Arithmetic Schemes Research Articles

THE WEIL-ÉTALE FUNDAMENTAL GROUP OF A NUMBER FIELD I

Lichtenbaum has conjectured (Ann of Math. (2) 170(2) (2009), 657-683) the existence of a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta function ζX(s) at s = 0. In this paper we consider the category of sheaves XL on this conjectural site for X = Spec(ΟF) the spectrum of a number ring. We show that XL has, under natural topological assumptions, a well-defined fundamental group whose abelianization is isomorphic, as a topological group, to the Arakelov-Picard group of F . This leads us to give a list of topological properties that should be satisfied by XL. These properties can be seen as a global version of the axioms for the Weil group. Finally, we show that any topos satisfying these properties gives rise to complexes of étale sheaves computing the expected Lichtenbaum cohomology.

Sur l'analogie entre le système dynamique de Deninger et le topos Weil-étale

We express some basic properties of Deninger's conjectural dynamical system in terms of morphisms of topoi. Then we show that the current definition of the Weil-étale topos satisfies these properties. In particular, the flow, the closed orbits, the fixed points of the flow and the foliation in characteristic $p$ are well defined on the Weil-étale topos. This analogy extends to arithmetic schemes. Over a prime number $p$ and over the archimedean place of $\boldsymbol{Q}$, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-étale topos. This morphism is compatible with the structure mentioned above.

A Lefschetz fixed point formula for singular arithmetic schemes with smooth generic fibres

In this article, we consider singular equivariant arithmetic schemes whose generic fibres are smooth. For such schemes, we prove a relative fixed point formula of Lefschetz type in the context of Arakelov geometry. This formula is an analog, in the arithmetic case, of the Lefschetz formula proved by R. W. Thomason in [31]. In particular, our result implies a fixed point formula which was conjectured by V. Maillot and D. Rössler in [25].

Swan conductors and torsion in the logarithmic de Rham complex

We prove, for an arithmetic scheme X/S over a discrete valuation ring whose special fiber is a strict normal crossings divisor in X, that the Swan conductor of X/S is equal to the Euler characteristic of the torsion in the logarithmic de Rham complex of X/S. This is a precise logarithmic analog of a theorem by Bloch [1].

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” Ext ˆ X i ( F , G ) we introduce are extensions by groups of analytic types of the usual extension groups Ext X i ( F , G ) attached to O X -modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group Ext ˆ X 1 ( F , G ) – the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X – and we especially consider the case when X is an “arithmetic curve”, namely the spectrum Spec O K of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers. Namely, for any two hermitian vector bundles F ¯ and G ¯ over X : = Spec O K , we attach a logarithmic size s F ¯ , G ¯ ( α ) to any element α of Ext ˆ X 1 ( F , G ) , and we give an upper bound on s F ¯ , G ¯ ( α ) in terms of slope invariants of F ¯ and G ¯ . We further illustrate this notion by relating the sizes of restrictions to points in P 1 ( Z ) of the universal extension over P Z 1 to the geometry of PSL 2 ( Z ) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K ′ ): when the base field K is Q , we establish that the size, which cannot increase under base change, is actually invariant when the field K ′ is an abelian extension of K, or when F ¯ ∨ ⊗ G ¯ is a direct sum of root lattices and of lattices of Voronoi's first kind. The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.

The Weil-étale topology for number rings

There should be a Grothendieck topology for an arithmetic scheme X such that the Euler characteristic of the cohomology groups of the constant sheaf Z with compact support at infinity gives, up to sign, the leading term of the zeta-function of X at s = 0. We construct a topology (the Weil-etale topology) for the ring of integers in a number field whose cohomology groups Hl (Z) determine such an Euler characterstic if we restrict to / < 3.

On different notions of tameness in arithmetic geometry

The notion of a tamely ramified covering is canonical only for curves. Several notions of tameness for coverings of higher dimensional schemes have been used in the literature. We show that all these definitions are essentially equivalent. Furthermore, we prove finiteness theorems for the tame fundamental groups of arithmetic schemes.

Smallness of fundamental groups for arithmetic schemes

The smallness is proved of étale fundamental groups for arithmetic schemes. This is a higher dimensional analogue of the Hermite–Minkowski theorem. We also refer to the case of varieties over finite fields. As an application, we prove certain finiteness results of representations of the fundamental groups over algebraically closed fields.

Tamely ramified covers of varieties and arithmetic schemes

Tamely ramified covers of varieties and arithmetic schemes

Adelic Approach to the Zeta Function of Arithmetic Schemes in Dimension Two

This paper suggests a new approach to the study of the fun- damental properties of the zeta function of a model of elliptic curve over a global field. This complex valued commutative approach is a two-di- mensional extension of the classical adelic analysis of Tate and Iwasawa. We explain how using structures which come naturally from the ex- plicit two-dimensional class field theory and working with a new R((X))- valued translation invariant measure, integration theory and harmonic analysis on various complete objects associated to arithmetic surfaces one can define and study zeta integrals which are closely related to the zeta function of a regular model of elliptic curve over global fields. In the two-dimensional adelic analysis the study of poles of the zeta function is reduced to the study of poles of a boundary term which is an integral of a certain arithmetic function over the boundary of an adelic space. The structure of the boundary and function determines the analytic proper- ties of the boundary term and location of the poles of the zeta function, which results in applications of the theory to several key directions of arithmetic of elliptic curves over global fields.

An interval arithmetic scheme for analyzing uncertain systems

AbstractThis contribution is dedicated to the numerical study of the global behavior of dynamical systems whose mathematical descriptions are imperfect due to uncertainties concerning involved parameters. At the beginning, we constitute themathematical model in order to provide a tailor‐made framework dealing with this problem. Here, uncertainties are regarded as external random noise or perturbations influencing an underlying deterministic system. Afterwards, an interval arithmetic scheme is presented which allows the numerical evaluation of the model. (© 2008 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Arakelov theory of even orthogonal Grassmannians

We study the Arakelov intersection ring of the arithmetic scheme \mathrm{OG} which parametrizes maximal isotropic subspaces in an even dimensional vector space, equipped with the standard hyperbolic quadratic form. We give a presentation of the ring \mathrm{CH}(\mathrm{\overline{OG}}) (when \mathrm{OG}(ℂ) is given its natural invariant hermitian metric) and formulate an ‘arithmetic Schubert calculus’ which extends the classical one for the cohomology ring of \mathrm{OG} . Our analysis leads to a computation of the Faltings height of \mathrm{OG} with respect to its fundamental embedding in projective space, and a comparison of the resulting formula with previous ones, due to Kaiser and Köhler [KK] and the author [T3], [T4].

Transcribing the Tableau Économique: Input-Output Analysis à la Quesnay

Since its publication in 1955, Almarin Phillips' article “The Tableau Économique as a Simple Leontief Model” has inspired an interesting line of interpretation of the arithmetical schemes presented two centuries earlier by François Quesnay. Subsequent attempts to represent the Tableaux économiques in the form of Input-Output transcriptions by Shlomo Maital (1972), Bernhard Korte (1972), Tibor Barna (1975), and Paul Samuelson (1986), among others, have underlined the brilliance of the French doctor's formal conception of the economy as a reproductive system.

P-adic étale Tate twists and arithmetic duality

In this paper, we define, for arithmetic schemes with semi-stable reduction, p-adic objects playing the roles of Tate twists in étale topology, and establish their fundamental properties.

Singular homology of arithmetic schemes

We construct a singular homology theory on the category of schemes of finite type over a Dedekind domain and verify several basic properties. For arithmetic schemes we construct a reciprocity isomorphism between the integral singular homology in degree zero and the abelianized modified tame fundamental group.

Class field theory for arithmetic schemes

Let X be a regular arithmetic scheme, i.e. a regular integral separated scheme flat and of finite type over Spec \({\mathbb{Z}}\) . Generalising classical class field theory for number fields, we define a class group CX and show there is a natural surjective map \({C_X\rightarrow\pi_1^{ab}(X)}\) whose kernel is the connected component of 0.

Sieve methods for varieties over finite fields and arithmetic schemes

Des méthodes du crible classiques en théorie analytique des nombres ont été récemment adaptées à un cadre géométrique. Dans ce nouveau cadre, les nombres premiers sont remplacés par les points fermés d’une variété algébrique sur un corps fini ou plus généralement un schéma de type fini sur ℤ. Nous présentons la méthode et certains des résultats surprenants qui en découlent. Par exemple, la probabilité qu’une courbe plane sur 𝔽 2 soit lisse est asymptotiquement 21/64 quand son degré tend vers l’infini. La plus grande partie de cet article est une exposition des résultats de [Poo04] et [Ngu05].

Low power and high-speed implementation of fir filters for software defined radio receivers

The most computationally intensive part of the wideband receiver of a software defined radio (SDR) is the intermediate frequency (IF) processing block. Digital filtering is the main task in IF processing. The computational complexity of finite impulse response (FIR) filters used in the IF processing block is dominated by the number of adders (subtracters) employed in the multipliers. This paper presents a method to implement FIR filters for SDR receivers using minimum number of adders. We use an arithmetic scheme, known as pseudo floating-point (PFP) representation to encode the filter coefficients. By employing a span reduction technique, we show that the filter coefficients can be coded using considerably fewer bits than conventional 24-bit and 16-bit fixed-point filters. Simulation results show that the magnitude responses of the filters coded in PFP meet the attenuation requirements of wireless communication standard specifications. The proposed method offers average reductions of 40% in the number of adders and 80% in the number of full adders needed for the coefficient multipliers over conventional FIR filter implementation methods

A construction of covers of arithmetic schemes

Let X be a regular arithmetic scheme, i.e. a regular integral separated scheme flat and of finite type over Spec Z . Assume that for all closed irreducible subschemes C ⊆ X of dimension 1 with normalisation C ˜ there are given open normal subgroups N C of π 1 ( C ˜ ) , which fulfil the following compatibility condition: For all x ˜ ∈ C ˜ 1 × X C ˜ 2 the pre-images of N C 1 and N C 2 in π 1 ( x ˜ ) coincide. If the indices of the N C are bounded, then these data uniquely determine an open normal subgroup of π 1 ( X ) , whose pre-image in π 1 ( C ˜ ) is N C for all C.

Finiteness of Abelian fundamental groups with restricted ramification

We define a certain quotient of the étale fundamental group of a scheme which classifies étale coverings with bounded ramification along the boundary, and show the finiteness of the abelianization of this group for an arithmetic scheme. To cite this article: T. Hiranouchi, C. R. Acad. Sci. Paris, Ser. I 341 (2005).