We classify all possible combinatorial geometries associated with one-dimensional difference equations, in any characteristic. The theory of difference fields admits a proper interpretation of itself, namely the reduct replacing the automorphism by its nth power. We show that these reducts admit a successively smoother theory as n becomes large; and we succeed in defining a limit structure to these reducts, or rather to the structure they induce on one-dimensional sets. This limit structure is shown to be a Zariski geometry in (roughly) the sense of Hrushovski and Zil'ber. The trichotomy is thus obtained for the limit structure as a consequence of a general theorem, and then shown to be inherited by the original theory.

2000 Mathematical Subject Classification: 03C60; (primary) 03C45, 03C98, 08A35, 12H10 (secondary)

Certain classes of smoothly approximable structures — the class of affine covers of Lie geometries — are shown to have the amalgamation property. In particular, this shows that any affine cover of a Lie geometry has the small index property.

2000 Mathematical Subject Classification: 03C45.

Given that available technology permits one to establish that almost all natural numbers satisfying appropriate congruence conditions are represented as the sum of three squares of prime numbers, one expects strong estimates to be attainable for exceptional sets in the analogous problem involving sums of four squares of primes. Let E(N)$ denote the number of positive integers not exceeding N that are congruent to 4 modulo 24, yet cannot be written as the sum of four squares of prime numbers. A method is described that shows that for each positive number $\epsilon$, one has $E(N) \ll N^{13/30 + \epsilon}$, thereby exploiting effectively the 'excess' fourth square of a prime so as to improve the recent bound $E(N) \ll N^{13/15 + \epsilon}$ due to J. Liu and M.-C. Liu. It transpires that the ideas underlying this progress permit estimates for exceptional sets in a variety of additive problems to be significantly slimmed whenever sufficiently many excess variables are available. Such ideas are illustrated for several additional problems involving sums of four squares.

2000 Mathematical Subject Classification: 11P32, 11P05, 11P55.

We compute the Euler characteristics of the individual connected components of the intersection of two opposed big cells in the real flag variety of type G2, verifying a conjecture of Rietsch [6].

2000 Mathematical Subject Classification: primary 14M15; secondary 20G20.

2000 Mathematical Subject Classification: 10B10.

According to the Church-Turing Thesis a number function is computable by the mathematically defined Turing machine if and only if it is computable by a physical machine. In 1983 Pour-El and Richards defined a three-dimensional wave $u(t,x)$ such that the amplitude $u(0,x)$ at time 0 is computable and the amplitude $u(1,x)$ at time 1 is continuous but not computable. Therefore, there might be some kind of wave computer beating the Turing machine. By applying the framework of Type 2 Theory of Effectivity (TTE), in this paper we analyze computability of wave propagation. In particular, we prove that the wave propagator is computable on continuously differentiable waves, where one derivative is lost, and on waves from Sobolev spaces. Finally, we explain why the Pour-El-Richards result probably does not help to design a wave computer which beats the Turing machine.

2000 Mathematical Subject Classification: 03D80, 03F60, 35L05, 68Q05.

We prove that the Novikov assembly map for a group $\Gamma$ factorizes, in ‘low homological degree’, through the algebraic K-theory of its integral group ring. In homological degree 2, this answers a question posed by N. Higson and P. Julg. As a direct application, we prove that if $\Gamma$ is torsion-free and satisfies the Baum-Connes conjecture, then the homology group $H_{1}(\Gamma;\,\mathbb{Z})$ injects in $K_{1}(C^{*}_{r}\Gamma)$ and in $K_{1}^{\rm alg}(A)$, for any ring $A$ such that $\mathbb{Z}\Gamma\subseteq A\subseteq C^{*}_{r}\Gamma$. If moreover $B\Gamma$ is of dimension less than or equal to 4, then we show that $H_{2}(\Gamma;\,\mathbb{Z})$ injects in $K_{0}(C^{*}_{r}\Gamma)$ and in $K_{2}^{\rm alg}(A)/\Delta_{2}$, where $A$ is as before, and $\Delta_{2}$ is generated by the Steinberg symbols $\{\gamma,\,\gamma\}$, for $\gamma\in\Gamma$.

2000 Mathematical Subject Classification: primary 19D55, 19Kxx, 58J22; secondary: 19Cxx, 19D45, 43A20, 46L85.

The main result is an upper bound for the Riemann zeta function in the critical strip: $\zeta(\sigma + it) \le A|t|^{B(1 - \sigma)^{3/2}} \log^{2/3} |t|$ with $A = 76.2$ and $B = 4.45$, valid for $\frac12 \le \sigma \le 1$ and $|t| \ge 3$. The previous best constant $B$ was 18.5. Tools include a variant of the Korobov–Vinogradov method of bounding exponential sums, an explicit version of T. D. Wooley's bounds for Vinogradov's integral, and explicit bounds for mean values of exponential sums over numbers without small prime factors, also using methods of Wooley. An auxiliary result is the exponential sum bound $S(N, t) \le 9.463 N^{1 - 1/(133.66\lambda^2)}$, where $N$ is a positive integer, $t$ is a real number, $\lambda = (\log t)/(\log N)$ and

$S(N,t) = \max_{0 < u \le 1} \max_{N < R \le 2N} \left| \sum_{N < n \le R} (n + u)^{-it} \right|.$$

2000 Mathematical Subject Classification: primary 11M06, 11N05, 11L15; secondary 11D72, 11M35.

Let ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ be the Hilbert scheme of closed 1-dimensional subschemes of degree 6 and arithmetic genus 4 in $\mathbb{P}^3$. Let $H$ be the component of ${\rm Hilb}^{6t-3}(\mathbb{P}^3)$ whose generic point corresponds to a canonical curve, that is, a complete intersection of a quadric and a cubic surface in $\mathbb{P}^3$. Let $F$ be the vector space of linear forms in the variables $z_1, z_2, z_3, z_4$. Denote by $F_d$ the vector space of homogeneous forms of degree $d$. Set $X = \{(f_2,f_3)\}$ where $f_2 \in \mathbb{P}(F_2)$ is a quadric surface, and $f_3 \in \mathbb{P}(F_3/f_2 \cdot F)$ is a cubic modulo $f_2$. We have a rational map, $\sigma : X \cdots \rightarrow H$ defined by $(f_2,f_3) \mapsto f_2 \cap f_3$. It fails to be regular along the locus where $f_2$ and $f_3$ acquire a common linear component. Our main result gives an explicit resolution of the indeterminacies of $\sigma$ as well as of the singularities of $H$.

2000 Mathematical Subject Classification: 14C05, 14N05, 14N10, 14N15.

If $G$ is a pro-$p$, $p$-adic, Lie group containing no element of order $p$ and if $\Lambda (G)$ denotes the Iwasawa algebra of $G$ then we propose a number of invariants associated to finitely generated $\Lambda (G)$-modules, all given by various forms of Euler characteristic. The first turns out to be none other than the rank, and this gives a particularly convenient way of calculating the rank of Iwasawa modules. Others seem to play similar roles to the classical Iwasawa $\lambda $- and $\mu $-invariants. We explore some properties and give applications to the Iwasawa theory of elliptic curves.

2000 Mathematical Subject Classification: primary 16E10; seconday 11R23.

Philip Hall proved that the group operation in a finitely generated nilpotent group can be described by polynomials in the coordinates with respect to a Mal'cev basis. In this paper we generalize this result to the class of polycyclic groups. Using the construction of a semi-simple splitting for a polycylic group as introduced by Dan Segal, we prove that the group operation is described by polynomial and exponential functions. Hall's result provided a powerful route to the understanding of the connection between the abstract nilpotent group and associated topological and algebraic groups. In the same manner, we apply our result for polycyclic groups to illuminate various constructions, due to Steve Donkin and Andy Magid, of topological and algebraic groups associated with a polycyclic group.

2000 Mathematical Subject Classification: 20F22, 20G15, 22E05.

A first aim of this paper is to answer, in the case of real ruled surfaces $X_l$ of base $\mathbb{C}P^1$, with $l \geq 2$, a question of V. A. Rokhlin: is it true that the equivariant isotopy class does not suffice to distinguish the connected components of the space of smooth real algebraic curves of $X_l$? A second aim is to prove that there exist in these surfaces some real schemes realized by real flexible curves but not by smooth real algebraic curves. These two results of real algebraic geometry are deduced from the following comparison theorem: when $m = l + 2k$, with $k > 0$, the discriminants of the surface $X_m$ are deduced from those of the surface $X_l$ via weighted homotheties. All these results are obtained from a study of a deformation of ruled surfaces.

The paper is written in French.

2000 Mathematical Subject Classification: 14H10, 14J26, 14P25.

We use Nielsen methods to study generating sets of subgroups of groups that act on simplicial trees and give several applications. In particular, we exhibit an explicit bound for the complexity of acylindrical splittings of a finitely generated group in terms of its rank. This is applied to JSJ-splittings of word-hyperbolic groups and 3-manifolds. As a last application we construct examples of amalgamated products that show that there exists no non-trivial rank formula for amalgamated products.

2000 Mathematical Subject Classification: 20E06, 20E08, 57M27.

The aim of this work is to expand Bushnell and Kutzko's theory of $G$-covers [Proc. London Math. Soc. 77 (1998) 582–634] up to a full description of the generalized principal series of the $p$-adic group ${\rm Sp}_4(F)$, with $p$ odd.

We start with a Levi component $M$ of a maximal parabolic subgroup $P$ of $G = {\rm Sp}_4(F)$ and an explicit type $(J_M, \tau_M)$ for the inertial class $S$ in $M$ of a supercuspidal representation of $M$. We compute the Hecke algebra of a $G$-cover $(J, \tau)$ of $(J_M, \tau_M)$ constructed in our previous work [Ann. Inst. Fourier 49 (1999) 1805–1851]: it is a convolution algebra on a Coxeter group (namely, the affine Weyl group of either $U(1,1)(F)$, in the case of the Siegel parabolic, or ${\rm SL}_2(F)$), described explicitly by generators and relations.

From this and Bushnell and Kutzko's work we derive the structure of the parabolically induced representations ${\rm ind}_P^G \pi$, for $\pi$ in $S$, and we find their discrete series subrepresentations if any, thus recovering, through the theory of $G$-covers, results previously obtained by Shahidi using different methods.

The paper is written in French.

2000 Mathematical Subject Classification: 22E50, 11F70.

We study finitely generated modules $M$ over a ring $R$, noetherian on both sides. If $M$ has finite Gorenstein dimension $\mbox{G-dim}_RM$ in the sense of Auslander and Bridger, then it determines two other cohomology theories besides the one given by the absolute cohomology functors ${\rm Ext}^n_R(M,\ )$. Relative cohomology functors ${\rm Ext}^n_{\mathcal G}(M,\ )$ are defined for all non-negative integers $n$; they treat the modules of Gorenstein dimension $0$ as projectives and vanish for $n > \mbox{G-dim}_RM$. Tate cohomology functors $\widehat{\rm Ext}^n_R(M,\ )$ are defined for all integers $n$; all groups $\widehat{\rm Ext}^n_R(M,N)$ vanish if $M$ or $N$ has finite projective dimension. Comparison morphisms $\varepsilon_{\mathcal G}^n \colon {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ )$ and $\varepsilon_R^n \colon {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ )$ link these functors. We give a self-contained treatment of modules of finite G-dimension, establish basic properties of relative and Tate cohomology, and embed the comparison morphisms into a canonical long exact sequence $0 \to {\rm Ext}^1_{\mathcal G}(M,\ ) \to \cdots \to {\rm Ext}^n_{\mathcal G}(M,\ ) \to {\rm Ext}^n_R(M,\ ) \to \widehat{\rm Ext}^n_R(M,\ ) \to {\rm Ext}^{n+1}_{\mathcal G}(M,\ ) \to \cdots$. We show that these results provide efficient tools for computing old and new numerical invariants of modules over commutative local rings.

2000 Mathematical Subject Classification: 16E05, 13H10, 18G25.

Let $G = SL_n(F)$ where $F$ is a non-Archimedean local field. This paper concerns the smooth (complex) representation theory of $G$, specifically, the construction of types in $G$, in the sense of Bushnell and Kutzko. The main result describes a type for each non-supercuspidal component of the Bernstein decomposition of $G$. If this is combined with earlier work of Bushnell and Kutzko, it follows that $G$ admits a complete set of types.

2000 Mathematical Subject Classification: 22E50.

We extend the range of observed behaviour among length functions of optimal asynchronously automatic structures. We do so by means of a construction that yields asynchronously automatic groups with finite aspherical presentations where the Dehn function of the group is polynomial of arbitrary degree. Many of these groups can be embedded in the automorphism group of a free group. Moreover, the fact that the groups have aspherical presentations makes them useful tools in the search to determine the spectrum of exponents for second order Dehn functions. We contribute to this search by giving the first exact calculations of groups with quadratic and superquadratic exponents.

2000 Mathematical Subject Classification:20F06, 20F65, 20F69.

It is shown that Teichmüller distance for analytically finite surfaces is C2. This extends Earle's classical result of 1977 that the distance is C1. Earle showed that the first derivative is given by a quadratic differential. In order to obtain the C2 result, first, a formula is derived for the second derivative for a generic choice of quadratic differential, and with respect to certain local coordinates on Teichmüller space. Then this formula is interpreted so that it can be seen that the limit exists at non-generic points.

2000 Mathematical Subject Classification: 30F60, 32G15.

We study second order elliptic operators with periodic coefficients in two-dimensional simply connected periodic waveguides with the Dirichlet or Neumann boundary conditions. It is proved that under some mild smoothness restrictions on the coefficients, such operators have purely absolutely continuous spectra. The proof follows a method suggested previously by A. Morame to tackle periodic operators with variable coefficients in dimension 2.

2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.

We define the scenery flow space at a point z in the Julia set J of a hyperbolic rational map $T : \mathbb{C} \to \mathbb{C}$ with degree at least 2, and more generally for T a conformal mixing repellor.

We prove that, for hyperbolic rational maps, except for a few exceptional cases listed below, the scenery flow is ergodic. We also prove ergodicity for almost all conformal mixing repellors; here the statement is that the scenery flow is ergodic for the repellors which are not linear nor contained in a finite union of real-analytic curves, and furthermore that for the collection of such maps based on a fixed open set U, the ergodic cases form a dense open subset of that collection. Scenery flow ergodicity implies that one generates the same scenery flow by zooming down towards almost every z with respect to the Hausdorff measure $H^d$, where d is the dimension of J, and that the flow has a unique measure of maximal entropy.

For all conformal mixing repellors, the flow is loosely Bernoulli and has topological entropy at most d. Moreover the flow at almost every point is the same up to a rotation, and so as a corollary, one has an analogue of the Lebesgue density theorem for the fractal set, giving a different proof of a theorem of Falconer.

2000 Mathematical Subject Classification: 37F15, 37F35, 37D20.