Etale cohomology and the weil conjecture pdf download

Weil conjecture, standard conjectures on algebraic cycles. It is now some sixteen years since delignes spectacular proof. They also play an important part in the etale cohomology of more general rings. Some years ago a conference on ladic cohomology in oberwolfach was held with. 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. Weil cohomology theo ries and the weil conjectures 3 proposition 1.

Y is a generically nite, surjective morphism of degree. Mircea mustata, weil cohomology theories and the weil conjectures pdf. There exists a functorial homomorphism from the group of algebraic cycles in of codimension into which maps the direct product of cycles to the tensor product and is nontrivial in the sense that, for a point, becomes the canonical imbedding of into. Weil tale cohomology over finite fields request pdf. In this section, we will see that the weiletale cohomology h w. 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. This is a neron model version of the corresponding result gei04, thm. Lectures on etale cohomology an introductory overview. Cohomology of algebraic varieties with coefficients in a field of characteristic zero, with formal properties required to obtain the lefschetz formula for the number of fixed points. The earliest antecedent of the weil conjectures is by carl friedrich gauss and appears in section vii of his disquisitiones arithmeticae, concerned with roots of unity and gaussian periods.

We give a reformulation of the birch and swinnertondyer conjecture over global function fields in terms of weil etale cohomology of the curve with coefficients in the neron model, and show that it holds under the assumption of finiteness of the tateshafarevich group. From chapter 3, we assume that the ground eld f is totally imaginary. Weil had no suggestions on how to define weil cohomology, but he knew what qualities cohomology should have to provide a proof of the weil conjectures. Etale cohomology course notes department of mathematics. Weiletale cohomology and deningers dynamical system wec. Eberhard freitag and reinhardt kiehl, etale cohomology and the weil conjecture article pdf available in bulletin of the american mathematical society 221990. Deweil i in june, 1973, of the riemann hypothesis for zeta functions of projective nonsingular. The hardest part is an analogue of the riemann hypothesis. The weil conjectures this seminar is a continuation of the course on etale cohomology taught in winter 20162017. In mathematics, the etale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by grothendieck in order to prove the weil conjectures. Every algebraic cycle of codimension gives a cohomology class in dimension, which makes it possible to construct a theory of chern classes cf. As a consequence, we get a long exact sequence relating weil etale cohomology to etale cohomology, show that for finite coefficients the cohomology theories agree, and with. A concise account of the weil conjectures and etale cohomology 3 for instance, if x is a hypersurface of degree d and dimension n over fq. The weil conjectures are a series of statements about the generating functions obtained by counting the number of points of smooth projective varieties.

Etale cohomology and the weil conjectures sommaire. 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. Etale cohomology and the weil conjecture book, 1988. The problem treated is exceedingly elementary counting solutions of polynomials over finite fields but the proofs require and indeed motivated the creation of. In comparison with my book, the emphasis is on heuristics rather than formal proofs and on varieties rather than schemes, and it includes the proof of the weil conjectures.

We study weil etale cohomology, introduced by lichtenbaum for varieties over finite fields. The necessity for such a theory was pointed out by a. Series of modern surveys in mathematics on free shipping on qualified orders. Learning path for the proof of the weil conjectures. We strongly urge the reader to read this online at instead of reading the old material below. We now turn to the proof of the functional equation, assuming all the good properties of. Etale cohomology has been brilliantly successful in explaining weil s observation. Weil, who showed that the rationality of the zetafunction and function of a variety over a finite field follow from the lefschetz formula, whereas the. Get your kindle here, or download a free kindle reading app. Etale cohomology and the weil conjecture eberhard freitag.

At the end of the 1940s nobody knew any cohomology which could solve the conjectural problem and thus unify the geometric aspect, related to the solution of equations and the arithmetic aspect. We give a conjectural description of the vanishing order and leading taylor coe. The conjectures of andr e weil have influenced or directed much of 20th century algebraic geometry. The story of the weil conjectures and the development of etale cohomology is the. 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. Etale cohomology of fields and galois cohomology theories. 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. Free algebraic geometry books download ebooks online textbooks. Trying to understand delignes proof of the weil conjectures. The most useful case of the conjecture is proved in proposition 1. The weil conjectures, as we sketch next section, led to the development of etale cohomology, as 4 above suggests that a certain cohomology theory is lurking in the background, and grothendieck realized that a suitable cohomology theory would be very useful in proving the weil conjectures. S, a is an integral model for the corresponding ladic and padic cohomology theory if is finite.

This note is based on the 3hour presentation given in the student seminar on winter 2014. The aim of our research proposal is to study the connections between lichtenbaums weiletale cohomology and deningers dynamical system. Eberhard freitag and reinhardt kiehl, etale cohomology and the weil conjecture. Sophie morelthe weil conjectures, from abel to deligne. In lic09b, lichtenbaum also made a conjecture for special values of lfunctions of 1motives over q correct up to a rational factors. Etale cohomology theory and a partially incomplete proof of the weil conjectures. Folge a series of modern surveys in mathematics eberhard freitag, reinhardt kiehl download bok. Weil etale cohomology and special values of lfunctions. Freitag, eberhard, 1946 etale cohomology and the weil conjecture. Weil cohomology theories and the weil conjectures 3 proposition 1. The weil conjectures and analogues in complex geometry jeremy booher the weil conjectures are a statement about the zeta function of varieties over nite elds. Weissauer, weilconjectures,perversesheavesandladicfouriertransform. Weiletale cohomology respectively deningers program is meant to provide an arithmetic cohomology respectively a geometric cohomology relevant for the study of motivic lfunctions.

Much of the material in these notes parallels that in, for example. 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 intended especially to provide a complete introduction to etale and ladic cohomology theory including the monodromy theory of lefschetz pencils. Definition and the basic properties, cohomology of curves, cohomological dimension, purity. In the first half of the paper we give an explicit description of. Etale morphisms, etale fundamental group, the local ring for the etale topology, sheaves for the etale topology, direct and inverse images of sheaves, cohomology. Introduction the conjecture of birch and swinnertondyer is one of the most important problems in arithmetic geometry. Let xbe a smooth, connected, ndimensional projective variety. Number theory learning seminar 20162017 the seminar will meet wednesdays 1. Hochschild cohomology and group actions, differential weil descent and differentially large fields, minimum positive entropy of complex enriques surface automorphisms, nilpotent structures and collapsing ricciflat metrics on k3 surfaces, superstring field theory, superforms and supergeometry, picard groups for tropical toric. Delignes proof of the weil conjecture springerlink. Given a shimura datum c, h of pel type, let p be an odd prime at which g is unramified.

In the first half of the paper we give an explicit description of the base change from weil etale cohomology to etale cohomology. We will basically follow milec, chapter i,ii,iii,v and millec, sections 114. The essentials of etale cohomology theory springerlink. I especially recommend the notes of james milne on etale cohomology that also contain a justification for the theory. Of course, all the central ideas are due to the people who. A weiletale version of the birch and swinnertondyer formula.

Here is a partial list of commutative algebra concepts. Questions and speculation on cohomology theories in. Etale cohomology and the weil conjecture springerlink. Eberhard freitag and reinhardt kiehl, etale cohomology and the weil conjecture nicholas katz. Weiletale cohomology over finite fields internet archive. Citeseerx document details isaac councill, lee giles, pradeep teregowda. In, we established a formula computing the ladic cohomology of the. Our exposition heavily relies on the etale cohomology theory of grothendieck so i included an overview in section 2. In article 358, he moves on from the periods that build up towers of quadratic extensions, for the construction of regular polygons. The desire to prove them motivated the development of etale cohomology, a process begun by grothendieck and nished by deligne.