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.

2000 Mathematical Subject Classification: 10B10.

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.

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.

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.

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 investigate in this paper the complementation of copies of $c_0(I)$ in some classes of Banach spaces (in the class of weakly compactly generated (WCG) Banach spaces, in the larger class $\mathcal{V}$ of Banach spaces which are subspaces of some $C(K)$ space with $K$ a Valdivia compact, and in the Banach spaces $C([1, \alpha ])$, where $\alpha$ is an ordinal) and the embedding of $c_0(I)$ in the elements of the class $\mathcal{C}$ of complemented subspaces of $C(K)$ spaces. Two of our results are as follows:

(i) in a Banach space $X \in \mathcal{V}$ every copy of $c_0(I)$ with $\# I < \aleph _{\omega}$ is complemented;

(ii) if $\alpha _0 = \aleph _0$, $\alpha _{n+1} = 2^{\alpha _n}$, $n \geq 0$, and $\alpha = \sup \{\alpha _n : n \geq 0\}$ there exists a WCG Banach space with an uncomplemented copy of $c_0(\alpha )$.

So, under the generalized continuum hypothesis (GCH), $\aleph _{\omega}$ is the greatest cardinal $\tau$ such that every copy of $c_0(I)$ with $\# I < \tau$ is complemented in the class $\mathcal{V}$. If $T : c_0(I) \to C([1,\alpha ])$ is an isomorphism into its image, we prove that:

(i) $c_0(I)$ is complemented, whenever $\| T \| ,\| T^{-1} \| < (3/2)^{\frac 12}$;

(ii) there is a finite partition $\{I_1, \dots , I_k\}$ of $I$ such that each copy $T(c_0(I_k))$ is complemented.

Concerning the class $\mathcal{C}$, we prove that an already known property of $C(K)$ spaces is still true for this class, namely, if $X \in \mathcal{C}$, the following are equivalent:

(i) there is a weakly compact subset $W \subset X$ with ${\rm Dens}(W) = \tau$;

(ii) $c_0(\tau )$ is isomorphically embedded into $X$.

This yields a new characterization of a class of injective Banach spaces.

2000 Mathematical Subject Classification: 46B20, 46B26.