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Editorial Board E.Bombieri, Princeton S.Feferman, Stanford N. H. Kuiper, Bures-sur-Yvette P.Lax, NewYork H.W.Lenstra, Jr., Berkeley R. Remmert (Managing Editor), Munster W. Schmid, Cambridge, Mass. J-P. Serre, Paris 1. Tits, Paris KK Uhlenbeck, Austin

Eberhard Freitag Reinhardt Kiehl

Etale Cohomology and the Wei! Conjecture With an Historical Introduction by lA. Dieudonne

Springer-Verlag Berlin Heidelberg GmbH

Eberhard Freitag Math. Institut, Universitiit Heidelberg Im Neuenheimer Feld 288 D-6900 Heidelberg Reinhardt Kiehl Lehrstuhl fiir Mathematik II Fakultiit fiir Mathematik und Informatik Seminargebiiude AS D-6800 Mannheim

Translated from the German manuscript by Betty S. Waterhouse and William C. Waterhouse

Mathematics Subject Classification (1980): 14-XX

ISBN 978-3-662-02543-7 ISBN 978-3-662-02541-3 (eBook) DOI 10.1007/978-3-662-02541-3

This work is subject to copyright. AII rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. Duplication of this publication Of parts thereof is only permitted under the provisions of the German Copyright Law of September 9, 1965, in its version of June 24, 1985, and a copyright fee must always be paid. Violations fali under the prosecution act of the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1988 Originally published by Springer-Verlag Berlin Heidelberg New York in 1988 Softcover reprint ofthe hardcover Ist edition 1988

2141/3140-543210

Preface

Some years ago a conference on l-adic cohomology in Oberwolfach was held with the aim of reaching an understanding of Deligne's proof of the Weil conjectures. For the convenience of the speakers the present authors - who were also the organisers of that meeting - prepared short notes containing the central definitions and ideas of the proofs. The unexpected interest for these notes and the various suggestions to publish them encouraged us to work somewhat more on them and fill out the gaps. Our aim was to develop the theory in as selfcontained and as short a manner as possible. We intended especially to provide a complete introduction to etale and l-adic cohomology theory including the monodromy theory of Lefschetz pencils. Of course, all the central ideas are due to the people who created the theory, especially Grothendieck and Deligne. The main references are the SGA-notes [64-69]. With the kind permission of Professor J. A. Dieudonne we have included in the book that finally resulted his excellent notes on the history of the Weil conjectures, as a second introduction. Our original notes were written in German. However, we finally followed the recommendation made variously to publish the book in English. We had the good fortune that Professor W. Waterhouse and his wife Betty agreed to translate our manuscript. We want to thank them very warmly for their willing involvement in such a tedious task. We are very gr