s proof}

of the Mordell

_conjecture

to be

given

in the next section. As was noted

above, Vojta’s Dyson

lemma is

extremely powerful

when

_applied

to a divisor like

_Yn

as it _says that no

section of

_H°(C

x

_C,

_vn)

can have

large

index _{at any}

_point.

3. THE MORDELL CONJECTURE

In this section we will

give

a sketch of

_{Vojta’s proof}

[V2, V3]

of the

famous Mordell

_conjecture

Theorem 3.1

_(Faltings).

_Suppose

C is a smooth

_projective

curve

of

_genus

&#x3E; 2

_defined

over a number

field

k. Then C has at most

finitely

_many

k-rational

_points.

Of course, Mordell

originally

conjectured

Theorem 3.1

only

for the case

where k =

Q.

This is in fact the case which we will

_{ultimately consider,}

not because it is

_{intrinsically}

easier than the

_general

case but because there

is less notational

_complication

as

Q

admits

only

one

_embedding

in C. How does one use

_Vojta’s

Theorem 2.10 to prove the Mordell

conjecture?

Vojta’s

beautiful idea is to take

_advantage

of the freedom to choose the line

bundle L in Theorem 2.10. If C is a curve of _genus at least one then there are more _{line bundles} on C x C than the

_pull-backs

of bundles from the

two

_projections

because the

_diagonal

is not

_{linearly equivalent}

to a sum of

fibres. This additional

_parameter

of freedom allows for the choice of a line

bundle L

_(given

in fact

_by

the divisor

_Vn

at the end of the last

_section)

such that

,

As was noted after the statement of Theorem

_2.10,

if we also _arrange for

di

»

_d2

then 2.12 tells us that no section s E

HO(C

x

C, L)

can have

large

index at _any

_point.

Thus all of the hard work we

_spent

in Roth’s

theorem

_trying

to show that the

_auxiliary

_polynomial

_{P(Xl, ... , Xm)}

has small index at

_p.,,Iq.,,,)

is

_{accomplished automatically, regardless}

of our _{choice of section s.}

This sounds too

_good

to be true and indeed it is. It will turn out that what was _easy in the case of Roth’s theorem

(namely

to

_produce

an

arith-metic reason

_{why P}

must vanish at the rational

_{approximating}

_point)

will