Etale cohomology and the weil conjecture ebook, 1988. This years seminar will focus on etale cohomology, the goal being to understand laumons proof of the main theorem of delignes weil ii paper that gave a powerful and vast generalization of the riemann hypothesis over finite fields. All schemes are taken to be noetherian and separated 1 introduction the development of etale cohomology was motivated by work on the weil conjectures, which state that localzeta functions x. Tom lovering june 22, 2012 for mathematical musings. Then weil cohomology with coefficient field is a contravariant functor from the category of varieties into the category of finitedimensional graded anticommutative algebras, which satisfies the following conditions.
Tamas szamuely etale cohomology and the weil conjectures periodo. Tom lovering june 22, 2012 abstract in this essay we brie. Trying to understand delignes proof of the weil conjectures a tale in two parts january 29, 2008 part i. Weil cohomology theories and the weil conjectures 3 proposition 1. The conjectures of andr e weil have influenced or directed much of 20th century algebraic geometry. We give a conjectural description of the vanishing order and leading taylor coe. The problem treated is exceedingly elementary counting solutions of polynomials over. Etale cohomology and the weil conjecture eberhard freitag.
We recall the construction, following the method of morel and voevodsky, of the triangulated category of etale motivic sheaves over a base scheme. We will describe the history of the padic cohomological program in the next section. Finite generation conjectures for cohomology over finite. Reinhardt kiehl this book is concerned with one of the most important developments in algebraic geometry during the last decades. Series of modern surveys in mathematics on free shipping on qualified orders. The pdf files are formatted for printing on a4letter paper. Weil conjectures and motivation september 15, 2014 1 the zeta function of a curve we begin by motivating and introducing the weil conjectures, which was bothy historically fundamental for the development of etale cohomology, and also constitutes one of its greatest successes. Weildeligne representations and padic hodge theory. The padic hodge theory of semistable galois representations.
Etale cohomology and the weil conjecture springerlink. He himself proved his conjectures by means of an algebraic theory of abelian varieties in the onevariable case. The prerequisites for these lectures are elementary. This note is based on the 3hour presentation given in the student seminar on winter 2014. Trying to understand delignes proof of the weil conjectures. Tamas szamuely etale cohomology and the weil conjectures. In 19, lichtenbaum introduced the weil etale topology in order to produce finitely generated cohomology groups for varieties over finite fields which are related to special values of zetafunctions. Weil etale cohomology and special values of lfunctions of 1. Reliable information about the coronavirus covid19 is available from the world health organization current situation, international travel. The aim of this ergebnisbericht is to develop as selfcontained as possible and as short as possible grothendiecks 1adic cohomology theory including delignes monodromy theory and to present his original proof of the weil conjectures. Eberhard freitag and reinhardt kiehl, etale cohomology and the weil conjecture nicholas katz.
The general theorems about etale cohomology allowed grothendieck to prove an. An introduction to bloch and katos conjecture two lectures at the clay mathematical institute summer school, honolulu, hawaii, 2009 prerequisites. We calculate the total derived functor for the map from the weil etale site introduced by lichtenbaum to the etale site for varieties over finite fields. Let xbe a smooth, connected, ndimensional projective variety. Weil conjecture, standard conjectures on algebraic cycles. The zeros and poles of the zeta function satisfy a suitable riemann hypothesis as also conjectured by weil but proved by deligne 19741980 using etale cohomology. An introduction to bloch and katos conjecture two lectures at the clay mathematical institute summer school.
In particular, there is a long exact sequence relating weil etale cohomology and etale cohomology. The cropped files have had their margins cropped may be better for viewing on gadgets. Doing etale cohomology just for etale cohomology can be dry. Here is a partial list of commutative algebra concepts. Etale cohomology tom sutherland october, 2008 note. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. Finally in section 3 we explain a proof due to serre of a complex analogue of the weil conjectures which is a simpli ed model of the proof using etale cohomology. There two main motivations of grothendieck to develop the etale cohomology. Cohomology of smooth projective curves over a separably closed field.
Next we generalize the weil etale cohomology to sintegers and compute the cohomology for constant sheaves z or r. Aside from the references that you mention, see the answers and comments here textbook for etale cohomology. Weil tale cohomology over finite fields request pdf. Etale cohomology has been brilliantly successful in explaining weil s observation.
It does everything in terms of the weil conjecture and it leaves most of the technical stuff for appendixes, and makes frequent comparisons to stuff that should be familiar, like group cohomology, cech. We go through the formalism of grothendiecks six operations for these categories. A guide to etale motivic sheaves joseph ayoub abstract. The weil conjectures are a series of statements about the generating. Deligne 1980 found and proved a generalization of the weil.
Fk etale cohomology and the weil conjectures by freitag and kiehl. Number theory and representation theory seminar analytic number theory, algebraic number theory, arithmetic geometry, automorphic forms, and even some things not beginning with the letter a. Research statement minhhoang tran 1 introduction my research focuses on the area of special values of lfunctions and their relations to the weil etale cohomology. Etale cohomology and the weil conjecture pp 255281. There are also milnes course notes on various subjects. Torsors and rational points were not going to cover everything in these books. Math 274 rational points on varieties mit mathematics.
Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf. Number theory learning seminar stanford university. Test case for an absent theory michael harris department of mathematics columbia university perfectoid prologue its not often that contemporary mathematics provides such a clearcut. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. One of the rst result of special values of lfunction is the analytic class number formula which i shall explain. We study also some nonminuscule cases and show that the kottwitz conjecture is. The etale cohomology of fields, or equivalently, galois cohomology, are the topic of famous problems in modern mathematics such as the milnor conjecture and its generalization, the blochkato conjecture, which was solved by vladimir voevodsky in 2009. Etale cohomology and the weil conjectures sommaire. Math, university of cambridge, 2010 a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy in the department of mathematics at brown university providence, rhode island.
The problem however, was to prove that the etale cohomology satisfies the requirements to be a weil cohomology. In mathematics, the weil conjectures were some highly influential proposals by andre weil. Numerous and frequentlyupdated resource results are available from this search. Mircea mustata, weil cohomology theories and the weil conjectures pdf. Weil etale cohomology and special values of lfunctions of 1motives by minhhoang tran b.
With the appropriate choice of the perversity, intersection cohomology gives the usual ladic cohomology, with or without compact support. Weil etale cohomology and special values of lfunctions. As an application, we show the kottwitz conjecture in many new cases including the cases for all inner forms of gl3 and minuscule cocharacters. They also play an important part in the etale cohomology of more general rings. Etale cohomology of constructible sheaves is used to set up adic cohomology and to prove the weil conjecture on the zetafunction. In this article i prove the weil conjecture about the eigenvalues of frobenius endomorphisms. Recent research has demonstrated a connection between weiletale cohomology and special values of zeta functions. This seminar is a continuation of the course on etale cohomology taught in winter 20162017. Number theory learning seminar 20162017 the seminar will meet wednesdays 1. For the convenience of the speakers the present authors who were also the organisers of that meeting prepared short notes containing. Etale cohomology theory and a partially incomplete proof of the weil conjectures.
We also prove a conjecture of lang and weil concerning the picard varieties and etale cohomology spaces of projective varieties. How the existence of good cohomology theory implies weil conjectures besides riemann hypothesis. First, understand how a good cohomology theory like in 1. The early history the story of the weil conjectures and the development of etale cohomologyis the story of one of the great triumphs of 20th century algebraic geometry. We study weil etale cohomology, introduced by lichtenbaum for varieties over finite fields. Then understand etale topology and definition of adic cohomology groups and how the frobenius morphism happens to act on them. I do this mostly to orient the reader, since upon rereading its somewhat nonobvious what the goal of this note is. We strongly urge the reader to read this online at instead of reading the old material below. Weil cohomology theories play an important role in the theory of motives, insofar as the category of chow motives is universal for weil cohomology theories in the sense that any weil cohomology theory factors through chow motives. The riemann hypothesis for curves over finite fields.
Acrobat reader etale cohomology and the weil conjecture some years ago a conference on ladic cohomology in oberwolfach was held with the aim of reaching an understanding of delignes proof of the weil conjec tures. In particular, lichtenbaum has shown that the order of vanishing and leading coefficient of the zeta function of a smooth, projective variety over a finite field has a weiletale cohomological interpretation in terms of certain secondary euler characteristics. Some years ago a conference on ladic cohomology in oberwolfach was held with the aim of reaching an understanding of delignes proof of the weil conjec tures. Using this, the bass conjecture on finite generation of motivic cohomology, and the beilinsontate on. The weil conjectures for the cohomology below the middle dimension follow from this by applying the weak lefschetz theorem, and the conjectures for cohomology above the middle dimension then follow from poincare duality. Let kbe a number eld with r 1 real embeddings and r. So i think the best resource is the book etale cohomology and the weil conjecture. The date of 2008 on the pdf file simply reflects that i recompiled it.
At that time alexander grothendieck introduced the concept of etale cohomology and proposed that it should play the role of the mysterious, unknown, but essential weil cohomology. We also prove a conjecture of lang and weil concerning the picard varieties and. Zeta functions of varieties and the weil conjectures in number theory, one studies the riemann zeta function s x1 n1 n s y. If you have interest in this area, then i think it would be useful to study etale cohomology.
In the first half of the paper we give an explicit description of. More precisely, one would like an analogue of weil ii i. Much of the material in these notes parallels that in, for example. Eberhard freitag and reinhardt kiehl, etale cohomology and the weil conjecture article pdf available in bulletin of the american mathematical society 221990 january 1990 with 52 reads.
We construct an intermediate cohmology between motivic cohomology and weil etale cohomology. What links here related changes upload file special pages permanent link page. Etale cohomology and the weil conjecture book, 1988. The key step in dworks proof is to show that zy,t is padic meromorphic. Note that, however, the category of chow motives does not give a weil cohomology theory since it is not abelian. Weiletale cohomology over finite fields internet archive. Etale cohomology of fields and galois cohomology theories. Weil cohomology theo ries and the weil conjectures 3 proposition 1. Delignes proof of the weil conjecture springerlink. If the pdf files are placed in the same directory, some links will work between files you may have to get the correct version and rename it, e. I tried to make the demonstration as geometric and elementary as possible and included reminders. The notes also discuss the proof of the weil conjectures grothendieck and deligne.