Etale Cohomology and the Weil Conjecture (Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics)

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 conjec­ tures. 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 self­ contained 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 grateful to the staff of Springer-Verlag for their careful work. This book is concerned with one of the most important developments in algebraic geometry during the last decades. In 1949 Andr? Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. He himself proved his conjectures by means of an algebraic theory of Abelian varieties in the one-variable case. In 1960 appeared the first chapter of the "El?ments de G?ometrie Alg?braique" par A. Grothendieck (en collaboration avec J. Dieudonn?). In these "El?ments" Grothendieck evolved a new foundation of algebraic geometry with the declared aim to come to a proof of the Weil conjectures by means of a new algebraic cohomology theory. Deligne succeded in proving the Weil conjectures on the basis of Grothendiecks ideas. The aim of this "Ergebnisbericht" is to develop as self-contained as possible and as short as possible Grothendiecks 1-adic cohomology theory including Delignes monodromy theory and to present his original proof of the Weil conjectures. Eberhard Freitag, Reinhardt Kiehl ; With An Historical Introduction By J.a. Dieudonné ; [translated From The German Manuscript By Betty S. Waterhouse And William C. Waterhouse]. Includes Index. Bibliography: P. [308]-314.