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. Recent title s in thi s series american mathematical society. In mathematics, padic cohomology means a cohomology theory for varieties of characteristic p whose values are modules over a ring of padic integers. We develop a cohomology theory for locally analytic representations of padic lie groups on. There are similar constructions for compactly supported. Compatibilities, correspondences, and integral structures in padic cohomology. April 30, 2011 abstract introduced by borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. Brian conrad stanford mathematics stanford university. Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. The point of this article is that, via the laplace transform, the results and methods of. It extends crystalline cohomology to schemes that need not be proper or smooth, and extends monskywashnitzer cohomology to nonaffine varieties. Grothendiecks conjectural theory of motives predicts that, in some sense, the choice of cohomology theory is irrelevant and that these dif. For example, we show the following noncompleted version of a comparison isomorphism of illusie.
Completed cohomology and the p adic langlands program 3 suitable surrogate for a space of p adic automorphic forms. On the p adic cohomology of the lubintate tower 3 is in a suitable sense a gl nftorsor. This yields a complete, purely p adic proof of the weil conjectures when combined with recent results on p adic di. In the process, we will define a theory of p adic absolute hodge cohomology with coefficients, which may be.
In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Our starting point is the wellknown formula relating the. As an application, i will show that the dmodule theoretic pushforward of an isocrystal along a proper smooth morphism is an isocrystal. Kedlayas estimates for ppowers in the reduction process on hyperelliptic curves 8 3. Editor of manuscripta mathematica and journal of the korean math. Syntomic cohomology as a padic absolute hodge cohomology. As a topological space, it is compact, hausdorff, and totally disconnected i. The etale site of a rigid analytic variety and an adic space. Professor brian conrad, chair professor karen smith professor stephen debacker. Completed cohomology and the padic langlands program 3 suitable surrogate for a space of padic automorphic forms. Pdf on the \etale cohomology of algebraic varieties with. The machinery of etale cohomology and its relative, ladic cohomology, is formidable. In mathematics, rigid cohomology is a p adic cohomology theory introduced by berthelot 1986.
For a scheme x of finite type over a perfect field k, there are rigid cohomology groups h i rig xk which are finite dimensional vector spaces over the. A comparison theorem for ladic cohomology compositio. The best understood theory is adic cohomology for 6 p. Some years ago a conference on ladic cohomology in oberwolfach was held. In this subsection, we always assume that a is a category with in. As far as i know, the aim was to prove the weil conjectures, which deligne.
In that case, we have etale cohomology groups hi et x k. Zd0, for zmust be given the discrete topology, and the image of any continuous map. Completed cohomology and the adic lang lands program. For example, we prove the following result see theorem 1. As far as i know, the aim was to prove the weil conjectures, which deligne did in 1974. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 arizona winter. The arizona winter school 2007, on which the current book is based, was a unique opportunity to introduce graduate students to this subject. Cohomology of projective varieties, excision and comparison with topological cohomology 5 3. Lan and suh, ls12, prove that if the level is hyperspecial at and one takes cohomology with coe cients in the local system l.
Bernard le stum, one century of p p adic geometry from hensel to berkovich and beyond, talk notes, june 2012. The aim of this book is to give an introduction to adic spaces and to develop. Moreover, every point is an accumulation point, and there is a countable basis of clopen sets a stone space with these properties must be homeomorphic to cantor space. X y determines a homomorphism from the cohomology ring of y to that of x. Bounding picard numbers of surfaces using padic cohomology. Cais a dissertation submitted in partial ful llment of the requirements for the degree of doctor of philosophy mathematics in the university of michigan 2007 doctoral committee. The third part sketches a particular algorithm for producing an approximate frobenius matrix on the cohomology of a smooth hypersurface, using p adic cohomology and a description of the cohomology of a smooth hypersurface due to gri. Peter scholze, p p adic geometry, icm report arxiv. Introduction to equivariant cohomology in algebraic geometry dave anderson. Unlike more subtle invariants such as homotopy groups, the cohomology ring tends to be. Furthermore, it may appear in the later sections of his chapter 111 however, i have thought it elements. Bernard le stum, one century of p padic geometry from hensel to berkovich and beyond, talk notes, june 2012.
This is a p adic analogue of a work of beilinson be1 which interprets beilinsondeligne cohomology in terms of absolute hodge cohomology. Its values h n xw are modules over the ring w of witt vectors over k. I am an editor for journal of the ams, algebra and number theory, and imrn. The purpose of this paper is to interpret rigid syntomic cohomology, defined by amnon besser bes, as a p adic absolute hodge cohomology. Ban international research station, conference \p adic cohomology and arithmetic applications with tomoyuki abe, ambrus p al, christopher lazda, october 2017. Of local or relative cohomology groups of shea ves on preschemes.
This chapter summarizes the theory of the etale topology on schemes, culmi nating in the results on ladic cohomology that are needed in. Received by the editors february 22, 2011 and, in revised form, november 16, 2011, and. Ais an ideal in a noetherian qalgebra a, then the i adic completion abcoincides with the. Dworks p adic cohomology theory was developed further in as3, as4, where the cohomology of a general class of twisted exponential sums was computed, and in as5, where the cohomology of smooth complete intersections over finite fields was computed. Canonical liftings and log structures in the context of mirror symmetry, the moduli space of complex calabiyau varieties acquires canonical local coordinates near a \large complex structure limit point. We then use these series to describe the p adic eisenstein classes. After this the basic results on the etale cohomology of adic spaces are. It was introduced by alexander grothendieck 1966, 1968 and developed by pierre berthelot crystalline cohomology is partly inspired by the p adic proof in dwork 1960 of part of the weil conjectures and is closely. However, here we wish to relay another, deeper, relation between the cohomology. See the history of this page for a list of all contributions to it. 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. Mar 06, 2020 etale cohomology of rigid analytic varieties and adic spaces. Automorphic forms and the cohomology of vector bundles on shimura varieties, michael harris. Exterior square lfunctions, herve jacquet and joseph shalika.
Most of the known and expected properties of motivic cohomology predicted inabs87andlic84canbedividedintotwofamilies. The padic hodge theory of semistable galois representations. In mathematics, padic cohomology means a cohomology theory for varieties of characteristic p. 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 p adic cohomologies. Problems arising from the tate and beilinson conjectures in the context of shimura varieties, dinakar ramakrishnan. The idea being that if one has information about the open sets that make up a space as. There is also a construction of rigid cohomology with supports in a closed subscheme, and of cohomology with compact supports. Etale cohomology of rigid analytic varieties and adic spaces.
Correspondences, integral structures, and compatibilities in padic cohomology by bryden r. This text is an extended version of a series of lectures given by the third and fourth authors during the 2016 arizona winter school. In recent decades, \p\ adic geometry and \p\ adic cohomology theories have become indispensable tools in number theory, algebraic geometry, and the theory of automorphic representations. On the l adic cohomology of siegel three folds, invent.
Introduction to equivariant cohomology in algebraic geometry. Etale cohomology and the weil conjecture eberhard freitag. Sug woo shin sug woo is my first name ucb mathematics. Results 1 10 of 10 etale cohomology of rigid analytic varieties and adic spaces by roland huber and a great selection of related books, art and collectibles. A con gruence arithmetic group is a group obtained by taking a semisimple qgroup. Steenbrink 31 crystalline cohomology luc illusie 43 conjectures on algebraic cycles in adic cohomology john tate 71 some remarks on the hodge type conjecture morihiko saito 85 independence of. Therefore the etale cohomology is as expected in the. The cohomology of locally analytic representations unidue. Classical algebraic geometry dealt with complex varieties defined by a system of polynomial equations in several variables.
In this note i try to give a very brief introduction to etale cohomology, especially to adic cohomology. Etale cohomology was invented by grothendieck and his collaborators in the 1960s. The university of michigan, 2007, supervised by brian conrad. In mathematics, crystalline cohomology is a weil cohomology theory for schemes x over a base field k. Kedlaya, p adic cohomology from theory to practice arizona winter school lecture notes. 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. This definition in top alone already goes a long way. See here for other expository notes and other mathematical writings. Correspondences, integral structures, and compatibilities in. There are many di erent cohomology theories one can associate to this situation. Z, and proper smooth base change theorems in etale cohomology date. The ring of p p adic integers has the following properties. We give a transcription into rigid p adic cohomology oflaumonsproofofdelignes weil ii theorem, using a geometric fourier transform in the spirit of dmodules. The obstruction to defining ladic cohomology using sheaves is a shortage of open subsets in the zariski topology.
187 445 3 457 534 1679 1497 1529 122 334 42 779 1510 452 133 1311 1479 1035 589 841 1534 517 1383 559 113 966 1382 572 229 1052 667 1073 1238