The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes
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264 William Messing Cranbury, NJ/USA
The Crystals Associated to Barsotti-Tate Groups: with Applications to Abelian Schemes
Springer-Verlag Berlin-Heidelberg - New York 1972
AMS Subject Classifications (1970): 14 K 05, 14 L 05
ISBN 3-540-05840-0 Springer-Verlag Berlin • Heidelberg • N e w York ISBN 0-387-05840-0 Springer-Verlag N e w York • Heidelberg • Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. © by Springer-Verlag Berlin • Heidelberg 1972. Library of Congress Catalog Card Number 72-79007. Printed in Germany. Offsetdruck: Julius Beltz, Hemsbach/Bergstr.
Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . Conventions
. . . . . . . . . . . . . . . . . . . . . . . . . . .
Chapter
I.
Definitions
Chapter
ii.
The R e l a t i o n Lie G r o u p s
and E x a m p l e s between
. . . . . . . . . . . . . .
Barsotti-Tate
iii.
Divided
Powers,
Chapter
IV.
The Crystals
Chapter
V.
The D e f o r m a t i o n
Exponentials
Associated Theory
10
11
and F o r m a l
. . . . . . . . . . . . . . . . . . . . .
ChaRter
I
and Crystals . . . . . .
to B a r s o t t i - T a t e
Groups.
and A p p l i c a t i o n s . . . . . . .
23 77 .112 150
Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
175
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . .
188
INTRODUCTION The concept of Barsotti-Tate group w a s introduced in [I] w h e r e the name
equidimensional h y p e r d o m a i n w a s used (actually for an equivalent
concept) and in [30] w h e r e the n a m e p-divisible group w a s used.
Following
Grothendieck, w e prefer the t e r m Barsotti-Tate group because the concept of "p-divisible group" has a m e a n i n g for anyabelian group object in an arbitrary category and does not indicate any relation with algebraic geometry, Barsotti-Tate groups arise in "nature" w h e n one considers the sequence of kernels of multiplication by successive p o w e r s of p on an abelian variety.
Also, as Orothendieck has observed, there are Barsotti-
Tare groups which are naturally associated with the crystalline cohomology of a proper s m o o t h s c h e m e which is defined over a perfect field of characteristic
p.
Since w e do not discuss crystalline cohomology no further
mention is m a d e of this example. Returning to the situation w h e r e
A
is an abelian variety over a
(perfect) field of characteristic p, let A(n) plicationby group.
p
n
on A.
The s y s t e m
denote the kernel of multi-
(A(n))n> 1
constitute a Barsotti-Tate
A s opposed to the situation w h e r e one looks at the kernel of mul-
tiplication by
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