Formal Groups
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74
A. Frohlich 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. © by Springer-Verlag Berlin' 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 formal groups which I gave during the session 1966-67 at King's College London.
They are based on a rou9h draft by A.S.T. Lue.
I have not included here the last part of the course. on formal complex multiplication and 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.TC.
ray
course.
CONTENTS CHAPTER I
CHAPl'ER II
CHAPTER III
PRELIMINARIES §l.
Power series rings...........................
1
§2.
Homomorphisms................................
16
§3.
Formal groups................................
22
LIE THEORY §1.
The bialgebra of a formal group..............
29
§2.
The Lie algebra of a formal group............
43
COMMUTATIVE FORMAL GROUPS OF DIMENSION ONE §l.
Generalities.................................
§2.
Classification of formal groups over a
§3.
CHAPl'ER IV
51
separably closed field of characteristic p...
69
Galois cohomology............................
86
COMMUTATIVE FORMAL GROUPS OF DIMENSION ONE OVER A DISCRETE VALUATION RING
96
§l.
The homomorphisms....................... ••••
§2.
The group of points of a formal group........ 104
§3.
Division and rational points ••••••••••••••••• 119
§4.
The Tate module •••••••••••••••••••••••••••••• 121
LITERATURE. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • ••
139
-1-
CHA.P:rER I. PRELIMINARIES
91.
Power Series Rings Let R be a commutative ring.
The power series ring
in n indeterminates Xl •••• ,X over R is a ring n whose elements are formal power series R
, •••
with component-wise addition 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 = (i •••• ,in)' with components it l
. N.
In other words M is the set of maps of n
u, •.• ,n}
define addition and partial order on
component-wise, i.e.
M
n
and
The zero element
a
on M is the n-tuple (0, ••• ,0). n
Now we can write
f(X)
=l' = L
i E M
n
(interpret
r
.
i i as Xl l •••X n
1), and define
into N.
We
-2-
(g +
=g.
f).J.
+ f., J.
J.
l, ••• ,xJl is a cOlIDl1Utative ring, which
With these definitions R[[X
contains R as a subring : identiry a E: R with the power series f, for which f of M ). n
O
= a and f
i
= 0 (the zero of R) when i > 0 (the zero
We shall write
for the inclusion map.
The a:uentation
is the ring homomorphism with
R
R [[JS.••••
& (f)
/
=
f • O
Note that the diagram
R
,x;JJ
commutes.
!2i!.:: of maps M
n
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