Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZUrich
74 A. FrShlich King's College, London
1968
Formal Groups
Springer-Verlag Berlin. Heidelberg. New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. 9 by Springer-Verlag Berlin 9 Heidelberg 1968. Library of Congress Catalog Card Number 68-57940 Printed in Germany. Title No. 3680.
These notes cover the major part of an introductory course on form~1 groups which I gave ~uring the session 1966-67 at King's College London.
They are based on a rough draft by A.S.T. Lue.
I have not included here the last part of the course, on formal complex multiplication end class field theory, as this subject is now accessible in the literature not only in the original paper but also in the Brighton Proceedings.
The literature list on the other
hand includes some papers published since I gave ~
A.F.
course.
CONTENTS CHAPTER I
CHAPTER II
CHAPTER III
PRELIMINARIES 51.
Power series rings ...........................
I
52.
Homomorphisms ................................
16
53.
Formal groups ................................
22
LIE THEORY 51.
The bialgebra of a formal group ..............
29
52.
The Lie algebra of a formal group ............
43
COMMUTATIVE FORMAL GROUPS OF DIMENSION ONE 51.
Generalities .................................
52.
Classification
of formal groups over a
seperably closed field of characteristic 53. CHAPTER IV
p...
Galois cohomology ............................
COMMUTATIVE
51
69 86
FORMAL GROUPS OF DIMENSION ONE
OVER A DISCRETE VALUATION RING 51.
The homomorphisms..., .......................
52.
The group of points of a formal group ........ 1 0 4
53.
Division and rational points .................
119
5~.
The Tare module ..............................
121
Lz~rRE
........................................
96
139
-I-
CHAPTER I. PRELD~NARIES
w
Power Series Rings Let R be a commutative ring.
R[[~,...,X~]
The power series ring
in n indeterminates Xl,...,X n over R is a ring
whose elements are formal power series
iI
in
~(xl,...,x) - ~ ~h"'"in xl """xn with component-wise s~dition and Cauchy multiplication as its operations. Denote by N the set of non-negative integers and let M n
be the set of n-tuples i = (il, o..,in) , with components is 6 N. In other words M n is the set of maps of {l,...,n} into N.
We
define addition and partial order o h M n component-wise, i.e.
i + k = (
+
,,.., i n
i -->k < >
i A _> k A
and
The zero element 0 on M n
for
A = l,,.,,n.
is the n-tuple (0,...,0).
Now we can write
f(7~,...,x)
iI (interpret ~
as X 1
- ~(x) = ~ =
i ...Xn~ ! )" and define
Z is
f -xi n
-2-
(g + f)i = gi + fi'
(g'f)i -- Z
k+j=i
~-
contains R as a subring : identify a ~ R with the power series f, for which fo = a and fi = 0 (the zero of R) when i > 0 (the zero
of Mn). We shall write 9
~-
for the inclusion map.
R[[Xl,...,Xn]] The augmentation
is the ring hcmomorphism with z(f) = fo" R
Note that the dia6ram
R
~[[&,...,xj:; co~tes
9
Note::
we can view t h e f o r m a l power s e r i e s r i n g as t h e s e t
o f maps Mn § R. are not e x p l i c i t l y
If
the particular neede~ we s h a l l
symbols f o r t h e n i n d e t e ~ a t e s s ~
~[[%,...,x]] -- ~.
write
-S-
It is cleat of course that the map
%?_, ;. %~i ieM
i&M
n
n
sets u~ an isomorphism
,[[%,...,x]] ~ R[[Y~,...,~]] com~tible ~ith both z a n d I~-
~ e diagram
Rn_ I ~
-
R
1
=
Rn_I
R
D,
, R
~R,n-i co~mntes, Denote by U(S ) the group of ~nits (invertible elements)of a ring S.
-4-
,PROOF
As U is a functor from rimgs to groups, f 9 U(R n) will imply
~(f) ~ oCt). Let n = I.
If ~(f) = f0 9 U(R) then one cam solve
successively the equations
f0g0 = i,
f0g I + flg 0 = 0,
... ,
f0g r + flgr_ I § ,.. + frg 0 = 0
for the coefficients of the power series gCx) = (fCx)) -I. settles the case n = i.
This
N~r proceed by induction, using Lemma I.
Filtrations of, Abelian ,Groups A filtration v of A is a
Let A denote am abeliam group. map v:A§ which satisfies
(1)
v(o)
(2)
vCx-y) > i n f
I t follows t h a t
=-
, zmvW
,
{vCx), vCy)}
vC-x) = v(x).
(Note : suppose that vCx) = -
Hausdorff f i l t r a t i o n
(-}
(see below)
only if x = O, i.e. that v is a 9
Then by t a k i n g I x I = (~)v(x)
e
-5-
we get a metric space since Ix-Yl
~ote ~ o ,
that v ( ~ )
§ -
Given a f i l t r a t i o n
~ sup (1~1, l y l } ~_ I~1 I%1 ~ 0).
i~lies
v, then f o r m e N, define
I v(x) im)
Am={XeA
A m is a subgroup of A ( = AO) , and A m ~
A
|
+ lyl
=
m~NAm
.
Am+ 1.
Defining
,
we have in fact
A
= {x s A
v is in turn determined h y t h e
Iv(x) ---=) ,
groups Am, for m ~ N, via the
equations v(z) =
sup
x~A
m. m
In fact if we are just given a decreasing sequence {Am} (m ~ N) of subgroups of au abelian group A = A0, then this last equation defines a filtration on A.
LEMMA 2
Suppose A is am S-module for some r i ~
e~e S-mgdules i_~f am_.~donl~ ~
S.
v(sx) ~_ vCx) for. ~
Then the A
x e A, s e So
~aau this is the case, we speak of S-filtrations. A filtration is IIausdorff if A @0
-- {0} .
If {an} is a sequence of elements of A, and n l ~ v ( ~ then we ~rrite lirav am = a. n-~eo
a) = |
For v Hausdorff, a sequence can only have one
-6-
limit.
A seq~ence with a limit is a limit sequence.
A sequence
{an} in A is a Cauchy sequence if
(%+I - %)
n-~=V Ever~ ~ m i t
= o.
sequence is a_ Cauchy sequence.
A filtration v is ~
(or, A is complete under the
filtration v) if it is Hausdorff and if eve~j Cauchy sequence in A has a limit in A. Example (i) If there exists k for which A is complete.
= A k = {0}, then A
The Cauchy sequences are the sequences which are
ultimately constant. @0
Examnle (ii)
A = rl A(k), where A(k) are S-modules, and a(k) k=O denotes the k-th component of a & A. Define
I r = {a e A I a(k) = 0 for .11 k < r},
Pr = {a s A I a(k) = 0 for all k ~ r } ,
v(a) =
inf
n =
a(n)#O
L~5.~. 3
sup a 6 I
r. r
With these definition9, (i) v i s an S-filtration o f
A with the I r as associated subgroups: (ii)
lira a(n) =
v i s a complete filtration, and n + V
a
(a (n) is defined to be the element of A with a(n)(k) = a(k) for k < n, and a(n)(k) = 0 for k > n); (iii) for each r,
A -is - the direct sum , I r + Pr 9
-7-
ExamPle (iii) A is an abelian group, v a filtration on A with associated subgroups A . Denote by ~ m
: A/Am+ 1 § A/Am the
m
natural quotient maps. Consider, in the direct product ~m (A/Am)' the submodule A of elements ~ for which ~m(a(m+l)) = a(m).
The filtration of [-Im(A/Am)
(of. 2nd Ex, page 6) defines a filtration ~ of ~, under which is Hausdorff and complete.
Also, Pm : A § A/Am defines a m
homomorphism A + ~qm (A/Am) whose image is contained in A. therefore a hcmomorphism p : A § A.
L ~
~CpCa))
(ii)
p is injective if and only if A is Hausdorff;
(iii)
p is bijective if and only if A is complete.
h
lira a b nn
vCa), for a 6 A;
If v is a filtration of A, and if A is a ring, then
~Cxy) >_ vCx) V
We have
(i,~
=
This gives
+
vCy) i~ ~ a ~
i~ A A
c An+m. In this case, ~ s o
= lirav an.lirav bn, and w_~es a y v i_~s_a ring filtration.
We leave t h e proofs as exercises. n
If i ~ Mn, define lil = of f to be ord(f) =
~
~.
For f ~ Rn, we define the order
inf k=l fi#0 lil 9 By taking f(k) =
~ . T f'X~ li =k ~
(homogeneous polynomial), we see ord(f) = f(k)#O inf k.
Denote by Rn(k)
t h e R-module of homogeneous polynomials of d e g r e e k in t h e
variables ~ , , ...Xno
Then
Rn
k-O
In Lemma 3, by taking A = Rn, S = R, and ACk) = RnCk) , we have
-8-
" r : (f I q
: o for lil
< r} ,
Pr = {polynomials in XI,...,X n of degree _< r-l} .
PROPOSITION 2 The function ord is ~ complete filtration of R n w i t h a
L
~
L
_
Jl
i
m
m
-
IL
i
~
L ii
~, of R-modules I. associated subgroups Ir, and R n = Ir + P r (direct sum ---
Also, ord(f.g) ~ ord(Z) + ord(g). and the I r are ideals --of I 0 = R n ~ Moreover, lqlp = Ip+q, -------PROOF
By Lemmas 3 dud 4. Note that I I = {f ~ R n i fo = 0} .
Therefore I I = Ker r , and
Rn/I 1 =~ R,
NOTATION : f = g rood dog q means f = g (rood lq), i.e., f-g ~ lq. PROPOSITION 3 I f R is ___~integral $o~-~n, then ord(f.g) = ord(f) + ord(g) an~ R n ---is---anintegr~l domain. PROOF
Verify directly for n = i.
Then by induction on n, using
Suppose mow that J is an ideal of R. %
~he power series f, with
6 J for all i, form an ideal J[[X]] = J [ [ ~ , . . . , X j ]
J[[X]] = Ker ( R m § (RIJ) n } .
of ~ ,
and
If K, J are ideals of R, them
K[[x]].j[[x]] c (K.J)[[x]]. PROPOSITION ~ Le._~tv b_~e_a r i ~ Jp.
filtration of R with associated ideals i
|
'|
V'(T) = inf v(f i) is a ring filtration of R with associated i . . . . n .... ideals %[[X]]. If v i_EsHausdorff/ccmplete then v' i_~sHausdorfflcgmple~e.
~OOF
Then
~,(~) ~_ p c--~ v ( q ) ~_ p for ~ u i < ~ q ~ Jp for ~ u i > f ~ J [Ex]]. P
-9-
%[[x]].q[[x]]c 89 q[[x]]c If V is Hausdorff then n
Jp[[X]] = O, and therefore v" is
Hausdorff. To show that v complete => v" complete, let {f(n)} Cauchy sequence,
Then v'(f(n+l)-f(n)) + -.
v(fi(n+l)-fi(n)) § -.
be
a
V ~"
Therefore, for all i,
Since R is ccmplete under v, then for
each i there exists lira fi(n) = f.. n_~ov
Let f = ~ f.X x.
1
9 1
Given say
1
positive number K, there exists n o such that v'(f(n+l)-f(n)) > K,
f o r a l l n ~ no. and hence
v(fi(n+l)-fi(n)) > K,
f o r a l l n >_no, and f o r a l l i .
v(fi(n) - fi) > K,
f o r a l l n ~ no, and f o r .11 i , and
v'(f(n) - f) > K,
f o r a l l n ~ n O.
So
Therefore
This shc~rs v" i s c o m p l e t e .
f = lim.f(n). n-~. v
THEORy4 1 Jq.
Suppose v i_~s~ring filtration o f R with associated ideg~
Define,
Jq = Jq[~X]] + Jq_l[[X]]Ii § ,. + Jq_r[[X]]I r + . , Iq ; ~Cf) = inf
{n + v'CfCn))} ,
n
(v" is the induced filtration of Prop 4, f(n) denotes the homogeneous component of f of d e g r e e n ) . ~en
(i) v~ i s a_ ring filtration _of _ R n with associated ideals Jq; ,uq .
(iii) if v i s Hausdorff/complete, then v lS Hausdorff/complete, S is a local ring if it has one and only one maxi~-~ ideal w%.
10-
(A ring is local if and only if the non-units form an ideal, which will then be the maximal ideal ~ by the powers ~ COROLLARY 1.
).
We obtain a filtration on S
of ~ .
If R i_~sa locs~l ring then so is R n.
If in addition
R is Hausdorff, so is Rn, and if further R is complete, so i s Rn. The corollary follows from the theorem and the observation that if ~t is the ~ I i~al~[[X]]
§ k
ideal of R then by Prop. i the complement of the of ~
consists of units.
For the proof of the theorem
L~@L% 5
~m
first need a number of lemmas.
If the Ji are ideals Of R, with J q C
Jq-i C
.-- c Jl'
and if K = Jq[[X]] + Jq_l[[X]]l I + ... + lq, then f ~ K if and onl~ if f(A) has coefficients i_~nJq_s , A = O,1,...,q-1. PROOF
The sufficiency is strs/ghtfor~ard.
Then f =
r= 0
gr' where g r 6 J
q-r
[[X]]I . r
For necessity, take f ~ K. For ~ < q-l, f(A) = --
and for r <_ ~, gr(~) ~ : _r[[X]] ~ J _~[[X]].
~~. grC~) r-O
Therefore f(A) h~s
coefficients in Jq_s
m~a
6
I_f J is an idea of ~, an~ K -- j[[x]] § II, then
K~ = J~[[x]] § J~-l[[x]]k , ... § J~-r[[X]]I r § o.o + I q 9 PROOF
By induction on q, and using Lemma 5 with J
r
= jr. ~hat f(~)
has coefficients in Jq-A i~plies that f(A) ~ K q, and hence f ~ K q. Therefore L.H.S. D R.H.S. PROOF --of Theorem 1
That R.H.S. ~ L.H.S. is clear.
(i) v~ is clearly a filtration of R n.
- 11-
<=>
i~f {n + v'(f(n))} >_ q
<---->
+ v" (f(z)) >_ a for 0 <_ ~ ! ~-l
has coefficients in Jq_A for 0 ~ ~ l q - i
<=9
f(r
< 9
f ~ J
q
(by Lamm~ 5).
Since also Jp.JqC Jp+q, then this proves (i). (ii)
This foll~rs from Lemma 6.
(iii)
Suppose v is Haus~orff.
Take f E ~
coefficients in Jq-A for a11 q ~ A . coefficients in f~Jq,
Then f(s
has
This implies that f(Z) has
which by our hypothesis is O.
for a11 s , and therefore f = O. v is complete.
Jq.
Hence f(~) = 0
Therefore v is Hausdorff.
Suppose
Demote by f(q) the polymomialpart of f of degree <
Then ~ ( f ( q ) ) A ~ ( f ) .
q.
Also,
~-1 + v'Ct Cq)) L~Cf (~3) = ~
Cn + v'CfCn))} ~v'CfC~)).
Let { f r } be a Cauchy sequence under ~ (throughout the rest of t h i s proof, suffixes refer to the numbering o f th e
sequences).
Given k 9 O,
there exists n o = no(k) , such that ~ ( f r - fs ) 9 k,
for =11 r,s ~ n O . Therefore
~(fr (q) - fs(q)) > k, for all r,s _9 n O . Hence v'Cfr (q) - fsCq)) > k - q + 1 for all r,s _9 n O 9 For a fixed q,
{fr (~) }
is a Cauchy sequence with respect to v %
Therefore there exists limv. fr (q) = f(q) Prop ~) and
(since v" is complete,
- 12-
v'(fr(q)-
f(q)) > k -
q+Z,
for
rLn o 9
Now, (fr(q+l)) (q) = fr (q) ' and taking limits we obtain (f(q+l))(q) __ f(q).
Therefore there exists a unique power series
f such that f(q) are the terms of f of degree < q. that lim~ r.~v
f
= f.
r
We shoe now
Now
~(f(A) - f) ~ A, and ~(fr CA) - f r ) ~ A, for all r.
Also v'Cf Cz)
-
f
r
CA))
A, for r L nl(A)
(v" Cauchy sequence).
Therefore ~(f C~) - fCA)) ~ A , for r ~ n l ( ~ ) . Hence ~(f - fr ) >_ A, for r >_.nl(~). TH~EO~I 2 PROOF
This completes our proof.
If R is n oetheriau, then so is R n.
It ~ill suffice to establish the theorem for n = l, for then
the general case follows by a trivial induction argument, using
Let J be an ideal of R I.
For any q >.0, J (~ lq = {f E J I ord(f) >.q}
is an ideal of R1, and its image in R
under the map f ~ * f
q
(we here
revert to the notation where f q denotes the coefficient of X q in f) is an ideal Aq of R. Aq+ 1 D Aq.
As f s J • l q implies Xf s
1 we have
The ring R being noetheriau, it follows that we can find
a k >.0, so that A k - Ak+ A for all A >.0. It ~rill suffice to prove that J ~ l k is finitely generated over R1, for J/J/%~. = J + ~ / ~ ,
as an R-submodule of E1/~, is finitely
generated over R, hence over ~ .
Therefore J ~rill then also be
finitely generated over R I. As Ak is finitely generated there is a finite set f(i) (i = l , , . . , s )
- 13
of power series in J /~ ~
(the (i) serving as enumerating index here),
so that the fk (i) generate A k over R. generate J (~ ~
~e contend that the f(i)
over R1.
Let g E J ~ ~ .
We shall construct inductively sequences
(g(i'm)} m (i = 1,...,s) of power series, so that firstly g(i,m) = g(i,m+l)
(mod degree m)
and secondly
s f(i)g(i,m)
g
z ~ i=l
(rood degree re+k).
By the first relation we obtain pc~rer series g(i) = limord g(i,m)
s~ f(i)g(i).
sad by the second one g =
Thus we see that in fact
i=l f(i) generate J ~ IkThe step from m to m+l goes as follows (put g(i,O) = 0 to apply this to the first step') : h = g - ~f(i) g(i,m) lies in
J n ~+=.
~i f(i)
~enoe h ~ = ~ A k ~ = ~ , i,e., h~+= = i=l
ii e R.
Put g(i,m+l) = I.Xm + g(i,m)
Then
1
g " .X f(i) g(i,=+z) = h !
1
: h+= x~+~- X f~(i) ~i ~ + = =- o (=d degree =§ 1
For the rest of this section we suppose R is a complete local ring, with maximal i d e a l s ,
and k = R / ~ .
its image in kn under the epimorphism Rn § ~ The Weierstrasse-order of f,
For f E Rn,
denotes
induced by R § k.
17-ord(f), is defined by
- 14-
w-oral(f)
Then W-oral(f) # ~
<=>
=
~ # 0
<=>
f has some unit coefficient.
Note t h a t as Ordk(~.~) = Ordk(~ ) + Ordk(E) , also W - o r d ( f . g )
=
~-ord(f) + ~7-ord(g). A di'stimguishe~ ~ fo + flx + f2~
f of ~
is a polynomial of the form
* ''" + fq-iXq-I + Xq' ~here ~11 the fi are in~t~
Note then that N-oral(f) = deg(f).
THEOREM3
(Weierstrasse preparation theorem)
i_~ff ~ R I a~d
~-ord(f) = p < ~, t h e n there exists a unique u ~ U(R I) and unique distinguished polynomial g such that f : u.g.
PROOF
Then of course
~e shall prove by induction on m that (Am) : There exists a
v (m) E U ( ~ ) and a distinguished polynomial g(m), so that
f.
v
This congruence determines v (m) (and hence g(m)) uniquely rood
Ass!~ng (Am) for all m, it follows from the uniqueness part that :
As R I is complete with respect to the filtration ( ~ m [ [ X ] ] }, we obtain in the limit a unit v of ~ , polynomial.
so that f.v = g is a distinguished
Moreover, v is determined uniquely rood ~m~[X]] for all
m, i~e., is unique.
Now multiply through by u = v "I to get the theorem.
To establish (Am) we may work over the residue class ring R / ~TM, i.e., we may suppose t h a t ~
- O.
First for m = i the hypothesis
-
15
-
states that f(X) = X p. u(X), where u(X) is a unit of R I.
But this is
in effect also the assertion. For the induction step write m = r + i.
By the induction
hypothesis there exists a power series @0
v(r)(x) = v(x) =
v.X x i=0
so that, writing f(X) =
i--0
f.~,
we have
:L
v o ~ u(R),
(1)
r
Vofp + ... + v p f 0 = i + up,
Vofs § "'" + V s f o = ~s'
~s s
Up~ ~ ,
(2)
(all s > p).
(3)
This is just the congruence for CAr ) expressed coefficient-wise. By the uniqueness part of (At) the coefficients v~ of v(r+l)(x) must be of the form
v [ = v i + xi'
xi 6 ~ r
~e have to show that the ~. can be chosen so that 1
(2') v&%
+ ... § ~ o
- o, f o r ,~1t ~ ,
p.
(B')
CRe~mber that ~r+l _ ~m _ Ol ). Note t h a t v~ w i l l c e r t ~ n ~ in U(~).
From ( 2 ) , ( 3 ) , ( 2 ' ) and ( 3 ' ) we get the equations
lie
-16-
Xof s + klfs_ I -~ ... + lsf 0 = The Xi are to be chosen in ~ r ,
IJs,
(s >_.p).
and we know that fk ~ ~
for k < p.
Hence we must have
As %
XO% =
- Pp,
Xo~+1
+ ~l~
XoS+
+
15+
= - ~p+1' "'" ' i
"'" +
§
- -
>- o).
* U(R) these equations have unique solutions for Xi in R, and
by induction on k one also sees that Xp+k must lie in r . can solve for the v:. 1
Then we
The uniqueness of the v i nod ~c r and of the X. 1
implies the uniqueness of the vf. 1
,w
Homomorphisms A and B are abelian groups with filtrations v, w respectively.
A continuous homomorphism
e : A § B is a homomorphism of groups
such that, given m e N, there exists A ~ I~ for which (A~)0 c B mHence, if v(a n) § ~, then W(an%) § =. bic0ntin.uous means that
To say that e is
e : A § B is an isomorphism of abelian groups,
mad both e and e-I are continuous. THEOP~4 1 (i) Suppose S i s a_ commutative ring, complete ring filtration v, and R is _a subring o_~fS. S with values vCa i) > i, homomorllhism
there
under...
Given al,,..,a n in
exists a_ uniaue
continuous ring
8 : R n § S (with respect to the order filtration on E n)
which leaves R elemen~rise fixed, and such that X. e 1
= a i.
- 17-
(ii) Explicitly, i_~ff(Xl,,..,~) s Rn, then
li~ fC~)c%,,..,%) - fCxl,,..,x) e . (Here fCq)(Xl,,..,X n) is again the polynomial of degree < q - i t~ich coincides with f rood degree q). (iii) Let T be a c ommutativ9 ring containing R, complete under a ring filtratio_...___.__nnn w, aud with elemauts ~l,...,~n for which w(~ i) L l, s_~othat : Given S and al,...,an as in (i), there exists a_ unique continuous ring homomorphism ~ : T § S with ~i r = ai and leaving R elementwise
.~ze~. Then the continuous hom~mor~hism Rn § T which ~
R elementwise
fixe___~dand maps X i into ~i is a bicontinuous isomorphism.
~oo~
if fCx) ~ ~n' then
fC~+1) Cx) - fC~) Cx) Therefore fCq+l) (al,o..,%) . fCq)
~
lil-q
biX~ , b i E R.
iI im (~U...,%) - [biu...,in e! " " %
and its value under v is at least iI + ,.. + in
vCfC~+!)C%,...,~), {fCq)Cal,,..,~)}
is therefore
fCa)c~,...,%) a
=
q.
Hence
) + | as ~+-.
Cauchy sequence under v.
We put
f C x u . . . , x n) e - !q..~o i = v f(~) C ~ 1 , . . . , % ) . It follows quite easily now that 8 is a continuous ring homomorphism, and the uniqueness of 6 then follows fram continuity.
-
18
-
The proof of (iii) is standard (uniqueness of universal objects). If there is no ambiguity involved, we shall write f(al,...,an)
for liz q_~v
f(~)(~l,...,~).
For a ring S with ring filtration v, define l(S,v) = {s 6 S I v(s) > 0}. Comsider the category YR' whose objects are the pairs S,v as in Theorem i, and whose morphisms are the continuous ring homomorphisms S,v § T,w which maps i(S,v) into !(T,w) " d R D
R' where~R is the
full subcategory with objects Rn,ord (order filtration).
Theorem i
then ss~vs HOm fR(Rn,S ) ~ I(S,v) n, by associating with each e the
el~t
(X1 e ,...,Xe). Co
ider
the
case S -
9 whe
.
the indeterminates of Rm as Y's, to distinguish them from those of Rm, which are still denoted by X's.
Let (r = l,,..,n).
x r e = ~r(Y1,...Jm),
Them
fCx1,,..,x) e - ~ ~
f(~) (glCY),...,~CY)).
shall derive another expression for this elemaut of Rm.
for k 6 M
Write
n kI
~(~)
k
= gl (y)...~n(~)
~ X
g~k
yA
9 k
Since orgy arCY) i 1 we have ordy gk(y) ~ Ik I = d therefore gA = 0 for
I~i
< I~l~
~
it makes sense to define
- 19 -
z(gz,...,@) = z(%,...,%) (Yz,...j=)
=
PROPOSITION I PROOF
~
(
X
k~
yZ 9
f(~)(gl(Y),...,~(Y))= f(gz,...,~) (Yz,...,z=).
Verify for polynomials f.
Then extend to power series f by
continuity. Let f = (fl,,..,fr) be a "vector" of r p~rer series in n indeterminates, and let g = (gl,...,gn) be a "vector" of n power series in m indeterminates.
C%Cgu...,gn),
TJe denote by f o g the vector
... , ~Cgl,---,~))
of r power series in m indeterminates.
With this multiplication the
vectors f = (fl,...,fr) with varying r and n form a category, whose objects are the positive integers, f being viewed as a "map" r ~ > n. In view of the preceding theorem and proposition, a homomorphism % 6 HO~R(Rn,Rm)
determines a vector g8
ge o~ = ge o g~ . categories.
: n + m.
In fact this map 8 ~-~ge
Moreover
is an isomorphism of
In other words we can either use the language of
homomorphisms e or that of vectors of power series. In Rn, consider the ideal I = Ker e, and denote by ~ the image of f under the natural epimorphism I * 1/12 = D(R n) . n
If
n
f =
Y C l ~ + terms of degree 9 2, then f = ~ ciX i. D(R n) is a i=l -i=l free R-module on ~l,,~ ~.~en e : Rn -* Rm is in ~ R ' then I(Rn)e c I(Rm) , and 12(Rn)e c 12(Rm), and so e induces a
homcmorphism D(e) : D(R n) + D(Rm) , of R-modules. Denote by @ R the category of finitely generated free R-modules, or "vector spaces over R".
- 20-
PROPOSITION 2 D is a fu~ct9r: ~ R §162 COROLLARY
!_~fR n is bi.coDtinuously isomorohic to Era, theD n = m.
For, a finitely generated free module over a co~utative ring R has a unique rank. If e is a homomorphism in ~ R
then x.e = ~ CikYk + terms l k=l
m
of degree _> 2, and ~.D(e), =
[ Cik~k . D(8) can be represented in k=l
the matrix form o(e) = (Cik) , and Cik = (~xie/~Yk)y= 0. By Prop. 2 , O defines a map : HO~R(Rn,R m) § HOm~RCD(En),DCR m)). We define a map E in the opposite direction as follc~s, X. onto ~Cigk, then take E(~) : R I
which maps X i onto ~CikY k.
T~O~
2
-
Let
n
§ R
m
to be the homomorphism
We have
e be a c o n t i n u o u s h o m o m o ~ m
'
.
.
.
if $ maps
.
.
.
R n
§ R .
~en
e
n
i_2.Sa_ bicontinuous isomorphism i_~_fs~d only i.f D(8) i~ an ~
of
R-modules. COROLL~
!/_fe is. sur.iectiw then i___ti_~san
PROOF OF COROLLARY isomorphism =>
e surjectlve =>
D(e) surjective =>
D(8)
e isomorphism.
The theorem can be rephrased to read : given fi(Y1,...,Yn), with f.(0,...,O)l = 0, i = l,...,n, then det (~fi/~Yk)Y=0 is a unit if and only if there exist gj(X1,...,Xn) such that fi(gl,...,gn) = X.l (IITVERSE FUNCTION THEOR~). PROOF OF THEOF~ Assume that
~0~eneed only prove the sufficiency of the condition.
~ = D(e) is an isc~orphlsm.
~Trite ~ = E(~).
- 21-
-1 Then D(e @ ~ ) = i. As ~ is an isomorphism, it will suffice to -1 show that 8 ~ ~ is an isomorphism. Without loss of generality we can therefore suppose that D(e) = i.
With this assumption,
X i ~ X.l e mod i2, where XI,... X '
(s gi
are the indetermlnates of R . We construct polynomials n
n
of degree ~ - i so that
x.~ - gi(~)(xe) moa x~, g~ ~+l)(x) ~ gl(~)(x)
rood I~.
By induction on ~ , suppose that
Then X i = gi(~)(xe) Take gi(s
+
X~ ikI=
ck(xe)~ rood I~+ I.
g~)(X) + Ikl- ~
sequence, ~ith limit gi(X), say.
ckXk.
Then {g~)(X)}
is a Cauchy
Also,
x~ = gi(Xe) = gi(X) e . Define ~ by the equations X.T~ = gi(X).
Then x(T~ o e) = gi(x)e = xi,
and so by the uniqueness part of Theorem l, W ~ e = i. 1 = D(~) ~ D(e) = D(W).
As before, there exists X so that X ~ W = I.
Therefore X = X " (~ " e) = e isomorphisms.
Therefore
, and hence ~ and e are inverse
- 22-
Altho!~sh H o ~
(Rn,Rm) is not a group, we can define some
/ %
sort of "filtration" on it by taking
ord(e) = inf (or~(fe) f#O
- or~(f))
inf (oraz(xie) - l). i=l,o..,n ~th
this definition,
ord (e o ~) _> ord(e) + ord(~). ~3.
Form~! Groups In this section ~re take R to be a fixed ring, and all p ~ e r
series are over R. A fo~~F(X,Y)
of dimension n is a system F.(X,Y) of n I
p@rer series in 2n indeterminates X = {X~,o..,X}~a , Y = {Y1 '''''Y-}a satisfying
(1)
;(x,o) = x,
(2)
;(F(x,z),z)
;(o,z) = z; = ;(x,;(Y,z)).
In view of (1), the substitution in (e) makes sense. immediately~r
have F(O,O) = O, and
Fi(X'Y) ~ Xi + Yi mod degree 2. ~[oreover, terms of degree greater than 1 are "mixed", i.e. X's and Y's only occur together. PROPOSITION 1
F is commutatlve if F(X,Y) = F(Y,X).
Given F, there exists aunique i(X) (n p~rer series in
n indeterminates) so that F(X,i(X)) = F(i(X),X) = 0. PROOF
Put gi(X,Y) = X i - Fi(X,y), i = l,...,n,
gi has no constant
-
23
-
term when viewed as a power series in Y.
(~gi/BYk)x=Y=O --(~Fi/~Yk)x=Y=O = - ~ik" By w
Prop. i, the determinant of (~gi/BYk)Y=0 is a unit of
RKKY~l,...,Xn] ].
Apply w
Theorem 2: there exist h i(X,Y), (i = 1,...,n)
such that gi(X,h(X,Y)) = Yi' i.e. X i - Fi(X,h(X,Y)) = Yi' or Fi(X,h(X,Y)) = X i - Y.~, (i = l,...,n).
Put Y = X : Fi(X,h(X,X)) = 0.
i(x) = h(X,X). The proof of the uniqueness of the inverse is a translation of the standard proof of group theory. Suppose n ~ respectively.
that F and G are formal groups of dimensions n and m
A homom0rphism f : F § G is a "vector" f = fl''"" 'fro
of m power series in XI,..~
~rith no constant terms, so that
fCFCxj) ) = GCfCx),fCY)). The homomorphism f determines a homomorphism ef : Rm § Rn, given by ZiG f = fi(X), ~rhere Z i are the indeterminates of R m and X i those of R n.
If f : F § G, g : G § H are homomorphisms of formal groups then
g ~ f : F § H is a homomorphlsm of formal groups. gives the identity homomorphism of F.
Also li(X) = X.
Hence :
PROPOSITION .2. The form~1 ~rou~s and their homo morphlsm form a cate~oznj ~R §
~R
(=~)' and f F-~ 8f defines a con travariant functo_r
(But as f is written on the left, e
on the right ~re still
have %f o g = ef o eg.) Remark : (in ~ R )
A homomorphism f : F § G of formal groups is an isomorphism if and only if ef is an isomorphism (in ~ R ) .
Moreover,
- 94
if f is amy "vector" of n power series with 8f an isomorphism, and if F is a formal group of dimension n, then there is a unique formal group G(= f ~ F ~ "l ) so that f is an isomorphism F +G. THEORF/{ i
(i) Le_~_tF be a f o r ~
grou~ of dimension n, and S,v 6 ~fR"
Then Ho~R(Rn,S)9 becomes a grotrp F(S) under the operation given by
is commutative, then FCs) is abelial. Cii) f
~ e HO~R(S,T) , then (~$) o ~ = Cc~ @ ?)9(~ ~ ~).
(iii) Let G b e ~ fur~er formal group of dimension m, then when f s Hong(G,F) , w_~ehaye
(iv) With the hyDothes.is o f(iii), and if i_~naddition F is commutative, then Hom~(G,F) i_s_sa sub~Toup of the abelian ~oup -
Re~F~:
(i)
Identifying
Ho R( n,S) = i(S,v) n (cf. w
Theorem I), the group operation becomes o~*~ = F(~,8),
m,8 e I(S,v)n. (ii) Again, if we express H o ~ (Rn,Rm) in terms of vectors f of power series we get the group operation
(x) (iii)
-- F C f ( X ) , g C X ) ) .
By the theorem, Ho~R(Rn,R n) is closed under composition
(multiplication) and ~ (addition), with a one-sided distributive law,
- 25-
i,e., it is a near ring. (iv)
The theorem, plus a f ~
is a functor ~ R (v)
X~R§
formal trivialities, tells us that F(S)
groups.
Let ~ a b be the full subcategor$~ of
the commutative formal groups.
~
=~%Rwhose objects are
Then the sets Hom (F,G) for F, G ~
ab
have the structure of abelian groups and the composition of homomorphisms is bilinear.
In particular End~F)
= Ho~(F,F)
is
now a ring. The proof of Theorem 1 is by a straightforward application of the definitions, and the ~:ioms for for~1
groups.
Suppose now that F is a commutative for~]
group of dimension n.
We define a function ~ on the abelian group F(S) = !(S,v) n by
~(~I,...,%) -- inf v(~i). i ~.le
state the following two propositions without proof :
PROPOSiT!~ 3 :
is a filtration of F(S).
(~le have not defined filtrations for non-abelian groups' ) PROPOSIT!OH 4: map Ho%fCG,F)
~,TithF and G as in Theorem i (iv), the composite ~
IIo~RCRn,P~m)
D ~ IIo~R(D(Rn),DCRm))
is a homo-
morphism o_~fgrougs (the compositio.n in Horn (G,F) bein~ ~). For a given prime number p, denote by
=
:R § n
R
n
the
homomorphism which fixes R and ts/~es X. into X p. Then 2 n l a :L o ~ = 7 (2) : x.. ~ x p ands(q) : x ; ~ x p - . Let ~+ denote the I
I
I
I
additive group of R. 'I"I-~,OTRt~..I
2
Let f : F '+ G b_~ea_ homomorphism of for~7 ~
dimensions n and m respectively) and let Of : Rm § R n the
(o_~f
- 26-
corresponding homomorphism of rings.
(i) ~ R
Then D(ef) = 0 i_~faud only i__fff = 0.
(ii) Suppose R + i~so__~fexponent
p (prime).
+ is torsion free.
Then D(ef) = 0 i_~faud onl~ if either f = O, or
ef = ~e_ ~ ~(q), where O(#f) # O__and q > O. PROOF
~[e use the notation 8fi/8~ = fik(~l,..,,Xn);
(3F/~)
(xJ1,...Jn) = F (X,Y); (~Gi/~V~) (U,V) = Gi~ (U,V).
N=r differentiating the equation f'z(F(X'Y) ) = Gi(f(X) ,f(Y) ) ~_th respect to Yk' ~e obtain (chain rule) n
m
; fij CFCxJ));jI~Cx,Y)
j=l
~=l ai~CfC',0,fCY))fz~C~).
Define the matrices
erCx)
[email protected]));~ a2FCx,'z) -- C~j~Cx,Y)); a2a(u,v) = (ai~(u,v)). Our equation, for ~11 i a u d k, then gives the matrix equation
~(F(X,Y)).a2F(X J) = a2G(f(X),f(Y)).af(Y). Hence
~(F(X,O)).a2F(X,O) = a2a(f(x),o).af(o) , i.e.,
af(X).a2F(X,O) = a~G(f(x),o).af(o). N~
df(O) = D(ef).
Hence by w
Also, det d2F(0,O) = i i.e., Cn(det d2F(X,O)) = 1.
Prop. !, det d2F(X,O) is a unit, and so d2F(X,0) is aa
-27-
invertlble matrix.
If D(ef) = O, i.e., df(O) = O,
and therefore 3f./BX. = 0 for ~11 i,j. l O this implies f = O.
~en
R + is torsion free
R + is of exponent p this implies
f(X) = g(xP), i.e., e~ = 8 o ~. g induction.
~en
then dr(X) = O,
In the latter case now proceed by
But we must sh~r that e comes from a homomorphism of g
formal groups, i.e., that g(X) is a homomorphism of formal groups.
Now
g(F(P)(xP,YP)) = gCFCX,Y) p) = fCFCX,Y)) = GCf(x),fCY))
=
where F (p) is obtained from F by raising each coefficient to its pth p~er.
}~ have then g(F(P)(x,Y)) = G(g(X),g(Y)).
Since F(P)(x,Y)
is a formal group (the map which sends each element of R into its pth p~er
is an endomorphism of R), g is indeed a homomorphism of formal
g%'OUpS. If
8 = r o =(h), and D(~) # O, then h = ht(8) is called the
height of 8 9
l~e define ht(O) = = .
For f a homomorphism of formal
groups, ht(Sf) = ht(f) is called the height of f.
If f # O, then
h ht(f) = h is the greatest integer so that f is a power series in X p . PROPOS!TION >.
(i) if f, g are homgmorphisms of formal groups an__~df 9 g
is defined , then ht(fe g) L h t ( f ) eo~mtatlve formal o ~ a n d
+ ht(g).
(ii) ~
G is ~
f,g ~ Hom(F,G), then
ht(f~g) ~ inf
{ht(f), ht(g)} .
ht(f) PROOF
(i)
If f is a power series in YP
, and g is a power series in
-
x
p
h
t
28
-
(
g
)
(where Y and X are the corresponding indeterminates) then ht(f) + ht(g)
clearly f e g is a power series in X p ht(f) + ht(g) ~_ ht(fo g).
, and therefore
(NOTE: if our formal groups are of
dimension i, and R is an integral domain, then ht(f) + ht(g) = ht(f 9 g), and the height function is a valuation). (ii) Since ef can be ~itten in the form ~f,
(ht(f)) where
D(~f) # O, then inf {ord Xi~ f} = i. Therefore ~(f) = inf{ord f.} =pht(f). i i The filtration property of ~, established in Prop. 3, n ~ gives pht(~g)
~ i n f { ht(f), pht(g)}
ht(f~g) ~ i n f
, which implies that
{ht(f), ht(g)) . ~Throughout this proof read "power series"
t o mean "vector of power series"]
- 29-
CHAPTER II LIE THEORY w
T h e bialgebra of a f o r ~ 1 group Throughout this chapter R is a fixed commutative ring with
identity and J ~ is the category of R-modules.
For ~I N ~
also M e RN and HomR(M,N) are R-modules. Ne shall need the notions of a coe.lgebra and of a bialgebra over R.
The definitions we shall give are ad~pted to our special
situation.
A coalgebra
is given by an R-module M
{~I, K, s, 8}
and homomorphisms of R-modules
(Z.Z)
(comultiplication),
RM
)M@
= :
7M,
8:M
yR,
so that the foll~ing diagrams commute:
M
(1.2)
.
~
IK
-
.~ M
|
ttM
[I@
<
<81
("associative law" - here we identify (M @ M) @ M = M ~ (M @ 14)).
M ~ RM
(here t is the "twisting map", t(x @ y) = y 8 x ;
"commutative law").
-30-
R
(~.~)
M
(i.~)
M @ E ~l~
(~.6)
/ ROEM
<
-"
M
--
>M@ER
A hi algebra is given by (i)
a coalgebra
{M,
<,
(ii)
the structure of an associative (but not necessarily commutative)
R-algebra on M with identity
a,
8} ,
[Exercise : describe by diagre~ ]
.
Here m is to coincide with the algebra structure map R § M, and and 8 are to be homomorphisms of R-algebras.
Note that together with
M also M @R M has the structure of an R-algebra ~rith identity, the product being given by (xI @ yl).(x 2 @ y2 ) = xlx 2 @ yly2. demand that ~(x.y) = ~(x).K(y).
We thus
Apart from the possible absence of the
commutative law for multiplication the axiom set for a bi~Igeb~-a is
-31-
self dual.
If the multiplication in
M is commutative we shall
spea2~ of a commutative b i _ ~ e ~ . He shall now consider the category J ~ w h o s e
objects are the
pcn~er series rings Rn, for varying n, v i m Ted as filtered R-modules under the order filtration, and whose morphisms are the continuous homomorl~hisms of filtered R-modules (i.e. not just the ring homomorphisms as in ~ R ).
We shall also view R = R 0 as a complete filtered R-module,
via the trivial filtration : v(a) = 0 if a # O.
Notation:
Write
If f e Rn, u E Um we shall use the symbol
for the image
in R of f under u.
Thus
= +
,
= + ,
and if r ~ R,
= = r < f,u>
.
~ne fact that u is continuous means that = 0 whenever ord f > m u where m u E N depends on u.
}Te identi~j U 0 with R, via
= rlr 2. %[e also need provisionally a notation for the action of a~u
eme=ent s
=ite
Is,u] for the
of
U u~der s. In the sequel let j~':" denote the full subcategoz-j of J ~ by the modules U . n
formed
-32-
PROPOSITION 1 (i) I_f.f8 E H O ~ R n , R m) then the equation
=
defines a e 6 IIo~Um,Un).
The maps Rn~-~ Un, 8 ~
additive contravariant functor j~-§ are ~rriting the maps U (ii)
§ U
m
n
~
Zl
define an
(IIote however that as we
on the left we sh~1
have (e ~ r
If m s Horn. (U ,U ) then the e~uation --
e
=e * or
[s,m u] = [sm',u]
n
defines a map ~' - Horn. (U ,R) + H O k U m , R ) . -JvU n (iii)
The equation [sf,u] =
for a ~iven f & R n,
and ~l] u e Un defines an sf e Ho~((Um,R).
The = ~
f ~-~ sf i s ~
homo morphism R n
(iv) sfe = (sf) (e*),. PROOF
Straightfor~rard and standard!
PRQPOSITIO}[ 2
U n i_.ss~ free R-moduS.e on ~k (k 6 Mn) , where
= fk' i,e.,
<~
PROOF
, ~k >
If u ~ Un, then u =
=
l, 0,
k = k # A.
~ < ~,u> kcM
~k (the right--hand sum is in
n
fact a finite sum : u is continuous and hence <X=, u> = 0 for all Ikl sufficiently large).
~ ck dk > = k
[ fkCk 9 k
U n is therefore spanned by ~k"
Now
Therefore ~ ck dk = 0 implies k
flck = 0 for all f, and so also ck = 0 for .11 k. k ~" is free on the ~k"
Hence in fact U
n
*
- 33 -
COROLLARY i PROOF
-The -
map f ~-~ sf i s an isomgrghism R n § Ho~(Un,R).
If sf = O, then 0 = [sf,u] = for all u ~ U n.
This
implies f = O, and therefore f~-~sf is in~ective.
Thus
-3~ n " ' is surjective.
f ~-~sf
COROLLARY 2. maps e ~-~ e
k
Th_e_ehomomorphism Homx,(Rn,R m) § Hom~(Um,U n) ~.thich i_~san isomorph"ism.
Thus the functor J ( §
of
Progosit.ion i is an antisomorphism o f .cateaories. PRQOF
Suppose 8
= O.
Then for all f, u,
= O.
0 = = [sfe,u] , which implies Sfo = 0 for all f. an isomorphism, and therefore f8 T h i s proves injectivity.
Therefore But s is
= 0 for all f, which means 8
= O.
If w ~ Ho~CUm,Un) , define 0 6 Ho~(Rn,R m) r
by f8
= ~ 2 . Then e k COROLLARY 3 . The map U n @ RUn
= ~ .
This proves surjectivity.
~ U2n
~iven by
~(kl,...,kn) 0 S(~Z,...,~n) i_s_sa n isom@rphis m ~ PROOF
R-modules.
Obvious. The significance of the last corollary lies in an interpretation
presently to be derived. Let
I = {f~
R
n
I ord
{(z o R~n
(f)
+
9
.
Then
Rn 0 RZ) § ~n 0 Ran}
is an ideal of the ring Rn e RRn .
Let v be the filtration of Rn
corresponding to the p ~ e r s ~q of the ideal ~. v(f @ g) = oral(f) § ord(g).
= -9
Then
Denoting the indeterminates of Rn by
- 34-
X = "" ~l,...,Xn and those of R2n by X',X" = X ~ ,...,~n,~l,..., " ~:" Xn
we
define a homomorphism R n @ RRn + R2n of R-modules by : f(X) @ g(X)
'
" f(X') g(X").
This turns out to be an injective
continuous homomorphism of R-algebras.
In fact the filtration v is
seen to be simply the restriction of the order filtration of R2n. A
Going over to the completion R n @ RRn of R n @ RRn we obtain a bicontinuous isomorphism R n @ HEn = R2n.
Thus we get an isomorphism
URn ~ Homj~(R n @ RRn, R), and the module on the right can be identified with U n @ RUn .
The resulting isomorphism U2n = U n @ RUn
is the inverse of that of the last corollary. The ring multiplication in R
is described by a map : n
R n @ RRn + R n.
As multiplication is continuous and R n is complete,
this map extends uniquely to a continuous homomorphism
Wn : Rn determineduniquely
@ R
7
n'
by the rule
(f(x) e g(X))
n
= f(x)g(x).
A
Identify from now on R n @ RRn = R2n.
In the previously introduced
notation for the indeterminates of R2n ,
h(X',X")
Wn is then given by
= h(X,X).
Let ~n : Rn § R be the augmentation, ~n : R § R n the ring embeddirg.
}[e then have, on identifying U2n = U n Q RUn,
(~iting ~
for ~ R )
- $5-
PRpPOSITIO 3
The o
~n : Un § U2n = U n @ RUn , cn : R §
n P
de f~De on U n the structure of a_ coel~ebr~. The .isomorphism '
~s
rise t_~oa_ bijection
Eom CRn. ) I~,00F
omCo g( ,Un).
For the first assertion we only have to show that ~n' en
and ~n enter into commutative diagrams dual to those postulated for 9~.
the maps
,
~ = ~n ' ~ = r
(see (1.1)-(1.6)).
.*
8 = ;in defining a coalgebra structure
For example (1.2) follo~m from the associ&tive
l~w for the product ~n' (1.3) from the commutative law, and so on. For the second part of the proposition note that a e ~ Homj~(Rn,R m) will actn~.11y lie in Hom~D(Rn,R m) if and only if the diagrams W n
888
R2m
"~
e
)
-36-
8
R
n
,
> R
ra
R
R
R/
e
~
rl
conmmte.
R
]~
The dual diagrams ( s t a r e v e r y t ~ n g and r e v e r s e .11 arrows)
g i v e p r e c i s e l y t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s f o r 8
t o be
a homomorphism o f c o a l g e b r a s . Let {C be t h e c a t e g o r y whose o b j e c t s a r e t h e c o a l g e b r a s {Un' ~n' r
~n } and whose morphisms a r e t h e homomorphisms o f
coalgebras.
The l a s t p r o p o s i t i o n t h e n t e l l s
j ~ § J ~ * y i e l d s an antlsomorphism ~
§ ~
us t h a t t h e f ~ n c t o r of categories.
Let F = F(X' ,X") be a power s e r i e s i n 2n i n d e t e r m i n a t e s x,
-
x"-
v,,
v.
,
w i t h zero c o n s t a n t t e r m .
Let
8F E Hom~ ( R n , ~ n ) be t h e c o r r e s p o n d i n g homomorghism o f power series rings.
Thus f(x)e F = f(F(X',X")).
Then F will be a formal
group i f and o n l y i f t h e f o l l o w i n g diagrams commute :
-3'/-
R
n
Y
(z.7)
/-
R2n @1
n
(~(o,x) :~ x
/R2n @
E
:
F(X,O))
n
R
n
(identii~ring
Rn @ RR = Rn) ,
eF Rn
-'2n
(z.8)
("Associative law"). i | eF R2n
~ R3n
In addition we know of course that eF is a homomorphism of rings, preserving identities.
R
Hence the following two diagrams also co~ute
e F ~ 8F
@ R Rn
^
~ R2n 8 R R2n
(~.9)
Rn
~
:
R2n
R
(~.I0) Bn
~
n
F Rn
> R2n
-38-
In the first diagram the vertical maps are those induced by multiplication, going over from @ to the completed tensor product @. Consider now the dual map
P = eF :
U2n = Un @R Un § Un"
This is an RIlinear multiplication on U (1.7)-(1.10) now show that U
n
and ~Jae duals of diagr~m.~
becomes a bialgebra.
n
E.g. the dual
of (1.8) is the associative law for multiplication and the dual of (1.9) tells us that the comultiplication ~ of U is an algebra n n homomorphism. We can sum up : If F is a formal group of dimension n, then p = eF defines the structure of a bialgebra on U m ( the coalgebra structure being fixed once and for all by ~m' an and ~n ) . Conversely if p : Um @R Un § Un is a map, defining the structure of a bialgebra then in particular p ~ HOmcoalg(U2n,U n) and hence .
p = eF for some unique power series F(X' ,X").
The axioms imposed on
p (as stated above) then imply that F is a formal group.
We have
thus proved the first part of THEOREM 1 ( i ) The map F ~-~ 8F = PF i s a b i j e c t i o n form~ ~
of dimension n onto
on the coalgebra (ii) fo~...1, ~
(iii)
{Un' ~n'
of. th__~es e t o f
the set o f structures o f b.i,algebra
en' ~n } "
The _algebra Um,PF is commutative, if and only if the F i s co~a~-atative. i
I
z_! F
.
.
.
.
]
fprm
res~ectiyely then the iSomorphiSm
of d/mensio=s n an__!m
- 39
HO~
IR n
-
n'
gives rise to a bijection
Ho CF,G)
(Cun,PF), (U ,pG)).
The proof of (ii) is quite analogous to that of (i). For (iii), comaider the vectors f = (fl,...,fm) of m power series in n indeterminates.
We know that these stand in biunique
correspondence under a map f ~ > ef with the e 6 Hom~(Rm,R n) . ,
Going over to the duals we obtain a bisection f ~-* pf = ef e HOmcoalg(Un,U m) . By dualizing the appropriate diagram one sees then that f is a homomorphism F § G precisely when the diagram
PF
U
U n @ RUn
n
Pf @ Pf Um @ RUm
U
-
m
PG commutes, i.e. when pf is a homomorphism (Un,PF) § (Um,PG) of bialgebras. L e t ~ be the category of bialgebras whose underlying *
*
coalgebra is one of the {Un, ~n' The maps F : > Un,PF ,
§
Zn' ~n ) "
We can then sum up
f ~ > pf define an isomorphism
Note that we end up with a covariant functor: ~;e shall now discuss the bialgebra Un,PF in some more detail. If k = (kl,...,kn) ,
A = (AI,...,~ n) are in L!n we define
-40-
=
, and ~.; = zl;...Zn!
\h
/...
.
n
We denote by ~i the element of I~ of the form mi = (O,...,O,l,O,..,O), ~rhlch has 1 in the i-th position and 0 elsa~there. The element Ai of U n is defined by the equation A i = ~
, i.e., 1
<Xi,A i >= I,
<Xj,A.1 9 = o f o r i # ~ ,
.-x-~, A.> = o if 1
Ikl # i .
For the formal group F(X,Y), we introduce the notations FkCX'Y) e Xk + Yk + Bk(X'Y) mod degree 3,
Bk(X'Y) = z0"!" li'j'k XiYj
'
(k = 1,...,n),
Xi'j'ks R.
Then we have PROPOSITION 4 W
! if k=O,
(i) c~(1) = ~o; ~n(~ ) = 0 if k#0.
~+j =k
0
(iii) pF(~O @ u) = PF(U @ ~0 ) = u. (iv) pF(~k @ 6Z) =
s
c.~. , c . ~
Ok+ s +
0 0
0
(~ # o ~ ~). n
(v) PF(Ai,A~)=
~ iJ
~
J
k=l
z,j,k~"
R,
- 41-
n
COROLLARY PROOF
- lj,i,k )A k'
(i) ~nd (iii) are obvious 9 (ii) ~,Tehave
,
%.
xJ,
But by the definition of ~n'
~§
l, i f k = ~ + j O, otherwise.
<X ~ | X j, ~n (~k)> = <X~+J,~k > =
Hence the result, (iv)
Let r = (rl,...,rn). n
F(X,Y) r =
r.
K ~ (X,Y) ~ i=l " i
n
=
=
~
i=l
Then
I".
(xi+Y)~
+ terms of order > Irl
m
Cx + y)r + terms of order
II~, <xr~, pF(~ 9 ~A)~ - ,~yA in F(X.Y) r.
Irl.
is the coefficient of
This is thus 0 when Irl>Ik + A 1 and also when
Irl - Ik § 41 but r ~ ~ + 4.
on the other hand if ~ -- k + 4
this coefficient is clearly (k + &). 4
(v)
>
Finally when r = 0 then
By (iv) we know already that
pF(Ai,Aj) = \
; ~i,j,k n k' % / 6 ~~ .j + k-l
then
-42-
and we have to show that ~i,j,k = Xi,j,k"
"i,j,k = < z ( x J ) %
'
In fact we have
Ai ~ n.>
= J
= ~k (xJ)' ~i e ~j> = ~i,j,k
% ~ + Yk' ~i e ~.> = o = d 3.
A. e
m
Aj> = 0, ~ = e v e r
This completes the proof of the Proposition.
We define T(Rn) to be the submodule of those u ~ U n for vhich = 0 =
.
T(R n) is thus the submodule generated by the A i.
The next
proposition gives an ~n~er characterisation of T(R n) in terms of the coalgebra structure of U n.
m0POSi~O~, ~ ,
ai,~
u ~. u n ~ e
fo~o~
st ~ t ~ n t s
equivalent
(ii)
(iii) (Recall that
~*(u) = u O
r + r ~u,
= c(f) + r
,
r = 60 is always the identity in amy bialgebra PF
structure of Un).
- 43 -
PROOF
(i) => (ii): By Prop. 4 (ii) holds for u = Ai, hence by W
R-linearity of ~
for all u E
T(Rn).
(ii) => (iii)
:
=
=
@
g,
u@
e +
r @ u>
= ~(g) + ~(f)
.
(iii) => (i) :
If f, g s I then ~(f) = r By linearity = 0.
= 0 and so = 0. Also
= = r
+ r
= 2 , i.e. = O.
Hence = O.
w
The Lie al~ebral of a formal ~rou~ First we list, without proofs, the definitions and results
on Lie algebras to be used. Throughout R is a fixed co~atative ring, and all "algebras" are algebras over R. a Lie algebra ~ Product [a,b] i n ~
For each associative algebra A there exists
(A), which coincides w~th A as a module, the Lie (A) being given in terms of the associative
product ab by
a,b] = ab - ba.
- 44
Each Lie algebra L has an enveloping algebra E(L).
More
precisely E(L) is an associative algebra with identity with an attached homomorphism j : L §
of Lie algebras so that the
map
fl
~f
@ j
(f & Homassoe(E(L),A)) is a bijeetion
(2.1) Note:
Homassoc(ECL),A )
> HomLieCL~CA)).
All associative algebras have identities, and Homasso c is
the set of homomorphisms preserving identities. By (2.1), taking A = R we get from the n1111 a homomorphism of assoclative algebras
: ECL)
§ R.
As E(L) has an identity we also have a homomorphism o : R § E(L). Next if L 1 and L 2 are Lie algebras, then their cartesian set product L I x L 2 has again a Lie algebra structure, and
E(Ti x L2) =~ ECLl)
eR
ECL2).
In particular
E(~ x L) ~ E(~) e R E(~) via
j(~l,~2),
~ j(zl)
e i
+,
e
j(~2).
-
45
-
The diagonal map L § L x L thus gives rise to a homomorphism
D
:
E(L) § E(T) eRE(L)
of associative algebras.
A.
The associative algebra structure on E(L) to~etherwith
the maps D, (of.
w
u, 9
define on E(L) the structure of a bialgebra
for the definition).
To prove this one would only have to verify now the co~utativity of the diagram~ (l.l) - (1.6), and this can be done by going back to the defining property of the enveloping algebra.
For the particular Lie algebrsswhich we shall have to
consider this also follows from the explicit description to be given below. From (2.1) we obtain a map
E : HOmLie(Li,L 2) § HOmBialg(E(Ll),E(L2) ). In fact given a homomorphism
~
: L I + L 2 of Lie algebras there
is one and only one homomorphism E(~) : E(L I) § E(L 2) of associative algebras so that
L1 -
) L2
F,(~) ECh) co~zm~es,
[ ~ECL2)
-46-
In view of the obvious functorial properties of the maps D,
a and ~ associated with each L,
with these. B.
E(m) will in fact commute
In other words
E is a functor frgm Lie algebras t o bialgebras. Now let L be a Lie algebra which as an R-module is free on
say generators ~ " ' ' ' ~ n " C.
Then :
("Poincar~-Birkhoff - Witt Theorem") L § E(L) is in~ective. We shall accordingly vi~v L as embedded in E(L).
~or ~
= (k1,..., ~ ) ~- ~% = ~
...%
(the order of the factors matters')
p_.
Write
(i) (ii) (iii)
c~~ _- ~) Then we have the ~escription
E(L) i_~sthe fr.ee R-module on the dk.
dkd ~ - d ~+~ +
X
a.d j
(k,~ # 0).
D(d i) = 1 g d i + d i g l,
and hence
i+j=k (iv)
~(i) = ~o = i.
~(~) _-
O, k#O i, k=O.
Now we return to the associative algebra U , p F defined in the preceding section, F being a formal group of dimension n.
We
-47
shall write
[']
-
F for the Lie product.
Thus
[ u,v ]F = PF (u'V) - PF (v'u)" In the notation of Ii
w
Prop. 4, we now see that
42.2.)
It follows that the submodule TCR n) of U n generated by the Ai (see w
is closed under ['IF"
In other words T(R n) is a Lie algebra
under ['IF' which we shall denote by LF - the Lie algebra associated with the for~a~ group F. Let f : F + G be a homomorphism of form_~l groups (dim F = n, dim G = m).
The homomorphism ef
: Rm § Rm
maps R
(viewed
as
a
subring) into itself, and maps I~m ) ( = {f e Rml ord f ~_ 2}) into I n)"
Hence the dual homomorphism ef : Un §
~rill map
T(Rn) § T(Rm) 9 Moreover, ef is a homomorphism of associative algebras, i.e., it takes the multiplication PF into PG"
Hence
also
Of u, 8f
G
for u, v @ U . It foll~s that ef gives rise, by restriction, to n a homomorphism Lf : ~ PROPOSITION cate~
I
§ L G of Lie algebras.
He sum up :
L F and Lf define a covariant functor from the
~ of formal groups to_ th_~ecategory o_~fLie algebras ,, and
LF preserves dimensions, i.e., L F is ~ fre___~eR-module on dim F generators.
-48-
Alternative Description: a = (al,...,a n) (ai (
Let R (n) be the module of n-tuples
R).
Let BECx,Y) be t h e homogeneous
quadratic component of Fk(X,Y) (k = l,...,n) (cf. ~(X,Y)
~--"
~Cx,Y)
t
BkCY,X).
w
and write
Define a multiplication on R (n) I
i,e., a map R (n) @ R (n) § R Cn) by
~(a,b) - (A1(a,b),...,%(a,b)). Then, if p is the isomorphism T(Rn) § R (n) of modules given by
pC~.aiAi) = (~,...,an), we get fr= (2.2) p( [u,v]F) -- ~(p(~),p(~) ). Thus R (n) , ~
is a Lie algebra, in fact isomorphic with LF.
For formal groups F and G of d~mensions n and m respectively, we denote by Ai, F and Ai, G the corresponding free module generators of ~
and LG.
If f : F § G is a homomorphism then fik6 R are
defined by
fir
) - ~ fik ~
r
de~ 2)
k--i
m
PROPOSITION 2
Lf(~,F) =
~
fik Ai,G
i=l m
PROOF. Suppose Lf(~,F) = i=l ~
Cik Ai ,G' say.
indeterminates of G by YI'"" ''Ym )
Then (we denote the
- 49 -
Cik = .:§
=
ef(~,F)>
= -
i
D
F>
= fik" COROLLARY 1
The homomorphism f : F § G is an isomorphism of formal
if and onl~ i_~fLf is an isomor,~ism of Lie algebras. PROOF
By I,
COROLLARY 2 PROOF
w Theorem 2. If R + is torsion free then Lf = 0 if and onl~ if f - O.
By I,
w Theorem 2.
For the rest of this section we assume that R is a Q-algebra (Q is the field of rational numbers).
Under this hypothesis we
shall prove that the category of formal groups and the category of Lie algebras which are free E-modules of finite rank are isomorphic. More precisely we have: THE O R ~ i
(i)
Let R b e a Q - ~ .
For each Lie_ algebra L which
is a free module of finite dimension over R, there exists a formal ~rou~ F such that L i s isomorphic to 2 " (ii) (iii) on ~
Hom~(F,G) § HOmLie(~,L G) is abijection. The formal ~rou9s F and G ar.e isomorphic i_/_fand
if the correspondin~ Lie al~ebras L F and L G a~e isomorphic. The proof of Theorem i requires three lemmms.
We take L
-50-
to be a Lie algebra which as module is free on generators ~,... ,dn. The module homomorphism C L : E(L) § U n is defined by the equation
c,..(d~) - kz ~ , where we sh-11 use throughout the description of E(L) given in D. CL is an isomorphism ofmodules.
L~I
E(~) ..
D
U
Moreover, the diagrams
E(~) eR ~(~)
n
>
U
n
n
@~U .t~
n
R
T
~n
/c E(~)
> u n
R COCm~t e. The multiplication E(L) @ E(L) ->E(L) defines through CT. a multiplication qL : Un @ U n § L~4A 2
n~
There exists a_ formal group F fo__~rwhich qL -- PF' i.e.,
qL defines on. U the structure of' a_,bialgebra.
Then also L = LF
-51-
Let now conversely F be a given formal group.
Since LFC~(Un,PF) ,
this inclusion map can be pulled back to a homomorphism : E(~) § LD~3
of algebras (universal property of enveloping algebra).
n : E(L F) § Un,PF i_~san isomorphism.
~.c; = Cn' ~n" ~ = ~
PROOF of Lemma i
Als O
Ca @ a ) . D = w n ~.
and..
Since R + is divisible and by II w
m
Prop. 2, the
k'~ k form a free basis for Un, and so CL is an isomorphism of modules. Also by D a n d II
w
Prop. 4
~.+j=k X
k'
i+j=k
CT '
cai)
e
CT,CdJ)
I i+j=k
=k'
n
z
(6k) = ~n CL(dk)~
By extending linearly to E(L), this proves that the first diagram is commutative.
Similarly for the second diagram.
PROOF of Lemma 2
qL is defined so that
~.C~)
o R ECT.)
Un @Un
-
-
~ ~ ECT.)
qT, -
~,
Un
-52-
is comnutative.
That qL defines a bialgebra structure on U n
is now trivial by Lemma i.
The isomorphism of categories ~
~
of the last section ensures the existence of a formal group F such that t h e Un,qL = Un,PF.
Since CL maps d i onto Ai, L and
are isomorphic under CL as modules, and since CL preserves the Lie product then this is an isomorphism of Lie algebras.
P~90F 9~ Lemma 3 the dkwith
Let E r be the submodule of E ( ~ ) generated by
Ikl ~ r
the ~k with Ikl ~ r.
and let V r be the submodule of U n generated by We shall then prove by induction on r the
assertions
r
When [k[ = r t h e n
(mod Vr_l) , 9 k a
unit of R ;
(Br)
maps E r bijectively onto V r.
As E(LF) is the union of the Er, bijectivity of
~ : E(~) §
U n the union of the V r the
n follows.
By the definition of ~ ,
n(a ~
= n(1) = Co"
n(a~i) = n(ai) = ~i = as. 1
where I%1 : l,
% has a i at the i-th place.
AS for I~l _<
the dk are free generators of E 1 stud the ~k are free generators
of v l, both (~) ~na (BI) are tru,. For the induction step f r o m r to r + 1 let lJl = r + 1,
and w ~ t e j = k + ~ where I~l _< r,l~l
_< r.
~ e n by D_ (ii),
- 53 -
dj = dkd A (rood Er).
Hence by the induction hypothesis, and by w
Prop. ~, ~(dJ) - P F (~(dk)' ~(dA)) - 9 PFC~k,~A) -eVdj (mod Vr) , where e,e' are units of R (recall here
that
R is
i.e., that we have unique division by integers).
a
Q-slgebra, We have thus
established (Ar+l). But now (Br+I) follows from (Ar+I) and (Br) by an easy
argument.
The first equation in Lamina 3 just tells us that ~ is a map of R-algebras.
All the maps occuring in the lest two equations of the
lemma are homomorphism of algebras preserving identities.
In each case
it then suffices to verify that the images of the generators di of the algebra E(LF) coincide, and this follows from the explicit description given earlier on. PROOF o_~fTheorem i
(D and w
Prop. 4).
(i) is just Lamina 2. (iii) follows from the
fact that F ~-~ LF is a functor, end from (ii). For (ii), we recall (cf. II w
Theorem i) that
Hom~(F,G) ~ HOmBialg(Un,PF ; Um,PG) Cn = dim F, m = dim G).
Recalling the way ~
and Lf were defined,
we see now that it suffices to prove that the map 9 HOmBialgCUn,PF ; Um,PG) § HOmLi e (~,L G) is bijective.
We consider the diagram Cof module homomorphisms)
- 54-
LG
JF , E(L G)
]~
Un 'PF
~m'PG
Here JF is the inclusion map,
~F as in Le~m~ 3 is the unique
homomorphism of algebras so that ~F ~ JF is the inclusion map = T(R n) § U n. ~(V) : ~
Let ~ be a homomorphism of bialgebras.
Then
§ LG is uniquely determined as the module homomorphism
for which
(a) 9 ~ ~o Next let
JF
e : ~§
=
nG
~
JG ~
~(~)"
a homomorphism of Lie algebras.
Then E(e) : E(L F) § E(LG) is the unique homomorphism of bialgebras so that
(b) ByLaw3,
E(e) o JF - JG ~ e. ~
A(e) : Un,PF §
(c)
is an isomorphism of bialgebras. by
~(e) o ~F = ~G ~ E(e).
Define
-55-
l(e) .
2F o ~F =
Thus (cf. (a) with W
= ICe)),
x(e)
:
e.
On the other hand, for given
~ ~
nG o JG o e.
V , we have by (a) and (b)
~ ~F = 2G ~ E(uCv)) ~ ~F'
hence
and so by (c) (with e = ~(~)), =
Thus A and ~ are inverse bijections. established.
The theorem has thus been
(Incidentally we have proved also that E is a
bij ection. ) COROLLAR.Y 1
I f R is a_ Q-algebra th.en every commutative formal group
F is isomorphic t_~oth__~eadditive ~
of dimension dim F.
The additive group G a of dimension n is given by (Ga) i (X,Y) = X.~ + Y.. PROOF .....
F is commutative
commutative
<=>
abeliazl. This ~
<=>
the multiplication PF in U n is
PF(Ai,Aj) = PF(Aj,Ai) for .~I~ i,j
<=> ~
is
is uniquely determined by its dimension, and the
dimension therefore determines tk~iquely the class of F.
-56-
C0ROL~, Y 2 comparative.
I! R i_~s~ Q-~A~ebra, ever~ form~, group of aimension i i_% Corollary 2 is also true if instead we suppose R has
no nil potent elements (Lazard).
-57-
CHAPTER II!.
w
9~ERALITIES
CO}~K3TATIVE FOP~La.L GROUPS OF DI~TSION ONE
Throughout this chapter all formal groups are
co~matative of dimension one.
We repeat the definitions, and a few
pertinent facts. A formal group F(X,Y) is a power series (over R) in two variables X,Y, satisfying (i)
F(O,X) = X = F(X,O)
(ii)
F(F(XJ),Z)
(iii)
F(X,Y) = F(Y,X).
= F(X,F(Y,Z));
A homomorphism f : F § G of formal groups is a power series (with zero constant term) in one variable satisfyingthe relation (iv)
f(F(X,Y)) = G(f(X),f(Y)).
We sha1~ write (f o g) (X) = f(g(X)).
We denote by HomR(F,G)
the set of homomorphisms F § G of formal groups.
If f, g ~ HomR(F,G) ,
(z + g) (x) = G(Z(X), z(X)). With respect to this addition HomR(F,G) is an abeliau group, and the composition
~ for homomorphisms is bilinear (cf. I,
w
Th.1).
We shall c~11 such a category "additive" (always with quotation marks, as the term add/rive category without quotation marks is now accepted to mean something more).
The "additive" category
of commutative formal groups of dimension 1 over R will in the sequel be denoted by ~ R" HomR(F,F) = End(F) is a ring with identity.
There thus
exists a unique homomorphism Z § End(F) which preserves
-58-
identities.
The image of the integer n will be denoted by [n]F.
Thus [I]F(X ) = X, I,
w
[-1IF(X) is the power series i(X) of
Prop. l, i.e., F(X, [-1IF(X)) = O, and [n+l]F(X) = F(Kn]F(X),X). HomR(F,G) is a left End(G) and a right En~(F)-module, and
the action of Z on HO~R(F,G) can be described in terms of either of its two embeddings in the rings of endomorphisms.
In other words,
for n e Z and f 6 HOmR(F,G ) In] o f = f o [n~ G F" Now we recall the definition of the map D (cf. I, w For dimension i we simply have D(f) = fl = coefficient of X in f(X). D is then a functor
~R § R, in other words
D(f ~ g) = D(f). D(g), D(f + g) = D(f) § D(g).
Moreover f is an isomorphism if and only if D(f) 6 U(R). PROPOS!T!ON 'I
!homomorphism
V : R § S o; ~ ( w i t h
identity)
gives r%s 9 t_~ofunctors ~ R § ~S o_~f"addltive" categories , which preserves the action o_~fD. PROOF
Obvious.
The desired map of objects and morphism is that
induced by V on the coefficients of the appropriate power series. PROPOSITION 2 (non-co~tative
If R is an integral domain, then End(F) is a__ inter)
domain, and HOmR(F,G) is a torsion-free
m~(F) (~d m~2(G)) mod~e.
- 59-
PROOF
If f = f X r + f
9
r
X r+l + .... r+l
'
sad
g = gs Xs + gs+l
then
f o g = frg ~
sad frgsr # 0.
xS+l
+ "''" '
fr'gs # 0,
X r+s + .... ,
Therefore f o g = 0 implies either f = 0 or g = O.
From this we deduce that E n d ( F )
is an integral domain, and that
HomR(F,G) is a torsion-free E n d ( F ) The image of Z + E n d ( F )
(and E n % ( G ) ) module.
is thus also an integral domain,
and its kernel must therefore either be 0 or pZ for some prime p. If the characteristic of the quotient field of R is 0, then (cf. I,
w
Th. 2) D : E n d ( F ) § R is an embedding.
E n d ( F ) is a commutative integral domain and ker
Therefore
{Z § EndR(F)}
= O.
C0}~?ARISON OF FOEt~L GROUPS A polynomial in R[X,Y]
is ~rimitive if the ideal in R
generated by the coefficients is the unit ideal. in Z[X,Y]
(A polynomial
is thus primitive if the highest common factor of the
coefficients is 1.)
The natural map Z § R can in the obvious way
be e ended to a map Z[X,Y]
then pri ti
po
o
als
are mapped onto primitive polynomials. We shall now introduce Lazard's polynomials B Bn(X,Y) = (X + y)n - X n _ yn.
If n is not a prime power, then
c(x,Y)
-
n
and C . n
Here
-60-
If on the other hand n = q r
where r > 0 and q is a prime, then
cn(x,Y) - lq Bn(XJ). Note that C
n
is always an integral polynomial.
Cn(X,Y) is a primitive polynomial i n Z IX,Y].
PROOF
Suppose that PlCn(X,Y).
If first n is a power of p, this
implies (by induction on m) that m n ~ m S
Next if n = p r, S
(Xp
(mod p2), which is false.
r > i, (p,r) = i then we get S
S
S
+ yp )r = (X + y)n = X n + yn _ xp r + yp r
(X + y)r ~ X r + yr
hence
(rood p);
(mod p),
which is false (coefficient of XY r-I I). The following theorem exhibits the relation between t~ro
formal groups which agree up t o a g i v e n d e g r e e . .TI~OP~,l.(Lazard)
Let F dud G be, f o r m ~ groups over ~ corrnut.a,tive
r i m g R w~th
F = G Then
(mod deg n).
F = G + aC
(mod deg n + i) n
fg.r some a ~ R. To prove this theorem we shall need also LEPTA 2
Assume the Same hypothesis as in Theorem i, and .furthermore
let F(X,Y) be the homogeneous Polynomial of degree n for which
F = G + F
(rood deg n + 1).
-61-
r(x,Y) -- r(Y,x) ,
~n
r(x,o) ~
o - r(o,x),
)I(P)
r(x,~) + r(x + Y,z) = r(x,~~ + z) + r(Y,z
PROOF The first two equations are trivial.
To prove the third
equation we observe that, working modulo degree n + l, (and using the notation G(X,Y) = X + Y + G2(X,Y) , which means that G2(X,Y) is the sum of terms of G of degree ~ 2)
~(F(X,Y),Z) ~ O(F(X,Y),Z) + r(F(X,Y),z) F(X,~) + z + a2(F(X,Y),z) + r(x + Y,z)
_= o(x,~) + rCx,~) + z + a2(Q(x,Y).z) + r(x + Y,z) o(G(x,Y),z) + r(x,~) + r(x + ~,z). Similarly one shows that
F(X,F(Y,Z)) ~ G(X,O(Y,Z)) + r(x,z + z) + r(~,z). This proves our assertion.
(The second equation can also be derived
from the third). To prove the theorem it will suffice to show that any homogeneous polynomial r of degree n, satisfying conditions (P) is of the form aC
@
n
Lazard's original proof is very tough and computational.
We
shall give here a simpler proof in which the computations are restricted to fields.
The basic idea is first to generalize the theorem
appropriately.
Instead of polynomials over a ring we consider
polynomials over an (additive) Abelian group A.
With these one can
compute in the same way as if A were a ring - except that there is no multiplication.
The advantage is that one can now use the structure
theory of Abelian groups.
To be more precise we define
- 62-
A[~,...,Xnl = A
@Z
Z[XI'''''Xn]'
aud call the elements of this module "polynomials over A". Z[Xl,...,X J
As
is a free Z-module, one may view the module of
polynomials over a subgroup B of A as contained in A[Xl,...,Xn] Theorem 1 is then a consequence of THEOR~la
~e_~.~homogeneouspolynomial
F o_f_fdegreen, satisfying
conditi0ns (P) of Lemma 2, is of the form
F = aCn with a 6 A.
Let us first assume I.
The theorem is true whe__~mR = A i_ss a field.
Then in view of L e m m a i, and by I with R = Q the rational field, we conclude II.
The theorem i~strue for R = Z.
Next one shows III.
The theorem is true for A = E = Z/(pr), p a prime, r > O.
In fact, for r = l, this follows from I. on r.
Now we proceed by induction
The induction hypothesis can be written as a comgruence
r(x,Y) ~ aCn(x,y) + prrl(X,y) where a s Z, and where F1 satisfies (P) mod p.
rl(X,Y) ~ bCn(X,Y)
(mod pr+l) But then
(mod p)
(b ~ Z), and hence
r(x,y) ~ (a + prb) Cn(X,Y) IV.
(mod pr+l).
l_ttsuffices to establish the theorem for finitely ~enerated
Abelian ~ o u n s A. In fact, amy polymomial F with coefficients in am Abelian group
- 63
A m a y b e viewed as a polynomial over the subgroup of A generated by the coefficients of F . V.
If the theorem is valid for groups A and B then it is valid
als___~ofor their direct sum__. This is obvious. N~r the theorem follows from II - V and the structure theory of finitely generated Abelian groups.
We still have of course to
establish I, and this requires some computation. PROOF of I.
Note that Cn(X,Y) viewed as a polynomial over the given
field R is non-zero (by L e n a (P).
l) and clearly satisfies conditions
It then suffices to show that the conditions (P) determine a
subspace S of d i m e n s i o n a l
of the vector space of homogeneous
polynomials of degree n. n Write F(X,Y) = ~ a xry n-r . r=O r
ar = a n-r , a o = a
n
Then by (P)
=0.
Moreover ~re get from the last equation in (P), on comparing the coefficient of XIY~Z n-~-u
(l > O, I
n-~
+ U < n) the equations
at+w (t~lJ)= an_~(n_t_l~) , i~
Take
t=1
(1)
s and
~=~
a + 1 (~ + l ) =
:
aI ( n ~ l ) .
- 64-
Next take
~ = i,
(2)
A = ~ :
a+l(m
+ i) = a ( n -
~).
If first the characteristic of R is zero then (I) shows that dim S ~ 1 as required.
From n~r on assume that the characteristic
of R is p # O. Suppose first of all that whenever i < ~ < n - i then either (~,p) = i or ( n -
~a
(n-u)
m,p) = i.
I n-i
= a1
m-i
am = ( n -
, when (~,p) = 1 ,
~)a n - ~
Thus again d/m S _ ~ I . n = p.
Then we have again, by (i),
= al ( nn-~- i ii , when (n - m,p) = I.
This covers the case (n,p) = i and the case
For the r e ~ n i n g
case n = m p
9 p, we can proceed by
induction. Let then n = mp, m > i. a +I = O w h e n e v e r
Now use (2).
This shows that
either Plm or when p~ m + i and a m = O.
arp+s = 0 for r > l ,
and i ~ s
Therefore
<_p - I.
In other words
a As
a
n-m
= 0 when (p,m) = i and m > p.
= am
, and as n > 2p, it follows that a m
In other words
r(xj)
= r1(xPjP)
,
= 0 whenever
65where FI is homogeneous of degree m < n and clearly satisfies conditions (P).
Hence
rl(x,Y) = aCm(xJ). It remains to be shown that
Cm(XP,YP ) = bCn(X,Y).
If m is not a power of p, then Cm = CBm,
Cn = B n and the result follows
from the corresponding result for the polynomials B k. power of p we work over~., BmCxP,YP) = [(X + Y ) P -
If m is a
We have BpCx,Y)] m - X pm - ypm
(X + y ) p m _ xl~n_ ypm +
~ (-l)Z<m) Bp(X,Y) r
(X + y)p(m-r).
r=l This is = Bn(X,Y)
(rood p2), as m = O,
each term under the s ~ a t i o n
Bp(X,Y) - 0 (rood p) and so
sign is = 0 (rood p2).
On dividing
through by p we finally get
c C x P J P) - CnCXJ)
(~d p).
This completes the proof of Theorem la.
T.~L~ 3 Suppose F and G are formal groups and F=G+aB n
(rood deg n § l).
Then there exists a_ power series f(X),
r(F(r-~X, r-~)) -- G(Xj)
f(X) = X
(rood deg n), so that
(~a ae~ n * i).
66-
PROOF
Put f(X) ~ X - bX n (mod deg n + I).
proper choice of b,
f(F(X,Y)) ~ G(fX,fY)
We show that,for a (mod deg n + 1).
We
work modulo degree n + 1 :
f(F(X,Y)) ~ F(X,Y) - b(X + y ) n G(X,Y) + a(X + y)n - el, n _
G(fX,fY)
=- G(X,Y)
ayn_
b(X + y ) n
- b X n - bY n.
The right congruence is obtained by taking b = a.
Suppose F and G are formal groups and ,
|
F ~ G + aC n
(mod deg n + i).
T h e n for m E Z,
~mlF(X) - [m~G(X) + a{r n(m n - m) }X n
Cmod deg n + i),
where ~n = i, when . .n .is .not a prime power,
cn
=i
-q , when n -is a_power of the prime q, (Note: Cn
n n ),
= c B
and where
{en(m n - m)} the integer
PROOF m.
stands for the element of R which is the image o_~f
en(mn - m).
The !emma is clearly true for m = i.
Proceed by induction on
Write Am(X) for the polynomial of degree _< n which is congruent
(modulo deg n + i) to [m]F(X) - Em~G(X).
Working modulo deg n + I,
-
67
-
we have
[ m + 1IF(X) = F([m]F(X),X)
- G([mSF(X),X) + a Cn([m]F(X) , X)
aC[m]G(X),X) + ~mCX) +
a
OnCe, x)
[m+ I]GCx) + %(X) + a CnC~,X)o Over Z,
Bn(mX,X) = (rex + X) n - (reX)n - X n,
- ((m + l) n - =n _ l)X~o Therefore Cn(mX,X) = Cn((m + i) n - m n - I)Xn
Hence Am+l(X) = ~m(X) + a{r ((re+l)n n
.
m n - 1)}X n.
By the induction
hypothesis, this is equal to a{~n(m n - m + (m + i) n - m n - 1)) X n,
which is a{~nC(m + I) n - (m + l)))X=. THEOREM 2
A f o r m ~ group F i s isomorphic over R to the additive
G a i_~fand oml[ ~
for all primes p,
[P]F has coefficients
in pR.
Recall that Ga(X,Y) = X + Y.
PROOF
If f : F § G is an isomorphism of formal groups, then
-
f-l
o
~
f"
That ~ ] F has coefficients in pR for all primes p is therefore a property of isomorphism classes of formal groups.
Since [n]G
(X) = nX, a
this shows that the condition is necessary.
- 68-
To prove the sufficiency of the condition we construct a sequence ~gn } of invertible p ~ e r
series such that
gn+l - gn
(rood deg n),
m
gn ~ F 9 gn I = X + Y (rood deg n + i).
The sequence (gn) is a Cauchy sequence, with limit g, say. Hence
g ~ F @ g-i = X + Y
for n arbitrarily large.
(rood deg n),
Therefore
g @ F ~ g-i = X + Y.
Construction of (gn }
:
constructed gl'""" 'gn-l'
Take gl = X.
Suppose we have ~lready
and suppose
gn_l o F - gn_-I = H - X + Y
(rood deg n).
It will suffice for us to construct a power series f such that
f ~ X
and
(mod deg n),
f o H @f-i ~ X + Y
(mod deg n + i),
for then the required gn can be taken to be f Q gn-l" By Lazard's theorem (Theorem !),
H for some a ~ R.
X + Y + aCn(X J )
(mod deg n + I),
If n is not a prime power, then aC n = aBn.
If n
69
is a prime power, n = pr, then by Lamma 4
(rood deg n + i),
[P]H (X) ~ ~ ] G (X) + aCp n-I - I)Xn a
pX + a(p n-I - I)X~a
But by our hypothesis
(mod deg n + i).
[pSH(X) h a s coefficients in pR.
a ~pR, and a = pb for some b e R.
Hence
This implies aCn = bB n.
We have shown then that (rood deg n + i).
H -- X + Y + mBn(X,Y)
By L ~ m a 3, there exists f with the required properties. COROLLARY I.
(Independent of Lie theory)
If R i_~sa Q-algebra
the_~n every commutative formal group o_~fdimension 1 is isomorphic ove_~rR t o the add%tire ~ . COROLLARY 2 .
group F defined over R is ~
a fo~l =
Let R be a ring with pR = O, p a prime number. t
o
G
Taen
if and only if
o.
COROLLARY 3.
Let R b e ~ local r i ~
prime characteristic p.
~hos 9 r esidu 9 class field is of
Then a fpr~Aal group F defined over R is
isomorphic to the additive ~
i_~fsgd 'only i_f_fthe coefficients o_~f
~ ] F lie inn pR. w
CLASSIFICATION OF COF2~JTATIVE F O P ~ L GROUPS OF ONE DI~ENSION OVER
A SEPARAB_LY CLOSED FIELD 0F, CK~A. CTERISTIC p. (p) 0) Let k denote our base field, of characteristic p.
For formal
-70-
groups F and G (over k) and f ~ HOmk(F , G), f is a power series in h X p , where h = ht(f). (cf. i, w Th. 2). More precisely, we have, h
h
f(x) = alxP
PROPOSITION I
+ a2x 2p
(i)
ht(f + g) ~ inf
(ii)
(i)
PROOF
(ii)
{ht(f),ht(g)} .
ht(f @ g) = ht(f) + ht(g).
has been proved already (I, Put n = ht(f),
n
f(x) = a x p
+ ... , a I # o.
m = ht(g).
w
Prop.5).
Then
n
+...
, g(X) = b x p n
+...
, aW0,
b#O.
n+m
n
Therefore f(g(X)) = ab p
Xp
COROLLAR_Y 1
if and omly i__ffu i_~san invertible powe r series,
ht(u) = 0
+ ... , and ab p
# O.
i_~nwhich case ht(u o f @ u -1) = hi(f).
COROLLARY 2
If we consider Z with the p-adic filtration, and
E n d ( F ) with the height filtration, then Z § E n d ( F )
i_~scontinuous.
We define the height Ht(F) of the f o r ~ l group F to be ht( ~ ] F ) .
By Cor. 1 to Prop. i, Ht(F) only depends on the isomorphism
class of F.
COROLLARY 3
PROOF
I_.ffHt(F) # HZ(G), then Homk(F,G) = 0.
If f s H
F,G), then f @
ht(f) + Ht(F) = ht(f) + Ht(G). ht(f) =
~
, aud f = 0.
F
-
G
o
f.
Hence
Since Ht(F) # Ht(G), then
-71-
PROPOSITION 2 PROOF
Homk(F,G)
is complete under the ~
filtration.
Let {f } be a Cauchy sequence under the height filtration. n
Then it is a Cauchy sequence with respect to the order filtration, and ord(g) = ?t(g).
Put f = limord(fn).
Then, working modulo degree n,
we have
fC;(xj)) qC;Cx,Y))--GCq(x), qCY)) oCfCx), fCY)). Hence f s HOmk(F,G) and f is the limit of {fn } under the height filtration. COROLLARY Zp § End(F)
The homo morphism Z § End(F) extends t_~oa homomorphism (where Zp denotes the p-adic integers).
The d i ~ , r , ~
k
of ring homomorphisms commutes, as all the maps preserve identities. Thus D(~]F) = O, i,e., ht([P]F) > O, i.e., Ht(F) > O. 2 to Theorem 2 of the last section, Ht(F) = isomorphic to the additive group G a.
if and only if F is
Thus if F is not isomorphic
to the additive group, i.e., if Ht(F) < ~ is an embedding.
By Corollary
then the map Zp § End(F)
-72-
The three main theorems which follow give firstly the existence of formal groups of prescribed height h > 0 over auy field of characteristic p, secondly the complete classification of formal groups over a separably closed field, and thirdly the determination of End(F).
THEOREM i
Given a positive integer h, there exists a formal group F h defined over GF(p), so that ~ F ( X ) = X p .
THEORF~ 2
(Lazard)
I fk
p, then formal ~ F
is ~separ~o!y closed field of characteTistic
and G defined oyer k are k-isomorphic if and
only i_~fHt(F) = Ht(G).
THEOR~ 3
(Dieudonnd - Lubin)
Suppose k is a separably closed
field of characteristic p, and let F b_~e~form~! group defined over k with Ht(F) = h < ~ 9rder~
.
Then Em~(F) i_Lsisomorphic to the maximal
in the central division a l g e b r a ~
o_~finvariant I/h and of rank h 2
over ~ .
The last theorem is due to Dieudonnd in the weaker form that Endk(F) is isomorphic to some order i n ~ .
That this is actually the
maximal order was proved by Lubin, using results on formA2 groups over discrete valuation rings.
We shall give a direct proof.
l~e shall need some l~mm~.
Let R be a discrete valuation ring
with finite residue cl~ss field of pS elements. m~ximal ideal of R, and take w in R so that
L~A
I
Denote by ~ the - wR.
Suppose f(X) s@d g(X) are power series over R satisfying
-73-
f(x) z g(x)
~x
(mod deg 2),
f(X) = g(X) = X q where q = pS~
(mod ~% ).
for some positive integer ~
. Let L(XI,...,X n)
Then there exists ~owgT series
be a linear form over R.
F(XI,...,X n) over R sat%sfying th_~econditloms
(i)
FCxI,...,~) z T.CXl,...,X)
(mod deg 2),
(ii) f(F(Xz,...,~)) = F(g(Xl),...,g(X)). These conditions determine F u~iquely over the quotient field of R. (This is a slight variant of a lemma of Lubin - Tate. ) PROOF
Our aim is to construct a sequence
over R in Xl,...,X n
with the properties
{Fm} :
F(X ! , . . . , X n) is of degree m Fm(XI,...,X)
---L(XI,...,X)
of polynomials
l,
(rood deg 2),
fCFmCx1,...,x)) : ~(gCXl),,..,gCx))
(rooddeg z),
=
where A(~,...,Xn) is a homogeneous polynomial of degree m.
These
conditions imply (here we work with congruences modulo degree m + i) that
~+zCg(Xl),...,~CXn)) = FmCgCXl),... ,gCXn) ) , AC~Cxz) ,... ,g(X)) -- FCgC~),...,gCXn) ) * AC~Xl,...,~X) = ~(gCX I),... ,g(~) ) + m ~(Xl,... ,x).
- 74-
If we write f(X) = ~X + f(2)(X), then we also have
fC~+lCXl,...,X)) = fC~Cxl,...,x) + ACXl,...,X))
,~c~,...,x~
+
-A(Xl,...,~
+
q2~c~(x,_,.o.,X~
- fC~Cx~,... ,x) ) + ,ACx 1,... ,x). We are therefore required to find A satisfying the congruence
FmCgCX1),...,gCXn) ) + ~m hCXl,...,Xn) _ fCFmCXl,...,Xm)) + ~ACxI,...,Xn). In other words, we must solve over the quotient field of R the congruence
ACx~.,...,x)
= 1
There clearly exists a unique solutiono unit of R.
Fm(g(Xl) ,.
coefficients in ~ ( i . e .
9
"IX .,~,
n;)~ . f(Fm(~,o..,Xn))
is divisible by ~
).
is a
c~c~,...,~
- c~cx~,...,~l I~ ~: ~$~,...,~
has
Since f(X) = g(X) = X q
), then
~cx~,...,~ ~
i
To shc~r that the solution has coefficients in R we
must show that
(rood ~
But ! - m -
C=d ].- ).
c ~ ~ ~.
-75-
(Fq
denotes the polynomial obtained from F
m
by raising all the m
coefficients to the q-th power).
As q is a p~rer of the cardinality
of the residue class field, we have F~ = Fm.
Hence
(mod ~ ), and therefore
Fm(gCXl),...,g(Xn))- f(FmCXl,~
-0
(mod ~ ) ,
as required. We are now in a position to prove Theorem i.
(We use here
au idea which plays a central role in Lubin - Tate.)
PROOF OF, T H E O,R ~ i
In L e m m a l , take R = Zp '
some positive integer h,
~ = p, q = p
h
for
f(X) = g(X) = pX + X q, sad L(X,Y) = X + Y.
There then exists a power series F(X,Y) over Z
such that P
F(X,Y) ~ X + Y
(mod deg 2),
and F(f(X),f(Y)) = f(F(X,Y)).
But the power series F(Y,X) is also a solution of our existence problem.
By the uniqueness of solutions therefore we have
F(X,Y) = F(Y,X). ~ith L(X,Y,Z) = X + Y + Z, we easily check that the corresponding existence problem of the lemm~ has both F(F(X,Y),Z) and F(X,F(Y,Z)) for solutions.
From the uniqueness of solutions we deduce
-76-
F(F(X,Y),Z) = F(X,F(Y,Z)).
Hence F(X,Y) is a commutative formal group over Zp, and f lies in En~
(F), with D(f) = p. Since D : E n ~ (F) § Zp i~ injective, P P h and D( ~ ] F ) = p, then f = ~ I F. In other words [plF = pX + X p The homomorphism Z
§ GF(p) induces a functor from the
P
category of form~1 groups over Z
to the category of formal groups P
over GF(p).
The image ~ of F is then a formal group over GF(p), h with [p]~ = X p . This then yields Theorem I. h From now on h is a fixed positive integer, and q = p .
L~SL~
Suppose k is a field of characteristic p, and F i_~sa
2
formal group o_~fheight h to a formal ~ G ,
define a
oyer k.
Then F i F k-isomorphic
where G : X + Y + CCq(X,Y)
(rood deg q + i)
and c # O.
PROO F
We kn~r that F = X + Y
F = X + Y
(rood deg 2).
(nod deg n) with n < q.
F - X + Y + cC n
By Th. I of
(mod deg n + i) for some c E k.
c Cn = bB n for some b E k (all primes p" n = p w
k
we
Suppose now that w
we have
If n ~ pk then
# p are units in k).
with k < h we assert that cC n = 0 (= Bn).
If
For by Lemma 4 of
have (Ga denoting the additive group)
[P]F (X) =- ~ ] G (X)
+
(rood deg n
c(-I)X n
+
i).
a
But
: 0
(nod deg
[PIG (X) = pX = O. a
(rood deg
n + I)
and
so
c = O.
Hence
cX n
-= 0
-77-
Thus we have shorn that if F - X + Y (rood deg n) with n < q then there is a b e k such that F = X + Y + bB Now apply Lemma 3 (w
n
(rooddeg n + 1).
to obtain an invertible f such that
f o F o f-1 = X + Y (mod deg n + I). F = X + Y (rood deg q).
We can therefore assume that
If we had F = X + Y (rood deg q + l) we could
apply Lemma 4 (again with m = p) to obtain [p]F(X) - 0
(mod deg
q + l), which contradicts the hypothesis on the height of F. We state next a lemma in ~hich (for the first time) essential use is made of the hypothesis that k be a separably closed field.
L~A
3
Let k b_~e~ separ,ably closed field of characteristic p.
sup~se g(X) = f(x q) with g(o) = 0 and fl # O.
~ e n there is Em
invertible power series u (over k) such that u -1 ~
g
o u =A-.
PROOF ' Let g(X) ~ aX q (mod deg q + i), ~here a # O. separably closed, there exists c e k, cl'q = a.
~2Cx) = C~[I
o
g
o
But g2Cx) is a p~r
v I) Cx). ~ e n ~2Cx) ~ x q
As k is
Put Vl(X ) = cX,
Czo~ de~ q + i).
series in X q and hence g2(X) ~ X q (mod deg 2q).
We now prove for r > 2 :
If grCX) is a power series in X q
gr(X) ~ X q (mod deg rq) then for a suitable choice of b in Vr(X) = X + bXr we have
(~!
gr ~ ~ )
(X) - X ~
( ~ d deg (r + 1)q).
Starting with g(X) and defining inductively -1 v
~176
-1 Vl ~ g
~ Vl~176
~--~r
- 78-
we obtain an infinite product v I @ ... @ Vr @ Vr+l o ... , which under the order filtration converges to a v(X) so that
(v-x ~ g
o v) ( x ) - - x q.
Let gr(X) = X q + aXrq throughout).
( we use congruences rood deg rq + I
Then
gr(Vr(x)) ---x q
+
(a
+
b~)~ ~
and Vr(Xq) ~ x q + bx rq.
We have then to solve
bq-b+a--O
in the unknown b. be solved in k,
This equation is separable, and hence can Thus
(~i o ~
o ~ ) (x) = x q
(rood aeg rq + i)
and hence also rood deg (r + i) q.
DEFINITION
A formal g r o u p F of finite height h over a separably
closed field k of characteristic p, is in n o ~
Ci) (ii)
~]rCx)
--
xq
_form if
(q -- ~ ) ,
F(X,Y) -- X + Y + cCq(X,Y)
(rood deg q + l)
for some c # 0 in k. Our next lemm~ shows us that for formal groups in normal form we can work over GF(q) rather than k.
-79-
l_~f [P]F = Xq then F is defined over GF(q) ap.d ever~
L~a~{A 4
eu%omorphism o_~fF i_~sdefine__.___~doyerGF(q).
PROOF
Since GF(q) = {a e k I a q = a}
we have the following
equivalence : g(X1,...,X n) = g(X) is defined over GF(q) g(X) q.
<=> g(X q) =
Now [P]F ~ F = F o [P]F so F is defined over GF(q) ~ e n
~ ] F : Xq"
Moreover ~ ] F is in the centre of E n d ( F ) so if f e En%:(F)
we have [ P ] F
o
f = f~
[P]F'
i.e. f is defined over G F ( q ) .
The next lemma is the crucial one for the proofs of Theorems 2 and 3.
Yt ties lenmms 2,3,4 together.
Each formal ~ o v e r
a separably closed field k i_~s
i sgmorphic to one in normal form.
PROOF
Let F be a formal group of height h over k.
Apply L e ~
to [P]F : there is a u(X) such that u -1 ~ [p] F - u = X q. U
u
~
-i
"
F
~
U =
F o u = F.
But
for all integers m ~ i, and
u -1 o F o
u
So we may assume F is such that
[p]F
by Lemma 2, there is an invertible v(X) in GF(q)[[X]] over GF(q) by Lemma 4) such that (v"! o
(rood deg q + 1), with c # O.
3
F
Then
(F is defined
@ v) (X) = X + Y + cC
Now ~] -i v ~ F"
(since v is defined over GF(q)).
= X q.
Thus v -I ~ F
= V -1
o
[~]
q F o v-
V " v is in normal form.
So we can now assume whenever it is convenient that all formal groups are defined over GF(q), where throughout q = ph. Define a category ~ as follows : objects, all formal groups F in normal form and of height h over k;
[p] F
morphisms, ~.1! homomorphisms
-80-
of form~1 groups (in ~ ) over k.
~ is "additive" (cf.
w
Next let M be the module, under ordinary addition, of h-i [ i=O
polynomials of form
i a. X p ~
= a(X),
a. 6 OF(q).
M is in
the obvious way an h2-dimensiona! vector space over GF(p). define a multiplication
~e
on M by
~
(a @ b) (X) m a(b(X))
(mod deg q).
This makes M into a ring q-1
if f
f(x)
-
=
Z
f
j=1
9
J
xJ
~rite ?
,
-"
?(x) -- [ j=l
f
9
Then we have PROPOSITION 3
The map f : > ~ defines a functor ~ §
"addi'tive" categories. + g, I = i.
ExpliCitly f ~Jg = ~ @ g, f ~ g =
Moreover, f ~ - ~ i.s asurjectigp Homk(F,G) § M,
for any pair o_~fformal ~ F , G PROOF
~e s h ~
first that ~ ~
homomorphism, f o form,
~]F
=
M of
[PIG
[P]F =
=
Xq"
in
M
~.
for f 6 HOmk(F,G).
Since f is a
[PIG o f, and since F and G are in normal We deduce that f is defined over GF(q).
Next by (ii) of the definition of normal form we see that f(X + Y) - f(X) + f(Y) polynomial in Xp.
(rood deg q), and so rood deg q, f is a
Thus ~ ~ M.
It is clear that f ~
g = ~ o ~, and ~ = 1.
f + g (x) = G(f(X), ~(X)) ---f(X) + ~(X)
(mod deg ~)
G
of the definition of normal form).
Now
(again by (ii)
Therefore f + g (X) = f(X) + g(X) = G
~(x) + ~(x) since deg ~ < q and ~ g ~ <
~.
-81-
Finally we show that the map is surjective. suffices to show that given a s M ~ i t h
For this it
first coefficient a ~ # O,
there is am f ~ HOmk(F,G) such that ~ = a, for these elements generate M (as an additive group).
~i~
As usual, we produce an f
the =mpleteness of GF(~)[[X11.
~e oonst~ct a sequenoe ~fn ~
of invertible power series (for n ~ q )
f
f
q
=
with the properties :
a I
o F - G
f
@
n
(nod deg n),
n
f+l
{ f
(mod deg n).
Suppose we have reached f .
Put H = f-i q G
m
F - H
(nod deg m).
m
By Lazard's theorem (w
c ~ GF(q) such that F - H + cC
(mod deg m + l).
o
f .
Then
m
there is a If m # p~ then
m
CCm - bB m for some b s GF(q).
ocm - o
Csi~ce
b s GFCq).
If m = p
[p]F -- ~]G -- ~]I~)" ~
Now apply Lemm~ 3 of
w
then by La~,~a 4 of w
F : H + bB~
to deduce the existence of an
invertible power series u over GF(q) such that u (mod deg m + 1), and uCX) - X (mod deg m). It is clear that fm+l = f m the induction step. f * HOmk(F,G).
@ F ~ u -1 -- H
Put fm+l = f m
" u.
(nod deg m), and so we have completed
Now put f = i ~
fn"
We see that ~ = a, and
This completes the proof of the proposition.
PROOF OF T H E O R ~ 2. form.
(zod d ~ z + i).
]~e can assume both F and G are in normal
Choose f % HOmk(F,G) such that ~ = i, (surjection of Prop.3).
This implies that f(X) = X (rood deg 2), and so f is an isomorphism.
- 82-
For literature on the arithmetic theory of division algebras over %
see M.
Deuring, Algebren, J,P. Serre, Local. class field
theory i Appendix (Brighton notes). PROOF OF THEORF~[ 3
We shal! ~ i t e E = Endk(F).
We split the proof
into five steps :
3)
E is a f r e e Z -module of rank h 2 ; P ~ = E | Z Q~ is a division algebra of rank h 2 over Q~ P E is the maximal order of ~ (over Zp) ;
4)
The centre cent (~) o f ~
5)
The invariant inv(~) of ~
i) 2)
is %
;
is I/h.
~Te first show that (A)
pn E = {f e E I ht(f ) ~ n h
Clearly if f =
[pin F 9g
= n.ht(~]F)}.
then hi(f) = n.ht
Conversely, let ht(f) ~ m h .
([p]F ) +
ht(g) i nh.
This means that there is a pm~er
series g(X) so that f(X) = g(xqn), i.e., so that f = g @ ~
must show that g e E.
(remark ~ e r L e m m a S )
Since F is defined over GF(q)
we have n
n
n
f(F(X,Y)) = g(F(X,Y) q ) = g(F(X q ,Yq )), and n
F(f(X,Y)) -F(g(X
n
), g(Y )).
Comparing the two expressions we deduce that
gCFCXJ)) = FC Cx), gCY)) as required.
F"
- 83
-
Now E is Hausdorff and complete under the height topology.
Hence
(~ pn E = 0 and E is a complete topological Z -module. As E contains P Zp and has no divisors of zero (cf. III, w Prop.2), E is a torsionfree Zp-module.
By the preceding Prop9 3,
finite dimension h 2 over GF(p). Z -module of rank h 2. P By i), As E has
~
E/pE
=
~'
i.e
,
E/pE is of
Therefore E itself is a free
This gives assertion 1).
= E ~ Zp~
no zero-divisors,
is an algebra over ~ ~
of dimension h 2.
is a division algebra.
~e shall denote multiplication in ~ write f.g for the product of f and g.
in the usual way, i.e.,
If f and g happen to be in
E then of course f.g coincides with the composite power series f o g. Thus i n p=-tio
= for f
E, pnf =
o f.
To establish 3) we first reck1 1 that the normalized p-adic valuation v of ~ , with v(p) = I, again to be denoted by v.
has a unique extension to
On the other hand ~ ht : f ~ +
ht(f)
is a valuation of E, whose restriction to Z
coincides wi~h v. P i Hence ~ ht can be extended to a valuation of ~ , and by uniqueness this is the same as v. (B)
In other words we have
ht(f) = h.v(f), The m a x i m a l order N o f ~
Thus clearly E C N9 element g of N. n >_ O.
for f ~ E. is t h e set
For the opposite inclusion consider an
Then a~ E spans J~ over 0~, png E E for some
Now v(png) >_ n, and so by (B),
ht(png) >_ nh, whence by (A)
- 84-
pg=
for some f e E. g ~ E.
o f=
As~
Thus N ~ E ,
f,
is torsion-free, this implies g = f, i,e.,
and hence N = E.
For 4) we first note that it suffices to establish the equation
cent(E)
= Z
P
As Z
c cent(E), it will suffice to show that the Z -rank of P P cent(E) is < 1. If f 6 E, pf 6 cent(E) then f ~ cent(E). Thus cent(E) is a direct summand of E, and therefore its Z -rank P coincides with the dimension over GF(p) of its image cent(E) By Prop9 3 the map E § M is surjective, whence
in the algebra M. cent(E) C
cent(M).
It thus remains to be s h ~ m that the dimension
of centCM) over GF(p) is at most i9 Let a(X) = a~ X ,
h-1
b(x) = X b
.xpj
j=O be two elements of M.
Then o
h-1 (a o b)
(x)-
X
a0bj xp~ .
j=0
h-i (b 9 a)
pj " % b. x p~.
(x) = j=O
For b(X) to lie in the centre of M it is thus necessary that for all a 0 e GF(q) a n d a l l
j = l,
.9
h - i,
pJ
b j ( a 0 - a~ ) = O9
But i f
-
85
-
pJ a 0 is a primitive element of GF(q) then a0 # ~0 and so we must have b. = 0 for these values of j. central element is of the form of a(X). a(X) E cent(M), ~ i l e
(J = i, ..., h - i), In other words a
So now suppose that
bCX) is arbitrary.
If we choose b. = I for
all j we get the equation a 0 = a~, i.e., a 0 ~ GF(p). cent(M) is of dimension
( 1.
Thus in fact
(of course one has equality here).
To prove 5) we first recall the definition of inv ( ~ ) most convenient for our purpose.
There exists an element g o f ~
so that
for all f ~ E
(c)
gfg-l : fp
(mod
) I
is the m a x i E ~
(~ro-sided) ideal of E.
g is of course not unique but
the values v(g) of such elements g form a unique coset mod Z, which is the invariant o f ~
.
One may, by multiplying through by elements of
cent (J~) = 0~, suppose that 0 ~ v ( g )
< 1.
One then has to show that
1 for such a g we have v(g) = ~ . Let then g satisfy (C), and assume that v(g) = K 0 ~ ~ ~h
- 1.
l~e sh~11 sh~r that
~ = 1.
,
From (C) we have,
on multiplying up by g,
N~
we can translate our statements into the language of power
series.
We have a power series g(X) of height ~
g(x) so that for all f(X) ~ E
(mod deg
pK+l)
,
, i.e., with
-86-
(g o f) (x) ~ (f(P) o g) (x)
(mod deg pK+!),
where
f(P) (x) = (f @ f o ... o f) (x) f
~
If f(x) = Zl x + .... , then f~P' (X) = ~ X hence, mod deg p
(p times). + ... , ana
K+I K
(go
f) (x)
(f(P) @
a~
x,
g) (X) ~ af~l X,
K
i.e., a ~1
= a ~1.
As fl can be any element of GF(q) (by Prop. 3)
and as a # 0 it follows that K
=
i.
This completes the proof of the theorem.
w
Galois cohomolo~, Let F and A be topological groups, and suppose A is a F
-group,
so that the elements of r induce automorphisms of A and so that the map
F x A §
is continuous.
For 7 ~ F , a e A we denote by
7a
the image of a under the map defined by Y 9 A cocycle of r in A is a continuous map a : r § A w h i c h satisfies the relation
aC~c~) = a(~c).
"%(~).
We denote the set of cocycles of r in A by zlCr,A)9
Note that
ZI(F,A) is a set with a base point, viz., the trivial cocycle which maps each element of P onto the identity of A. b 9 A, the equation
For a ~ ZI(F,A) and
-87-
= b -1
aC ).
defines a cocycle a1 * zl(r,A).
Yb
~;o cocycles which are related by
such an equation for some b ~ A are said to be associated.
This is
an equivalence relation, and the equivalence classes in zl(F,A) thus defined are called the cohomo!ogy classes.
The set of
cohomology classes is denoted by ~(F,A), which again is a based set with base point the class of the trivial cocycle.
The cocycles
associated with the trivial cocycle are called splitting coc~cles, and they are given by
aC )
=
b-1.
for some b ~ A. Consider now a field k of characteristic p, and let K be a norm~Ll separable extension of k.
Denote by r the Galois group Gal(K/k).
For k I a finite field extension of k in K, we write
Ak! = {Y ~ F I Y
leaves k I fixed elementwise} .
A topology on F is defined by taking as basis of open neighbourhoods of the identity the subgroups Akl for all finite field extensions k I of k in K.
With this topology, a continuous ~ e m e m e ~
map
a : r § A of topological groups has the follc~Ting interpretation : TaP~e y ~ r , and U a neighbourhood of a(Y).
There exists a
finite extension field k I of k so that whenever 8 * F effect on k I as
has the same
Y , then a(6) ~ U.
We state the following two 'le~,~' i~ithout proof.
(K+ denotes
the additive group of K, and K* denotes the multiplicative group
-88-
of the non-zero elements of K).
~sa I
HI( r,K+) -- o.
(This is a consequence of the Nor~1
basis theorem. )
~m
2
(mlbert's
Satz 9 0 ) .
RA(r,K ) - 1.
Let S be the group, with respect to o composition, of power series f(X) defined over K of the form f(X) = flX + f2~
+ ,.. , fl # 0.
A topology on S is defined by the order filtration i.e., by viewing S as a subset of K[[X]].
The action of F on S is defined by
the action of F on the coefficients of the pc~zer series in S.
~lith
this structure we have
PROPOSITION 1
PROOF
HI(F,S) = 1.
Define S (n) = {f(X) s S I f(X) - X (rood deg n+l) }.
is a normal subgroup of S Cexercise for the reader). 1 §
(I) §
§
Then S (n)
The sequence
§
w
where S § K
maps f(X) onto fl' is an exact sequence of
Also, if f(X) f S (n), then f(X) = X + ~ u + l map f(X) ~-~ ~
F -groups.
(rood deg n+2).
The
defines a homomorphism S (n) § K + of r-groups, and
m§
(n+l)§ (n)§ + §
is exact 9 ~,Temust show that, if a ~ Z1 (r,S), then there exists b t S such that a(7) = b -1 ~ A .
- 89-
Then a(y) = [ ar(Y)X r , ar(7) s K. r=l
Take a E ZI(F,S). The map
Y ~ ~alC7) is a cocycle r § K , and hence by Lemma 2
there exists b I 6 K
and define b(1)(X) map
such that blal(Y) = 7b I.
~ a(7)
~ (7b(1)(X))-i
Y ~-~ a(1)(Y) is a cocycle r § s (I).
aCz)(~) =
x
then the map y ; ~ a~ 1)
+ a~ z) ~
,
Take b(1)(X) = blX
= a(1)(7) e S (1).
The
If we write
...
,
i s a c o c y c l e F + K+.
By Le~m~ I there
exists c 2 E K + such that
a~z)(~) = Yc 2 -
Take cCX) = X + c2X2.
c 2.
Then cCx)
o aC1)cy)
This way, we get a Cauchy sequence (b(n)(x)}
b(n)(x) Put b = nl~
bCn) (X).
a(~)
=b
@
(Yc(X)) -1 = a(2)Cy) E S (2) such that
@ aCy) o (YbCn)cx)) - 1 E
S (n).
Then
-I
o
~b.
Let F be a formal group of height h defined over k and fixed once and for all. k
If G is another formal group defined over
and f : F § G is an isomorphism defined over K then
and so Yf : F + G is an isomorphism, a(y) = f - 1
It follows then that
7f is an automorphism of F (defined over K).
Also,
- 90-
a(y6) = f-to
~f
= f-l o ~f o ~f-1 o y6f _ a(v)~ ~ a(~).
Let k I = kCfl,,..,f n) be t h e field obtained by adjoining the first n coefficients of f to k.
Then if Y and ~ have the
same effect on k 1 we have 7f ~ ~f (mod deg n + 1), and hence a(y) = a(6)
(mod deg n + 1).
Thus a is continuous.
Hence
if f : F § G is an isomorphism (over K), then a(7) = f-l~
yf
defines a cocycle of r in AutK(F). Every other isomorphism F § G is of the form f o g for g ~ AutK(F).
Since
,h Cy) _- ( f .
~)-s.
u
o g) _ g-1 o f - ,
. ~.f o ~,g _ g - l o
aC'r) ~ "rg,
then we can associate uniquely with G the cohomology class of
z-I ~ ~f -- a(~c). Suppose now further that G and H are isomorphic formal groups over k and that 9f : F § H
s : G § H is an isomorphism defined over k.
Then
is an isomorphism defined over K and
= f - i o 7f,
since
7A = A .
G and H are therefore associated with the same
class of ~(r,AutK(F~). Denote by ISOK/k(F) the set of k-isomorphic classes of formal groups which become isomorphic to F over K. defined a map ISOK/k(F) § HI(F,AUtK(F)).
THEOP/~4 I
ISOK/k(F) § ~(F,AUtK(F)) i_~sa_bi~ection.
We have then
-91-
PROOF
Suppose G and H are smsociated with the same cohomology
class.
Let f : F § G,
~ : F § H be K-isomorphisms.
Then there
exists g s AutK(F) so that f-i o yf = g-I o A-I ~ YA o yg.
Rearranging we get
o g"
f-l=yz
~ ~gO
~f-l=v(~.
g ~ f-l).
The K-isomorphism A o g o f-i is therefore fixed for all Y s F. It follows that A 9g 9
f-i : G § H is a k-isomorphism.
Thus
the map ISOK/k(F ) § HI(F,AUtK(F)) is an injection. Let a s ZI(F,AUtK(F)).
Since ZI(F,AUtK(F)) c
ZI(p,S),
then by Proposition i there exists f s S such that
a(~)--z - I o
~f
,
for all Y & F.
But if
GCxJ) = ~Cf-iCx),f-iCY)) then f : F § G is an isomorphism of formal groups.
Moreover
~G(X,Y) = f o a(~) F(a(~)-I~ f'l(X), a(~)-I ~ f-1(y)) zFCz'iCx), z-ICY)) - GCx,Y), as a(y) ~ AutK(F).
This being true for all T s F we conclude that
G is defined over k.
It maps onto the cohomology class of a.
Thus we do
have a surjection. Let now I(k,h) be the set of k-isomorphisms classes of form~1 groups of height h.
By
w
Theorem l,
-92-
I(k,h) - IsoK/k(F) when K is a separable closure of k.
We thus obtain a full
classification of formal groups of height h from the theorem. COROLLARy
I f K is a separable closure of k, then
I(k,h) ~ ( r , A u t K ( F ) ) . Note that by w
Let h = I.
Theorem 2 ~e know the group AutK(F).
Then by w
Th. 3 En~(F) = Zp for any K
and hence AutK(F) = Up, the group of p-adic units.
The Galois
group r leaves Z C EndK(F) element~ise fixed, hence leaves the closure Z
P
of Z fixed.
Thus r leaves AutK(F) fixed.
But then
Hl(r,AutK(F)) = Hom(r,AutK(F)) is just the set of continuous homomorphisms r § AutK(F) , i.e. of continuous homomorphisms r+U.
P Now let the field k of definition of F be a finite field
and let K be its algebraic closure.
Then, as a topological
group, the Galois group r is generated by the Frobenius substitution : ~ ~-~ mr , where k = GF(r) (r a power of p).
Moreover, for each
element ~ of Up there is one and only one continuous homomorphism r § Up which takes ~ onto ~ .
Thus we can identify Ham(r,Up) = Up
and ~le end up with a bijection
I(k,1) = !SOK/k(F) ~
~ Up
~henever k is a finite field, and the height of F is i.
-
93
-
~[e can use the preceding Theorem 1 to derive a classification of formal groups of finite height h over a finite field k, which is due to J-P. Serre.
Let F be such a group, fixed once and for all, and write
E = EmdK(F) , K being an algebraic closure of k. Cs
If a e E denote by
its conjugacy class (under inner automorphisms).
Let w be
the normalized p-adic valuation of E @ ~p0~Which__ takes as its set of finite values precisely~, If k has pS elements then ~ i t e
in other words, for f * E, w(f) = ht(f). T
for the set of conjugacy classes S
Cs
of elements with value w(~) = s. Now let G be another formal group of height h defined over k.
Choose an isomorphism
(i) over K.
g : F ~ G
Then
defines an isomorphism EndK(G) = E of ZZ -algebras. Moreover to P ~Tithin an inner automorphism this O is uniquely determined by G. Nine clearly the power series S
t = t(X) = X p is an endomorphism of G, and so
~(G3 = Cs
solely depends on G, and not on the choice of g in (1). THEOREM 2.
(Serre).
The map ~ gives rise to a bijection
I(k,h) m~ T S .
- 94-
S
PROOF r
Let a be the Frobenius automorphism a ~-* ap
of K/k.
As
= GalCK/k) is free profinite on the single generator o it
follows that the map
is a bijection zlcr,u(E)) ~ U(E)
Hence a :
~ a(~) ~ t
(the group of units of E).
is a bijection
(3) zl(r,u(E)) Observe now that if g =
tog=
~ gn Xn then n=l
Og @ t ,
and so when gl # 0 then
(4)
g
-i ~
t " g = c~
C = g
-I
e
a
g
Thus in the map (3) cohomologous cocycles correspond to conjugate elements, i.e. we get a bijection
If now g and e are as in (I) and (2) then, by (4), e(t) = a(a)
~ t, where a is a cocycle corresponding to the isomorphism
class of G under the bijection of Theorem i. through ~(r,U(E)),
i,e.
class of G, and moreover
r
Thus the map ~ factorizes
solely depends on the isomorphism
$(G) ~ T s.
Hence finally r induces a map
I(k,h) § Ts, which factorizes into the product of the bijection of Theorem i and the bijectlon (5) and thus is a bijection. We also note
-95-
PROPOSITION 2.
With e as i_~n(i), (2), End(G) i_~sSsomor~hic to the
subring o_~fE of elements commuting ~rith PROOF byt
e(t)
In EndK(G) the ring Endk(G) is characterized by qm = a, i.e. o
a =
COROLLARY i
m~ Endk(G) is al~ays the maximal order of the O~-a!gebra
i_t_tspans. COROLLARY 2.
There exists a group G defined over k, and of height h
wit__~hEnd(G) = EndK(G), if and only i_~fh divides s. For, the set of values of w on the centre ~
P
of E is the set
of positive multiples of h. COROLLARY 3.
If k is the prime field then End(G) is commutative and
its field of ~uotients i_~s~
ramified of degree h over ~ .
In fact in the algebra 0~ @ Z
Endk(G) = D, the field 0~(t) P 1 has ramification index at least h, as w(t) = ~ w(p). But as a subfield of a central division algebra of rank h 2,
0~(t) is of degree at most h.
Thus in fact h is its degree and ramification index, and moreover ~(t) is then a maximal commutative subfield of D.
-96-
CHAPTER IV. CO~Iu~ATIVE F O ~
GROUPS OF
D~4EI~ION OI~E OVER A DISCRETE VALUATION RI~G
w
~ e homomorphisms. Throughout this chapter we limit our consideration to
co~.,tative formal groups of dimension I. PROPOSITION i
Let L b_~ea field o f characteristic O, and F
a form~1 r o ~
(commutative, of dimension i) over L.
Then
there exists a unique isomorphism AF : F + G a (the additive group) defined over L, so that ZF'(O) = i.
Suppose now that S i_~s
inte~rsl domain w i.th quotient field L, and that F is defined over S.
Th.en s
~ S[[X]],
(We denote the inverse of the isomorphism AF by eF). (Motivation for notation :
If F is the multiplicative group @@
Gm(X,Y) - X + Y + XY, then s
= log(l + X) =
~ (-i) m-I Xn/n, n=l
and _ eF(X)
-- e x
| X
i =
sT.
n=l
.PROOF By II, Corollary i, g : F § G a.
w
Theorem i, Corollary i, or II! w
Theorem 2,
we see that there exists an isomorphism (over L) Also, D(g) - g'(O) # O.
Now D : EndL(G a) § L is an
isomorphism, since the elements of EndL(G a) are the monomials aX. We can therefore find gl ~ EndL(Ga) such that D(g I) = D(g) -I. Thus AF = glg 9F § G a
is an isomorphism with A'F(O) = i.
uniqueness, suppose f, g : F § Ga are isomorphisms with
To show
97
f'(O) = g'(O).
Then f o g-i 6
Therefore f o g
--I
Aut(G a) and D(f ~ g-l) = i.
.
~s the identity on G a and so f = g.
To prove the second p a r t o f the proposition ( ~ i t e
~F = ~)
we differentiate, with respect to Y, the equation
~(F(XJ)) = ~(X) + ~(Y). We obtain
~'(F(XJ)) F2(XJ) = ~,(Y), where F2(X,Y) denotes the derivative of F(X,Y) with respect to Y. Put Y = 0 : ~'(X) F2(X,O) = 1.
From our assumption on F, F2(X,0)
has coefficients in S and leading coefficient 1.
Therefore ~'(X) has
coefficients in S, being the inverse of F2(X,0). COROLLARY i
Ho~L(F,G)
COROLL~RY 2
D : HOmL(F,G) § L i_~s~bijection.
COROLLARY 3
I f F and G are formal groups defined Over S
= eG ~ Endi(Ga)
o
AF.
then D : HomS(F,G ) § S i_~sinjective. PROPOSITION 2
With the same hypothesis as in Prop. i_~ and if
in addition q i s a ~ the set o f ~
intgger, q > i, then HOmL(F,G) i_~s
power series f (with zero constant term) defined
over L so that
f
PR00F
o
F
G
of.
(*)
By Prop. i, Cor. ~ we only have to shc~r that (.), together
- 98 -
We shall
with the equation D(f) = a, determines f uniquely. establish uniqueness of the partial series f(n)
= qx +
. . .
+ q_l
by induction on n. Suppose that
fCn) o
[4F =- [4o ~ f(n)
f(n) o
[q]F" ~ G
(moddeg n),
i,e., that ~ f C n ) _ cx n
(mod
d e g n + i).
Then
r(n+z) ~ [q]F- [~]G ~ f(n+l) ~ c~ + q ( C as D([q]) = q.
Here we must
f
n
~.
c,
,
q-q n
Note that c l e ~ l y
s
q)~
(rood deg n + i),
have "
is a "polynomial" in fl"'" ~s
!~re precisely we see by iteration that there exist polynomi~!~ n(T)' depending on F, G and n so that fn = Cn (fl).
The unique
f satisfying (.) in Proposition 2, with D(f) = a, is thus
o@
~
n=l
~nCa)~.
Suppose from now on that R is a discrete valuation ring with quotient field K of characteristic O, maxims/ ideal ~ residue class fiel~ k of characteristic p # O.
, and
Let v denote the
-99-
valuation on K given by ~ v(p) = I).
(Note:
. (we take v normalized, so that
we are now e l]~ing filtrations whose values
are real, but not necessarily integral. ) l
COROLLA2Y
Let F and G be defined over R.
Then D(HomR(F,G))
is closed in R. PROOF
D : HomK(F,G) § K is a bijectlon by Prop. i.
D-l(a) in @@
HOmK(F,G) has leading c o e f f i c i e n t a, and hence D-l(a) =
~
~n(a)xn.
n=l
R is a closed set (with respect to the valuation topology) in K, and since ~ is a polynomial it is continuous. n
~he elements a E K
for which # (a) e R therefore form a closed subset C n
n
D(H~
= (~n Cn'
of K.
Since
then D(HOmR(F,G)) is closed.
~e denote by ~ the separable closure k. R § k induces a functor ~ R §
(cf. III,
The homomorphism
w
Prop. 1), under
which F : ~ F PROPOSITION 3
I_~f~ is not isomor~%c t_~o~a then Hom2CF,G) § Hom~ (~,~)
i~sinjectlve. PR00 F
Suppose f : F § G is a non-zero homomorphism so that ~ = 0.
Let (~) = ~ o
Then f(X) = r
g(X) where r > 0 and g # 0.
~rg(F(X,y)) = G(~rg(x),
He have
~rg(y))
= wrg(x) + ~rg(y)
(rood ~ r+l R[~X]]).
- i00
-
Hence gCFCx,Y)) - g(X) + g(Y) and
=
§
r
-
-
g ~
Therefore
=
o g = 0.
Since g # O, then [p]~ = O, i . e . , ~ = ~ a (Iii, w In v i ~ of llI, COROLLARY
w
Th.2).
Cor 3 to Prop. i, we have
If Ht(~) # = , Ht(~) # Ht(~), then HomRCF,G) = O.
Suppose n ~
that F and G are formal groups over R, and
~, G are of finite height.
Then we can define three different
filtrations on HomE(F,G) , viz., the filtration induced by the normalized filtration v on R and the injection D : HomR(F,G) § R (again to be denoted by v); the filtration induced by the height filtration ht on Hom~(~,~) and the injection of Prop 3 (again denoted by ht); the p-filtration where the associated subgroups are
{[~1~ Ho~(F,G)} (denotedby Up). Recall now that two filtrations on a group are said to be equivalent if they give rise to the same topology.
Before
stating Theorem l, which gives the relation between v, ht and Up,
we make the following definition.
A filtration w on a free
Zp-module A of finite rank is called a norm if, for some valuation v' of Zp, equivalent to the p-s~ic one,
w(ca) = v'(c) + wCa),
c e Zp, a 9A.
Any t~ro norms are then equivalent. THEORY4 1
Suppose R is complete.
(i) v, ht and up are equi.'.valent
- 101-
f i l t r a t i o n s o_~nHomR(F,G), au~dHOmR(F,G) is complete uuder these filtrations h = Ht(~).
(ii) HomR(F,G) is a free Zp-module of rank < h 2 (iii) (Lubin) End(F) is a
whose quOtient fiel8 has degree over ~ Remark: h.
commutative Zp-order d~'vid/ngh.
One can in fact shoe that the rank of HomR(F,G) divides
See below (Corollary 3 to Theorem 4 in
w
A Zp-order is a Zp-algebra which is free of finite rank as a Zp-module.
Recall that we alrea~ly know End(F) to be an integral
domain. Note that v and ht are in fact valuations on End(F). Hence
~
v(f)
ht[p] F = V(~]F )
CORO~Y
= v(f)
~uswe
Im EndR(F) , h t ( f ) = v ( f ) .
PROOF OF THEOR~ i
have the
Ht(~).
It follows from the Corollary to Prop. 2 that
HOmR(F,G) is complete under the v-topology.
With respect to the
p-adic topology on Z and v-topologies on End(F) and R,
Z
R is a commutative 0/agram of continuous maps.
z § m
CF) to Z P
R
We may therefore
- 102
commutative. (III, w
-
Since HomR(F,G) is a torsion-free En~(F)-module
Prop.2), then HO~R(F,G) is a torsion-free Z -module. P
If g 6 Im {Zp §
, i.e. D(g) ~ Zp, then for f ~ HomR(F,G)
we have
(*~)
vCf o g) = vCf) + vCDCg)),
where v(D(g)) is the p-adic value of D(g). Now consider End(F) and En~(~) with the height filtration, and Z, Zp again with the p-adic topology.
~Te get a diagram of
continuous maps Z P
-L
which is commutative when Z is replaced by Z, hence remains P commutative now. It now follows that C***) --
is a homomorphism of Zp-mOdules.
m
But Kom~(F,G) is a free
Zp-mOdule of rank 0 or rank h 2 (cf. Lemma i, given after this proof).
Since (***) is an embedding (Prop.B), then HomR(F,G)
is a free Zp-module of rank --
But the
restriction to Z of the height function is a valuation equivalent P to the p-ad/c one. Hence ht is a norm on Ho~R(F,G). In view of
(-:§
v is also a norm on H o ~ ( F , G ) .
Trivia11~, up is a norm on
-
HomR(F,G).
Since all
norms
are
103
-
equivalent,
then
v, ht and Up
are equivalent. We know that, to within isomorphism, ZpC EndR(F) c End(F). Since En~(~) is isomorphic to the maximal order of a central division algebra ~
of rank h 2 over %
(III, w
then the
quotient field of End(F), which contains % , is contained in ~
.
But every subfield of a central division algebra over a field is contained in a maximal such subfield, and if h 2 denotes the rank of the ~/vision algebra then every such maximal subfield has degree h.
This gives the rest of the theorem.
He have still to prove the lemma promised above, viz.,
I~,~l
I_f,f~ is of finite heiF~ht h, then Ho~(~,~) is a free
Zp-mOdule of rank 0 or h 2. PROOF
If Ht(~) # Ht(~), then Hom~(~,~)= 0 (III, w
Suppose then that Ht(~) = Ht(~)= h.
By IIl,
w
Prop l, Cor.3).
Th.2, there
exists an isomorphism ~ : ~ + ~, and hence Hom~(~,~) = ~ o En~(~). The maps g~-~ ~ o ~ define an isomorphismEn~q(~) § Hom~(~,~) of Z -modules. The !emm~ n~r foll~s from IIl, w Th.3. P Suppose now that L is a finite field extension of the quotient field K of R, and denote by S the ring of integers of L. Then Theorem 1 holds for S substituted in place of E. the quotient fields of End(F) and Ends(F) by
He denote
~R(F) and ~s(F)
respectively. PROPOSITION h
l_~fK contains 811 al~ebraic extensions of 0~ i_~nL
-
of ~ I ~ PROOF
104
-
dividing h, then EndR(F) = Ends(F). Since D : Ends(F) + S is injective (Prop.l, Cor 3), then
~s(F) is isomorphic to the quotient field is a subfiel~ of L. of ~
~ of D(Ends(F)) which
By Theorem i, &s(F) is an algebraic extension
of degree dividing h, and by our hypothesis on K,
Consider f a End(F). f'(0) E K.
~le have f'(0) = D(f) E @
~ C K.
, and therefore
By Prop. i, Cor. 2, there exists g ~ End(F) such that
g'(0) = f'(0).
Regarding g as being in End(F), then g'(0) = f'(0)
implies f = g (Prop. i, Cot.2).
Therefore f E EndK(F ) 6~ Ends(F )
= En CF). Note:
Let ~h) denote the composite field (inside some algebraic
closure ) over ~ dividing h.
of all algebraic extensions of ~
One knows that the number of these extenqions is
finite and hence that [~h) : 0]~] < ~ all extensions of ~ w
of degree
. If K does not contain
of degree dividing h, then K ~ h) does.
The ~rgup of points of a f o ~ l
~roup F
In this section R is a complete discrete valuation ring with quotient field K of characteristic 0, max/m~1 ideal residue class field k of characteristic p ~ 0.
~
, and
~le assume the
-valuation v on K is norm~llzed so that v(p) = I. All formal groups, unless othe~rise mentioned, are defined over R, and are assumed to be commutative, of dimension i. is the al~ebraic clos.ur,e of K.
The integers in K (i.e.
the elements of the integral closure in K of R) form a local ring (not Noetherian), i.e. the non-units of ~ form an ideal which is the
-
unique ~-xi~! ideal ~
of ~.
105
-
The unique extension to K of the
valuation v of K, will again be denoted by v.
Note that K is no_~t
complete. Suppose L is a finite field extension of K, and let S denote the integers in L. el,...,% 6
~
Take f e S[[Xl,...,XJ].
Then for
, f(el,...~n) makes sense and converges in m
(and if the constant term in f is O, fCel,...,~n) lies in ~ Note that el,...,mn
).
and all the coefficients of f are integers
in L I = L(al,... ,~n ) , and LI, being a finite extension of a complete field K, is complete.
~e then apply I.
w
Theorem i.
Let now F be a formal group (defined over R). PROPOSITION I
(i)
The elements of ~
form an abelian grou~
F(~) = P(F) under the operation
and
v(a*8)_> inf {v(~), vCB)} . The e l,ements o.f P(F) of finite
order ~orm a subgroup
A(F), the torsion subgroup O f P(F).
(ii) P(F) and A(F) are modules over (iii)
If f : F § G is a homomorphism of formal groups defined
ove~ R then the map P and
F = GaI(K/K).
e ~-~ f(e) is ahomomor2hism P(f) : P(F) + P(G).
A are covariant functors from the cate~olV
c ate~or~ o_~f r-modules.
In particular P(F) and
~ R to the A(F) are modules
over Em~(F), and these endomorphisms commute with PROOF
(i)
If L/K
r .
is finite, and SL the valuation ring of L, then
F(S L) is defined as in I, w
Theorem i and is an abelian group.
We
-
106
-
have then
P(F) = l ~ m F ( S L) = L (ii)
If
UL
F(SL)
y 6 F, then Y F ( % 8 )
= F(Y%YS),
since F is defined
over R and its coefficients are therefore fixed by
7 9
This part
of the proposition is then easily verified. (iii)
If f : F + G is a homomorphism defined over R, then
m
f maps~into
f(;(xj))
itself (since f has zero constant term).
= QCf(x),f(Y)),
Since Yf(~) = fCY~), f commutes ~ t h Remark
Since
y
o
I f F i s the a d d i t i v e group GaD then PCF) i s j u s t ~ w i t h
the ordinary addition.
ACF) = 0.
If F is the multiplicative group Gin, Gm(X,Y) = X + Y + XY, then P(F) is isomorphic to the group U of those units u of ~ for which u - 1 ( m o d e ) .
The isomorphism P(F) §
U is given by m ~ - ~ l + m.
A(F) is isomorphic ~clth the group of pn-th roots of unity for all n. An iso~eny f : F § G is defined to be a non-zero homomorphism defined over R.
Since f' (0) is algebraic over K then f'(0) lies in
some finite extension L of K.
By Prop. l, Cor.2, there exists
g e HOmL(F,G) such that g'(0) = f'(0).
Since f,g ~ Hom~(F,G), then
the same proposition tells us that f = g. L ~
~.
f is thus defined over
IIence every isogeny is defined over some finite extension of R.
From now on all formal groups to be considered are assumed to be of finite
-
107
-
height, unless otherwise mentioned. i
(i)
the map P(f) : P(F) § P(G) is sur~ective;
(ii)
the kernel of P(f) is a finite group of order ht(f).
PROOF
(Lubin,Serre)
Let ~ 6 ~ .
extension S of R.
I@t f : F § G be an isogemy.
Then
THE O R ~
Then f(X) -
is defined over some finite
For the Weierstrass order we have the equation
~-ord(f(X) - ~) = W-ord(f(X)) = pht(f),
and ht(f) is finite by w Theorem (I,
w
Prop. 3.
By the ~eierstrass Preparation
Th.3) therefore, f(x)
-
= u(x).g(x),
where u(X) is an invertible power series and g(X) is a distinguished polynomial: xpht(f)
g(X)
+
O
Take a E [ so that g(a) = O. then a ~ ~.
As gi E ~ ,
' gi
Since the coefficients of g lie in S
then also ~ s ~
f(X) - ~ are precisely the zeros of g(X). some s E ~.
But the zeros of Hence we have f(a) = u
for
This proves (i).
For (ii), take ~
= 0.
Now g(X) has pht(f) distinct roots,
provided that g(m) = 0 implies g'(=) # O. roots i n ~
Ys"
Thus f(X) = 0 has p
ht(f)
, provided f(a) = 0 implies f'(a) # 0, (mE ~).
Differentiating the equation f(F(X,Y)) = G(f(X),f(Y)) with respect to Y, we obtain
-
108
-
f'(F(X,Y))F2(X,Y) = G2(f(X),f(Y)).f'(Y) (here, the suffix 2 denotes the derivative with respect to the second variable).
Put X = c, Y = O.
If f(m) = O, then
f'(~)F2(~,0) = O2(0,0).f'(0). = f'(0) # 0 (by
~l, ~op. l, Cot.2).
Therefore f'(~) # O. The following theorem is really a Corollary of Theoraml. THEOEI~4 2
(Lubin, Serre). (i) P(F) is a divisible ~roup, and the
integers prime t o p induce automorphisms o~fP(F).
(ii) A(F) = (O~/Zp)(h)
h - Ht(F).
((h) denotes h-fold product) PROOF (i)
For n prime to p, i.e. n a unit of R,
automorphism of F.
~n]F is an
Hence P([n~F) is an automorphism of P(F).
Apply Theorem I to f = ~p]r F"
The surjectivity of
P ( ~ ] F ) : P(F) § P(F) implies that P(F) is divisible. (ii)
A(F) is a torsion subgroup of the divisible group P(F)
hence divisible.
Also
A(F) is p-primary.
Hence A(F) =~ (~/Zp) (c) , dim{Ker ~] F }
where c = dim{Ker[P]F}.
But the cardinality of K e r ~ ] ~ is p
which by Theorem 1 is ph.
Therefore c = h.
For each real number p is a fractional ideal of ~. in particular, Jo = ~ "
, the set J
= {~ ~ ~Iv(u) > P} P If p > 0, then Jp is an ideal of ~, and
For 0 -->0, the elements of J0
form a
-
subgroup F(Jp) of P(F) By w
109
(Prop.l).
-
(Abuse of Notation).
Prop, I, there ez~sts a unique isomorphism ~F : F § Ga,
defined over K, such that Z~(X) is defined over R a n d
~(0) = I.
As before, we denote the inverse of ~F by eF. THEOR~ $ (ii) P(F) §
(Serre)
(i)
converges o n e ;
The map ~, ) AF(~)
e F 9onverges on Jl/p-!
(s s ~) defines a homomor~hism
o.~fF-modules and of En~(F)-modules.
0 § A(F) § P(F) § (iii)
s
The sequence
§ 0 is exact.
~F and eF de;ine inverse isomorph~sms
F( Jl/p _ i ) ~= J+ l/p-1 (where the group operation on
j+i/p-i
is the usual addition).
For the proof of Theorem 3 the following lemm~is needed. L~E~A 1
For each real number p
>
O, there exists an integer
n = n(p,F) such that, for ~11 ~ e
with v(m) _> p
, we have
) 9 i/p-l.
P~OOF
We may assume p < i, since othe~rise we may ta~e n = O.
~ow __[P]F(X)
~
(mod deg 2).
If v(~) > O, then
~]F(m) = p m + for some r ~ R.
m2r,
Thus if v(~) _> p, then
v(~]F(~)) A inf(1 + v(~), 20) A in~ (i,2~). We deduce then by induction that
- ii0 -
v([pl~(~))
> inf (1,2np),
and we t h e n choose n so t h a t 2rip 9 l / p - 1 . PROOF OF THEgPa2~ 3
~ite
~F(X) = n=l ~ anXu"
over R and A~(O) = i, then v(na n) > 0
v(%) _~- vCn).
Since ~'F is defined
and aI = i.
~e thus have
Put n =p-Cn) ; then vCn) _<~(n).
~ov
V(an ~n) = nv(u) + v(a n) _> t-(n) v(u) - vCn) ~ pu(n) v(~) - u(n), which tenas to ~ a s n § ~,
provided that v(~) 9 O.
Hence ~F (m)
converges if v(u) > O. Q@
Choose ~ ~ ~ so that v(~) = I/p-l, n=l
e,~e
Bp-I = p.
Then
V(an sn-1) _> ~
- ~(n)
which is 9 0 when v(n) = O.
n-i . v(n) 9 I>-I
--
If v(n) 9 0 we continue
pv(n)-1
- v(n)
p-I
= I + p * p2 + ,,. + p v ( n ) - i . v(n) > _ 0 .
Therefore (S-lo
~ anSn-i xn has coefficients in ~, n-1 Its inverse under composition
AF o 8) (X) =
and leading coefficient i.
@@
(s-I ~ eF ~
=
X bn sn-lxn n=l
is thus also a p ~ e r series ~rith integral coefficients and leading coefficient i.
Hence V(bn Bn'l) > O.
Tahe ~ e Jl/p-l' i.e., such that
- iii-
v(a) 9 I/p-l.
Then
V(bn~=V(bn~-I = V(bn~'l)
(~)n-i ~) as n § ~,
+v((~) n-l) + v ( ~ ) § ~
8
since v(~) > O.
Thus eF(m) converges if m ~ J1/p-l"
V(bn~n) > v(m), if n > 1.
Moreover
Therefore eF(~) = m + ~' ~rhere v(m')>
Hence ~r deduce that, if a ~ J1/p-l' then v(eF(a)) = v(m). if a e J1/p-l' then v(~(m)) = v(m).
v{m).
Similarly,
The maps a ~-~ eF(~) and
~-~ ~(~) thus define inverse bijections J1/p-1 § J1/p-l"
Under
therefore the subgroup of points F(J1/p_ l) becomes isomorphic to the additive group of J1/p-l' and the inverse isomorphism is given by eF.
We have thus established (1) and (iii). Since E + is torsion free, then A(F)C Ker ~ .
i,
F(J1/p-1) for
gF C ~]F (m)) = 0, then by (iii),
Let m & Ker
integer n > 0
~ F (~) = 0.
~.
Since
Therefore m 6 A(F).
Thus in fact Ker ~(F) = A(F). Suppose a E E + 9 Since E+/Jl/p-i p
m
a ~ Jl/p-! for some m.
such that ~F(m) = p a.
is a torsion module, then
Thus by (ill) there exists a ( J!/p-I
But P(F) is divisible (Theorem 2) so there
exists 8 e P(F) such that ~]F(8) = ~
. Since proOF(8) = pma, then
~F (8) = a.
: P(F) § E+ is sur~ective
He have thus sho~m that s
and so that the sequence 0 § A(F) § P(F) § E + § 0 of groups is exact. Since ~
is defined over K this is a sequence of F -modules.
If f ~ ~u~(F), then both A~ ~ f and f'(O) o ~F are homomorphisms F * Ga ~ith derivative f' (0) at O.
They therefore coincide.
From
- 112
-
the commutative diagram
IP(f) I P(F)"
we deduce that P(F) ~ The foll~r
~+
-
is a homomorphism of E n d ( F ) - modules.
theorem is a converse of Theorem i.
that every finite subgroup of
It shc~m
A(F) arises as the kernel of some
isogeny.
THEORI~I I~ (Lubin)
Let r be a finite subgroup o f A(F).
Let L be t h e
fixed field of the stabilizer o f r i n GaI(~/K), and let S denote the integers o f L.
Then ther_~eexist_____~s a formal group G and an
f : F § G, both defined over S, so that
(i)
Ker f = r
(ii)
Ifg
,
: F +H
exists a u n i q u e ~
(we ~ i t e
Ker f for Ker P(f)),
i s an isogeny with Ker g D h
: G §
suc h that g = h
r , then there Q
f.
Ifg
and H
are defined over the integers S I of some finite extension L I o_~fL then so is h.
COROLLARY ,,,m
J
SI,
I
,,
I_f.fthere exists an isogeny F § G defined over some
then there also exists an isogeny G § F defined over S I. OF
PRooF coRo
mY 1
If f
exponent of Ker f is pr.
:
F § G is an isogeny, then suppose the Then Ker f C
Ker ~p]r F"
By Theorem 4
-
113
there exists an isogeny h : G §
-
such that h o f = ~]F' and h
is defined over S1. COROLLARY 2
Either HomsCF,G) = 0, o_rHo~sCF,G) s@~___En~(F)-module
i_~sisomorph~c to a non-zero ideal of Ends(F) , .a~d as an Ends(G)-module i__sisomo~,rph,~ic t~o~nonrzero ~deelo_~fEnds(G). PROOF
Suppose HOms(F,G) # O.
g : G § F over S.
By Cor. 1, there exists an isogeny
The map f ~-~ g 9 f is
an injective homomorphism
Homs(F,G) +Ends(F) of En~(F)-modules, whose image is a non-zero ideal.
Analogously for the map f ~-~ f o g.
CORQLI~. y 3
(1) (ii)
If HOms(F,G) # 0 then the quotient fields of DCEn~CF))
and of DCEndsCG)) coincide;
the rank of Homs(F,G) over Zp i_L th_~erank of End(F)
(and of EndsCa)). PROOF
(ii) fo!l~s immediately from Corollary 2 and from the fact
that any non-zero ideal of an integral dom~A~n !, which is a Z -order, P has the same Z -rauk as I. P For (i) ~rrite ~ and let ~
= D(Ends(F)) , H = HomE(F,G) , T F = En~F(H) ,
be the quotient field of E F (viewed as a subfield of ~).
Define similarly EG, T G and LG.
By Corollary 2, H is isomorphic
to a non-zero ideal of the integral domain EF, and therefore T F is a subring of ~ , c
similarly
containing E F.
Clearly E G C
TF, hence
L G.
For the proof of Theorem 4 we shall need some lemmas. If A is a complete local ring (this always to imply that it is
- 114 -
Hausdorff) then so is A[[X]].
~[e vrite ~u for the maximal ideal
and w for the associated filtration of the latter ring. T ~ A, then we = y L~A
2
If T s
vim, A[[T]I as a subring of A[[X]].
Suppose that X is a root of the polynomial in U
n-i
P(U) -- ~
-
i
Z i=l
pi u
,
with coefficients in A[[T]], an_d s~ that
w(p i ) _ > n -
A[[x]] PROOF
i .
g er te as
l,
"l.
For each non-negative integer m and for i = 0,1,...,n - i,
there are unique elements r
. in A[[TJ], so that in A[[T]][U]
mjl
n-i
~m.
Z rm,i ui
(rood P(u)).
i--O Here rm, i - ~m,~" when m_< n - i, and rn, i = Pi" W(rm, i) _9 m -
i.
For m 9n
Thus, for m_< n,
one easily establishes the same
inequality by induction, using the iteration formulae for the rm, i. o@
}fence if am ~ A, the series
~
Zmrm, i
converges under the
m=o
w-topology and hence
as we were required to show. Reck11 now (of. I,
w
Th.l) that if B is a commutative
ring containing A, complete under some filtration u, and if 8 E B,
- 115
u(8) > 0 then there is a unique continuous homomorphism of rings
e : A[[X]], order § B,u with 8(X) = 8, which leaves A elemen~rise fixed. B = A[[X]], u = w.
Let in particular
Then the resulting 8 is continuous also for
the w-topology. Let now F be a formal group (commutative, of dimension l) defined over A, and let # e *~A (the maximal ideal of A).
Then by
the preceding argument we obtain a continuous automorphism 8r of A[[X]] over A which maps X into F(X,r
Let F(A) be the group
of points, i.e. of elements o f ~ A under the product ~ If r
is the inverse of r in F(A) then
automorphism of
8r .
Hence 8@
e~_1
= F(m,8).
is the inverse
is bicontinuous.
The map r § er
is then an im~ective homomorphism of F(A) into the bicontinuous automorphism group of A[[X]]/A.
Let now r be a finite subgroup of F(A),
and suppose that A is an integral domain. l~.~,[& 3 PROOF
The fixed rinE of r i_~nA[[X]] is A[[T]], where T = ~
r
F(X,r
We consider the T~ierstrass order in U on the power series
ring A[[T]]
[[U]].
We have, with n = card r
W-ord ( ~ F ( U , r
- T) = W-ord(r~F(U,r
r
=
ord F(U,r
,
r
= n,
r
as W-ord F(U,r (I,
w
Th.3)
= I.
Therefore, by the Weierstrass Preparation Theorem
-
116
-
r-] FCu,~) - T = PCU).Q(U), 0 where P(U) is a monic polynomial in U of degree n over A[[T]] and Q(U) is an invertible power series in U.
Clearly the FCx,r
and so in particular X = F(X,O) are roots of
[-I F(U,#) - T, hence
of P(U).
Counting degrees and number of roots we see that
C1)
eCu) =
~
(u - r(x,r
ThUs P(U) satisfies the conditions of Lemma 2, and hence
(2)
A[[X]] is generated by I,X,,..,X n'l
as an A[[T]]-module.
Now let E = quotient field of
A[[X]].
Z 0 = quotient field o f A[[T]], E 1 = fixed field of ~ in E. E"aen (3)
E0 C
El,
and by Galois theory
:
-n.
But by (2), E is generated over E 0 by i, X,..,X n'l.
In view of
(3), (~) it follows firstly that E 0 = El, and secondly that the I,X,... ,Xn-1 are independent over E O. E 0.
Thus P(U) is irreducible over
By (2) therefore AK[x]] is a free A[[T]] -module on 1,X,...,X n-l,
i.e., every element ~ of A[[XI] has a unique representation in the form
-
117
-
n-1
(~)
, =
Suppose n ~
I
a.x',
i=o
~
A[[~]].
a.:
that m is fixed under *
, i.e., in E 1 9
As
E 0 = E 1 and as (5) is the representation of a as an element in terms of a basis over EO, it follows that
A[[x]] n ~l
c
Thus
A[[~]]. The o~site ino~usion ~s t r i ~
PROO____FFOF THEOR~I ~ of L'.
m = a 0 s A[[T]].
Let L' = K(*), and let S' denote the integers
Then f(X) = F] F(X,~) is a power series over S' with
vanishing constant term. Therefore K e r f _~ @ F(m,~) = O.
.
For ~ * *
, f(~) = [~ F(~,~) = O.
Also, if a ~ P(F) and f(~) = O, then
This means that F(m,~) = 0 for some
is the inverse of $ under +. F that Ker f = ~ .
Hence a * ~.
~ ~ ~, sad
Thus we have s h ~ m
Let A = S' [~X]], and define f*(Y) = f(F(X,Y)) ~ A[[Y]]. Then
f*(~) -- ~
r(r(xj),~)
For ~ ~ ~ , f*~(Y) = ~ -- ~
-- ~r(x,r(Y,~)).
F(F(XJ(X,~)),
~)
r(r(x,Y),z(~,~)) - f*(Y).
By Lemma 3, the fixed ring of A[EY]] under ~ is A[[f(Y)]]~
Hence
f*(Y) ~ A[[f(Y)]] = S'[[f(Y),X~] = B[[X]], where B - S'[[f(Y)]]. Consider f**(X) = [7 F(F(X,Y), $). action of ~ on B[[X]] given by X @-- F(X,~).
This is fixed under the We may apply the lemma
again, and deduce that the fixed ring of B[[X]~ under ~ is
-
118
-
B[[f(x)]1 -- s, [[f(x),fcY)]]. NOW we s ~
up : f(F(X,Y) = f*(Y), when considered as a
power series in Y over A = S' [[X]], and we saw that
f*(Y) 6 A[[f(Y)]] = B[[X]], where B = S' ['[f(Y)]]. Thus f(F(X,Y) = f**(X) is an element of B[[f(X)]] = S' [[f(X),f(Y)]].
Hence there exists
G(X,Y) E S' [[X,Y]] so that
(6)
1~ow f,(x) -
f(F(X,Y)) = G(f(X), f(Y)).
[ {F,(x,~) N
F(X,~)} .
~e put X - 0 and observe that
0 s ~, and we get f'(O) = F'(O,O) F-] ~ # O.
Thus, working over L'.
we conclude that
(7)
a-
fo
F o f-1
is a formal group. Let A be the stabilizer of * in Gal(~/K). this is a subgroup of finite index, ~ o s e L.
If ~ ~ A , then f(X)~ =
As , is finite,
fixed field we denoted by
U-~F(X,$~) = F-] F(X,~) = f(X).
f(X) is defined over L, and by (7), so is G.
Thus
Thus finally G is a
formal group defined over S = S' 6~ L and f is an isogeny F § G defined over S so that Ker f = *
.
~Te have thus established (i).
Let now g : F * H be an isogeny with Ker g m
* , defined
over the ring S I of integers in some finite extension ~ may suppose that L ! D
L'.
l.~e see that for
gePCx) = gCFCx,~)) = HCgCx),gCr
of K.
~Te
~ ~
= HCgCX),O) = gCX).
Thus gCX) lies in the fixed ring of # in Sl[[X]] , i.e., gCX) = h(f(X))
-
by Lemma 3, with h(X) s S1 [Ix]]. neither has h(X). w
-
As g(X) has no constant term,
One nc~r verifies easily that h is an isogeny G § H.
Diyision.and Rational Points. ~,R, K, v , ~ , ~, ~
between K and ~, s=
119
{~e~
etc., are as in
w
L is a field
S is the domain of integers of L, i.e,,
[vC~)A0}
.
~=G~C~I~).
Let F be a formal group over R, whose reduction mod is of finite height.
Write
P(F,L) = P(F) ~
L,
A(F,~) = A(F) O L (subgroups of points, and of torsion points in L).
Let moreover
~(F,L) be the subgroup of P(F) of points m which are of finite order rood L, i,e., for which ~ ] F (m) s L for sufficiently large n. Thus ~ (F,L)IP(F,L) is the torsion group of P(F)IP(F,L). TEEOR~4 1
AF gives .rise t_~oa cg~utative diagram with exact
r o w oz h o = = o r ~ % s = 0
§ ACF,L) § PCF,L) ~ L + § ~Cfl,ACF)) § ~(fl,PCF)) § 0
1 0
§ A(F)
COROT.T,~RY 0
o z ~n~(F)-=~dules.
~
I §
O § L+
§
0
get an exact sequence
§ A(F)IA(F,L) § ~(F,L)IP(F,L) § L+llm ~ §
O.
- 120-
PROOF OF ~ PROOF
COROLL~Y
In view of w
Immediate. Theorem 3, we get an exact sequence
0 § HOcn,ACF)) § HOcn,P(F)) § HOcn,~+) § ~Cn,ACF)) § ~C~,PCF))§ But ~C~,~ +) = o.
~e thus get the top roar of the diagram.
that I is given by the restriction of AF. cohomology groups are F~CF)-modules,
~(n,~+).
Note also
It is clear that the
as the operation of En~CF)
on P(F) and on ~+ commutes with the Galois group (~+ is an En~(F)-module via the map End(F) § R § ~).
The proof of the theorem ~-ill be complete
once we have shown that ZF(e) s L + if and only if m e ~
(F,L). Here
we use
~ 1
Let fb_~e an i s o ~ e ~ F
Then f(~) ~ P(GJ) ~ ~
o~
§
~ e ~ n e ~ o v e r S, and ~ ~ P ( F ) .
~
for ~
~ ~ ~ . ~ ~ ~ Ker P(f).
(~ i_~sthe ~fference in_ P(F)). Taking the le~
f o r g r a n t e d a t t h e moment we n o t e t h a t
if
~ ~t (F,L) then ~ ] F (~) ~ L for some n, hence for that n and for all also ~]F(~m ~ m) = O, i,e., us ~ ~ ~ A(F) = Ker ~F"
Thus
mAF(m) = AF(~m) = AF(a).
Conversely,
In other words ~F(a) s L +.
~F(m) & L + implies that ~ . m ~Ker AF for all ~ . But there are only a finite number of elements ~ ~
~.
Hence for all ~ and for some n,
~ s Ker ~]F " This implies that [p]F(e) = ~ ] F ( m )
i.e.,
fp]F(S) ~ L, whence ~ s ~(FtL).
<=>~f(=)~ f(~) - 0 for all < = 9 f(~ ~
=) = 0 for all ~
.
for all m ,
- 121-
SUGGESTION
In the following discussion (Theorem 2 and 3) consider the
particular case when F = G
m
and its relation to Kummer theory.
In the next theorem A c stands for the product of c copies h is the height of F.
of a group A.
ACF)IA(F,L) ( Izp)cI
THEOREM 2
,
:
L*Iz=
C89
=
~(F,L)IP(F,L)
c2
=~ (~/Zp) c,
, c = cI
+
c 2.
Here cI < h, and if the valuation on L i s discrete cI = h. c 2 <_ [L : Qp]
(the de~ree), and if the valuation on L
is discrete c2 = [L : g . COROLLARY
If L is algebraic of, finite d e g r e e over % ~(F,L)/PCF,L)
PROOF OF THE COROLLARY PROOF
:~ (~/Zp) [L : 0~] + h
then 9
Immediate.
The groups A(F), ~(F,L) and L + are divisible.
the same is true for their respective quotient groups. ~(F,L)/P(F,L)
Hence Moreover
is a p-primary torsion group, by definition.
it is of the form (%/Zp) c.
Hence
The other two isomorphisms and the
equation c = cI + c2 now follow from the Corollary to Theorem !. Clearly c2 cannot exceed the dimension O~-space L.
~L : 0~]
of the
On the other hand the isomorphism A(F) ~= (%/Zp) h
implies that cI _9 h.
Moreover if
A(F,L) is finite then c! - h.
-
122
-
Suppose now that the valuation of L is discrete.
Let p
be the least properly positive value of v on L, and apply w Lemma i with this p .
This yields a positive integer n, so that
[P]F (~) 6 Jllp-I whenever v(~) _.> p .
[P']F (ACF,L)) C J'l/p-1 By w
Thus
~
ICer
Theorem 3, the latter group is null.
A(F,L) C Ker [P]F
and hence is finite.
In other words
Thus in fact cI = h.
Let ~F(X) : ~ % x n. ~e already ~o~ that v(%)~ - v(~). n=l If the valuation of L is discrete, and p is as above then for all 6 P(F,L),
V(s
> inf vC~_ mn) > inf n vCcl) - vCn) > i~ n
~ere
~
= inf
:11.
"
{np - v(n)} > -
n
~ .
I
Thus vCIm A) > t for some
n
integer t, i.e., the fractional ideal
~ t contains Im ~ , and d
80
we have a surjection
,.+/p " (%/z)I:,.. =
Let nc~r @ be a subgroup of A(F,L). isogenies g over S originating from F.
I
Define ~ % = set of
(i.e., g : F § G for some G),
so that Ker P(g) C 0 . ~
= set of a 6 P(F), so that for some ge ~ ~, gCa) s L.
Note that gl(al) ~ L,
~
is a subgroup of P(F). g2(a2) E L.
For suppose gl' g2 6
~
Then the subgroup of A(F) generated by
,
-
123
-
Ker P(gl ) and Ker P(g2 ) is finite, hence of form Ker P(f) where f = fl ~ gl = f2 9 g2" to be defined over S.
As Ker P(f) C ~ we may suppose f, N o w w e see that f t
fl and f2
~ # , say f : F § G.
On the o t h e r h a n d f(al~ a2 ) = fl(gl(al)) ~ f2(g2(a2)) 6 L, as fi(gi(ai)) 6 L. W_~ehave ~cg~nutative diagram with exact rc~m 0 §
r
o § ACFJ)
§
e
PCF,L) § PCF,~)
§
L
§
~(a
(~11§
t~ ( e , P ( r ) )
(Homc = continuous homomorphisms). (ii)
Define for a e ~ r
, ~ efi,
e(a + F Then
Note:
= 0 for all m ~
) ae
= 0 for all a <
> ~
PCF,L), leave_~sL C ~ 0) elementwise fixed.
The last result gives a perfect pairing
Unfortunately this does not in general allow us to determine GalCLC6~@)/L)uniquely. COROLLARY finite, ,
,,,
GslCLC~#)/L) is an Abelian pro p-~oup.
then the ~
that of r .
But we evidently have
o_~fGaI(L(~r
I5" O i_~.s
i_~sfinite and divides
-
124
-
The diagram comes from the diagram
PROOF OF THEORI~{ 3
0 --~ #
§ PCF) §
0
§ P(F) §
§
~
r §
0
§
0
on taking cohomology, provided that we show that
(i)
- H=cCn, on
Cii) i,e.,
and
HOcn,P(F)Ir
a~ a s #
Now if a 6 ~ r g e
r
which is true as ~ acts trivi~11y
),
= ~/r ,
for ~.11 m <
r
a s
~r
.
then by Lemma l, ~ a~ a s Ker P(g) for some
, i.e.,
a ~ a 6 r , for all ~ .
a ~ a 6 * for all
Conversely, if
m, then these elements (finite in number)
lie in a finite subgroup of Hence by LA~.~a i, a 6
~,
r
i.e., in Ker P(g) for some g s ~ r 9
.
Note that = ~a ~ a for the proof of the second part of the theorem.
N~r let f : F § G be a fixed isogeny over S and let
r = Ker P(f).
T h e n C e = [a ~ P ( F ) if(a) s L] ,f defines a
homomorphism P(f,L) : P(F,L) § P(G,L) mud we have
(.)
coker PC f,T )
The second isomorphism f o l l ~
/ P(F,L) = I m e
.
from Theorem 3, the first from the
commutative diagram ~rith exact r~rs
-
125
-
o + e * P(;j.) * z= P(f,L) § o 0§
~#
§f
P(G,L) § 0
From (,) we get a homomorphism ~ , the composition
PCG,L) §
Coker
PCf,L) § Homc(~,r
Explicitly this is given by the usual construction of Ku~mer theory : Let b ~ P(G,L).
Choose
a
~bC~) = ~a ~
so that f(a) = b.
Then
a.
Write
~(~)
= {b,~) .
We aerive a pairing
Coker PCf,L) x GalCLC~#)IL) § Ker PCf) =
#
,
with zero kernels. If L is a local field we can use the symbol
{b,~}
and
the norm residue symbol to define a symbol
[b,c] ~ Ker P(f)
b 6 P(G,L),
c E L .
J~11 this applies in particular to f = [P]F' assuming Ker P( ~ ] F ) C in (,).
A(F,L).
Then we can determine the group
In fact this is the kernel of p in the group ~ (F,L)/P(F,L).
Hence by Theorem 2, we get :
(amd ~ r
~#/P(F,L)
If the valuation on L is discrete
P(~IF) C A(F,~)) then
- 126 -
~,/P(F,L) is a vector space over ZIpZ O..~.f dimension (**) [L : % ]
+h.
Now let
A = GalCLC~Ke r PC ~p]F ))/L)" By the lost corollary this is a vector space over Z/pZ.
Class
field theory .I lows us to give an upper bound on dim A when
[~ : ~
is finite, n=ely
(~ = I or O, depending on whether L does or does not contain the p-th roots of unity).
We also get a lower bound.
d/mCHomcCA , Ker PC~]F))) = Cd/m A)h.
dimCIm 8) = h + [L : % ] i
Hence
h + [~ : 89 ~ C~,. A~.h, and thus
d/mA >i+ --
(case
h
= 1
~).
h
For,
-
w
127
-
The Tate Module The notation is the same as that of
w
We shall frequently
write ~ ] F in place of A( ~ ] F ). We know that ~ ] F y i e l d s
a
homomorphism pn+m:m Ker ~]F +m (he
st=
=
=
§ Ker [P]F
bre
tion f o r
These maps, a n d the groups Ker ~]F define an inverse system of Abeliau groups, whose inverse l~m~t is the Tare module T(F) of F.
Thus the
elements of T(F) cam be ~ i t t e n as sequemces
(%,a 2, ,..),
a.
~]F (al) - O,
~]F (ai+l) = ai"
1
e
A(P)
Similarly we have an inverse system, indexed by the integers m~O,
whose groups all coincide with A(F), the map from
A(F)n+mtO A(F) m being the endomorphism ~ ] nF" inverse limit.
Let V(F) be the
The elements of V(F) can bewritten as sequences a . e A(F),
- (ao,5,a2,...) , ~F
1
(ai+z) = ai
The map ~ , ~ a 0 is a homomorphism V(F) § A(F), whose kernel may clearly be identified with T(F), i.e., we get an exact sequence (4,1)
0 § T(F) §
§ A(F) § O.
-
128
-
Equivalent description : We start with the isomorphism HornZ
n (~ Z p / Z p , A(F)) ~Ker [P]F P P
'
"1
which ta~.es f into the image f(~mod~ Zp). The direct system P lh Zp/Zp ~zith limit O?/Zp Gives rise to an inverse system by means P of the fUnctor HomZ ( ,A(F)), which under the above isomorphism P goes over to the inverse system (Ker ~]~, ~m n+m )" Hence in fact
Hom Z CO~/Zp, ACF)) = TCF). P Similarly from the direct system l---Z n p with limit P isomorphism (I~.3)
%
one obtains an
Hom Z (%, A(F)) ~ V(F), P
and of course we have the natural isomorphism HomZ (Zp, A(F)) ~ A(F). P By means of these isomorphisms the sequence (4.1) cam now be interpreted as being obtained by applying the functor Homz ( P to the sequence O § Zp
*~
§
,A(F))
Qp/Zp+O.
Alternatively (4.1) maybe viewed am obtained from this sequence by tensoring over Z ~r T(F). P Another consequence of (~.2) and (4.3), together ~ t h the isomorphisms
129
P
and w
Theorem2, is
PROPOSITION i
= Z
(h)
P
V(F) - 89
(h).
We shall in fact view T(F) as a lattice (= free Z -module of P maximal rank) in the vector space V(F). The g r o u p s and maps of (4.1) are clearly fuuctorial.
Hence
in particular T(F) and V(F), as well as A(F), are En~(F)-modules, and the maps of (4ol) are homomorphisms of EndR(F)-modules.
~reover,
an isogeny f : F § G gives rise to a commutative digram T(F) § VCF)
§ ACF)
T(G) § V(G)
§ A(G).
PROPOSITION 2 V(f) i_~san isomorphism and T(F) is inj~ctive, with Coker T(f) ~ Ker A(f) finite. PROOF
If d i m ~ Ker V(f) = s, then Ker A(f) contains the submodule
Ker V(f)IEer TCf) ~ (~IZp) s. s = 0 and so Ker vCf) -- 0.
As Ker A(f) is finite
Similarly, as Coker
sama theorem), we conclude that Coker V(f) = 0.
(cf.
w
Th.l),
ACf) = O, (again by the Now it follows that
130
Ker T(f) = 0 and Ker A(f) ~ Coker T(f)
(Snake Lemma).
From this proposition it follows that Im T(f) is a lattice in V(G), a sublattice of T(G).
(The term lattice L in a vector
space V is always to imply that L is of maximal rank, i.e., spaus V). We shall write L(f) for the inverse image of T(G) under V(f), i,e., for V(f) "I (T(G)).
This is a superlattice of T(F) in V(F).
The Galois group
r = GaI(~/K) acts on V(F) and T(F) as well
as on A(F) and the maps of (4.1) are homomorphisms of r -modules. ~,le are assuming throughout that the given formal group F is defined over R, but we do not assume other formal groups G,H,.... to be necessarily defined over R - they m~y be defined over the integers in some finite extension of R.
If however G as well as the isogeny
f : F § G are defined over R, then the diagram (h.4) is one of F -module homomorphisms and so both Im T(f) and L(f) are F -modules. THEORD~ 1
(Lubim~ (i)
Then ther____eexists an ~
Let L be a sublattice of T(F) i_~nVCF). f
: H § F so that L = Im T(f), and
if L i_~sstable under F then H and f may be chosen t_~ob_~edefined over R. Im TCf I) C Im TCf), fl bein~_sa iso~eny ~ i_~san is oge%y h : H I § H with fl = f o h.
§ F
then there
l_~mparticular Im T(f)
determines H an__ddf to within isomorphism. (ii) ez/sts an ~
Let L b_~eA_ su~erlattice of T(F) in V(F). g
: F § G with L(f) = L.
Then there
If L i~stable under F
then G and g may b_~echosen t_~ob_~edefined over R. l._ffL(g) C L(g I) , gl bging am iso~en~ F + GI, then there
-
131-
i_~san isoge~v h : G § G I so that h @ g = gl"
In particular
L(g) determines G an__d_dg t_~owithin i.somor~hism.
PR.OOF
First that of (ii).
V(F)/T(F) = A(F).
L/T(F) is a finite subgroup of
Ts/~ing quotients rood T(F) we thus get an
order preserving bijection from the set of superlnttices L to the set of finite subgroups of A(F), which also preserves stability under
F.
Note also that if g : F § G is an isogeny, then
Ker ACg) = L/T(F) precisely when VCg)L = T(G), i.e., L = LCg). (ii) now follows from
w
Theorem ~.
Next the proof of (i). with pn T(F) C
L, L being now the given sublattice of T(F).
Then p'nL = L' D g : F §
Let in the sequel n be an integer
T(F) and so, by (ii), there exists an isogeny
L' = L(g), i.e., with V(g)L' = T(H),
Now
paL' = L C T(F) implies that pn Ker A(g) = O, i.e., that Ker A(g) C Ker [P]F"
By
w
f : H § F with f o g = ~ F "
Theorem 4, there is an isogeny But then Im T(f) = V(f @ g)L'
= paL' = L, as required. Note that in the above constructions the choice of n is ~m~terial
(of course within the stated conditions).
m ->- n , then gl = g ~ ~jm-n F
replaces g and still f o gl = [p]m F"
Note secondly that if L is F -stable then so is L'. g to be defined over R.
Choose then
Hence g-1 (inverse under substitution) is
defined over K, and thus f = ~ ] F over R.
If say
" g-i is defined over K, hence
- 132
Let fl : HI § F be an isogeny with Ll -- Im T(fl ) C Im T(f) - L, We may suppose that Im T(f I) ~ pn T(F). hypothesis, pn Ker
Let g be as above.
As, by
A(fI) = 0 there is an isogeny gl : F § HI
with gl o fl = ~ ] H I"
But then also fl ~ gl = ~ ] F = f @ g"
Now we have
Ker A(g1) = p-n Im T ( f l ) / T ( f ) C p-n Im T(f)/T(F) = ~ r Therefore, by w
g-ho
,
A(G).
Theorem ~, there is an isogemy h : HI §
i.e., f o b o
--flo gl, =
soq = f ' h .
This completes the proof of the theorem.
We can extena the injective map
Ho (F,G) § Hom (V(F), V(a)) to a map %
HOm~(F,G) § Hom(V(F), V(G))
e Z
P which we shall still denote by V, and which remains injective. Vi~ng
Hom~(F,G) as contained in %
Z Hom~(F,G) we have P Then g ~ HOm~(F,G) if
.and only if V(g) ~ PROOF
T(F)Pinto T(G).
"Only if" is trivial. "If" : Let pn g = h e Ho~(F,G).
Then Im T(h) C pn TCG),
whence by Theorem i, h = ~ ] G ~ hl' hi ~ Ho~(F,G).
But then
g = h 1. ~rite n~r E F = D(En~(F)) and let ~
be the quotient field
-
of E F in E.
133
-
Then of course D induces an isomorphism
%
@ Z End(F) ~ L F. P
We vi~r T(F) as an EF-mOdule and so V(F) as an LF-mOdUle.
By
Theorem 2 = {a e L F
[
Let g : G § F be an isogeny, that ~
a T(F) C TCF)}. l,[ek~ow (w
Theorem 4
Corollary 3)
= LG, and in fact V(g) is an isomorphism of B-modules.
Hence COROLL&RY (Lubin)
EG = {a e nF I a Iz ~ ( g ) c zz ~(g)}.
Now one has
T~OP~,[ 3
(Lubin)
Let. @ b e an order .oyer Zp (contained i n ~).
Then t h e r e i s a_ formal groun F w i t h ht(F) = [ 6~ : Zp] so that E F =
@ .
We first find an F so that ht(F) = [ • : Z ~
and so that
is the quotient field of ~ . Let K be the quotient field of g7 , R the valuation ring of K.
We then have
PROPOSITION 3
so that ~ PROOF ~of
There is a formal group F of h e i g h t h - [ K :
-- R.
(Construction of Lubin-Tate), R mud let q = card(R/~) = pS.
unique F(X,Y) ~ R[[X.Y]] with
Let ~ generate the ~ x i m a l ideal By III,
w Le~m~ I there is a
-
;(xj) ~ x + Y
134
-
(=od des 2)
and with
F(ZCX), ZCY)) = zCFCxJ)) where f(X) = ~X + X q.
We shall then show below that F is a form~!
group, so that the map D : End(F) § R is surjective, hence bijective. Moreover [P]F : fe o u, where e is the ramification index of K / %
is a ~ t
of ~ % ( ~ ) .
~erefore ht(~]F) : e.s : [K : % ) : h.
it follc~,s that K : ~
and R : E F.
Let a ~ R and construct, along the lines of IlI w a power series
and u
Lemma i,
[a] (X) over R with
[a]
(x) - ex
(rood degree 2)
and f " [a] = [a~ ~ f. We have then to show that
F(xj)
- F(Y,x),
F(F(X,Y) ,Z) = F(X,F(Y,Z)),
[a] (F(xj)) - F( [a] (x) , [a] (Y) ) , and it will follow that F is indeed a co ~-~tative for.~ group sad [aS is an endomorphism of F with D( [a] ) = a. via the uaiqueness part of III w
In each case this is done
Lemm~ i.
Thus e.g. the two sides
in the last equation are both solutions of the problem of finding G, so that
Q(z(x), f(Y) ) - z(s(xj)), G(X,Y) = aX
+ aY
(nod degree 2).
-
PROPOSITION 4
-
Let F b_~ea formal group o_~ffinite height and let
b_~e an o r d e r ~ t h
auotient field L F.
group G isogeneous t o F so that
PROOF
135
Then there i_~sa formal
@ = E G.
Let L be any sublattice of T(F) so that
~
= {a6 L F I aL C L } .
Such sublattices exist, e.g,, L = @ x with 0 # x ~ T(F). Theorem l, there is an isogeny g : G § F so that L = I m the Corollary to Theorem 2, E G = @
By T(g).
By
9
Theorem 3 now follows from the last two propositions.
The Tare .mgdule as a_ module over r
= GaI(~/K).
We already F~aow that TCF), and hence VCF) is a
r -module.
An element Y of r will leave T(F) and hence V(F) elementwise fixed if and only if y leaves A(F) fixed.
But A(F) is just a
subset of K, and so we see that the representation of r by V(F) (or by T(F)) is a faithful representation of its quotient group
GalC (ACF) )Ix). Let t : r § GL(T(F)) (automorphism group of T(F)) be the
homomorphism with xt(y) = x7
for x e T(F).
GL(T(F)) is a topological
group, a typical open neighbourhood of the identity being the subgroup of automorphisms a : i (nod pn) (i.e., of form i + sp n, s an end~morphism of T(F)).
t is continuous.
i = identity,
To see this we only have
to note that t(Y) - 1 (nod pn) if and only if Pn t(Y) = Pn t(1), where Pn is the map T(F) § Ker ~ ] nF"
l =it).
(definition of T(F) as inverse
= p Cz), if and o=z We n ~
consider the
if
F -module V(F).
Zeaves Ker
f ed.
- 136
,'i"KEOREM3 V(F) is an irreducible
-
r-module, over ~
(i,e., the only
~-subspaces of V(F) which are F -modules are V(F) and O). This is a version of a result given by Serre, watered down to fit in ~rith the tools we have available. PROOF
Denote by Fs the orbit under r of an element s in a r -set S.
l~aat we have to s h ~
is that if 0 # x ~ V(F) then the subspace
generated by rx is the whole of V(F). consider an x ~ T(F), with x $ p T(F). of V(F) generated by
Fx.
It clearly suffices to Let then M be the Zp-SUbmodule
M is a free Z -module of rank s < h and we p
have to show that s > h. ~Trite Pm for the surjection T(F) § Ker ~ ] F associated with the inverse limit T(F) = lim Ker ~]F" defined.
M C T(F) and so Pn(M) is
It is the direct product of at most s cyclic subgroups, and
so the number of elements in Pn(M), not in ppn(M) is at most p
ns
(n-l)s
- p
9 Write ~n = Pn (x)"
Pn(M), and not in ppn(M).
Then each element of Fmn lies in
Therefore
card(r%) <_ p(n-1)s (ps_ ~). The left hand side is the number of conjugates of mn over K, and so equal to the degree [K(mn) : K] .
~Te thus get the inequality
[K(en ) : K] _< p(n-1)s (los - 1)
(~.5) holding for all n. Now note that
(~.6)
[;]r (%) -- o,
% # o;
[;]r(%+l ) = %.
-
137
-
We shall show that this implies the existence of a positive constant c so that
(4.7)
[K(%) : K] I cp nh , for all n.
Comparison of (4.5) with (4.T) as n § = yields then the required inequality s ~ h. To get (I~.7) from (4.6) we require a lemma,to be proved later. LEI~4A
o~,
Let a,6 s P(F), [P]F (s) = 8 9
(a)
I~f v(8) < l , then v(~) < v(S)/p.
(b)
I fv(6) ~ l/e, e b e i ~ t h e
r~fication
index of K over
then v(~) ~ vC~l/ph. ~;e apply the l e ~
to (4.6).
By
to complete the proof of the theorem.
w Theorem 3, v(a l) _< l/p-1 _< 1.
From the l e ~ ,
Return form
(a), we obtain by induction the inequality v(m n) < i/p n-1. Therefore for some n o , V(mno ) _< i/e.
Now use form (b) in the lemma to get
for n > n o the inequality v(m n) < 1/e p (n'no)h. en be the ramification index of K(an)/K.
On the other hand let
Then certainly en v(a n) >_ l/e,
1/e being the least strictly positive value of v on K.
[K(~n) : K] --> en--> ~ It remains to prove the lemma.
_9 pnh c,
Hence finally
c = p-n~h.
Let ~]F(X) =
~
anXn.
n=! Here aI = p.
Apply I, w Theorem 2 to the ring RIpR mud t/he
reduction of ~]F(X) rood pR. whenever p $
This tells us that v(a n) >__v(p) = i
n, i.e., in particular
-
(~.8)
v(~) .9 z
138
for 0
-
<
n
<
p.
S~.~larly, applying the same resaoning to the residue class fiel~ of R, one gets
(~.9)
vC~) _~ lle
for 0 < n < ph.
~t now v(aj=j) = i=f vCa=~),
C~.xo)
me=
vCB) _~vC~j~j)9 ~
jvC~) _~ vCaj) + jvC=) - v(~ ~) _~ vCB).
If first v(8) _< i then for 0 < n < p, we have by C~.8)
and so j _> p, whence by (;4.10) pv(~) < v(8).
If next v(~) <_ i/e, then we ~e~uce similazly that j _> h , whence again by (4.10) ph v(~) <_ v(8).
,o
-
139
-
Literature A. i)
Gauera! Reading
For Lie theory see
J.-P Serre, Lie Algebras and Lie Groups,
1964 Lectures at Havard University, published by Banj~m~n, and the literature quoted there. 2)
For commutative form~ groups over fields of characteristic p 90 see
the article by Yu. I. ~ n ,
Ups. Mat. Nauk. 18, 193,
3-91, translated in Russian Mathematical Surveys, 18, (1963) 1-84. B. i)
This contains an extensive literature list. References in the notes (under the author's names).
M. Lazard, Sur les groupes de Lie formals ~ un param~tre, Bull. Soc. math. France, 83 (1955) 251-274.
2)
J. Lubin, One parameter formal Lie groups over p-adic integer rings.
Ann of Math. 80 (1964), 464-484, and Correction,
Ann. of Math. 84 (1966), 372. 3)
J. Lubin, Finite subgroups and isogenies of one-parameter formal Lie groups, Ann. of Math 85 (1967), 296-302.
4)
J. Lubin and J. Tate,
Formal complex multiplication in local
fields, Ann. of Math, 81 (1965), 380-387. 5)
J,P. Serre, Lectures at the College de France (unpublished), 1965-66. C.
I)
Further literature
The theory of form~1 complex multiplication in its relation to local class field theory (cf. B. [4] ) is also treated in detail in Chapter VI by J.-P. Serre, in the Brighton Proceedings (Algebraic Number Theory, Academic Press, 1967).
-
2)
140
-
J. Lubin and J. Tare, Formal moduli for one-parameter formal Lie groups, Bull. Soc. Math. France, 9~ (1966), ~9-60,
3)
JeT. Tare, p-d/visible groups, Driebergen Proceedings (Local fields, Springer 1967)
~)
J.-P. Serre, Sur les groupes de Galois atta~hds aux ~oupes p-divisibles, Driebergen Proceedings (Local fields, Springer 1967).
The last two papers deal with a generalization.
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