Weil cohomology theories and the weil conjectures 3 proposition 1. It is a very intricate subject that thrives on its interaction with algebraic ktheory, arithmetic geometry, number theory, motivic cohomology, hodge theory and mathematical physics. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves. In this paper we consider period domains over local fields for quasisplit reductive groups. A comparison theorem for ladic cohomology compositio.
L9 o is nilpotent, then its second cohomology group is not equal to 0j. Our proof is purely local and does not require global automorphic methods. The lefschetz numbers were used by deligne and lusztig to construct virtual characters of. Of local or relative cohomology groups of shea ves on preschemes. Rigid cohomology does seem to be a universal padic cohomology with. It has been known for many years that the stabilization of the arthurselberg trace formula would, or perhaps we should write will, have important consequences for the langlands functoriality program as well as for the study of the galois representations on the ladic cohomology of shimura varieties. Hida, modules of congruence of hecke algebras and lfunctions associated with cusp forms, amer. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ladic cohomology. As a second year graduate textbook, cohomology of groups introduces students to cohomology theory involving a rich interplay between algebra and topology with a minimum of prerequisites. In this paper, we study the l adic cohomology of the rapoportzink tower for gsp4. On padic lseries, padic cohomology and class field theory. Structure of various lambdaadic arithmetic cohomology groups. Lefschetz trace formula and ladic cohomology of lubintate.
Completed cohomology and the adic lang lands program. This material has since appeared in expanded and generalized form in his paris seminar of 1962 16 and my duality seminar at harvard in 196364 17. If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article. We investigate the action of the weil group on the compactly supported ladic cohomology groups of rigid spaces over local fields. The purpose of this paper is to investigate the relations between the cohomology groups of g, k, and gk. This book aims first to prove the local langlands conjecture for gl n over a padic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the ladic cohomology of the simple shimura varieties. Completed cohomology and the padic langlands program 3 suitable surrogate for a space of padic automorphic forms. Introduction the elliptic polylogarithm was introduced by beilinson and levin in their seminal paper bl. The remaining talks, given in the category theory seminar at chicago, were more advanced. That is, when the l2 stokes theorem holds, the reduced l2 cohomology is simply the space of l2 harmonic forms. We also discuss how this representation can be computed in several simple cases. Then we reprove behrends lefschetz trace formula for stacks, and give the meromorphic continuation of the lseries of stacks defined over a finite field. Etale cohomology, taking values over a ring of ladic integers for l.
Hida, a padic measure attached to the zeta functions associated with two elliptic modular forms ii, ann. The geometry and cohomology of some simple shimura varieties. We establish several close links between the galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of p adic artin l series, the. A comparison theorem for l adic cohomology volume 112 issue 2 r. The etale fundamental group and etale cohomology with compact support.
Some questions from the audience have been included. We study the structure of the ordinary parts of the arithmetic. The other important technical advantage is the ease with which one. Totally degenerate reduction over padic fields to j. Posts about l adic cohomology written by alexyoucis. The fundamental theorems in etale cohomology apply to adic cohomology of constructible adic sheaves. Etale cohomology of algebraic varieties with totally. Also, in weibel, a functor category need not be a category. In this article, we investigate the alternating sum of the l adic cohomology of the lubintate tower by the lefschetz trace formula. 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. Furthermore, it may appear in the later sections of his chapter 111 however, i have thought it elements. Posts about ladic cohomology written by alexyoucis. We compute the continuous ladic cohomology with compact support of them in the case of a basic isocrystal.
On p adic l series, p adic cohomology and class field theory david burns and daniel macias castillo abstract. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 arizona winter school. Motivation the motivation here comes from observing the set of all functions f from basis elements of cn to some selected group g. An introduction to ladic sheaves and the functionsheaf dictionary. Galois representations can be constructed by taking the. We develop the notion of stratifiability in the context of derived categories and the six operations for stacks in the work of laszlo and olsson. Finally, we show following cartan that the third cohomology group of a semisimple algebra ljo never reduces to 0, which proves that the third betti number of a compact connected semisimple lie. This book covers the main materials in sga 1, sga 4, sga 4 12 and sga 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, galois cohomology, etale cohomology, derived categories, base change theorems, duality, and l adic cohomology. On padic lseries, padic cohomology and class field theory david burns and daniel macias castillo abstract. Etale cohomology is an important branch in arithmetic geometry.
An introduction to bounded cohomology michelle bucher april 24, 2008 abstract lecture notes for a 4 x 1 hour minicourse on bounded cohomology given in april 2008, at kth. Trying to understand delignes proof of the weil conjectures people. On the action of the weil group on the ladic cohomology of. Jul 15, 2014 moreover, a pull back of such a cohomology tensored by the field of l adic numbers is isomorphic to the l adic cohomology. We compute the continuous l adic cohomology with compact support of them in the case of a basic isocrystal. 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 establish several close links between the galois structures of a range of arithmetic modules including certain natural families of ray class groups, the values at strictly positive integers of padic artin lseries, the. The same is true if you ask the system sage for the padic regulator of an elliptic curve over q, for pa good ordinary prime.
The idea being that if one has information about the open sets that make up a space as. Xy be a quasicompact and quasi separated morphism of algebraic spaces over s. New edition available here etale cohomology is an important branch in arithmetic geometry. We prove that every eigenvalue of the action is a weil number when either a rigid space is smooth or the characteristic of the base field is equal to 0. L theorie des topos et cohomologie etale des schemas.
Bh17 bergdall, john and hansen, david, on padic lfunctions for hilbert modular forms. There the elliptic polylogarithm appears as an element in the motivic cohomology of a relative elliptic curve ebminus the zero section. The proper setup for this would be schemes, but for sake of simplicity i will only consider varieties. In this article, we investigate the alternating sum of the ladic cohomology of the lubintate tower by the lefschetz trace formula. Furthermore, it may appear in the later sections of. The geometry and cohomology of some simple shimura. Topics include nonabelian cohomology, postnikov towers, the theory of nstu, and ncategories for n 1 and 2. Available formats pdf please select a format to send.
Number theory learning seminar 20162017 the seminar will meet wednesdays 1. If is the field of rational adic numbers, then the spaces are called the rational adic cohomology of the scheme. Then we reprove behrends lefschetz trace formula for stacks, and give the meromorphic continuation of the l series of stacks defined over a finite field. To accomplish these tasks, we define and study the proetale topology, which is a. Finally, we show following cartan that the third cohomology group of a semisimple algebra l jo never reduces to 0, which proves that the third betti number of a compact connected semisimple lie group is always not equal to 0. Y is a generically nite, surjective morphism of degree. In this paper, we study the ladic cohomology of the rapoportzink tower for gsp4.
Our presentation of the formalism of weil cohomology theories follows with small. The padic hodge theory of semistable galois representations. For example, the first cohomology group of a complex elliptic curve is a free module of rank 2 over the integers, while the first. In this post we discuss the galois representation associated to a projective scheme, where is a number field. Etale cohomology of rigid analytic varieties and adic spaces. This book aims first to prove the local langlands conjecture for gl n over a p adic field and, second, to identify the action of the decomposition group at a prime of bad reduction on the l adic cohomology of the simple shimura varieties. Finite polynomial cohomology for general varieties pdf.
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