Hodge Cycles, Motives, and Shimura Varieties
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Pierre Deligne James S. Milne Arthur Ogus Kuang-yen Shih
Hodge Cycles, Motives, and Shimura Varieties
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Authors Pierre Deligne Institute for Advanced Study School of Mathematics Princeton, NJ 08540, USA James S. Milne Mathematics Department, University of Michigan Ann Arbor, MI 48109, USA Arthur Ogus Mathematics Department, University of California Berkeley, CA 94720, USA Kuang-yen Shih 3511 W. 229th Street Torrance, CA 90505, USA
1st Edition 1982 2nd, Corrected Printing 1989 ISBN 3-540-11174-3 Springer-Verlag Berlin Heidelberg New York ISBN 3-540-11174-3 Springer-Verlag New York Berlin Heidelberg CIP-Data applied for This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use ofillustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law.
© Springer-Verlag Berlin Heidelberg 1982 Printed in Germany SPIN: 10492186
46/3140-54321 - Printed on acid-free paper
PREFACE This volume collects six related articles. The first is the notes (written by J.S. Milne) of a major part of the seminar "Periodes des Int~grales Abeliennes" given by P. Deligne at I'.B.E.S., 1978-79. The second article was written for this volume (by P. Deligne and J.S. Milne) and is largely based on:
N
Saavedra
Rivano, Categories tannakiennes, Lecture Notes in Math. 265, Springer, Heidelberg 1972. The third article is a slight expansion of part of: J.S. Milne and Kuang-yen Shih,
Sh~ura
varieties: conjugates
and the action of complex conjugation 154 pp. (Unpublished manuscript, October 1979). The fourth article is based on a letter from P. De1igne to R. Langlands, dated 10th April, 1979, and was revised and completed (by De1igne) in July, 1981. The fifth article is a slight revision of another section of the manuscript of Milne and Shih referred to above. The sixth article, by A. Ogus, dates from July, 1980. P. Deligne J.S. Milne A. Ogus Kuanq-yen Shih
Note to the second printing 1989 For this printing an Addendum and Erratum were added (pp. 415, (16).
CONTENTS
Introduction • . • • • • • • • • • • . • • • • • • • • • •
I
Notations and conventions
8
P. Deligne(Notes by J. Milne): Hodge cycles on abelian varieties • • • • · • • • • . . . •
9
II
P. Deligne, J. Milne: Tannakian categories.
101
III
J. Milne, K.-y. Shih: Langlands' construction of the Taniyama group • • • • • • • • • • • • • • •
229
IV
P. Deligne: Motifs et groupe
261
V
J. Milne and K.-y. Shih: Conjugates of
varieties.
280
VI
A. Ogus: Hodge cycles and crystalline cohomology • • • • •
357
de Taniyama
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