Poincare duality for etale cohomology tony feng february 1, 2017 contents 1 statementofpoincareduality 1 2 thetracemap 3 3 derivedcategories 10 4 thedualitytheorem 16. The subscript l is in honor of steve lichtenbaum, who. Compatibility of local and global langlands correspondences. We study the ladic cohomology of the dual lubintate tower by using the exterior power of a pidivisible omodule to relate it to the cohomology of the lubintate tower. Topics include nonabelian cohomology, postnikov towers, the theory of nstu, and ncategories for n 1 and 2. Thus, studying a family of covers of g m q gave us a natural representation of g q, which is in fact wellde ned one can. Etale cohomology was invented by grothendieck and his collaborators in the 1960s. 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. Lefschetz trace formula and ladic cohomology of lubintate tower, mathematical research letters 19 2012, 95107. This years seminar will focus on etale cohomology, the goal being to. 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. The papers center is the solution of davenports problem. By using a result of harristaylor on the cohomology of the lubintate tower, we show that the supercuspidal part of the cohomology of the dual lubintate tower realizes the local langlands and the jacquetlanglands. The main tools used are the machinery of l adic cohomology and the converse theorem for automorphic representations.
If x is a smooth, projective variety over a number field k, then the absolute galois group gk gal k. Completed cohomology and the adic lang lands program. In this paper, we study the l adic cohomology of the rapoportzink tower for gsp4. Every lworks equally well, they are nite dimensional q lvector spaces whose dimension is independent of l. By matthew emerton and mark kisin let p be a prime number, fp the. Completed cohomology and the p adic langlands program 3 suitable surrogate for a space of p adic automorphic forms. Geometry of shimura varieties in positive characteristics. The idea being that if one has information about the open sets that make up a space as. The other important technical advantage is the ease with which one. On p adic l series, p adic cohomology and class field theory david burns and daniel macias castillo abstract.
In this case, there is an ladic galois representation associated to. One assertion of the theory, a simple statement of existence, is that an lfunctions for gl 1,aq associated to a grossencharakter. Y is a generically nite, surjective morphism of degree. Weil cohomology theories and the weil conjectures 3 proposition 1. In the second part of the paper, by using berthelot, esnault and r\ullings trace morphisms in ber, we first prove the case of coherent cohomology, then we extend it to the witt vector cohomology, and we then get a result on the frobeniusstable part of the witt vector cohomology, which corresponds the slope 0 part of the rigid cohomology. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 arizona winter school. Given any prime l6 p, one has the ladic etale cohomology groups hix. In this article, we investigate the alternating sum of the ladic cohomology of the lubintate tower by the lefschetz trace formula. We prove the homotopy invariance of coarse cohomology, and of the ktheory of the c.
We strengthen the compatibility between local and global langlands correspondences for gl n when n is even and l p. These notes outline the fundamental theorems of etale cohomology, following. The fundamental theorems in etale cohomology apply to adic cohomology of constructible adic sheaves. If is the field of rational adic numbers, then the spaces are called the rational adic cohomology of the scheme. An element f 2c k is exact or a coboundary if f is in the image of, i. Automorphy and irreducibility of some ladic representations. Lefschetz trace formula and ladic cohomology of lubin. Z pup z qons x lasqns uado asooqo kq paaouap aq x uo san vaqs uo sdno. Variables separated equations, contrasts the contributions of the simple group classification and of the branch cycle lemma for figuring the defining. Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space.
Completed cohomology and the padic langlands program 3 suitable surrogate for a space of padic automorphic forms. Translating the preceding in terms of ladic cohomology, serre has conjectured the exis tence, for each l, of a representation of galklq in a. One the padic local invariant cycle theorem caltechthesis. Lefschetz trace formula and ladic cohomology of lubintate. On padic lseries, padic cohomology and class field theory. This definition in top alone already goes a long way. Choosing m lkfor lsome xed prime, and kvarying over all positive integers, and making noncanonical, but a compatible system of identi cations zmz. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be. First cohomology theory of the kind described by weil was the etale cohomology, constructed in early 1960s by alexander grothendieck and michael artin. Thus, its generic bre x is a proper smooth rigidanalytic variety over c. Some questions from the audience have been included. A homotopy invariance theorem in coarse cohomology and ktheory nigel higson and john roe abstract. On the rigid cohomology of certain shimura varieties.
We study the l adic cohomology of the dual lubintate tower by using the exterior power of a pidivisible omodule to relate it to the cohomology of the lubintate tower. In this paper, we study the ladic cohomology of the rapoportzink tower for gsp4. Here we shall use the fact that the curve a if not its complex multiplication is already defined over. All links are to dvi files or to pdf files, as indicated. In this article, we investigate the alternating sum of the l adic cohomology of the lubintate tower by the lefschetz trace formula. In complete analogy with the case for absolute hodge cohomology, rigid syntomic cohomology h1 synx,m with values in an admissible. The main tools used are the machinery of ladic cohomology and the converse theorem for automorphic representations. Istrodlction in this paper i shall describe a method of associating a spectrum, and hence a cohomology theory, to a category with a compositionlaw of a suitable kind. Let xbe a smooth, connected, ndimensional projective variety. The field of definition of function fields and a problem in the reducibility of polynomials in two variables, illinois journal of math. We say that an element f 2c k is closed or a cocycle if f 0.
Bernstein bei linson, deligne, gabber faisceaux perverse asterisgue too constructible sheaves let x be a variety over ke and l be a prime different from char k then an ladic sheafon x is an inverse system of etoile sheaves ful such thateach fu is a illzlxmodulefut, fu induces anisomorphism fine. The authors do motivate the definition through the consideration of ordinary vector calculus, which serves to ease the transition to the more. Cuspidal representations in the ladic cohomology of the. Integral padic hodge theory, and qde rham cohomology. Our proof is purely local and does not require global automorphic methods. Morrow let c be an algebraically closed complete extension of q p. A family of calabiyau varieties and potential automorphy.
X y determines a homomorphism from the cohomology ring of y to that of x. Xy be a quasicompact and quasi separated morphism of algebraic spaces over s. The padic hodge theory of semistable galois representations. Unit lfunctions 1 unit lfunctions and a conjecture of katz. With appendix a, summarizing unpublished work of russ mann, and appendix b by mariefrance vigneras. Automorphy for some ladic lifts of automorphic mod l representations.
Richard taylors home page ias school of mathematics. The projective limit is naturally equipped with the structure of a module on which acts continuously with respect to the adic topology. 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. Tensoring up to q l, this gives a 1dimensional galois representation, the socalled ladic cyclotomic character. The remaining talks, given in the category theory seminar at chicago, were more advanced. In algebraic geometry, we have something called weil cohomology theories, which formalize the notion of a good cohomology theory of smooth projective varieties. We introduce a notion of homotopy which is appropriate to the coarse geometry and topology studied by the second author in 7. It is called the th adic cohomology of the sheaf on. By the brown representability theorem all cohomology theories that are called generalized eilenbergsteenrod cohomology theories are of this form, for a a a topological space that is part of a spectrum. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext.
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