Fonctions L associées aux F-isocristaux surconvergents II: Zéros et pôles unités

Résumé Nous démontrons la conjecture de Katz concernant la méromorphie et la caractérisation des zéros et p?les unités des fonctions L associées aux représentations p-adiques lorsque celles-ci se prolongent sur une compactification du schéma de base. Comme cas particuliers importants, on obtient celui de la fonction zêta d’un schéma quelconque et celui d’une représentation p-adique quelconque sur un schéma propre.

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If X is a smooth variety over a finite field ? _q of characteristic p > 0 and ℱ is a p-adic sheaf associated to a representation of the fundamental group of X, N. Katz conjectures, in his Bourbaki talk 409, that the L function L (X, ℱ, t) has its p-adic unit roots and poles given in terms of p-adic étale cohomology. We prove this conjecture in the case of the structure sheaf ℱ = ℤ _p , that is for the Zeta function, and also more generally when the p-adic sheaf ℱ extends to a smooth sheaf on a compactification of X: as a consequence we get the Unit-Root Zeta function of Dwork and Sperber as an L function. The idea of the proof is to get the p-adic étale cohomology with coefficients and compact support as the fixed points of Frobenius acting on rigid cohomology with compact support. For this purpose, we first build a crystalline Artin–Schreier short exact sequence on the syntomic site of a scheme which is separated of finite type over a perfect field k: this naturally generalizes the work of J.M. Fontaine and W. Messing in the proper smooth case. Then getting rigid cohomology with coefficients as a limit of crystalline cohomologies of variable level we deduce a long exact sequence connecting p-adic étale cohomology (with compact support) to rigid cohomology (with compact support). When X is smooth and affine over an algebraically closed field, the former exact sequence splits into short exact sequences that identify the p-adic étale cohomology with support of X to the part of its rigid cohomology invariant under Frobenius. We can then describe the p-adic unit roots and poles of the Zeta function of X; as a corallary we get the Unit-Root Zeta function of Dwork and Sperber as an L function. In the appendix we show that the characteristic spaces of Frobenius in rigid cohomology commute with isometric extensions of the base, and that isocrystals associated to p-adic sheaves with finite monodromy are overconvergent: we thus obtain a p-adic proof of the rationality of the corresponding L-function.

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Oblatum 8-XII-1994 & 30-IV-1996     

The Weil-��tale fundamental group of a number field II

We define the fundamental group underlying the Weil-étale cohomology of number rings. To this aim, we define the Weil-étale topos as a refinement of the Weil-étale sites introduced by Lichtenbaum (Ann Math 170(2):657–683, 2009). We show that the (small) Weil-étale topos of a smooth projective curve defined in this paper is equivalent to the natural definition. Then we compute the Weil-étale fundamental group of an open subscheme of the spectrum of a number ring. Our fundamental group is a projective system of locally compact topological groups, which represents first degree cohomology with coefficients in locally compact abelian groups. We apply this result to compute the Weil-étale cohomology in low degrees and to prove that the Weil-étale topos of a number ring satisfies the expected properties of the conjectural Lichtenbaum topos.     

Affine analog of the proper base change theorem

We prove a rigidity property for the étale cohomology with torsion coefficients of affine Hensel pairs.     

Potential semi-stability ofp-adic étale cohomology

We extend the methods of Faltings and Tsuji, and prove that ifK is a field of characteristic 0 with a complete, discrete valuation, and a perfect residue field of characteristicp, then thep-adic étale cohomology of a finite typeK-scheme is potentially semi-stable. We prove a similar result for cohomology with compact support, and for cohomology with support in a closed subspace ofX. We establish a relationship between these cohomology groups, and the de Rham cohomology ofX.     

Motivic Milnor fibre of cyclic <Emphasis Type="Italic">L</Emphasis><Subscript>∞</Subscript>-algebras

We define the motivic Milnor fiber of cyclic L _∞-algebras of dimension three using the method of Denef and Loeser of motivic integration. It is proved by Nicaise and Sebag that the topological Euler characteristic of the motivic Milnor fiber is equal to the Euler characteristic of the étale cohomology of the analytic Milnor fiber. We prove that the value of Behrend function on the germ moduli space determined by a cyclic L _∞-algebra L is equal to the Euler characteristic of the analytic Milnor fiber. Thus we prove that the Behrend function depends only on the formal neighborhood of the moduli space.     

Inverting the Motivic Bott Element

We prove a version for motivic cohomology of Thomason's theorem on Bott-periodic K-theory, namely, that for a field k containing the nth roots of unity, the mod n motivic cohomology of a smooth k-scheme agrees with mod n étale cohomology, after inverting the element in H⁰(k,(1)) corresponding to a primitive nth root of unity.     

A projection formula for log smooth varieties in log étale cohomology

We prove a formula for the log étale cohomology of a generically finite morphism of proper vertical log smooth varieties.     

On the Frobenius stable part of Witt vector cohomology

For a proper (not necessarily smooth) variety over a finite field with q elements, Berthelot?CBloch?CEsnault proved a trace formula which computes the number of rational points modulo q in terms of the Witt vector cohomology. We show the analogous formula for Witt vector cohomology of finite length. In addition, we prove a vanishing result for the compactly supported étale cohomology of a constant p-torsion sheaf on an affine Cohen?CMacaulay variety.     

Purity for Hodge-Tate representations

We prove that a kind of purity holds for Hodge-Tate representations of the fundamental group of the generic fiber of a semi-stable scheme over a complete discrete valuation ring of mixed characteristic with perfect residue field. As an application, we see that the relative p-adic étale cohomology with proper support of a scheme separated of finite type over the generic fiber is Hodge-Tate if it is locally constant.     

A Nisnevich local Bloch-Ogus theorem over a general base

We prove the exactness of the Nisnevich Gersten complex over a Noetherian irreducible base of finite type under some conditions. We also obtain, as a consequence, a Nisnevich analogue of the Bloch-Ogus theorem for étale cohomology in this setting.     

On the Galois cohomology of ideal class groups

We use étale cohomology to prove some explicit results on the Galois cohomology of ideal class groups. Received: 3 May 2007     

Construction inconditionnelle de groupes de Galois motiviquesAn unconditional construction of motivic Galois groups

We attach to any “classical” Weil cohomology theory over a field a motivic Galois group, defined up to an inner automorphism. We also study the specialisation of numerical motives and the behaviour of motivic Galois group by specialisation. To cite this article: Y. André, B. Kahn, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 989–994.     

Some finiteness results for étale cohomology

We prove some finiteness theorems for the étale cohomology, Borel-Moore homology and cohomology with proper supports with divisible coefficients of schemes of finite type over a finite or p-adic field. This yields vanishing results for their l-adic cohomology, proving part of a conjecture of Jannsen.     

Higher Intersection Theory on Algebraic Stacks: I

In this paper and the sequel we establish a theory of Chow groups and higher Chow groups on algebraic stacks locally of finite type over a field and establish their basic properties. This includes algebraic stacks in the sense of Deligne–Mumford as well as Artin. An intrinsic difference between our approach and earlier approaches is that the higher Chow groups of Bloch enter into our theory early on and depends heavily on his fundamental work. Our theory may be more appropriately called the (Lichtenbaum) motivic homology and cohomology of algebraic stacks. One of the main themes of these papers is that such a motivic homology does provide a reasonable intersection theory for algebraic stacks (of finite type over a field), with several key properties holding integrally and extending to stacks locally of finite type. While several important properties of our higher Chow groups, like covariance for projective representable maps (that factor as the composition of a closed immersion into the projective space associated to a locally free coherent sheaf and the obvious projection), an intersection pairing and contravariant functoriality for all smooth algebraic stacks, are shown to hold integrally, our theory works best with rational coefficients.The main results of Part I are the following. The higher Chow groups are defined in general with respect to an atlas, but are shown to be independent of the choice of the atlas for smooth stacks if one uses finite coefficients with torsion prime to the characteristics or in general for Deligne–Mumford stacks. (Using some results on motivic cohomology, we extend this integrally to all smooth algebraic stacks in Part II.) Using cohomological descent, we extend Bloch's fundamental localization sequence for quasi-projective schemes to long exact localization sequences of the higher Chow groups modulo torsion for all Artin stacks: this is one of the main results of the paper. We show that these higher Chow groups modulo torsion are covariant for all proper representable maps between stacks of finite type while being contravariant for all representable flat maps and, in Part II, that they are independent of the choice of an atlas for all stacks of finite type over the given field k. The comparison with motivic cohomology, as is worked out in Part II, enables us to provide an explicit comparison of our theory for quotient stacks associated to actions of linear algebraic groups on quasi-projective schemes with the corresponding Totaro–Edidin–Graham equivariant intersection theory. As an application of our theory we compute the higher Chow groups of Deligne–Mumford stacks and show that they are isomorphic modulo torsion to the higher Chow groups of their coarse moduli spaces. As a by-product of our theory we also produce localization sequences in (integral) higher Chow groups for all schemes locally of finite type over a field: these higher Chow groups are defined as the Zariski hypercohomology with respect to the cycle complex.     

Motives and Etale Motives with Finite Coefficients

We prove that for suitable base fields, inverting the Bott element in Voevodsky’s category of motives with finite coefficients yields the category of étale motives with finite coefficients. Mathematics Subject Classifications (2000): 19E15, 14F42, 14F20. The first author was partially supported by the Clay Mathematics Institute.     

K-Theory of Surfaces at the Prime 2

We prove that for smooth surfaces over real closed fields, and a class of smooth projective surfaces over a real number field, the map between mod 2 algebraic and étale K-theory is an isomorphism in sufficiently large degrees. For a class of smooth projective surfaces over a real closed field, including rational surfaces, complete intersections and K3-surfaces over the real numbers, we explicate the abutment of the mod 2 motivic cohomology to algebraic K-theory spectral sequence.     

Congruences between derivatives of geometric <Emphasis Type="Italic">L</Emphasis>-functions

We prove a natural refinement of a theorem of Lichtenbaum describing the leading terms of Zeta functions of curves over finite fields in terms of Weil-étale cohomology. We then use this result to prove the validity of Chinburg’s Ω(3)-Conjecture for all abelian extensions of global function fields, to prove natural refinements and generalisations of the refined Stark conjectures formulated by, amongst others, Gross, Tate, Rubin and Popescu, to prove a variety of explicit restrictions on the Galois module structure of unit groups and divisor class groups and to describe explicitly the Fitting ideals of certain Weil-étale cohomology groups. In an Appendix coauthored with K.F. Lai and K.-S. Tan we also show that the main conjectures of geometric Iwasawa theory can be proved without using either crystalline cohomology or Drinfeld modules.     

Stable étale realization and étale cobordism

We show that there is a stable homotopy theory of profinite spaces and use it for two main applications. On the one hand we construct an étale topological realization of the stable A¹-homotopy theory of smooth schemes over a base field of arbitrary characteristic in analogy to the complex realization functor for fields of characteristic zero.On the other hand we get a natural setting for étale cohomology theories. In particular, we define and discuss an étale topological cobordism theory for schemes. It is equipped with an Atiyah-Hirzebruch spectral sequence starting from étale cohomology. Finally, we construct maps from algebraic to étale cobordism and discuss algebraic cobordism with finite coefficients over an algebraically closed field after inverting a Bott element.     

Rigidity for orientable functors

In the following paper we introduce the notion of orientable functor (orientable cohomology theory) on the category of projective smooth schemes and define a family of transfer maps. Applying this technique, we prove that with finite coefficients orientable cohomology of a projective variety is invariant with respect to the base-change given by an extension of algebraically closed fields. This statement generalizes the classical result of Suslin, concerning algebraic K-theory of algebraically closed fields. Besides K-theory, we treat such examples of orientable functors as etale cohomology, motivic cohomology, algebraic cobordism. We also demonstrate a method to endow algebraic cobordism with multiplicative structure and Chern classes.