Kedlaya, and david roe january 17, 2006 abstract motivated by an application to ldpc low density parity check algebraic geometry codes described by voloch and zarzar, we describe a computational procedure for. Rigid cohomology for algebraic stacks by david michael brown doctor of philosophy in mathematics university of california, berkeley professor kenneth ribet, chair we extend le stums construction of the overconvergent site ls09 to algebraic stacks. 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. Rigid cohomology for algebraic stacks by david michael brown a dissertation submitted in partial satisfaction of the requirements for the degree of. There is a classical relation between the padic absolute value of the eigenvalues of frobenius on crystalline cohomology and hodge numbers, for a variety in characteristic p. Integral padic hodge theory, and qde rham cohomology. A comparison theorem for ladic cohomology springerlink. As far as i know, the aim was to prove the weil conjectures, which deligne. 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. Xy be a quasicompact and quasi separated morphism of algebraic spaces over s. Etale cohomology of rank one ladic local systems in. Morrow let c be an algebraically closed complete extension of q p.
M then the equivialence class that contains is denoted and called the cohomology class of 3. This is a long theory to do everything, but one computation of ext r goes as follows. Completed cohomology and the adic lang lands program. Of local or relative cohomology groups of shea ves on preschemes.
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. Lectures on local cohomology university of illinois at. It is called the th adic cohomology of the sheaf on. If is the field of rational adic numbers, then the spaces are called the rational adic cohomology of the scheme. We give a transcription into rigid padic cohomology oflaumonsproofofdelignes weil ii theorem, using a geometric fourier transform in the spirit of dmodules.
Introduction in this paper we investigate the theory of padic families of automorphic forms for the group gsp 4q whose component at in nity has singular harishchandra parameter and is a nondegenerate limit of discrete series. There is also a construction of rigid cohomology with supports in a closed subscheme, and of cohomology with compact supports. If an internal link incorrectly led you here, you may wish to change the link to point directly to the intended article. Furthermore, it may appear in the later sections of his chapter 111 however, i have thought it elements. On p adic lseries, p adic cohomology and class field theory david burns and daniel macias castillo abstract. The intention of the book is to give a fairly elementary introduction into the subject and to show that it yields interesting applications to number theory. A con gruence arithmetic group is a group obtained by taking a semisimple qgroup. It is useful to combine this result with the observation that xhet xget is a. Bernard le stum, one century of p padic geometry from hensel to berkovich and beyond, talk notes, june 2012. Bounding picard numbers of surfaces using padic cohomology. One could begin with homology and perhaps continue with cohomology before turning to homotopy. One can use existence of exotic torsion in padic cohomology to give examples of surfaces over f p with no lift to characteristic zero ill79, ii. In sch88, sch93 scholl gave examples of d 1 cases with the ap being eigenvalues of congruence cuspforms by showing the l adic scholl representations associated to those cases are \modular.
In order to prove these theorems, we establish some new results about the padic hodge theoretic properties of the etale cohomology of varieties over a number eld or padic eld with coe cients in a eld of characteristic p. Introduction in this paper we investigate the theory of p adic families of automorphic forms for the group gsp 4q whose component at in nity has singular harishchandra parameter and is a nondegenerate limit of discrete series. On padic lseries, padic cohomology and class field theory. If is the field of rational adic numbers, then the spaces are called the rational adic cohomology of. Introduction the elliptic polylogarithm was introduced by beilinson and levin in their seminal paper bl. Rigid cohomology for algebraic stacks emory university. Kedlayas estimates for ppowers in the reduction process on hyperelliptic curves 8 3. For each group gand representation mof gthere are abelian groups hng,m and hng,m where n 0,1,2,3. In 1904 schur studied a group isomorphic to h2g,z, and this group is known as the schur multiplier of g. Etale cohomology was invented by grothendieck and his collaborators in the 1960s.
Etale cohomology, taking values over a ring of ladic integers for l. Bounding picard numbers of surfaces using p adic cohomology timothy g. On the adic cohomology of varieties over number fields. It was first conjectured to exist by alexander beilinson and stephen lichtenbaum in the mid 1980s, and it was then defined by vladimir voevodsky in the mid 1990s. Among many other attributes, local cohomology allows one to answer many seemingly di. B is agmodule homomorphism, then for each i 0, there is an induced. In order to obtain a padic cohomology on algebraic vari eties over k stable. A congruence arithmetic group is a group obtained by taking a semisimple qgroup g. Notes on the ainfcohomology of integral padic hodge theory.
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. On the cohomology of rapoportzink spaces of eltype sug woo shin abstract. We prove a finiteness result for the padic cohomology of the lubintate. In the rst section we establish results about the mod p etale cohomology. They continue with a presentation of a stable homotopy theoretic approach to the theory of di erential extensions of generalized cohomology theories including products and. As an application we obtain that, for an algebraic variety x and associated rigid analytic variety x rig, the ladic cohomology of x and x rig agree. To understand this we need to know what a representation of gis. On the adic cohomology of varieties over number fields and. Motivic cohomology is a cohomology theory for schemes which in many ways plays the role of singular cohomology in algebraic geometry. Wiese, corresponding to the forschungsseminar on padic galois representations that took place during the winter term. In this subsection, we always assume that a is a category with in. However, not wishing to assume homological algebra at this point, we take a different tack.
The fundamental theorems in etale cohomology apply to adic cohomology of constructible adic sheaves. Journal of the american mathematical society volume 25, number 3, july 2012, pages 715738 s 089403472012007292 article electronically published on january 27, 2012. If this latter strategy is pushed to its natural limit, homology and cohomology can be developed just as branches of homotopy theory. One of the main strategies for approaching padic cohomology is to extract information from the case of smooth proper varieties. In this article, we are interested in determining the ladic cohomology of rapoportzink spaces associated to gln over an unrami ed extension of qp called unrami ed \eltype in connection with the local langlands correspondence for gln. Tsuzuki october 23, 2018 abstract forasemistable family of varieties over a curve in characteristic p, weprove the existence of a clemensschmid type long exact sequence for the p adic cohomology. On thecohomology of algebraic varieties clairevoisin. Bounding picard numbers of surfaces using padic cohomology timothy g. Higher coherent cohomology and padic modular forms of. In this paper i study some properties of these gkmodules. For this, in homological language, we need the fact that cig. In general, if j is an ideal of a ring r, the radical of j is the ideal v. Theory of weights in padic cohomology project muse johns. In the other direction, one could postpone homology and cohomology until after parts of chapter 4.
The third part sketches a particular algorithm for producing an approximate frobenius matrix on the cohomology of a smooth hypersurface, using padic cohomology and a description of the cohomology of a smooth hypersurface due to gri. Di erential cohomology ulrich bunke august 20, 20 abstract these course note rst provide an introduction to secondary characteristic classes and di erential cohomology. One of the most important uses of cohomology is that it converts short exact sequences of gmodulesto long exact sequences of abelian groups. Cohomology of projective varieties, excision and comparison with topological cohomology 5 3. 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. A good example of such a problem, where local cohomology provides a partial answer, is the question of how many generators ideals have up to radical. We prove that etale morphisms are morphisms of cohomological descent for nitely presnted. Thus, its generic bre x is a proper smooth rigidanalytic variety over c.