The fundamental theorems in etale cohomology apply to adic cohomology of constructible adic sheaves. On the adic cohomology of varieties over number fields and its galois cohomology uvve jannsen if x is a smooth, projective variety over a number field fc, then the absolute galois group gk galfcfc acts on the etale cohomology groups hlx, qi1en, where x x xf c jc for an algebraic closure jc of k. In order to prove these theorems, we establish some new results about the p adic hodge theoretic properties of the etale cohomology of varieties over a number eld or p adic eld with coe cients in a eld of characteristic p. The main innovation is a new approach to the automorphy of nonminimal lifts which is closer in spirit to the methods of tw than to those. A cohomology operation is a family of morphism between cohomology groups, which is natural with respect to the base space equivalently, if the cohomology theory has a classifying space as it does for all usual notions of cohomology, in particular for all generalized eilenbergsteenrod cohomology theories then, by the yoneda lemma, cohomology operations are in natural bijection with. We discuss classical questions concerning traces of elements of galois groups or correspondences in. Huber skip to main content we use cookies to distinguish you from other users and to provide you with a. In this paper, we prove that the syntomic realizations of. For l p, proper base change in etale cohomology gives a canonical. Far hills inn is famous for its crosscountry ski trails park, its guest services and its cuisine. Completed cohomology and the adic lang lands program. Automorphy for some ladic lifts of automorphic mod l.
Posts about l adic cohomology written by alexyoucis. Compatibilities, correspondences, and integral structures in padic cohomology. The workshop takes place at the far hills inn, located in the laurentians, 1 hours drive from montreal. Fourier,grenoble 68,52018217722 overconvergent cohomology of hilbert modular varieties and p adic l functions by daniel barrera salazar abstract. Morrow let c be an algebraically closed complete extension of q p. In this post we discuss the galois representation associated to a projective scheme, where is a number field. Syntomic cohomology and padic regulators for varieties over. Transportation to and from downtown montreal for people arriving january 1 and leaving january 4 will be provided by the organizers. Integral padic hodge theory, and qde rham cohomology. Lectures from the 2017 arizona winter school mathematical surveys and monographs 242, american mathematical society, providence, ri. Thus, its generic bre x is a proper smooth rigidanalytic variety over c. Integral p adic hodge theory bhargav bhatt, matthew morrow, and peter scholze abstract. Syntomic cohomology and padic regulators for varieties over padic fields jan nekova. If x is a smooth, projective variety over a number field k, then the absolute galois group g k galk.
Etale cohomology pro and con todays lecture has a somewhat schizophrenic goal. Counting the number of points on an algebraic variety over a finite field is one of the oldest topics in algebraic geometry, dating back to the work of gauss. Notes on the ainfcohomology of integral padic hodge theory. If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article. It is called the th adic cohomology of the sheaf on. There is also a construction of rigid cohomology with supports in a closed subscheme, and of cohomology with compact supports. Let e be a cmelliptic curve overq with good reduction at a prime p.
We need a cohomological formalism for constructible torsion sheaves. We show that the logarithmic version of the syntomic cohomology of fontaine and messing for semistable varieties over padic rings extends uniquely to a cohomology theory for varieties over padic. We use cookies to offer you a better experience, personalize content, tailor advertising, provide social media features, and better understand the use of our services. On motivic cohomology with zlcoefficients 405 proof. Therefore we will apply the etale cohomology of adic. Phrased like this, the construction works in ktheory as well. Weil cohomology in practice this page is due to be taken down in summer 2020.
In section 2 we will note some properties of this cohomology. On combining the two definitions, we obtain the notion. Some versions of cohomology arise by dualizing the construction of homology. Overconvergent cohomology of hilbert modular varieties and. Etale cohomology theory nankai tracts in mathematics.
This is a long theory to do everything, but one computation of ext r goes as follows. To understand this we need to know what a representation of gis. One could begin with homology and perhaps continue with cohomology before turning to homotopy. 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. Much of the material in these notes parallels that in, for example. We need to consider hodge structures together with adic representations in order. Let y be a finitely generated ztsubmodule of hg, t. In particular, it is an isomorphism for k 2nwhich is a direct sum of tate motives by, cor. In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups associated to a topological space, often defined from a cochain complex.
It is customary to use p for a prime number regarded as a point in s, and for the coe. This is close to but not the same as describing the l adic cohomology of rapoportzink spaces. Given an arbitrary finitedimensional vector space f over r, we denote by cm, v the vector space of all glinear alternating functions defined on vim with values in v. Etale cohomology, taking values over a ring of ladic integers for l. Completed cohomology and the padic langlands program 3 suitable surrogate for a space of padic automorphic forms. On p adic l series, p adic cohomology and class field theory david burns and daniel macias castillo abstract. For each group gand representation mof gthere are abelian groups hng,m and hng,m where n 0,1,2,3.
The subscript l is in honor of steve lichtenbaum, who. Galois representations in arithmetic geometry ii 3 glq vgl nq, where v is an ndimensional vector space over the adic number. Bernard le stum, one century of p padic geometry from hensel to berkovich and beyond, talk notes, june 2012. The kunnet homomorphism is an isomorphism for all spaces whose motives are direct sums of tate motives. We establish several close links between the galois structures of a range of arithmetic modules including certain natural families of ray class. The specializations of the motivic elliptic polylogarithm on the universal elliptic curve to the modular curve are referred to as eisenstein classes. He laid the modern foundations of algebraic number theory by. A adic local monodromy theorem annals of mathematics. The best conceivable analogue of grothendiecks theorem. Here, the strategy is to combine proetale localization with. Etale cohomology is an important branch in arithmetic geometry. An introduction to the cohomology of groups peter j.
Motivic cohomology is usually thought of as the universal blochogus cohomology, while the universal weil cohomology should probably be pure motives with respect to rational equivalence probably, because it depends on what exactly you mean by universal and weil cohomology. In the sequel, we say an l adic galois representation is \modular when it arises from a congruence automorphic form. 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. Cohomology can be viewed as a method of assigning richer algebraic invariants to a space than homology. In the rst section we establish results about the mod p etale cohomology. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense.
We now have to compute the degeneration of eisenstein symbols with residue zero. Tom lovering june 22, 2012 for mathematical musings. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 arizona winter school. Moreover, little attention is paid to the precise running times of algorithms, the. Let i be a prime number, z t the ring of l adic integers and t a topological gmodule which, as topological group, is a finitely generated z c module with the natural topology, and on which g operates zrlinearly. Returning to the proof, we may combine our inverselimit constructions to. The goal of this paper is to show that the cohomology of compact unitary shimura varieties is concentrated in the middle degree and torsionfree. Cohomology of varieties with totally degenerate reduction 265 we denote by b dr the ring of p adic periods of fontaine. For each hilbert modular form of noncritical slope we con.
Studied the riemann zeta function, and made the riemann hypothesis. 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. 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. Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles weibel american mathematical society clay mathematics institute clay mathematics monographs volume 2. On padic lseries, padic cohomology and class field theory. Etale cohomology was invented by grothendieck and his collaborators in the 1960s. In the other direction, one could postpone homology and cohomology until after parts of chapter 4. If is the field of rational adic numbers, then the spaces are called the rational adic cohomology of the scheme.
Crystalline cohomology, from the point of view of number theory, fills a gap in the l adic cohomology information, which occurs exactly where there are equal characteristic primes. The aim of this paper is to study the p adic analogue of the elliptic polylogarithm. These choices determine an embedding of b st, the ring of periods of semistable varieties, in b dr. A comparison theorem for l adic cohomology volume 112 issue 2 r.
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. Therefore, in order to define ladic cohomology of a variety x, we can not. We construct a new cohomology theory for proper smooth formal schemes over the ring of integers of cp. Adic categories for algebraic stacks 3 second problem is an. In this paper i study some properties of these gkmodules. We will discuss recent progress by many people in the program of representing motivic cohomology with torsion coe. A profound synthesis of this topic was made by weil. Number theory learning seminar stanford university. A comparison theorem for ladic cohomology compositio. The main innovation is a new approach to the automorphy of nonminimal lifts which is closer in. We report on the progress that has been made on this topic during the past ten years. Much of the material in these notes parallels that in, for example, iversen, b. 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 l adic fourier transforms in the sheaf setting.
On the adic cohomology of varieties over number fields. In the l2 case, when a group is acting, one can assign a dimension in 0. On the generic part of the cohomology of compact unitary shimura varieties ana caraiani and peter scholze abstract. Notions of purity and the cohomology of quiver moduli. On the adic cohomology of varieties over number fields and. Degeneration of ladic eisenstein classes and of the. When bis basic, there is a precise conjecture rap95, conj 5. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in. As far as i know, the aim was to prove the weil conjectures, which deligne did in 1974. If this latter strategy is pushed to its natural limit, homology and cohomology can be developed just as branches of homotopy theory. All of the p adic algorithms i discuss rely upon rigid cohomology in some incarnation, and the alternative approaches pioneered by mestre and satoh are not touched upon.
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