Lecture notes on motivic cohomology carlo mazza vladimir voevodsky charles weibel american mathematical society clay mathematics institute clay mathematics monographs volume 2. In mathematics, padic cohomology means a cohomology theory for varieties of characteristic p whose values are modules over a ring of padic integers. Comparison of p adic representations in the perfect and imperfect residue field case by using phi, gammamodules. For each group gand representation mof gthere are abelian groups hn g,m and hng,m where n 0,1,2,3. Exact sequences, chain complexes, homology, cohomology 9 in the following sections we give a brief description of the topics that we are going to discuss in this book, and we try to provide. We give a transcription into rigid padic cohomology oflaumonsproofofdelignes weil ii theorem, using a geometric fourier transform in the spirit of dmodules. The proper setup for this would be schemes, but for sake of simplicity i will only consider varieties. For the p adic case, we dont use hubers definition of etale cohomology with compact support as orlik did since it seems to give spaces that are much too big. On the generic part of the cohomology of compact unitary shimura varieties ana caraiani and peter scholze abstract. The goal of this paper is to show that the cohomology of compact unitary shimura varieties is concentrated in the middle degree and torsionfree.
Skip to main content open access publications from the university of california. Rham cohomology for general varieties, we will need some machinery of sheaf co homology. Number theory learning seminar 20162017 the seminar will meet wednesdays 1. Etale cohomology is an important branch in arithmetic geometry. As an application, we deduce the p adic monodromy theorem of fontaine in the imperfect residue field case. Algorithms for p adic cohomology and p adic heights abstract in part i, we present a new algorithm for computing the zeta function of a hyperelliptic curve over a. 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.
Cohomology is a very powerful topological tool, but its level of abstraction can scare away interested students. Integral p adic cohomology of drinfeld halfspaces ii i will compute the integral p adic etale cohomology of drinfeld halfspaces of any dimension. Indam program on serre conjectures and the padic local. Artinschreier extensions and the adams spectral sequence for elliptic cohomology. As far as i know, the aim was to prove the weil conjectures, which deligne did in 1974. To understand this we need to know what a representation of gis. In particular on singular varieties, this kind of l adic homology behaves much better that the classical l adic cohomology. Plymen editors, p adic methods and their applications, oxford university press 1992. So it is natural to ask if there is a good p adic cohomology theory that can satis es the above properties. 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. The padic hodge theory of semistable galois representations.
Galois representations in arithmetic geometry ii the. Test case for an absent theory michael harris department of mathematics columbia university perfectoid prologue its not often that contemporary mathematics provides such a. Using grothendiecks theory of ladic etale cohomology of varieties over a field of characteristic p l, deligne proved weils conjectures on the numbers of points. Moreover, since adic cohomology only encodes the betti numbers and torsion, there is no way to read any information about ptorsion from the adic. Sheaf theory etale cohomology is modelled on the cohomology theory of sheaves in the usual topological sense. Much of the material in these notes parallels that in, for example, iversen, b. This re nes the existing computation of the rational p adic etale cohomology. In the rst section we establish results about the mod p etale cohomology. As for padic hodge theory, it owes its origin to the works of tate 74 and grothendieck 26. New edition available here etale cohomology is an important branch in arithmetic geometry. Solutions to exercises from kenneth browns cohomology of.
This years seminar will focus on etale cohomology, the goal being to understand laumons. Galois representations can be constructed by taking the. Weil cohomology in practice this page is due to be taken down in summer 2020. 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. Around 1883, poincar e discovered that any nice connected space bhas a connected coveringspace that covers all others. There is also a construction of rigid cohomology with supports in a closed subscheme, and of cohomology. Structure of various lambdaadic arithmetic cohomology groups. This has the biggest deck transformation group of all. Nevertheless, the cohomology of a space, which is obtained by dualizing its simplicail chain complex, carries important additional structure. This yields a complete, purely padic proof of the weil conjectures when combined. Number theory learning seminar stanford university.
As an application, we give an much easier approach to construct the cycle maps for arbitrary algebraic schemes over fields of finite 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. Etale cohomology theory nankai tracts in mathematics. Theory of weights in padic cohomology project muse johns. A gentle introduction to homology, cohomology, and sheaf.
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