Weil conjecture, standard conjectures on algebraic cycles. 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. We study also some nonminuscule cases and show that the kottwitz conjecture is. An introduction to bloch and katos conjecture two lectures at the clay mathematical institute summer school, honolulu, hawaii, 2009 prerequisites. This note is based on the 3hour presentation given in the student seminar on winter 2014. In the first half of the paper we give an explicit description of. As an application, we show the kottwitz conjecture in many new cases including the cases for all inner forms of gl3 and minuscule cocharacters.
Weiletale cohomology over finite fields internet archive. We go through the formalism of grothendiecks six operations for these categories. An introduction to bloch and katos conjecture two lectures at the clay mathematical institute summer school. Etale cohomology and the weil conjecture eberhard freitag. Tom lovering june 22, 2012 for mathematical musings. 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. Jan 24, 2017 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. Zeta functions of varieties and the weil conjectures in number theory, one studies the riemann zeta function s x1 n1 n s y. The problem treated is exceedingly elementary counting solutions of polynomials over. Weil cohomology theo ries and the weil conjectures 3 proposition 1. Etale cohomology and the weil conjecture book, 1988.
Regardless of whether or not you get to the weil conjectures, the etale topology has all kinds of uses, so the effort would not be wasted. Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf. 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. There two main motivations of grothendieck to develop the etale cohomology. Learning path for the proof of the weil conjectures.
Using this, the bass conjecture on finite generation of motivic cohomology, and the beilinsontate on. Weildeligne representations and padic hodge theory. 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. 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. Delignes proof of the weil conjecture springerlink. There are also milnes course notes on various subjects.
If you have interest in this area, then i think it would be useful to study etale cohomology. Weil etale cohomology and special values of lfunctions of 1. 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. This seminar is a continuation of the course on etale cohomology taught in winter 20162017. 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. Weil etale cohomology and special values of lfunctions of 1motives by minhhoang tran b. Weil tale cohomology over finite fields request pdf. 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. 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.
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. We strongly urge the reader to read this online at instead of reading the old material below. Math 274 rational points on varieties mit mathematics. 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.
They also play an important part in the etale cohomology of more general rings. The key step in dworks proof is to show that zy,t is padic meromorphic. The notes also discuss the proof of the weil conjectures grothendieck and deligne. Let kbe a number eld with r 1 real embeddings and r. Etale cohomology and the weil conjectures sommaire.
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. The date of 2008 on the pdf file simply reflects that i recompiled it. 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. Then understand etale topology and definition of adic cohomology groups and how the frobenius morphism happens to act on them. Cohomology of smooth projective curves over a separably closed field. The conjectures of andr e weil have influenced or directed much of 20th century algebraic geometry.
The weil conjectures are a series of statements about the generating. Tom lovering june 22, 2012 abstract in this essay we brie. I do this mostly to orient the reader, since upon rereading its somewhat nonobvious what the goal of this note is. In this article i prove the weil conjecture about the eigenvalues of frobenius endomorphisms. The padic hodge theory of semistable galois representations. 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. Trying to understand delignes proof of the weil conjectures a tale in two parts january 29, 2008 part i.
Here is a partial list of commutative algebra concepts. Torsors and rational points were not going to cover everything in these books. First, understand how a good cohomology theory like in 1. Tamas szamuely etale cohomology and the weil conjectures. We also prove a conjecture of lang and weil concerning the picard varieties and etale cohomology spaces of projective varieties. Series of modern surveys in mathematics on free shipping on qualified orders. 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. Doing etale cohomology just for etale cohomology can be dry. A guide to etale motivic sheaves joseph ayoub abstract. How the existence of good cohomology theory implies weil conjectures besides riemann hypothesis. Much of the material in these notes parallels that in, for example. More precisely, one would like an analogue of weil ii i. Deligne succeded in proving the weil conjectures on the basis of grothendiecks ideas.
Weil cohomology theories and the weil conjectures 3 proposition 1. The pdf files are formatted for printing on a4letter paper. Numerous and frequentlyupdated resource results are available from this search. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. 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. What links here related changes upload file special pages permanent link page. Etale cohomology theory and a partially incomplete proof of the weil conjectures.
Etale cohomology has been brilliantly successful in explaining weil s observation. 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. 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. One of the rst result of special values of lfunction is the analytic class number formula which i shall explain. In particular, there is a long exact sequence relating weil etale cohomology and etale cohomology. 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.
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 give a conjectural description of the vanishing order and leading taylor coe. Finite generation conjectures for cohomology over finite. Note that, however, the category of chow motives does not give a weil cohomology theory since it is not abelian. Tamas szamuely etale cohomology and the weil conjectures periodo. Mircea mustata, weil cohomology theories and the weil conjectures pdf. We also prove a conjecture of lang and weil concerning the picard varieties and. We will basically follow milec, chapter i,ii,iii,v and millec, sections 114.
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. I tried to make the demonstration as geometric and elementary as possible and included reminders. Recent research has demonstrated a connection between weiletale cohomology and special values of zeta functions. We construct an intermediate cohmology between motivic cohomology and weil etale cohomology. We recall the construction, following the method of morel and voevodsky, of the triangulated category of etale motivic sheaves over a base scheme. Let xbe a smooth, connected, ndimensional projective variety. Eberhard freitag and reinhardt kiehl, etale cohomology and the weil conjecture nicholas katz. 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. Number theory learning seminar stanford university. The cropped files have had their margins cropped may be better for viewing on gadgets.
Weil formulated his famous conjectures about the numbers of solutions of diophantine equations in finite fields. Deligne 1980 found and proved a generalization of the weil. The prerequisites for these lectures are elementary. Etale cohomology tom sutherland october, 2008 note. Fk etale cohomology and the weil conjectures by freitag and kiehl. Number theory learning seminar 20162017 the seminar will meet wednesdays 1. 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. Weil etale cohomology and special values of lfunctions. In mathematics, the weil conjectures were some highly influential proposals by andre weil. We study weil etale cohomology, introduced by lichtenbaum for varieties over finite fields. So i think the best resource is the book etale cohomology and the weil conjecture. The general theorems about etale cohomology allowed grothendieck to prove an. For the convenience of the speakers the present authors who were also the organisers of that meeting prepared short notes containing. Etale cohomology and the weil conjecture ebook, 1988.
Etale cohomology and the weil conjecture springerlink. Oclcs webjunction has pulled together information and resources to assist library staff as they consider how to handle coronavirus. Aside from the references that you mention, see the answers and comments here textbook for etale cohomology. Etale cohomology and the weil conjecture pp 255281. Etale cohomology of fields and galois cohomology theories. The problem however, was to prove that the etale cohomology satisfies the requirements to be a weil cohomology. We will describe the history of the padic cohomological program in the next section. Etale cohomology of constructible sheaves is used to set up adic cohomology and to prove the weil conjecture on the zetafunction. He himself proved his conjectures by means of an algebraic theory of abelian varieties in the onevariable case. Reinhardt kiehl this book is concerned with one of the most important developments in algebraic geometry during the last decades. Trying to understand delignes proof of the weil conjectures. 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. 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. With the appropriate choice of the perversity, intersection cohomology gives the usual ladic cohomology, with or without compact support.
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