Michael harris eisenstein cohomology and special values of lfunctions. The trace cohomology of an algebraic variety is an abelian subgroup of the real line, obtained from the canonical trace on a calgebra attached to the variety. This definition in top alone already goes a long way. Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. Choosing k m to be a sheaf of zl vector spaces we get an action of.
For each group gand representation mof gthere are abelian groups hng,m and hng,m where n 0,1,2,3. However as in the analogous ladic situation, there is a key difference between the case z. We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general p adic l functions to nearly ordinary hilbert automorphic forms. If xq is a smooth projective variety and we choose an embedding q. Thus this theorem is illustrative that our techniques allow one to study the l function of varieties for which the image of galois acting on the l adic cohomology is large. On p adic l series, p adic cohomology and class field theory david burns and daniel macias castillo abstract. Integral padic hodge theory, and qde rham cohomology. An introduction to adic sheaves and the functionsheaf. Much of the material in these notes parallels that in, for example. In the winter we will delve further into the cohomology theory especially to duality theorems and kunneth formulas, and then move on to laumons technique of ladic fourier transforms in the sheaf setting.
Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. The obstruction to defining ladic cohomology using sheaves is a shortage of open subsets in the zariski topology. One might expect the ladic cohomology groups in the more general case to similarly be usefully defined by the inverse limit of an appropriate. In this paper, we prove that the syntomic realizations of. Q l satisfy finiteness, poincar e duality, lefschetz trace formula, and riemann hypothesis. In particular, it is an isomorphism for k 2nwhich is a direct sum of tate motives by, cor.
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. On padic lseries, padic cohomology and class field theory. We will discuss recent progress by many people in the program of representing motivic cohomology with torsion coe cients of arithmetic schemes by various arithmetic padic cohomologies. Etale cohomology is an important branch in arithmetic geometry. But even though we get to this at the very end, the bulk of the paper is devoted. Galois representations in arithmetic geometry ii the. In section 2 we will note some properties of this cohomology. However, here we wish to relay another, deeper, relation between the cohomology. Etale cohomology, taking values over a ring of ladic integers for l. As far as i know, the aim was to prove the weil conjectures, which deligne did in 1974.
Degeneration of ladic eisenstein classes and of the. Some questions from the audience have been included. Let xbe a smooth, connected, ndimensional projective variety. These examples are all special cases of the true source of adic representations, which is adic etale 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. A gentle introduction to homology, cohomology, and sheaf cohomology jean gallier and jocelyn quaintance department of computer and information science university of pennsylvania philadelphia, pa 19104, usa email. The subscript l is in honor of steve lichtenbaum, who. The idea being that if one has information about the open sets that make up a space as. Our next aim is the definition of the etale or more precisely of the ladic realization of a simplicial variety.
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. X y determines a homomorphism from the cohomology ring of y to that of x. Thus, its generic bre x is a proper smooth rigidanalytic variety over c. Exterior square lfunctions, herve jacquet and joseph shalika. 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. Aug 04, 20 automorphic symbols, p adic l functions and ordinary cohomology of hilbert modular varieties by mladendimitrov abstract.
Rational homotopy and ladic cohomology mathoverflow. Number theory learning seminar stanford university. If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article. Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles weibel american mathematical society clay mathematics institute clay mathematics monographs volume 2.
Automorphy for some ladic lifts of automorphic mod l galois. The specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as eisenstein classes. Etale cohomology theory nankai tracts in mathematics. A gentle introduction to homology, cohomology, and sheaf. On motivic cohomology with zlcoefficients 405 proof. Kedlaya department of mathematics, massachusetts institute of technology.
Etale cohomology was invented by grothendieck and his collaborators in the 1960s. Structure of various lambdaadic arithmetic cohomology groups. It is called the th adic cohomology of the sheaf on. This is an adic gkrepresentation of dimension 2g, where g is the dimension of a. Automorphic symbols, padic lfunctions and ordinary. An introduction to adic sheaves and the functionsheaf dictionary michael groechenig lausanne, march 20 1 the quest for the right topology the underlying zariski topological space jxjof an algebraic varietyscheme xdoes. A nice explanation of what is a smooth ladic sheaf. Automorphic symbols, padic lfunctions and ordinary cohomology of hilbert modular varieties. The etale homology and the cycle maps in adic coefficients arxiv. We introduce the notion of automorphic symbol generalizing the classical modular symbol and use it to attach very general padic lfunctions to nearly ordinary hilbert automorphic forms. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016. The kunnet homomorphism is an isomorphism for all spaces whose motives are direct sums of tate motives.
Phrased like this, the construction works in ktheory as well. Automorphic symbols, padic lfunctions and ordinary cohomology of hilbert modular varieties by mladendimitrov abstract. 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. If is the field of rational adic numbers, then the spaces are called the rational adic cohomology of the scheme. In this article, we are interested in determining the ladic cohomology of rapoportzink spaces associated to gln over an unramified extension of asp called unramified a. We establish several close links between the galois structures of a range of arithmetic modules including certain natural families of ray class. An introduction to the cohomology of groups peter j. Then for each prime l p, we have ladic cohomology groups. That will get us through the important smooth and proper base change theorems, as well as the basic formalism of ladic cohomology. Completed cohomology and the adic lang lands program. Automorphic forms and the cohomology of vector bundles on shimura varieties, michael harris. Olsson, chair we develop the notion of strati ability in the context of derived categories and the six operations for stacks in 26, 27. We report on the progress that has been made on this topic during the past ten years. The fundamental theorems in etale cohomology apply to adic cohomology of constructible adic sheaves.
Weil cohomology theories and the weil conjectures 3 proposition 1. 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. In this thesis, we study the structure of various arithmetic cohomology groups as iwasawa. We also wanted to see but ran out of time that the zlncohomology groups had a very strong poincar e duality formalism. 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. We discuss classical questions concerning traces of elements of galois groups or correspondences in.
C then the natural action of gq on the cohomology hixc,q l. Im currently trying to understand the construction of the category of ladic constructible sheaves as in sga5, and it seems that quite a lot of machinery the mlar condition, localization of the category of projective systems, etc. This fact, the first of the celebrated weil conjectures, follows from dworks proof using padic analysis 12, or from the properties of etale. Lectures on the cohomology of finite groups 3 2 using joins, we may construct a model for egwhich is functorial in g, namely eg colim i g. Completed cohomology and the padic langlands program 3 suitable surrogate for a space of padic automorphic forms. Weil cohomology theory when chark p 0, the ladic cohomology with k ql, for.
Problems arising from the tate and beilinson conjectures in the context of shimura varieties, dinakar ramakrishnan. In this note i try to give a very brief introduction to etale cohomology, especially to adic cohomology. A nice explanation of what is a smooth l adic sheaf. The remaining talks, given in the category theory seminar at chicago, were more advanced. That will get us through the important smooth and proper base change theorems, as well as the basic formalism of l adic cohomology. The machinery of etale cohomology and its relative, ladic cohomology, is formidable. In the hodge theoretic analogue this can be done by direct. We will show that a similar calculation is possible for hilbert modular forms over real quadratic fields using the second cohomology group. Eisenstein cohomology and special values of lfunctions. In a paper published in 1959, shimura presented an elegant calculation of the critical values of lfunctions attached to elliptic modular forms using the first cohomology group. We will assume that p 1 mod l and that the representation has unipotent. I will be glad if moreover someone can point out a reference about this i have looked at milnes notes and book and lei fus book, but they dont talk about this. Morrow let c be an algebraically closed complete extension of q p.
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