COMPOSITIO MATHEMATICA

GREGORY S. CALL JOSEPH H. SILVERMAN

Canonical heights on varieties with morphisms

Compositio Mathematica, tome 89, n^o 2 (1993), p. 163-205

<http://www.numdam.org/item?id=CM_1993__89_2_163_0>

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

http://www.numdam.org/

1993.

Mathematica 89: 163

163-205,

Compositio

©

1993 Kluwer

Publishers. Printed in

the Netherlands.

Academic

Canonical

on ^varieties with

heights

morphisms

GREGORY

Mathematics

S. CALL*

Amherst

Amherst,

College,

MA

USA

01002,

Department,

and

JOSEPHH. SILVERMAN**

Mathematics

Brown

RI

Providence,

USA

02912,

Department,

University,

Received 13

in final form 16 October 1992

1992;

May

accepted

Let

A

an

be

abelian

defined over

a

field

K

D

number

have

and let

be

a

a

a

variety

^divisor on A. Néron and Tate

the existence of

symmetric

canonical

proven

on

characterized

and satisfies

the

that

is

height hA,D

A(k)

by

properties

hA,D

Weil

for the divisor

D

for all

height

A,D([m]P)

m2hA,D(P)

Silverman

that on certain K3

surfaces

S

P ~ A(K). Similarly,

[19] proved

with

a

non-trivial

S ~

S

there are two canonical

~:

automorphism

height

for

functions

characterized

the

that

are

Weil

hs

by

properties

they

heights

+

certain divisors

E

and _{satisfy S(~P) = (7}

for all

P

E

43) 1 S(P)

S(K) .

In

we will

this

these

to

a

construct

canonical

paper

on an

generalize

examples

a

V

V -

V

and

a

height

divisor

arbitrary variety

~:

morphism

eigenvalue

strictly greater

possessing

which is an

with

class q

for 0

eigenclass

than 1. Wewill also

a

numberof results about these

canonical

prove

which should be useful for

heights

compu-

arithmetic

and

numerical

applications

tations. We^now describe the contents of this

in

more detail.

paper

Let

V

be

adefined over

a

number field

let

a

V ~

V

• be
• K,

variety

~:

and

that there is

a

divisor _{class q} E

main result

~ R such

morphism,

suppose

Div(V)

that

0*,q

some 03B1 &#x3E; 1.

Our

first

aq^for

that

says

(Theorem 1.1)

there is

a

canonical

function

height

characterized

the two

that

is

a

Weil

function for

by

properties

V,~,~

height

the divisor class

and

~

*

Research

_byNSFROA-DMS-8913113andan

supported by

Amherst

Trustee

supported

Faculty

Fellowship.

**

Research

and

NSFDMS-8913113

a

Sloan

Foundation

Fellowship.

164

of the

that

is

canonical

we show

then

As an

if 17

height,

points

ample,

application

contains

which ^are

only

finitely many

[13]

0-periodic, generalizing

V(K)

Narkiewicz

and Lewis

results of

[10].

has shown howto

intoasumoflocal

the canonical

on

Néron

[8])

height

(see

decomp_Ao_,s_De_{(P) = 03A3}

hl,D

anabelian

_v), where

variety

nVÂA,D(P,

heights,

^is over the distinct

of

^the sum

canonical

sum

We likewise show that the

places v

K(P).

constructed

in Theorem 1.1 can be

as

a

height

decomposed

V,~,~

Here

E

is

divisor

• in
• class

the divisor

and

q,

any

E

with the additional

is

a

Weil local

if f is any

function for

the divisor

height

property

constant

+

aE

then there is

a

that

function

~*E

satisfying

div( f ),

a

so that

in Theorem

proven

The existence of the canonical local

and the fact that the canonical

is

2.1,

is

height

V,~,~

is the sumof the local

height

heights

V,~,~

in Theorem 2.3.

two Weil

given

for

a

divisor differ

a

bounded

and

so

amount,

Any

heights

given

by

in

the difference of the canonical

Weil

particular

height

any given

V,~,~

is

a

on

and

For

bounded

constant

V, q,

0

height

by

important

hV,~

depending

hV,~.

it is

to have an

bound for this constant.

manyapplications

explicit

Such aboundwas

and Zimmer

for Weierstrass

given byDem’janenko [4]

curves,

[25]

families of

elliptic

for Mumford families

Manin and Zarhin

by

[12]

of abelian

and

Silverman and Tate

for

families

of

varieties,

by

[15]

arbitrary

abelian varieties.

V - T

Wefollowthe

in

andconsider

a

approach

[15]

family

of varieties

with

a

V ~ V ^over

T

and

a

divisor

0:

class 7y

satisfying

map

Then on

almost all

there is

a

canonical

~*~ =

^andwe

03B1~.

fibers ’V,

height

Vt,~t~t,

can ask to

boundthe difference between this

a

Weil

and

height

given

in

terms

t.

In Theorem 3.1 we show that there

of the

height

hV,~

parameter

are

constants

so that

ci, C2

165

a

similar

between

This

We

also

the difference

estimate for

3.2)

give (Theorem

the canonical local

and

a

Weil local

height AV,E .

height Vt,Et,~t

given

result

for abelian

varieties.

generalizes Lang’s

Suppose

Weil

[8]

a

now ^that the base

T

of the

a

V- T is

a

let

be

curve,

hT

family

T_{corresponding}

and let P: T ~ V

on

to

divisor of

1,

the

height

section. The

degree

over

be

a

fiber

V

of Yis

a

function

and

generic

variety

global

e

field

the section P_corresponds to

a

rational

Pv

K(T),

V(K(T)),

point

Theorem 1.1

the

a

canonical

which can be evaluated at

gives

height

Vt,~V,~V
There ^are then three

and

which

hT

we show in

Pv.

point

be

heights

V,~V,~V,Vt,~t,~V,

a

result of Silverman

may

compared. Generalizing

[15],

Theorem 4.1 that

Silverman’s

result has been

and

in various

original

[2],

generalized

Silverman

strengthened

[20]

Call

Green

and Tate

We

ways by

have not

[6], Lang [8, 9],

[22].

been able to

of these

results ⁱⁿ our

yet

prove any

stronger

general

situation.

In the fifth section we ^take

thecanonical local

the

of how one

up

question

and

might efficiently

thecanoni-

compute

heights V,E,~,

therebyeventually

cal

In the case that

V

^is an

Tate

curve,

global height

(unpub-

that the

hV,E,CP’

elliptic

a

series for

lished) gave

completion

rapidly convergent

v)

provided

V,(O),[2](P,

of

K

at v is not

and Silverman

Kv

closed,

algebraically

[18]

give

described

a

modification of Tate’s series which works for all ^v. We

series ^for our canonical local

the series of Tate

heights

generalizing

and

 V,E,cp

and Silverman

how such

discuss

5.1)

(Theorem 5.3)

practice.

briefly

(Proposition

in

series could be

implemented

The final section is devoted to

aof

canonical local

for

description

heights

non-archimedean

in termsofintersection

In the case ofabelian

intersection

places

varieties it is knownthat the local

theory.

computed

general

every

extends

canbe

heights

using

on the Néron model. We show in

that

if

V

a

model V

has

theory

a

local

such that

Ov

over

rational

a

extends to

complete

ring

point

that the

V

section and such

V ~

to

a

finite

0:

• morphism
• morphism

V ~

then the canonical local

is

(D:

a

certain

intersection

V,

height

the

given by

question

V. Weleave for future

index on

exist.

To

of whether

such models

study

in this

we

a

of

and

local

summarize,

paper

develop

theory

global

166

on ^varieties

with non-unit divisorial

canonical

heights

possessing morphisms

We describe how these

in

families and

heights vary

be used for

algebraic

eigenclasses.

which

may

of canonical

the

computational purposes.

give algorithms

on abelian varieties has had

of

arithmetic

The

theory

heights

profound

several

field

Likewise,

throughout

applications

geometry.

applications

and

K3 canonical

for

arithmetic

many

It ^is our

questions

open

heights

in

that the

likewise

of canonical

are described

[19].

hope

general theory

in

will

in

useful

this

the

described

paper

prove

studying

heights

arithmetic

of varieties.

properties

1. Global canonical

heights

In this ^section we ^fix the

data:

following

K

a

field with

a

set of

absolute values

proper

satisfying

since

global

complete

formula. Wewill call such

a

field

aa

global heightfield,

function on

product

a

it is for such fields that one can define

height

pn( K).

could be

a

number ^field or a one ^variable function

(For example, K

field. )

V/K

a

smooth,

variety.

projective

a

a

a

V ~

V

defined over K.

0

~

~:

morphism

R.

divisor _{class q} E

Weil

_Pic(V) 0

~ R

to

function

~.

height

V(K)

corresponding

hV,~

hV,~:

for details about

fields and

2, 3, 4,

(See [8],

global height

height

with

Chapters

is an

functions on

Wefurther assume

that q

we assume ^that

for 0

eigenclass

varieties.)

than 1. In other

words,

eigenvalue greater

It follows from

of

functions

that

functoriality

height

[8]

Weil

and the choice of

The

the

on the

V,

0,

map

Ov(1)

variety

depends

first

Our

P

E

function

but it is bounded

of

height

independently

V (K).

which

hV,~,

result

the

that there exists

a

Weil function associated

for

says

to ~

Ov(1)

entirely disappears.

THEOREM

1.1.

a

With notation as

there

exists

above,

uniquefunction

167

the

two conditions :

satisfying

following

in Theorem 1.1 the canonical

Wecall the function

described

V,~,~

height

^If one or more

divisor

associated to the

and the

on

V

class ~

q,

0.

morphism

it

is

will sometimes omit

of the elements of the

we

clear,

0)

triple (V,

from

notation.

the

1.

A

an

let

_{[n] :} A -

A

be the

Let

be

abelian

variety,

multiplica-

divisor.

Example

tion-by-n map

is,

for some n

and _{let q} E

be

a

2,

Pic(A)

symmetric

As

is

one then has the

relation

well-known,

= q.)

[-1]*~

[n]*=

(That

n2TJ,

a

canonical

so one obtains

height

satisfying ~(nP) =

h~

hA,~,[n]

andTate.

Néron

This is the classical canonical

constructed

height

by

n2~(P).

of Theorem 1.1 ^is an

for

Our

easy

(See,

[8, Chapter 5].)

argument

proof

more

example,

a

extension of Tate’s

to

slightly

general setting.

X

C

• P2
• P2be

a

smooth K3surface

The two

the intersec-

Example^{2. Let}

given by

S

a

S~P2

are

tion of

a

and

(2, 2)-form

(1, 1)-form.

projections

_0-2: S~

S. Let double

so

each

an involution

covers,

on S,

o-1,

gives

say

and

be

sections of

Ç1, e2 F-

let 0

Pic(S)

hyperplane

type (1, 0)

(0, 1) respectively,

+

2

and define divisor classes on S

the formulas

B13,

by

Then one can check that

Further

0’2 0 03C31.

let 0 =

obtain two canonical

and

It

is worth

so we

heights

noting

hS,~,~+

hS,~-1,~-.

in

of the ^effective cone

each ^lie on ^the

that

and

~ R,

~+

~-

Pic(S)

is useful

for

boundary

+

is

This

but that ^their sum

the arithmetic of S.

observation

this

~+
~-

ample.

studying

a

For more

details

concerning

example, including

stated

of the ^facts we have

and

formulas that can be used to

proof

explicit

the canonical

see

and

height,

compute

[19]

[3].

IPn

~ Pn be

a

3.

Let

of

d : 2. Then

any

Example

0:

degree

so we can

morphism

~*~

7L

divisor _{class q} E

satisfies

Theorem

Pic(Pn) ~

d~,

but never

apply

1.1.

had

been observed

earlier

Notice

Tate,

(This

just

by

published.)

that

as in the case of abelian

one can

to be

varieties,

take q

ample,

which means ^that

1.1.1 below is

Corollary

applicable.

a

the canonical

we can

result

Using

height,

easily

prove

strong rationality

168

for

P

is called

We

recall that

a

points.

point

pre-periodic for ~

pre-periodic

if its orbit

P

is

^if some iterate

is

is finite.

~i(P)

(Equivalently,

pre-periodic

a

periodic.)

bound for

Lewis

The

corollary gives

strong rationality

following

pre-periodic

and

It

results of Narkiewicz

who

generalizes

[13]

[10],

prove

points.

in

K-rational

Lewis’

the cases

that there are

only ^finitely many

pre-periodic points

An and

V

Pn

in

uses basic

V

respectively.

functions,

few lines.

proof,

particular,

pro-

reduces

but our use of the canonical

of Weil

to

height

height

perties

the

a

just

proof

COROLLARY

1.1.1. Let

V/K ~VIK be

a

over

a

0:

morphism defined

divisor class

number_field

and

that

there is an

suppose

K,

ample

q

satisfying

Let

P

E

(1).

V (K).

P

0.

^is pre-periodicfor 4J

contains

only if

_ifand

only ^finitely many

(a)

hV,7J,f/J(P)

More

gen-

0.

pre-periodic points for

(b) V (K)

erally,

is

a

set

bounded

so in

it contains

of

points defined

height,

particular

onlyfinitely many

degree.

over all extensions _ofK_ofa bounded

If

P

is

for

then

^takes on

0,

Proof. (a)

many

only finitely

pre-periodic

hV,~(~n(P))

distinct

and so

values,

if

then

0,

Conversely,

hV,~,~(P)

{P,

Hence the set

• is
• is

a

set of bounded

so ^it

finite.

is

...}

O(P), ~2(P),

height,

Therefore

this is where we use the

fact that

is

P

(Note

~

ample.)

0

pre-

periodic.

If

P

is

for

then

from

so

~,

(b)

pre-periodic

is bounded.

(a),

V,~,~(P) =

hV,~(P) =

form

+

This shows that the

a

0(l)

V,~,~(P)

pre-periodic points

set of

bounded

The rest

of

is then

since

a

set

of