Lubin-Tate formal groups and module structure over Hopf orders

J

OURNAL DE

T

HÉORIE DES

N

OMBRES DE

B

ORDEAUX

W

ERNER

B

LEY

R

OBERT

B

OLTJE

Lubin-Tate formal groups and module structure

over Hopf orders

Journal de Théorie des Nombres de Bordeaux, tome

11, n

o

_{2 (1999),}

p. 269-305

<http://www.numdam.org/item?id=JTNB_1999__11_2_269_0>

© Université Bordeaux 1, 1999, tous droits réservés.

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

Lubin-Tate formal

_{groups and}

module

structure

over

Hopf

orders

par WERNER BLEY et ROBERT BOLTJE

RÉSUMÉ. Ces dernières années les ordres de

_Hopf

ont

_joué

dans

des situations diverses un rôle

_important

dans l’étude de la struc-ture des module

_galoisiens

en

_{géométrie arithmétique.}

Nous

intro-duisons ici un cadre

_qui

rend _compte des situations

précédentes,

et

nous étudions les

propriétés

des

algèbres

de

Hopf

dans ce contexte

général.

Nous insistons en

_particulier

sur le rôle des résolvantes dans les calculs

_explicites.

Nous illustrons cette étude en

appli-quant nos résultats à la détermination de la structure de module

de

_Hopf

de l’anneau des entiers d’une extension de Lubin-Tate relative.

ABSTRACT. Over the last years

Hopf

orders have

_played

an

im-portant role in the

_study

of

_integral

module structures

_arising

in

arithmetic _geometry in various situations. We axiomatize these

situations and discuss the

_properties

of the

_(integral)

_Hopf

alge-bra structures which are of interest in this

general setting.

In

particular,

we

_emphasize

the role of resolvents for

explicit

compu-tations. As an illustration we

apply

our results to determine the

Hopf

module structure of the

_ring

of

_integers

in relative Lubin-Tate extensions.

1. INTRODUCTION

In this article we axiomatize the situation considered in

_Taylor’s

article

(T2) .

1.1. Assume the

_following

situation:

. C7 is a Dedekind domain of characteristic

0,

K its field of

fractions, K

an

_algebraic

closure of

_K,

and Q :=

Gal( K / K)

the absolute Galois group of K.

. G _is a finite _group on which Q acts from the left _{via group}

automor-phisms.

. r is _a finite _{set on which} G and f2 _{act from the} left such that G _acts

simply

transitive

_{(thus IGI}

= with

’(9y) =

("’9)( cu’Y)

for all w E

_Sl,

9 E

_{G, and 7}

E r

_(the

last condition

_just

means that the semidirect

product

G x Q acts on

r).

We

_always

write the action of w E 0 on G and r

_{exponentially}

from the left

(like

CUg

and

_{cu¡ for g}

E G

_{and 7}

E

_r)

and its action

on K

as for x E

K.

For _any intermediate field K C L C

_K,

we set

OL

:=

0,

and

denote

_by

_OL

the

_integral

closure of 0 in L.

There is a

_variety

of natural

_examples

for the

_general

situation described

in 1.1.

1.2.

_Examples.

For _any field of characteristic 0 let _{Mn, n} E

_N,

denote the

multiplicative

group of n-th roots of

unity

in its

algebraic

closure.

(a)

Let and set K := G := _I-tm. Fix a

primitive

r-th root of

_{unity (3}

E and set r E

K ~

_{I xl =,81.}

Then r C _itr+m and G acts on r

by

multiplication.

(b)

More

_generally,

let K be a number

field,

let m e N and G := _11m. For

(3

E KX set r :=

Ix

E K ~

(3}.

Then G acts on r

by

multiplication.

(c)

For K as in

_1.1,

let

L/K

be a finite Galois extension with Galois

group

G,

and let 0 act on G

by conjugation. Moreover,

let

_{qo e L}

be a

primitive

element

_(L

=

K(~yo)),

and let r be the set of Galois

conjugates

of _70.

(d)

Let

_{E / F}

be an

elliptic

curve with

_{complex multiplication by}

_C~M,

where F is a finite extension of a

quadratic

_imaginary

number field M. For any

integral

ideal a C

nM

write

E[a]

for the

subgroup

of

points

of

E(F)

that are annihilated

_by

all elements a E a. For x E

_nM

let

_[x]

E

_End(E)

be

the

_{corresponding endomorphism}

of E.

_Let,

for

_simplicity,

_(a)

= _a C

C~~

be a

_principal

ideal and set G :=

E[a].

For P E

E(F)

define K :=

F(P)

and r :=

_{Q

E

E(F) ~

_I

_[a](Q)

=

Pl.

Then G _{acts on} r

by

translation.

This kind of

_example

is

_extensively

studied in

_{[A], [ST]}

and

_[T2].

(e)

Let

_~

C F be a finite field extension of the field

of p-adic numbers,

and let p F =

(7r)

be the maximal ideal of the

ring

dF

of

_integers

in F.

Let F be a Lubin-Tate formal _group attached to a Lubin-Tate _power series

f (X )

E

_OF[[X]].

Let

_[-]:

_{OF -}

_End(.F)

be the usual

_ring

_isomorphism

with

_[7r](X)

=

f (X ).

For n _E N set

Gn

:=

~a~

E

_pF

_I

_{(~rn~ (x)}

=

01,

the

subgroup

of 7rn-torsion

_points

in the

_dF-module

_pp

endowed with the

F-group law and the

OF-action

aac :=

[a] (a?)

for a E

_CF

and x E

_PP,

and let

Fn

:=

F(Gn)

denote the field obtained

by adjoining

the elements of

G,~.

For fixed

_{r, m e N,}

set K :=

Fr,

G :=

G~,

choose

~i

E

_{Gr ,}

_{Gr-i ,}

and set

r :=

~x

E

_{p p}

_{I [7rm](x) = (3} ç}

Then G acts on r

by

translation.

For more details about this

_example

in the case m r see

[CT,

Ch.

1.3.

_Following

_[T2]

we define in the situation described in 1.1

B :=

_{Map(G, K) ,}

C :=

_{Map(r, K) ,}

the _group

_algebra

of G over

_K,

the set of _maps from G and from r to

K,

respectively.

Then A is a

K-Hopf

_algebra,

B is its

K-dual

_{Hopf algebra,}

C is a

K-algebra

and a

right

A-module

by

the

K-linear

extension of the

action

_{( f ~ g)(’Y)}

:= of G on

_C,

where _g E

_{G, f}

E

_C,

and _y E r. Note that SZ acts on

_{A, B,}

and C

_by

for w E r. This is an

action via

_{K-Hopf algebra automorphisms}

on A and

B,

and via

K-algebra

automorphisms

on C.

Hence,

we _may take fixed

_points

with

_respect

to the

subgroup

OL

=

Gal(K/L)

for _any intermediate field K C L C

K:

We omit the index L for L = K. Then

AL

and

BL are

L-Hopf subalgebras

of

A and

_B,

and

_they

are L-dual to each other.

_{Moreover, CL}

is an

L-algebra

and an

_AL-module

_by

restriction of the A-action on C. In

_Proposition

5.2

we show that

CL

_together

with its

AL-module

structure is a Galois

_object

in the sense of

[CS],

and,

if G is

abelian,

that

CL

is a free

AL-module

of

rank 1. In the abelian _case, we also define a resolvent

(, f , x)

E

I~,

for

_f

E C

and X

E

G

:= and show

that,

for

given f

E

_CL,

one has

, f - AL

=

CL

if and

only

if

( f ,

x) ~

0 for

all X

_E

G,

_see

Proposition

5.3.

1.4. For _any intermediate field K C L C

_K,

the

_L-algebras

_BL

and

CL

are commutative. Let us assume that G is abelian. Then also

AL

is

commutative,

and we _may define the maximal

OL-orders

- -- , F - - I -- I

of

_AL,

_BL,

and

_{CL, respectively.}

_Moreover,

for

_{arbitrary G,}

set

which is an

O L-order

of

AL. Then, CL

is an

.A.L-module

and we _may define

the associated order

of

_CL

in

_{AL ,}

which cont ains If G is

_{ab elian,}

then we have inclusions

1.5. Remark.

_(a)

_Suppose

that the action of 0 on r is transitive. Fix an

element _qo and denote its stabilizer

_by

_S2L

_{corresponding}

to an intermediate

field K C L C

K.

Then one has a

bijection

SZ/StL

G,

where is

mapped

to g

E

_G,

if

_’-yo

=

Moreover, associating

_to

f E C

the value

f (-yo) E

K,

defines

_isomorphisms

C

~

L and C

~

C~L

of L-

_{(resp. OL-)}

algebras.

(b)

The absolute Galois

_{group Q}

acts

_trivially

on G if and

_only

if A =

KG.

1.6. If one assumes suitable Kummer conditions in

_Examples

1.2

_{(a), (d)}

and

_(e),

then A = KG.

Moreover,

in _many

interesting examples

SZ acts

transitively

on r

and,

in the notation of Remark 1.5

(a),

the fixed field L

of the stabilizer is an abelian Galois extension of K such that

«

the

_bijection

_Gal(L/K)

=

St/SZL

=~ G is a _group

isomorphism

(see

_e.g.

Lemma

_6.1).

In this _case, the A-module structure of C

_corresponds

to the KG-module structure of

_L,

and we are in the classical situation of Galois

module structure

_{theory. Then,}

there are results due to

_{Cassou-Nogues,}

Schertz,

and

_{Taylor, stating}

that

_OL

is free over its associated order in

KG,

see _e.g.

[CT,

Ch.

_X,

Ch.

XI], (S).

In Section 6 we will

_study

a

_slight

generalization

of

_Example

1.2

_(e),

_namely

relative Lubin-Tate extensions. We will show that also without any Kummer

condition,

the maximal order

C ~

_OL

is a free rank one module over its associated order A~’

(see

The-orem

6.11).

But note that in

_general

the

_{algebra A}

is not the _group

_ring

KG.

_Combining

ideas of

_[Tl]

and

_[T2]

we will

_give

an

_{explicit description}

of as the Cartier dual of the

OK-Hopf

order which

_represents

the

OK-group scheme of 7rm-torsion on Y

(see

Proposition

6.5 and

Corollary

6.9).

In contrast to the methods used in

_[CT,

Ch.

_X]

the main tool for our

proof

of Theorem 6.11 will be the factorization of a suitable resolvent function

which takes values in

_Op[[X]]

_(see

Theorem

_6.10).

This will be of

_great

importance

for further

applications

of these local results to fields obtained

by

the division of

_points

on

elliptic

curves as in

Example

1.2

(d).

These

examples

are dealt with

_by

the first author in his habilitation thesis

[B].

1.7. The article is

_arranged

in the

_following

way. Sections 1-5 are devoted to the

_general

situation as stated in 1.1. In more

details,

the sections 2 and

3 summarize

_properties

of the

_{Hopf algebras}

_AL

and

_BL,

_{respectively,}

for intermediate fields K C L C

K.

Most of these

_properties

can

already

be

found in

_[T2]

without

_proofs.

For the reader’s convenience and for later use we

_provide

_proofs

of all assertions. In Section 4 we show that

AI,

(resp.

AL

if G is

_abelian)

is an

OL-Hopf

order in

AL

if and

only

if the discriminant

ideals

_dBL

of

_BL

_(resp.

_dAL

of

_AL)

over

OL

are

_trivial,

see

_Proposition

4.6

order is

_completely

described

_by

an arithmetic

_invariant,

the discriminant.

Moreover,

we

_compute

the index ideal

[AL :

in terms of

_dAL

and

dBL ,

see Lemma 4.3. Section 5 is concerned with the

_L-algebra

structure

and the

_AL-module

structure of

_CL.

If G is abelian we

determine,

for _any

f

E

_CL

with

_{f -}

_AL

=

CL,

the index ideal

[CL :

f -

A’

in terms of

_dBL,

and the resolvents

_{( f , x), x}

E

G,

see

Proposition

5.3.

Finally,

in Section

_6,

we

apply

the results from Sections

1-5,

in

particular

the index formula of Section

_5,

to the

_Example

1.2

_(e).

2. FORMS OF GROUP ALGEBRAS

We assume the situation defined in 1.1 and the notation introduced in

Section 1.

In this section we

_study

the

_properties

of the

K-Hopf

_algebra

A and its

subalgebras

AL

=

A OL,

for intermediate fields K C L C

~K.

For a

K-Hopf algebra

H we write as usual

_{L1 H :}

H - H

_{fi9K H}

for

the

_diagonal,

_{SH : H -}

H for the

_antipode

and EH : H --~ K for the

augmentation.

If there is no

danger

of

confusion,

then we omit the index.

We view A as a

_{Hopf algebra}

with

A(g) = 9

_{(8) g,}

S(g) =

g-1

and

=1,

for g

E G.

2.1. Lemma. Let K C L C

K

be an intermediate

fields,

and

_{let gl, ... ,}

_{gr E}

G be a set

_of

_{representatives}

_for

the

_of

G. For each i

_{E Ili ... , r~}

let

_Li

be the

_{fixed field of}

_{stabOL (9i)}

_{and let Xi,1,} ... , xi,,.; be an

L-basis

_of

_Li.

_{Finally, for}

₁

i r and

_{1 j}

_ri, set

and assume that L C M C N C

K

are

f urther

intermediate

fields.

Then

the

_following

assertions hold:

(i)

The elements

₁

i _{~ r,}

_{1 j}

_ri,

_{f orm}

an L-basis

_of

_AL.

_If,

for

each i E

_{~l, ... , rl}

_{the elements zj,i , ... , xi,r¡}

form

an

OL-basis

_of

_OL¡

then the elements i _ri,

_{f orm}

an

OL -basis

_of

Aî.

(ii)

One has

_{dimL(Ay) _ IGI.}

(iii)

The elements

₁

i r,

1 j

ri,

f orm

also an M-basis

of

AM;

in

_particular,

_{the y f orm}

a

K-basis

of

A.

(iv)

The _map

is an

_{injective L-algebra}

_map such that

iN

0 if

=

if.

In

particular, AL

_~y

AL

can be

_regarded

as an

L-subalgebra of

A A.

(v)

Under the

_{identification}

AL

(g)L

AL

9

A A

_of

_(iv),

AL

is an

(vi)

The _map

is an

isorrtorphisrn of M-Hopf algebras

such that the

diagrams

commute

_for

each w E

_0,

where can: N

_{®M M -~}

N is the

_{multiplication}

map.

Proof.

(i)

It _{is easy} to see that the elements i _r,

_{1 j}

_ri, lie in

AL.

On the other

_hand,

for

_arbitrary

a =

EGEG

Àg9

E

AL,

the coefficients

Ag

is fixed under

_stabOL

_(g),

for each _g E

_G,

and so

_Agi

E

Li

for each

i

_{6 {!,... r } .}

_Moreover,

for e ach i

_{E 11, - .. , ,y}}

and each w E one

has = Now it follows

easily

that the elements

1 ~

i

r,

1 j

ri, form an L-basis of

AL.

The second assertion is shown in a

similar _way.

(ii)

Indeed, by

(i)

we have

- - - -

-(ill)

First we show that the elements

_{1 i r, 1 j ri,}

form a

X’-basis

of A.

_Expressing

the elements

_by

the basis G of A

_produces

a block

diagonal

matrix with r blocks indexed

_by

the

_OL-orbits

of

_G,

where the i-th block is

_{given by}

whose determinant does not

_vanish,

since is

_separable.

This shows the result for M =

.I~.

For

_arbitrary

_M,

it suffices

_by

(ii)

_to show that the

elements

_{(1 i r,1 j}

_ri)

are

M-linearly

independent.

But this

follows from the case M =

I~’.

(iv)

This follows

_immediately

from

_(iii),

since

_2L

maps an L-basis to an

(v)

It is _easy to see that

S(AL)

_C

AL,

since

taking

inverses in G com-mutes with the f2-action on A. Let a E

AL. Then,

by

(iii),

we _{may write}

with

_uniquely

determined coefficients E

K.

We

_apply

an

_arbitrary

element E

_OL

on both sides. Since A

_respects

the

fraction,

we obtain

Thus, by

their

_uniqueness,

the coefficients are contained in

L,

and

E

if(AL

_0L

_AL)·

(vi)

It is _{easy to} see that

~L

is a

homomorphism

of

M-Hopf

algebras

and

that the two

_diagrams

commute.

_Moreover,

_by

_(iii),

_ølf

is an

isomorphism.

L

For the rest of this section we assume that G is abelian. Let L be a

subextension of

_K/K.

In order to understand the

_L-algebra

structure of

_AL

we work with the Wedderburn

_{decomposition}

of A. Let

G

:=

Hom(G,

Ï(X)

denote the abelian _group of

K-characters

of G. Then n acts on

G

by

for X

E

G,

w E

_f2,

_{and g}

E G. It is well-known that the _map

is an

isomorphism

of

,K-algebras.

_{For X}

E

_G,

we denote

by

_ex E A the element

_{corresponding}

to the

_primitive

_{idempotent Ex}

of

_{IlxEG K}

which has

_entry

1 in the

_{component X}

and 0

_everywhere

else. Then

Note that

_10(ex)

=

e~ ~x~

for each w

and X

E

G.

The f2-action on A is

transported

via _{p to} the action

for

_(ÀX)XEG

E

fiXEd ff

and w E f2.

_Thus,

the

_application

of w moves the

x-component

_Ax

to the

_{cux-component}

while

_{simultaneously}

_applying

w to

2.2. Lemma. Assume that G is abelian and let K C L C

K

be an

in-termediate

_field.

Let _{x1, ... , Xs} E

6

be a set

_{of representatives for}

the

fIL-orbits

of

6. For

each k E

_{~1, ... , s~,}

let

lk

denote the

_,fixed

_{field of}

stabnl

SZL,

and _{let Yk,l,.. -} be an L-basis

of

Lk-

For

_{1 ~}

k

s

and

1 ~

I

_~

_{sk, set}

Then the

_following

assertions hold:

(i)

The

_composition

where _p denotes the

_projection

to the _{xl, ... ,}

_{X, -components,}

restricts to

an

L-algebra isomorphism.

In

_{particul ar,}

the maximal ord er

_AL

_{of AL}

is

_{given b y}

_{~L )’}

(ii)

The elements

_ak,l,

1 ~

k

s,

1 t

sk,

form

an L-basis

of AL.

If,

,for

each k E

_{(I, ... , ~}~}

the _{elements ~i)’"}

_form

an

OL-basis

of

_Otic

then the elements

_ak,l,

_{1 h s, 1 t sk,}

_form

an

OL-basis

of

AL-Proo, f .

( i)

This follows

_immediately

from

_{( 1 ) .}

(ii)

The elements are contained in

AL

and the elements

1~&#x26;~~,1~~~~

form an L-basis of

LIc.

Thus

(ii)

follows from

(i) .

The

_integral

version can be seen in the same _way C7

3. DUALITY

We assume the situation described in 1.1 and the notation from Section 1.

It is well-known that the

_{Hopf algebra}

dual of A is the

_K-Hopf

_algebra

B :=

Map(G,

K)

consisting

of all set _maps from G to

K.

The

duality

is

The K-linear structure of B is obvious.

_{Multiplication}

is

_{given by}

_{( fl f2)(9)}

=

fi(9)f2(9),

for

_{fl, f2}

_E B and

g E

_G,

with the constant _map with value

1 as

_unity.

The

_{diagonal A, augmentation}

_e, and

_antipode

S are

_{given by}

respectively,

for

_f

E

_B,

g, gi, g2 e

G,

where in the definition of A we

identify

B

_{0 K}

B with

_Map(G

x

_G,

K)

in the obvious _way. For the

K -basis

elements

_{lg, g E}

_G,

with := for h E

_G,

we have

Note that 11 acts on B

_by

for w E

_{0, f}

E

_B,

_{and g}

E G. This action

_respects

the

_K-Hopf

_algebra

structure,

as is

easily

verified. For a E

_A,

_f

E

_B,

and w E 11 one has

For each intermediate field K C L

ç K

we set

Then

_BL

is an

L-subalgebra

of B.

The next lemma shows that

_BL

is an

_L-Hopf

_subalgebra

of B and is the

L-dual of

_AL

with

_respect

to the restricted bilinear form

_{(-, -).}

_By

_BL

we

denote the maximal order in

BL

which is

_{given by}

3.1. Lemma. Let K C L C

K

be an intermediate

_field,

and let _{gl, ... , gr} E

G, L1, ... ,

L,.,

and xZ,l, ... , Xi,r¡ E

Li

₍₁

i ~

_r)

be

_given

as in Lemma ~.1.

Moreover, for

1

i r and

_{1 j}

_ri, let

_bi,j

E B be

_{defined by}

I

Assume that L C M C N C

K

are

_further

intermediate

_fields.

Then the

(i)

The _map

..-..

is an

L-algebra isomorphism.

In

particular,

oL restricts to an

isomorphism

(ii)

The elements

_bi,j,

i _{r, ri,}

_form

an L-basis

of

BL.

If, for

each i E

_{~1, ... , r),}

_{the elements Xi,!, ... , Zi,r¡}

form

an

OL -basis

_of

C7L=

then the elements

_bi,j,

_{1 ~}

i _r,

_{1 j}

_ri,

_form

an

OL -basis

o, f

~L .

(iii)

One has

_{dimL (BL ) _ ~ IGI.}

(iv)

The elements

₁

i _r,

_{1 j}

_{rZ, ,}

_form

also an M-basis

_of

BM .

(v)

The _map

is an

_{injective L-algebra homomorphism}

such that

jM

0

jL -

jf.

In

par-ticular,

BL

OL

BL

can be

regarded

as

L-subalgebm of

B

_Ok

B.

(vi)

Under the

_{identification}

_BL~LBL

C

_of

_(v),

_BL

is an

L-Hopf

subalgebra of

B.

(vii)

The _map

is an

_{isomorphisrn,}

_{of M-HoPf algebras}

such that the

_diagrams

commute

_for

all w E f2.

(viii)

The _map

_AL

_®L

BL H AM

®M

Bwt,

a _®L b H a _0M

b,

is

injective;

in

_{particular, AL}

_®L

_BL

can be

regarded

as an

_{L-subspace of}

A

_®K

B .

Moreover,

the restriction

_of

_(-,

_-)

to

_AL

®L

BL

takes values in L and is

non-degenerate.

(ix)

The

_L-Hopf

_algebras

_AL

and

_BL

are dual to each other with

_respect

to

_{(-, -) :}

_AL

_®L

_{BL --~}

L.

Proof.

(i)

For

_{b E BL}

and i

_{E ~1, ... , ,r},}

the elements =

w E

_OL,

are determined

_by

b(gi).

Hence,

_aL is

_injective.

On the other

by

b(dgi)

:=

W(Ài),

for i =

1, ... ,

_r and _{w E}

SZL,

is well-defined and is

obviously

in

BL.

(ii)

This follows

_immediately

from

_(i).

(iii)

This follows from

_(ii)

and the

_equation

(iv)

This is

_proved

in a similar _way as Lemma 2.1

(iii)

_by

reduction to

the case M

= k

and

_using

the basis

_{Ig, 9}

E

_G,

of B which leads to the

same transition matrix as in the

proof

of Lemma 2.1

(iii).

(v)

This follows from

_(iv).

(vi)

This is

_proved

in a similar _way as Lemma 2.1

(v)

_using

the basis

(vii)

By

(iv),

is an

_isomorphism

of

_M-spaces.

The

_remaining

asser-tions are

easily

verified.

(viii)

The

_injectivity

of the

_{map AL}

_0L

_{BL --&#x3E;}

_AM

_~M

_BM

follows from Lemma 2.1

_(iii)

and

_part

_(iv).

_Moreover,

_{(AL, BL) C}

L

_{by Equation}

_(2).

Using

the bases _ai,j of

_AL

from Lemma 2.1 and of

_{BL, 1 ~ i ~}

_r,

1 ~ j ~

ri, with their

property

from Lemma 2.1

(iii)

and from

(iv),

we see that

_{(AL, BL)}

= L and that the restricted

_pairing

AL

_0~ L is

non-degenerate.

(ix)

This follows from

_part

_(viii)

and from the existence _{of bases ai,j}

of

_AL

and of

_BL

with the

_properties

from Lemma 2.1

_(iii)

and from

(iv).

~

0 4. INDICES AND HOPF STRUCTURES

We assume the situation described in 1.1 and the notation from Section 1.

Let K C L

C k

be an intermediate field. We _may take duals of

OL-lattices in

_AL

and

_BL

with

_respect

to the

_{non-degenerate pairing}

More

_precisely,

if 7Z

_{C AL}

and s C

_BL

are

OL-lattices,

then

are

OL-lattices

in

_BL

and

AL

_{respectively.}

Note that ?Z is an

OL-order

(resp.

OL-subcoalgebra)

of

_AL

if and

_only

if ~Z* is an

OL-subcoalgebra

(resp.

OL-order)

in

_{BL .}

A similar statement holds for S.

Moreover,

7~ is

an

O L-Hopf

order in

AL

if and

_only

if R* is an

OL-Hopf

order in

BL,

and

similarly

for S.

Recall that for

_OL-lattices

X C Y of

_equal

_C?L-rank

the

OL-order

ideal

ly :

is defined as the

product pi ..

_{pt of} non-zero

_prime

ideals

More

_generally,

for

_{OL-lattices X}

and Y in a finite dimensional L-vector

space, the order ideal

[Y : X] OL

is defined as the fractional ideal

[Y :

X fl X For the

_following

_properties

of order ideals

see for

example

[R, §4].

If L C L’

C k

is a finite extension field of L then

If p is a non-zero

_prime

ideal of

_OL

then the

_following

localization

_property

holds:

If Y and X are free

OL-modules

and M is the matrix of coefficients

_arising

from

_expressing

an

C7L-basis

of X

by

an

OL-basis

of Y,

then

Finally,

if L C L’

C K

is as above and X’ C Y’ are

OL,-Iattices

then

If G is abelian we denote

by

dAL

the discriminant ideal of the maximal

or-der

_,A.L

over i.e.

_{dAL -}

n~=1

in the notation of Lemma 2.2.

Sim-ilarly,

for

_{arbitrary G,}

we denote

by

_dBL

the discriminant ideal of the

max-imal order

_BL

over

OL,

i.e.

_{dL¡/L in}

the notation of Lemma 2.1

and Lemma 3.1. If L C L’

C K

is as above then we write

_{DL, /L}

for

the different of

_{0 L’}

over

0 L.

4.1. Lemma. With the notation

_{of Lemma}

2.1 and Lemma 3.1 one has

Moreover,

_Bi,

is an

OL-order

in

AL

if

and

onl y if

dBL =

OL.

Proof.

Each element a E

AL

can be written in the form

with

_uniquely

determined

Ai

E La, i

=

1, .’.. ,

r,

by

Lemma 2.1

(i).

Then

a E

₈₁

if and

_only

if

_{(a, f )}

E

_OL

for all

_f

E

_{BL. By}

Lemma 3.1

_(i),

each

_f

E

BL

is a sum of elements of the form

~~

_{OL¡, i}

=

_{1, ... ,}

r,

where the sum runs over coset

representatives

of

OL/stabnL

(g~ ) .

Hence,

for all i

_{~ {1,... r)}

and all Xi e

_OLp

which is

_equivalent

_{to Ài}

E

for all i

_{E ~ 1, ... , r ~ .}

If

_{dBL =}

OL

then

_OL¡

for all i =

_{1, ... ,}

_r, and therefore

81

=

,A.L

is an

_{Suppose, conversely,}

that

₈₁

is an

OL-order

and let

i E

_{~1, ... , ,r}}

be

_arbitrary.

Let i’

E

_{~1, ... , r)}

and v E

_QL

be such that

=

Vgi’.

Note that and that v :

Li,

-

_Lj

is

an

L-isomorphism.

Let

Aj

E and

_az~

E be

_arbitrary.

Then

are elements of

_81.

Hence,

also their

product

lies in

81.

In

particular

the

coeflicent at 1 e G of this

_product

is an element of

(’)L:

If

_az,

runs

through

then runs

through

and

(3)

implies

that C

_Of,.

But this is

_{only possible}

OL.

· Since

this holds for all i E

_{~1, ... , r}}

we obtain

OL.

0

4.2.

_Corollary.

One C

₈₁

and

_{[81 :}

_{dBL .}

Proof.

The inclusion

_AL

C

₈₁

is clear from Lemma 4.1. From the definition of

_AL

and from Lemma 4.1 we have

LJ

We remark

_that,

for G

_abelian,

_B1

is not

_necessarily

contained in

For the

_following

lemma we assume that G is abelian. Let L C L’

C K

be a finite extension field of L

containing

L1, ... , Lr

and

L1, ... ,

in the

notation of Lemma 2.1 and Lemma 2.2. Then

_AL~

=

L’G,

EL’

=

Map(G, L’)

and L’

_{via PL’ as}

_{L’-algebras.}

Such a finite extension L’ will

4.3. Lemma. Let G be abelian. Then

for

each intermediate

_field

K C L C

K.

Proof.

Let p be a non-zero

_prime

ideal of

OLe

Since

(,AL)p

is the maximal

(OL)p-order

in

_AL

and

_(Al)p

is the set of elements in

_AL

whose coefficient with

_respect

to G are

integral

over we _may as well assume that

_OL

is local with maximal ideal _p. Let L’ be a

splitting

field for

Ay

and

BL.

We determine

using

the

_isomorphism

_{øf’ :}

_{L’ fi9L}

_{AL, ,}

the maximal

_OLI-order

_AL,

of and the

_isomorphism

_IIxEG

L’. More

_precisely,

_{[AL :}

is the

_quotient

of the

_squared

order ideals

and

For the

_computation

of the

_squared

order ideal

₍₅₎

let _{Xk, yk,t,} and

1 ~

k s,

1

1 fi

Sk, be

given

as in Lemma 2.2 such that yk,l, ... , is an

OL-basis

of

_Or

for each k =

1, ... ,

_s. Then the elements

form an

_OL-basis

of

AL

and

(PLI

o

AL)

has as

OLI-basis

the

elements

Expressing

this basis

_by

the

_OL,-basis

_{ex, x} E

G,

_fIXEd

7

we obtain a block

diagonal

transition matrix with blocks indexed

by

G/SZL .

The block

_belonging

_{to xk} is

_{given by}

The _square of the determinant of this block

_generates

the

C7L-ideal

Thus,

the

_squared

order ideal in

₍₅₎

is

_{given by}

For the

_computation

of the

_squared

order ideal

₍₄₎

let _9i,

_Li,

_xi,j, and

1 ~

i _r,

_{1 j}

_rz, be

_given

as in Lemma 2.1 _{such that xZ,l, ... ,} is

an

_C7L-basis

of for each i E

_{1, ... , r.}

_Then,

the elements

form an

O L-basis

_Thus,

_{the elements}

form an

OLI-basis

of

_{( pL’}

o

_Expressing

_{this basis}

_by

_the

basis E

G,

of = _we obtain the transition _matrix

which is the

_product

M =

Ml M2

of the block

_diagonal

_matrix

Ml

with blocks indexed

_{by i}

=

1, ... ,

_r, the i-th block

_{given by}

and the matrix

Now,

_det(M1,i)2

generates

the

_OL-ideal

_dL¡/L;

thus

_{dBL .}

Moreover,

since

for _{Xl, X2 E}

_G,

we have

Altogether

this

_yields

which _expresses the

_squared

order ideal in

_(4).

_Now,

_dividing

the ideal

in

₍₈₎

_by

the ideal in

_(6),

the result

_follows,

since

_extending

_OL-ideals

to

4.4. Next we

investigate

under which circumstances

AL

and

(in

the abelian

case)

AL

are

_Hopf

orders over

OL

in

AL.

The crucial

_point

is to

_get

hold of the

_image

of the

_diagonal.

Since the

_{following might}

be of

_general

interest we

place

ourselves for

the moment in a more

general

setting.

Let H be a finite _group,

_E/F

a field

extension in characteristic _zero, and R C F a

_subring

such that F is the field

of fractions of R.

_Suppose

that ,A C EH is an

R-subalgebra

of EH with

R-basis _{at, ... ,} which is also an E-basis of EH. Then the

_{multiplication}

map E 0~ ,A. --~

EH,

A 0R

Aa,

is an

_{E-algebra isomorphism}

and the

map is

injective,

so that

can be

_regarded

as an

R-subalgebra

in

EH0EEH.

We would like to decide whether

_A(A)

g _{,,4 OR} or not. We write

for i =

_{1, ... , n}

with _E

_E,

and _we consider the matrix

together

with its inverse

Then we have the

following

criterion:

4.5. Lemma.

_Keeping

the notation

_of

_4.4,

the

_following

assertions are

equivalent:

Proof.

We write

with

_uniquely

determined coefficients E E.

_Then,

for i E

_{~1, ... , n ,}

we have E A _0p .~4 if and

_only

if

’Yj3,

E R for all

_{j, j’}

_~{!,...

_,~}.

Using

the

_expansion

_{0(as) -}

_h,

we obtain the

_equivalent

Hence,

the coefficients are

_uniquely

determined

_by

the

_system

of linear

equations

’

one

_equation

for each i E

_{{1, ... , n}}

and each

_pair

_{(hl, h2)}

E H x H.

Now _{it is easy} to

_verify

that these

_equations

are satisfied for

and the result follows. 0

In the

_following

_proposition

we

apply

Lemma 4.5 to the

OL-order

,A.L

in

AL .

4.6.

_Proposition.

The

_{OL -order}

_Al

is a

Hopf

order in

AL

if

and

only if

dBL -

OL .

Proof.

Clearly,

is stable under the

_antipode

of

_AL.

The inclusion

C

_AI,

can be tested

by

localization. Thus we _may assume that

_C~L

is a local

_ring.

In this case we choose _gi,

_Lz,

and aij as

in Lemma 2.1 such

_that,

for each i E

_{~1, ... , r},}

the _{elements a?tj,.... 7 xi,ri} 7

form an

OL-basis

of Then the elements

form an

OL-basis

of Al,

and,

in the notation of

4.4,

the coefficient matrix

,S’ is a block

_diagonal

_matrix,

the blocks indexed

_by

i =

_{1,... ,}

r, and the

i-th block

_given

_by

If

_{xz~l, ... ,}

E

_Li

is a dual basis of Xi 1, _{... , Xi r. with}

_respect

to the

trace form

_TrLi/L’

then the inverse T of S is

_{given by}

the block

_diagonal

matrix with i-th block

Jet.1,...

Now,

taking

into account the block

_diagonal

structure of ,S’ and

T,

Lemma 4.5 states that if and

_only

if

for all i e

_{~1, ... , r~}

and all

_{j’, j"}

E

_{~1, ... ,}

But since

1,...

rZ, form an of the condition in

(9)

holds for

given

But this is

_equivalent

to

_ÐLi/L

=

OLi

and to =

OL.

Now the result

follows. 0

4.7.

_Corollary.

The

_following

statements are

equivalent:

(i)

The order

_AL

is a

Hopf

order in

Ay.

(ii)

The discriminant ideal

_dBL

is trivial.

(iii)

One

_{has .AL =}

81-(iv)

One has

_(,.4L)*

=

BL .

(v)

The maximal order

BL

of BL is

an

OL-Hopf

order.

If

(i)-(v)

hold then

_AL

is the smallest

_OL-Hopf

order

_of

AL-Proof.

The

_equivalence

of

_(i)

and

_(ii)

is the content of

_Proposition

4.6. The

statements

_(ii)

and

_(iii)

are

equivalent by

Corollary

4.2 which asserts that

v

A’

I

=

dBL .

Obviously,

(iii)

and

_(iv)

are

equivalent,

since

OL

is a

Dedekind domain.

_Moreover,

_(i)

and

_(iv)

_imply

_(v).

_Finally,

_(v)

_implies

that

_B~

is an order in

AL,

and then Lemma 4.1

implies

(ii).

If

_(i)-(v)

hold then

_BL

is

_certainly

the

_largest

_OL-Hopf

order in

_BL.

Therefore its dual

_AL

is the smallest

_OL-Hopf

order in

_AL.

0

4.8.

_Proposition.

Let G be abelian. Then the maximal

_{C7L-order AL}

_of

AL

is a

Hopf order if

and

_{only if}

_dAL

=

OL.

Proof.

Since the

_antipode

S is an

L-algebra automorphism

of

AL,

the

max-imal

OL-order

of

_AL

is stable under S. As in the

_proof

of

_Proposition

4.6

we _may assume that

_OL

is local. Now let _Xk,

_Lk,

_{7 yk,,, and}

_âk,l

be

_given

as in

Lemma 2.2 such

_that,

for each k E

_{{1, ... ,}

_{the elements t/~,1? -" ~}

form an

OL-basis

of Then the elements

_{1 ~}

_{k fi}

_{s7 Sk7}

form an

OL-basis

of

,AL.

The matrix S in the notation of 4.4 is

_{given by}

We can write S =

Si S2 ,

where

Sl

is the block

diagonal

matrix with blocks

If,

for each k e

_{~1, ... , ~},}

we denote

_{by yk~l, ... ,}

E

Lk

the dual basis of yk,l, ... , yk,sk with

respect

to then

is the inverse of

_Moreover,

_by

the

_{orthogonality}

relations for

irre-ducible

_characters,

the inverse of

_S’2

is

_{given by}

Thus,

the inverse of S is

_{given by}

Now,

Lemma 4.5 states that

_{A(AL) 9}

_AL

AL if

and

_only

if

for all

_{( 1 ~, t ) ,}

_{( J~’ , l’ ) , (k" , I") ,}

where the

_triple

sum runs

independently

over

all w E and w" E

By

the

_{orthogonality}

relations of irreducible characters the last sum over g E G reduces to

IGI

if = 1 and vanishes otherwise.

If

_OL

then for all

_pairs

_{(k, l ),}

and the above condi-tion is

_certainly

satisfied.

_Conversely,

if the above condition is

satisfied,

then we choose k’ E

_{(1, ... , s~}

_arbitrarily

and remark that =

Let k"

_{E ~1, ... , sl}

and K E

_OL

be such that

_XWl

=

’~~~~ .

Moreover let k E

_{~1, ... , s)}

be such that xk = 1. Then

Lk -

L and the

above sum reduces further to

Since

_{cu’ Xk’ cu" Xk"}

= 1 if and

only

if = The elements

(t/ji),

an

OL-basis

of =

D. .

Since the last term in

form an

°L-basis

of

Lk" IL)

=

I)Lk,IL’

Since the last term in the above

_equation

lies in

_OL

for all I’ and

_,

we have

eh.

But this

_implies

_{d -Lk’IL}

=

elL.

This holds for all ~ _E

{1,... sl.

Thus,

~

=

OL-

~

5. SOME A-MODULES AND RESOLVENTS

We still assume the situation described in 1.1 and the notation from

Section 1.

Let C := be the

K-algebra

with

pointwise multiplication.

Then,

SZ acts on C via

_{K-algebra automorphisms by}

for úJ E 0,

For an intermediate field K C L C

.K

let

be th

_L-algebra

of

_OL-fixed

_points

of C.

_Moreover,

let

_CL

be the maximal

O L-order

of

_{CL .}

_Thus,

Note that C is a

right

A-module via the G-action on r:

for

_Ag

E

K,

_f

E

_C,

cv E O. This action of A on C satisfies

for all a E

A, f

E

_C,

w ~ fl.

_Thus,

the A-module structure on C restricts to an

AL-module

structure on

CL

for _any intermediate field K C L C

jK~.

Moreover,

as

_apparent

from

(10),

CL

is an

AL-module

by

restriction.

Similar to Lemma 3.1 for the

_algebra

B we have the

_following

lemma for

C.

5.1. Lemma. Let K C L

ç K

be an intermediate

field,

and E

r be a set

_of

_{representatives}

_for

the

_o/r.

For each m E

_{(1, ... , 1 tj}

let

_im

denote the

_{fixed field}

OL,

and let ~m,l)" ’ ?

zm,tn be

an L-basis

of

im.

For

_{1 ~ ?7t ~}

t and

_tm,

let E C be

_defined

by

for

y E r. Assume that L C M C N C

K

are

_further

intermediate

_fields.

Then the

_following

assertions hold:

is an

L-algebra isomorphism.

In

particular,

T restrict to an

_isomorphism

o f OL -algebras.

(ii)

The _{elements cm,n,}

_{1 m}

_{t, 1 ~ n}

_tm,

_form

an L-basis

_of

CL.

If, for

each m

E {I, ... , tl,

the elements z.,,,, 1, ... , Zm,tm

form

an

OL - basis

of

_{ClLm ,}

then the el ements

₁

_{t, 1}

n

t,.,.t, ,

form

an

OL-basis

o f

CL

7

(iii)

One has

(iv)

The elements Cm,n,

1

m ,

t, 1 n tm,

form

an M-basis

_of

CM.

(v)

The _map

is an

_{isomorphism of M-algebras}

such that the

_diagrams

commute

_for

all w E fl.

Proof.

All assertions are

proved

in a similar _way as the

analogous

assertions

of Lemma 3.1. 0

Next we show that the

_L-algebra

_CL

is an

_AL-Galois

extension in the

sense of

[CS].

Let us

_shortly

recall the relevant notions in a

_{general setting.}

Let R be a commutative

ring,

let H be an

R-Hopf algebra

which is

finitely

generated

and

_projective

as R-module.

_Furthermore,

let S be a

commu-tative

_R-algebra,

_{finitely generated}

and

_projective

as

R-module,

which is

also a

_right

H-module. Then S is called an H-Galois extension of R if and

only

if the

_following

conditions are satisfied:

(G1) (i) (S) ° l

_{_ ~~h~(S ~}

_{~~1~)(t ~ h(2»)’}

ii

for all h E

H,

s, t

E

_S,

where

_{A(h) =}

_r,(h)

_{h(l) (9 It~2~}

is the Sweedler

notation and the module structure of S over H is denoted

by

a dot.

(G2)

The _map

In

_fact,

it is well-known that

_(Gl)

is

_equivalent

to S

_being

an

H*-object,

and that

_(G2)

is

_equivalent

to the condition that the

_H*-object

S is a

Galois

_H*-object

in the

_terminology

of

_[CS,

_§7].

5.2.

_Proposition.

Let K C L C

K

be an intermediate

field.

Then the

L-algebra

CL

is an

AL-Galois

extension

of

L.

Moreover, if

G is

abelian,

then

_{CL is}

a

_free

AL -module

of

rank 1.

Proof.

It suffices to

_verify

_(G1)

in the case L =

K.

So

let 9

_E

G, f,

f’

_E

C,

and 7

E r. Then

thus,

( f f’) ~

g =

( f -

g) ( f’ ~

g)

which is the statement in

(G1)

(i).

Moreover,

g)(,)

= =1=

1 c (-y),

for all _{’Y E} r.

Hence,

also

(G 1)

(ii)

holds.

In order to _prove

_(G2),

we use Lemma 2.1

(vi)

and Lemma 5.1

_(v)

to

reduce the assertion to the case L =

K. Moreover,

by

Lemma 2.1

(ii)

_and

Lemma 5.1

_(iii),

it suffices to show that the _{map in}

_(G2)

is

_injective.

So let

_{~9EG 9 ~}

_fs

E A C with

_arbitrarily

chosen

_fg

E C

_{for g}

E G such that it vanishes under the map in

(G2).

Then

Let _{go E} G

_{and 7}

E r.

_Then,

_{choosing f}

E C in such a _way that

f( 9°y)

= 1

and

_{f (y’)}

= 0 for

V

# 9°y,

the

equation

in

(11)

implies

= 0. Thus

fgo

= 0 for all

go E G. This shows that

_CL

is an

AL-Galois

extension of L.

If G is

_abelian,

_AL

is commutative and

_{B L}

is cocommutative.

_By

_[CH,

Prop.

2.3,

Thm.

_3.1],

_CL

and

_BL

are

_isomorphic

as

_AL-modules,

where the

AL-module

structure

_{of BL}

is

_{given by}

_(b~

_LgEG

:=

¿9EG

Agb(gg’)

for all b E

_BL,

_{g’ E}

G.

_{Moreover, AL}

and

_BL

are

_isomorphic

as

_AL-modules

by

sending

to the element

₁₁

E

_BL

with =

8s,1 ~

0

Let G be abelian. Since

CL

is free over

AL

for _any intermediate field

K C L c

K,

the

_{following question}

arises

_naturally:

Is

_CL

free over some

OL-order

in

_AL.

It is well-known

_that,

if this is the _case, it is

_{only possible}

for the associated order

of

_CL

in

_AL.

_Obviously,

_Al

C C

_{.A.L. We}

will _prove

_that,

in the situation we consider in Section

_{6, CL}

is free over

_ALs.

In the

_proof

we

attached to

_{f E C}

_{and X}

E

â

_{for fixed qo e} r.

For an intermediate field K C L C

K,

let L’

C k

now denote a finite

extension of L such that

_{Q L,}

acts

_trivially

on

â

and r. We call such a field a

_{spli t ting}

field for

AL

and

CL .

Note

_that,

since

for all cv E

_{0, f}

E

_{C, X}

E

å,

and _1’0 E

_r,

we then have

(f,x),,/o

E L’

for all ,

_f

E

_CL,

_X E

å,

and _yo E r. We denote

_by

_dcL

the discriminant ideal of

_CL

over

OL. Thus,

in the notation of

Proposition

5.1 we have

dCL =

111ri=1

5.3.

_Proposition.

Assume that G is abelian and

_fix

_~o E r. Let K

_C

L

_C

K

be an intermediate

_field,

let L C L’ C

K

be a

_{splitting field for AL,}

_BL,

and

_CL,

and let

_f

E

_CL.

Then

_{f -}

_AL

=

C~

if

and

only if

=1=

0

for

all _X E

6.

_Moreover,

_AL

=

CL

then

Proof.

(a)

Let _Xk,

Lk,

_{yk,,, and the L-basis}

of

_AL

be

_given

as in Lemma 2.2. Then the elements

generate

(rL,

o

7rf’)(L’

_OL

f -

AL)

₉

TI7Er

L’ over L’. We _express these

generators

by

the

_{primitive idempotents}

E

_r,

of

_{ll7Er L’}

and obtain

a transition matrix

so that

f -

AL

=

CL

if and

only

if

det(M) #

0. We determine

det(M).

For

We can write M as M =

MlM2,

where

M¡

is a block

_diagonal

matrix,

the

blocks

_M¡,1c

indexed

_by

k =

1, ... ,

_s, with

and where

by

(7).

Thus,

This shows that

_{f -}

_AL

=

CL

if and

only

if

(, f ,

7~

0 for

all X E

G.

(b)

Next we show the assertion about

[CL f -

Since both sides

of the

_equation

behave well under

_{localization,}

we _may assume that

C~~

is

local. We

_transport

the two

_OLI-lattices

_{®oL CL}

and

_OL’

_AI,

by

the

_{isomorphism TL’}

o

7rt’

into

II7Er

L’.

Retracing

the calculations in

_(a)

one observes

that,

is an

(7L-baSlS

of

₀₁

_, for each k =

_{1,... ,}

_s, then the elements

(12)

form

an

C7L-basis

of and the elements in

(13)

form an

OLI-basis

of

(TL’

o

Thus,

the calculation in

_(a)

shows that

Moreover,

we

_already

know from

_Proposition

4.3 that

Now,

it suff ces to show that

Let _{1m, zm,n, and Cm,n be}

_given

as in Lemma 5.1 such

_that,

for each m

E ~ 1, ... , t~,

the elements _{zm,i , ... , Zm,tm form} an

OL-basis

of

0~ .

Then,

the elements form an

OL-basis of

_{CL. Thus,}

the elements o form an

OL,-basis

of

o

_®nL

_{CL ) .}

_{We express} this

_by

_{the canonical}

basis of

_{primitive idempotents}

E

_r,

of = and

obtain a transition matrix M. This is a block

_diagonal

matrix with blocks

M1,... , Mt,

where

for m E

_{{1,... t~.}

Since we obtain

(17),

and the

proof

is

_complete.

~

D

As an immediate _consequence of

_Proposition

5.3 we obtain:

5.4.

_Corollary.

In the situation

_{of Proposition}

5.3 with

_/

E

_CL

such that

(I,

X)70

~

0

_{for all x}

E

G

one has

for

each

_{OL -order}

_of

_{AL .}

contained in In

_{particular, if}

and

_{i, f . f}

E

_CL

is such that

then

_CL

=

f -

AL

is a

_free

-AL -module

of

rank _one with basis

{ f ~,

and

consequently,

6. RELATIVE LUBIN-TATE FORMAL GROUPS

Our main reference for the

_general

_theory

of relative Lubin-Tate formal

groups is

[dS].

Throughout

this section _p denotes a rational

_prime

number. For _any

extension field

_Q

C L

_{9 %}

we write

OL

for the

integral

closure of

Zp

in

L and pL for its maximal ideal. We fix a finite extension field F of

_~

and we _{set q} =

Let d &#x3E; 0 be a fixed

_integer.

We denote

by

F’/F

the unramified extension

of

_{degree d}

&#x3E; 0 and

_{write 0}

for the Frobenius

_automorphism

of

_F’/F.

We fix an

_{element ~}

E F such that

_{v p (g)}

=

d,

where

vF denotes the normalized

As in

_[dS,

Ch.

_I]

we set

~={/CC~M !

I

mod X2,

NF’/F(7r’) = ~

and

_{f -}

X~ mod

_{p p’ [[X]]).}

For _{any power}

_{series g}

in one or more indeterminates over

C?~~ [[~]]

let

g~

arise from

_{applying §}

to the coefficients of g. From

[dS,

Ch.I,

Th.1.3]

we

know that for _every

_f

there exists a

unique

one-dimensional

commu-tative _{formal group}

_7j

defined over

Op,

such that

f

E Note that

_{f ~}

E

_F~

and

~- =

_.~’fm.

Of course, classical Lubin-Tate formal groups

correspond

to the case d = 1.

In order to introduce all the _necessary notation we recall

6.1.

_Proposition.

_([dS,

_Ch.I,

_(1.5)])

Let

_{f(X) =}

7rlX

+ ... ,

g(X) =

+... be in Let a E

OF’

satisfy

= Then there exists _a

unique

power series E

dF~ (~X~~

such that

Moreover,

E

_Hom(F/,Fg).

The _map

is a _group

_isomorphism

and in the ccrse

_f

_{= g} a

_ring

_isomorphism

Furtherntore, if

h(X) _

ir3X

+ ... and

b46-1

=

~3/~2,

then

ab

In the

_sequel

we shall

always

write

[all

for

[a]/,/.

For

f (X ) _

7r’ X +...

E

F~

and

_{i &#x3E; 0,}

we

_put

I(i)

= o o

_{f .}

Then

I(i)

E

and if

_~(~)

_{= d,}

then

_{f~d~ _}

_(~~f

E

_Moreover,

for a E we

write

a~i~

= _a. It follows that

f ~z~(X) _

(X).

As usual we endow the set

_pF

with the structure of an

_OF-module,

denoted

_by

_{by setting}

for

_{x, y E pF}

and a E

_OF.

We now

fix ~

E

_C~F

with

_{vF (~)}

= d and a _{power series}

f (X ) _

~r’X +...

E

Let 7r E

_C~F

be _any

_prime

element. For n E

_No

we define the _group of

pF-torsion

points

of

_:1=1

_by