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Пусть $X$ - связное топологическое пространство с точкой $x\in X$, и пусть $K$ - поле с дискретной топологией. Изучаются категория Таннаки конечномерных (плоских) векторных расслоений на $X$ и ее двойственная по Таннаке групповая схема $\pi (X,x)$ относительно функтора слоя в точке $x$. Максимальная проэтальная групповая фактор-схема групповой схемы $\pi (X,x)$ является этальной фундаментальной группой пространства $X$, которая изучалась Кухарчиком и Шольце. Для хороших топологических пространств групповая схема $\pi (X,x)$ является проалгебраическим пополнением обычной фундаментальной группы. Получены некоторые структурные результаты о групповой схеме $\pi (X,x)$ для очень общих топологических пространств при помощи изучения (псевдо)торсоров, связанных с ее фактор-группами. Этот подход использует идеи Нори из алгебраической геометрии и результат Делиня о категориях Таннаки. Также вычисляется групповая схема $\pi (X,x)$ для некоторых обобщенных соленоидов.

We develop a theory of étale parallel transport for vector bundles with numerically flat reduction on a p p -adic variety. This construction is compatible with natural operations on vector bundles, Galois equivariant and functorial with respect to morphisms of varieties. In particular, it provides a continuous p p -adic representation of the étale fundamental group for every vector bundle with numerically flat reduction. The results in the present paper generalize previous work by the authors on curves. They can be seen as a p p -adic analog of higher-dimensional generalizations of the classical Narasimhan-Seshadri correspondence on complex varieties. Moreover, they provide new insights into Faltings’ p p -adic Simpson correspondence between small Higgs bundles and small generalized representations by establishing a class of vector bundles with vanishing Higgs field giving rise to actual (not only generalized) representations.

It is well known that all torsors under an affine algebraic group over an algebraically closed field are trivial. We note that under suitable conditions this also holds if the the group is not necessarily of finite type. This has an application to isomorphisms of fibre functors on neutral Tannakian categories.

We extend our previous theory of étale parallel transport to a larger class of slope zero vector bundles on $p$-adic curves. The new class is stable under pullback by ramified coverings. We also construct $p$-adic representations of a central extension of the fundamental group for certain bundles of non-zero slope.

In this paper we develop a novel approach to Witt vector rings and to the (relative) de Rham Witt complex. We do this in the generality of arbitrary commutative algebras and arbitrary truncation sets. In our construction of Witt vector rings the ring structure is obvious and there is no need for universal polynomials. Moreover a natural generalization of the construction easily leads to the relative de Rham Witt complex. Our approach is based on the use of free or at least torsion free presentations of a given commutative ring $R$ and it is an important fact that the resulting objects are independent of all choices. The approach via presentations also sheds new light on our previous description of the ring of $p$-typical Witt vectors of a perfect $\mathbb{F}_p$-algebra as a completion of a semigroup algebra. We develop this description in different directions. For example, we show that the semigroup algebra can be replaced by any free presentation of $R$ equipped with a linear lift of the Frobenius automorphism. Using the result in the appendix by Umberto Zannier we also extend the description of the Witt vector ring as a completion to all $\bar{\mathbb{F}}_p$-algebras with injective Frobenius map., Appendix by Umberto Zannier; added the construction of a functorial noncommutative Witt vector ring with Frobenius Verschiebung and Teichm\"uller maps for noncommutative rings extending the commutative theory

Given a countable residually finite group $\Gamma$, we write $\Gamma_n \to e$ if $(\Gamma_n)$ is a sequence of normal subgroups of finite index such that any infinite intersection of $\Gamma_n$'s contains only the unit element $e$ of $\Gamma$. Given a $\Gamma$-module $M$ we are interested in the multiplicative Euler characteristics \begin{equation} \chi (\Gamma_n , M) = \prod_i |H_i (\Gamma_n , M)|^{(-1)^i} \end{equation} and the limit in the field $\mathbb{Q}_p$ of $p$-adic numbers \begin{equation} h_p := \lim_{n\to\infty} (\Gamma : \Gamma_n)^{-1} \log_p \chi (\Gamma_n , M) \; . \end{equation} Here $\log_p : \mathbb{Q}^{\times}_p \to \mathbb{Z}_p$ is the branch of the $p$-adic logarithm with $\log_p (p) = 0$. Of course, neither expression will exist in general. We isolate conditions on $M$, in particular $p$-adic expansiveness which guarantee that the Euler characteristics $\chi (\Gamma_n , M)$ are well defined. That notion is a $p$-adic analogue of expansiveness of the dynamical system given by the $\Gamma$-action on the compact Pontrjagin dual $X = M^*$ of $M$. Under further conditions on $\Gamma$ we also show that the renormalized $p$-adic limit in the second formula exists and equals the $p$-adic $R$-torsion of $M$. The latter is a $p$-adic analogue of the Li--Thom $L^2$ $R$-torsion of a $\Gamma$-module $M$ which they related to the entropy $h$ of the $\Gamma$-action on $X$. We view the limit $h_p$ as a version of entropy which values in the $p$-adic numbers and the equality with $p$-adic $R$-torsion as an analogue of the Li--Thom formula in the expansive case. We discuss the case $\Gamma = \mathbb{Z}^N$ in more detail where our theory is related to Serre's intersection numbers on arithmetic schemes.

We associate with the ring $R$ of algebraic integers in a number field a C*-algebra $\cT[R]$. It is an extension of the ring C*-algebra $\cA[R]$ studied previously by the first named author in collaboration with X.Li. In contrast to $\cA[R]$, it is functorial under homomorphisms of rings. It can also be defined using the left regular representation of the $ax+b$-semigroup $R\rtimes R^\times$ on $\ell^2 (R\rtimes R^\times)$. The algebra $\cT[R]$ carries a natural one-parameter automorphism group $(\sigma_t)_{t\in\Rz}$. We determine its KMS-structure. The technical difficulties that we encounter are due to the presence of the class group in the case where $R$ is not a principal ideal domain. In that case, for a fixed large inverse temperature, the simplex of KMS-states splits over the class group. The "partition functions" are partial Dedekind $\zeta$-functions. We prove a result characterizing the asymptotic behavior of quotients of such partial $\zeta$-functions, which we then use to show uniqueness of the $\beta$-KMS state for each inverse temperature $\beta\in(1,2]$., Comment: 38 pages

We determine the universal deformation over reduced base rings of the Witt ring scheme enhanced by a Frobenius lift and Verschiebung. It agrees with a q-deformation earlier introduced by the second author, for which we also give a simpler description. In the appendix we discuss a Witt vector theory for ind-rings which may be of independent interest.

In 1990, Lind, Schmidt, and Ward gave a formula for the entropy of certain Z n \mathbb {Z}^n -dynamical systems attached to Laurent polynomials P P , in terms of the (logarithmic) Mahler measure of P P . We extend the expansive case of their result to the noncommutative setting where Z n \mathbb {Z}^n gets replaced by suitable discrete amenable groups. Generalizing the Mahler measure, Fuglede–Kadison determinants from the theory of group von Neumann algebras appear in the entropy formula.

We construct real polarizable Hodge structures on the reduced leafwisecohomology of Kahler–Riemann foliations by complex manifolds. As inthe classical case one obtains a hard Lefschetz theorem for thiscohomology. Serre's Kahlerin analogue of the Weil conjectures carriesover as well. Generalizing a construction of Looijenga and Lunts oneobtains possibly infinite-dimensional Lie algebras attached toKahler–Riemann foliations.

Nous construisons un feuilletage a feuilles denses et de codimension deux sur la variete de Heisenberg tel que la decomposition de Hodge feuilletee ne soit pas verifiee. Nous demontrons aussi qu'un certain type de formules de trace dynamique reliant les orbites periodiques aux traces sur la cohomologie feuilletee, ne sont pas vraies pour des flots arbitraires.

We define a class of analytic functions which can be obtained from their values at almost all positive integers by a canonical interpolation procedure. All the usualL-functions belong to this class which is interesting in view of the extensive investigations of special values of motivicL-series. A number of classical contour integral formulas appear as particular cases of the interpolation scheme. The paper is based on a formula of Ramanujan and results of Hardy. An approach to the problem via distributions is also presented.

For a coherent sheaf M on the split n-torus Gn A = spec A[7n] over a commutative ring A there is a natural En-action on its group of global sections F(M) = F(G AA, M). If we give F(M) the discrete topology, we obtain a En-action on the compact abelian Pontrjagin dual F(M)*. The dynamical properties of this action for A = Z have been investigated for some time, especially by K. Schmidt. His book [Sch] gives a comprehensive account of the theory developed so far. In particular, one may ask for the entropy 0 < h(M) < oc of the En-action on F(M)*I which measures to what extent repeated application of the action "scatters around points". Note that for any action of Zn on a compact abelian group, topological entropy and metric entropy with respect to Haar measure coincide. The results of Lind, Schmidt and Ward [LSW] (see also [Sch], Chap. V) can be formulated as follows (A = Z): The entropy of M is finite if and only if M belongs to the category T of coherent torsion sheaves on Gn,. By Yuzvinskii's addition formula [Sch], Th. 14.1, entropy defines a homomorphism from the Grothendieck group of T to the reals: h: Ko(T) RI so that it suffices to calculate h(Oz) for irreducible closed subschemes Z C ?n with Z G n. If Z is not defined by a Laurent polynomial, then h(Oz) = 0. On the other hand, for any

The paper [T] by Taniyama contains several results in the arithmetic theory of CM abelian varieties which have become well known. Moreover the notion of a compatible system ρ = (ρl) of l-adic representations is introduced there for the first time. Under suitable conditions on ρ, Taniyama proves an interesting formula for the alternating product of the L-functions of the exterior powers of ρ: it is given by an infinite product of Artin L-functions each of which modified by a change of finitely many Euler factors. As an application, Taniyama obtains such a formula for the Hasse–Weil zeta function of an abelian scheme over a localized number ring. These infinite product formulas of Taniyama seem to be little known but they motivated the works [JY] and [JR]. In [JY] Joshi and Yogananda show that the Hasse–Weil zeta function ζGm(s) = ζ(s−1)/ζ(s) of Gm = specZ[T, T −1] is an infinite product of Dirichlet L-series. This is a special case of Taniyama’s formulas but they give an elementary direct argument. In [JR] Joshi and Raghunathan express quotients of very general Dirichlet series with Euler products as infinite products of “twisted Dirichlet series”. The method is purely local. It applies in particular to the Hasse–Weil zeta function of X×Gm where X is a scheme of finite type over Z. In this case the formula has a geometric proof which was suggested by Serre, c.f. [JR] § 3. Joshi and Raghunathan also obtain certain infinite product formulas for quotients of automorphic L-functions. Recently these have been related to canonical bases by Kim and Lee, [KL] Remark 2.21.

For a smooth projective variety X over a global field k consider the completed L-function of H w(x). It is defined as the Euler product over all places v of k of the local L-factors L~(Hw(X), s) introduced in [Se]. In the function field case the generalized Lefschetz trace formula in /-adic cohomology gives a global cohomological expression for the L-function. This has been used on the following purposes

Using periodic points, we study a notion of entropy with values in the p-adic numbers. This is done for actions of countable discrete residually finite groups \(\Gamma\). For suitable \(\Gamma = \mathbb{Z}^d\)-actions we obtain p-adic analogues of multivariable Mahler measures. For certain actions of more general groups, the p-adic entropy can be expressed in terms of a p-adic analogue of the Fuglede–Kadison determinant from the theory of von Neumann algebras. Many basic questions remain open.

We study invariant functions on the reductions mod p^n of p-divisible groups. The proof of the main result, which applies to one-dimensional groups, combines results of Tate with van der Put's solution of his p-adic Corona problem. For higher dimensional groups a generalization of the p-adic Corona problem would have to be solved.

For an ergodic measure preserving action on a probability space, consider the corresponding crossed product von Neumann algebra. We calculate the Fuglede-Kadison determinant for a class of operators in this von Neumann algebra in terms of the Ljapunov exponents of an associated measurable cocycle. The proof is based on recent work of Dykema and Schultz. As an application one obtains formulas for the Fuglede-Kadison determinant of noncommutative polynomials in the von Neumann algebra of the discrete Heisenberg group. These had been previously obtained by Lind and Schmidt via entropy considerations., Final version

For a certain class of vector bundles E on abelian varieties A over local fields containing all line bundles algebraically equivalent to zero we define a canonical representation of the Tate module of A on the fibre of E in the zero section. This extends an old construction of Tate for line bundles to vector bundles of higher rank. We also compare this construction to the theory of parallel transport for vector bundles on p-adic curves developed in mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.

We define functorial isomorphisms of parallel transport along etale paths for a class of vector bundles on a p-adic curve. All bundles of degree zero whose reduction is strongly semistable belong to this class. In particular, they give rise to representations of the algebraic fundamental group of the curve. This may be viewed as a partial analogue of the classical Narasimhan-Seshadri theory of vector bundles on compact Riemann surfaces., Comment: The main result is now valid for arbitrary reduction; Theorems 5, 16, 17, 18 and 20 are either improvements of results in the first version or new. The article will appear in Annales Sci. de l'ENS 56 pages

For algebraic Anosov diffeomorphisms we first express the reduced leafwise cohomology with respect to the unstable foliation in terms of finite dimensional Lie algebra cohomology. We then prove a dynamical Lefschetz trace formula for the induced action on this cohomology using a result of Nomizu. We also consider generalized Anosov maps for which the cohomology in question is no longer finite dimensional in general. Using the representation theory of nilpotent Lie groups we still arrive at a meaningful definition of traces on cohomology. Again we establish a dynamical Lefschetz trace formula., Comment: This is a much improved version of our earlier preprint

We interpret the “explicit formulas” in the sense of analytic number theory for the zeta function of an elliptic curve over a finite field as a transversal index theorem on a 3-dimensional laminated space.

In the papers [D1], [D5], [D3] a cohomological formalism for algebraic schemes X 0 over spec ℤ or spec ℚ was conjectured which would explain many of the expected properties of motivic L-series. All consequences of this very rigid formalism that I could imagine turned out to be either provable [D2], [D4], [DN], [Sa] or to amount to some well known conjectures on L-series of motives, as for example the Riemann hypotheses and the Artin conjecture—both generalized to the context of motives—and the Bloch Beilinson conjectures on vanishing orders.

The first two sections of this note are an expanded version of the lecture at the Congress. We first explain how a cohomological formalism might look, one that would serve the same purposes for zeta functions of number fields and etale or cristalline cohomology for zeta functions of curves over finite fields. After indicating some consequences of the hypothetical formalism that can actually be proved, we explain how very naturally the Riemann hypotheses would ensue. These matters are also discussed in [De2] but in the much more general framework of motives. We thought that it might be useful to make the simple basic ideas available to non experts for motives as well.

The Beilinson conjectures describe the leading coefficients of L-series of varieties over number fields up to rational factors in terms of generalized regulators. We begin with a short but almost selfcontained introduction to this circle of ideas. This is possible by using Bloch’s description of Beilinson’s motivic cohomology and regulator map in terms of higher Chow groups and generalized cycle maps. Here we follow [Bl3] rather closely. We will then sketch how much of the known evidence in favour of these conjectures — to the left of the central point — can be obtained in a uniform way. The basic construction is Beilinson’s Eisenstein symbol which will be explained in some detail. Finally in an appendix a map is constructed from higher Chow theory to a suitable Ext-group in the category of mixed motives as defined by Deligne and Jannsen. This smooths the way towards an interpretation of Beilinson’s conjectures in terms of a Deligne conjecture for critical mixed motives [Sc2]. It also explains how work of Harder [Ha2] and Anderson fits into the picture.

We prove an entropy formula for certain expansive actions of a countable discrete residually finite group $\Gamma $ by automorphisms of compact abelian groups in terms of Fuglede-Kadison determinants. This extends an earlier result proved by the first author under somewhat more restrictive conditions. The main tools for this generalization are a representation of the $\Gamma $-action by means of a `fundamental homoclinic point', and the description of entropy in terms of the renormalized logarithmic growth-rate of the set of $\Gamma_n$-fixed points, where $(\Gamma_n, n\ge1)$ is a decreasing sequence of finite index normal subgroups of $\Gamma $ with trivial intersection., Comment: Using a recent result of Y. Choi the discussion of expansiveness was strengthened and shortened. The paper will appear in: Ergodic Theory and Dynamical Systems

On presente une famille naturelle et explicite des varietes referencees qui representent des elements d'ordre arbitrairement grand dans les groupes d'homotopie stables des spheres

We show that under certain additional hypotheses the proper base change theorem holds for arbitrary abelian sheaves.