We consider an arbitrary Riemann surface $X$, possibly of infinite hyperbolic area. The Liouville measure of the hyperbolic metric defines a measure on the space $G(\tilde{X})$ of geodesics of the universal covering $\tilde{X}$ of $X$. As we vary the Riemann surface structure, this gives an embedding from the Teichmüller space of $X$ into the Fréchet space of Hölder distributions on $G(\tilde{X})$. We show that the embedding is continuously differentiable. In particular, we obtain an explicit integral representation of the tangent map.

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.

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.

Bertini's theorem on variable singular points may fail in characteristic $p$; there are algebraic fibrations that are pathological in the sense that all fibres are non-smooth though the total space admits only a finite number of singular points. We prove that if an algebraic fibration by plane projective quartic curves in characteristic 7 is pathological then, after an eventual cyclic base extension of degree 3, it is, up to birational equivalence, obtained by a base extension from the pencil of quartic curves cut out by the equations $ zx^3 + xy^3 + t yz^3 = 0$.

More generally, we classify pathological fibrations by canonical curves of arithmetic genus $g = \frac{p - 1}{2}$ in the projective space of dimension $\frac{p - 3}{2}$. We prove that the gap sequences of the smooth Weierstrass points of the fibres have the property that a positive integer $\ell$ is a gap if and only if $2g + 1 - \ell$ is a non-gap. In analogy to the Kodaira–Néron classification of special fibres of minimal fibrations by elliptic curves, we construct minimal proper regular models of pathological fibrations, determine the structure of the bad fibres, and study the global geometry of the total spaces.

We complete and complement our recent work on Drasin-Shea-Jordan theorems for Fourier and Hankel transforms. In improving on the methods of our previous work, we were led to certain ratio Mercerian theorems for general kernels; these yield definitive versions of our earlier results for Fourier and Hankel transforms.

1991 Mathematics Subject Classification: primary 44A15; secondary 47B38.

We prove global existence and uniqueness of solutions to the Rarita-Schwinger evolution equations compatible with the constraints. We use a gauge fixing for the Rarita-Schwinger equations for helicity 3/2 fields in curved space that leads to a straightforward Hilbert space framework for their study. We explain how these results might be applied to the global analysis of the full Einstein vacuum equations and provide a complete analysis as a basis for such applications. These and a programme for developing a scattering/inverse scattering transform for the full Einstein equations are discussed.

1991 Mathematics Subject Classification: 83C60, 35Q75, 83C05, 35L45.

The leading asymptotic term for the function that counts the eigenvalues of the Stokes operator is determined for fairly general underlying bounded domains. Moreover, the remainder is estimated in terms of the fractality of the boundary of the domain. The results obtained resemble corresponding ones for the Dirichlet Laplacian.

1991 Mathematics Subject Classification: 35P20.

A Banach space $X$ is said to be sequentially unconditional if every Schauder basic sequence has an unconditional subsequence. We provide an example of a reflexive Banach space $X_{isu}$ which is sequentially unconditional and such that every $T \in \mathcal{L}(X_{isu})$ is a strictly singular perturbation of a multiple of the identity. The space $X_{isu}$ belongs to the class of Banach spaces with saturated and conditional structure. Constructing $X_{isu}$ we follow the general scheme invented by W. T. Gowers and B. Maurey in their celebrated construction of a Hereditarily Indecomposable Banach space. The new ingredient concerns the definition of the special (conditional) functionals. For this we use a non-injective coding function $\sigma$ in conjunction with a Baire-like property of a sequence of countable trees stated and proved in the present paper. It is worth mentioning that $X_{isu}$ admits a richer unconditional structure than classical spaces like $L^1(0,1)$ and it remains unsettled whether $X_{isu}$ is a subspace of a Banach space with an unconditional basis.

We study the exit position $S_{T(r)} $ of a random walk $S_n$ from the interval $[-r, r]$, showing that the tightness of $\vert S_{T(r)}\vert / r$ is equivalent to a generalised kind of stochastic compactness of $S_n$ which we call $SC^\prime$.

This property is in turn equivalent to another kind of compactness property, which we call $SC^{\prime\prime}$, of the maximal sum ${S_n^\ast = \max_{1 \leq j \leq n}\vert S_j \vert}$.

The classes $SC^\prime$ and $SC^{\prime\prime}$, and a related class $SC_0$, which so far seem unexplored, are related to, but different from, the class of stochastically compact $S_n$ studied by Feller, and are similarly of interest in the study of the weak convergence properties of $S_n$ and $S_{T(r)}$.

We give equivalent characterisations of $SC'$ and $SC''$ in terms of the domination of $S_n$ and $S_n^*$ over their maximal increment, and also some analytic characterisations in terms of functionals of the underlying distribution. As a corollary we obtain an equivalence for the stochastic compactness of $\vert S_ {T(r)} \vert / r$.

This paper proves conditional existence results for non-trivial solutions of the equation

\begin{equation} \sum_{i=1}^{n}a_{i}X_{i}^{3}=0 \quad (n=4\mbox{ or }5), \tag{$*$} \end{equation}

where the coefficients $a_{i}$ and the unknowns $X_{i}$ are taken to be rational integers.

No such results were previously known for $n\leq 6$. The proofs use elementary facts about the 3-descent procedure for elliptic curves of the form $E_{A}: X^{3}+Y^{3}=AZ^{3}$.

Thus, when $n=4$, and the $a_{i}$ are each prime, and are all congruent to 2 modulo 3, it is shown that ($*$) will have non-trivial solutions, providing that the Selmer conjecture holds for the curves $E_{A}$. One may replace the Selmer conjecture by an appropriate form of the Generalized Riemann Hypothesis, when $n=5$ and the $a_{i}$ are again taken to be primes, all congruent to 8 modulo 9. Finally, when $n=5$, one may require only that the $a_{i}$ be square-free and coprime to 3, providing one assumes both the Selmer conjecture and a special case of Schinzel's conjecture (on the representation of primes by cubic polynomials).

1991 Mathematics Subject Classification: 11D25, 11G05, 14G05.

We classify smooth ${\rm SO}(n)$-equivariant principal bundles over $S^n$ in terms of their isotropy representations over the north and south poles. This is an example of a general result classifying equivariant $(\Pi, G)$-bundles over manifolds with cohomogeneity 1.

2000 Mathematical Subject Classification: 55R91.

Let $A$ be a non-commutative normed algebra with centre $Z(A)$. A simple application of the triangle inequality shows that if $a \in A$ is close to $Z(A)$ then $a$ almosts commutes with elements of the unit ball of $A$. The extent to which the converse holds is a fundamental question in the study of non-commutativity and has been previously investigated for various classes of Banach algebras, including von Neumann algebras and other C*-algebras. In this context, $K(A)$ is defined to be the smallest number in $[0, \infty]$ such that $$ {\rm distance}(a, Z(A)) \leq K(A) \Vert {\rm ad}(a) \Vert $$ for all $a \in A$, where ${\rm ad}(a)$ is the inner derivation $x \to xa-ax$ for $(x \in A)$. For a C*-algebra $A$, the constant $K_s(A)$ is defined similarly, restricting to self-adjoint $a \in A$.

For a unital C*-algebra $A$, it is already known that the only possible values for $K_s(A)$ are $ \frac{1}{2}, 1, \frac{3}{2}, \ldots, \infty $, that $K(A) \in \{\frac{1}{2}, \frac{1}{\sqrt{3}}\} \cup [1, \infty]$ and that these cases are distinguished by purely topological conditions on the primitive ideal space ${\rm Prim}(A)$. In this paper, further progress is made on the question of possible values for $K(A)$ in $[1, \infty]$. It is shown that $ K(A) \leq \frac{2}{\sqrt{3}}K_s(A)$ and that either $ K(A) \leq \frac{4}{\sqrt{15}} \sim 1.033$ or $ K(A) \geq \frac{1}{2} + \frac{1}{\sqrt{3}} \sim 1.077$. The values $\frac{4}{\sqrt{15}}$ and $\frac{1}{2} + \frac{1}{\sqrt{3}}$ are attained and each of them arises as the maximal bounding radius of a constrained subset of the plane which is obtained from the structure of ${\rm Prim}(A)$.

It is also shown that if $A$ has trivial centre then $K(A)$ is further restricted: $$ K(A) \in \{\frac{1}{2}, 1, \frac{1}{2} + \frac{1}{\sqrt{3}}\} \cup [\frac{3}{2}, \infty].$$ Again, the various cases are distinguished by purely topological conditions on ${\rm Prim}(A)$, some of which are expressed in terms of primal ideals.

Using results of Browkin and Schinzel one can easily determine quadratic number fields with trivial 2-primary Hilbert kernels. In the present paper we completely determine all bi-quadratic number fields which have trivial 2-primary Hilbert kernels. To obtain our results, we use several different tools, amongst which is the genus formula for the Hilbert kernel of an arbitrary relative quadratic extension, which is of independent interest. For some cases of real bi-quadratic fields there is an ambiguity in the genus formula, so in this situation we use instead Brauer relations between the Dedekind zeta-funtions and the Birch–Tate conjecture.

We construct a Banach space X whose algebra of bounded linear operators has K0-group the additive group of rational numbers. We achieve this using a Gowers-Maurey space.

2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10

In this paper we investigate how the volume of a hyperbolic manifold increases under the process of removing a curve, that is, Dehn drilling. If the curve we remove is a geodesic, we show that for a certain family of manifolds the volume increase is bounded above by $\pi \cdot l$ where $l$ is the length of the geodesic drilled. We further construct examples to show that there is no lower bound to the volume increase in terms of a monotonic unbounded function of length. In particular, this shows that volume increase is not bounded below linearly in length.

1991 Mathematics Subject Classification: primary 51M10, 51M20; secondary 51M25, 52C25.

This paper describes parabolic induction, for smooth representations of finite length of the general linear group $\mbox{GL}(N, F)$ of a non-archimedean local field $F$, in terms of a functor between categories of modules over certain affine Hecke algebras, using the categorical equivalences developed in the Bushnell-Kutzko classification of the admissible dual of $\mbox{GL}(N, F)$. This result is used to prove that the Zelevinsky automorphism, which is an involutive automorphism on the representation ring of the category of smooth representations of finite length of $\mbox{GL}(N, F)$, preserves the irreducible representations. The result also implies a simplification of the study of multiplicities of composition factors of induced representations.

1991 Mathematics Subject Classification: 11S37, 22E50.

Let $\mathcal{B}$ be an irreducible spherical Moufang building of rank at least $2$. Thenthe group $G$ is called a group of Lie type$\mathcal{B}$ if it is generated by theroot subgroups corresponding to the rootsof some apartments of $\mathcal{B}$. This notion includes:\begin{enumerate}\item[(1)] classical groups of finite rank,\item[(2)] simple algebraic groups over arbitrary fields,\item[(3)] the `mixed' groups of Tits.\end{enumerate} General structure theorems and a general presentation type theorem for such Lie-type groups, which in a way generalize well-knowntheorems of Seitz and Curtis and Tits, are obtained. 1991 Mathematics Subject Classification:20G15, 20E42.

We construct explicit examples of elementary extremal contractions, both birational and of fiber type, from smooth projective $n$-dimensional varieties, with $n \geq 4$, onto smooth projective varieties, arising from classical projective geometry and defined over sufficiently small fields, not necessarily algebraically closed.

The examples considered come from particular special homaloidal and subhomaloidal linear systems, which are usually degenerations of general phenomena classically investigated by Bordiga, Severi, Todd, Room, Fano, Semple and Tyrrell and more recently by Ein and Shepherd-Barron.

The first series of examples is associated to particular codimension 2 determinantal smooth subvarieties of $\mathbf{P}^{m}$, with $3 \leq m \leq 5$. We get another series of examples by considering special cubic hypersurfaces through some surfaces in $\mathbf{P}^5$, or some 3-folds in $\mathbf{P}^7$ having one apparent double point. The last examples come from an intriguing birational elementary extremal contraction in dimension 6, studied by Semple and Tyrrell and fully described in the last section of the paper.

As shown by the author in {\em Proc.\ Amer.\ Math.\ Soc.} 115 (1992) 345–352, for every metric space $(K,d)$ with compact closed balls one has $(\mbox{lip}\,\varphi(K))^{**} = \mbox{Lip} \,\varphi (K),$ where $\varphi$ is any majorant (that is, non-decreasing function on ${\Bbb R}_+$ with $\varphi (0+) = \varphi (0) = 0$) such that $\varphi(t)/t$ monotonically tends to $+\infty$ as $t\rightarrow 0.$ Here $\mbox{Lip}\,\varphi(K)$ is the Lipschitz space on $K$ with respect to the metric $\varphi(d),\ \mbox{lip}\,\varphi(K)$ is the corresponding ’little‘ Lipschitz space of functions vanishing ’at infinity‘, and ’=‘ means ’canonically isometrically isomorphic‘. The main idea of the proof consisted of finding a normed space $M$ such that $M^* = \mbox{Lip}\,\varphi(K)$ and $M^c = (\mbox{lip}\,\varphi(K))^*,$ where ’$c$‘ stands for the completion, and identifying $M$ with the space of Borel measures on $K$ equipped with the Kantorovich norm. In the present paper, this argument is carried over to generalized Lipschitz spaces on ${\Bbb R}^n$ defined in terms of higher order differences. For an integer $k$ and for a majorant $\varphi$ with $\lim_{t \rightarrow 0}\varphi(t)/t ^k = +\infty,$ define $\Lambda ^k_\varphi$ to be the space of all bounded functions $f$ on ${\Bbb R}^n$ such that for some constant $C,\ \omega_k(f\,;t) \leq C\varphi(t)$ for all $t\geq 0$, where $\omega_k(f\,;\cdot)$ is the $k$th modulus of continuity of $f$. Let $\lambda ^k_\varphi$ be the (closed linear separable) subspace in $\Lambda ^k_\varphi$ which consists of functions $f$ vanishing at ’infinity' and such that $\lim_{t \rightarrow 0}\omega _k(f\,; t)/\varphi (t) = 0.$ We introduce an appropriate analogue $\Vert \cdot \Vert_{k,\varphi}$ of the Kantorovich norm on the space $M$ of finite Borel measures on ${\Bbb R}^n$ with compact support. The properties of this norm for $k \geq 2$ are significantly different from those of the Kantorovich norm ($k = 1$) which reflects the difference in analytic nature of generalized Lipschitz spaces versus classical ones. However, the core of the duality theory survives, and it is shown that $(M, \Vert\cdot \Vert _{k, \varphi})^* = \Lambda ^k _\varphi,\ \ (M, \Vert \cdot \Vert _{k,\varphi})^c = (\lambda ^k _\varphi)^*$ and consequently, $(\lambda ^k _\varphi )^{**} = \Lambda ^k _\varphi$. Several applications of these results are discussed, and a few open problems are formulated.

1991 Mathematics Subject Classification: 46E15, 46E35.

We prove a bounded decomposition for higher order Hankel forms and characterize the first order Hochschild cohomology groups of the disk algebra with coefficients in the space of bounded Hankel forms of some fixed order. Although these groups are non-trivial, we prove that every bounded derivation is inner and necessarily implemented by a Hankel form of order one higher. In terms of operators, this result extends the similarity result of Aleksandrov and Peller. Both of the main structural theorems here rely on estimates involving multilinear maps on the $n$-fold product of the disk algebra and we obtain several higher order analogues of the factorization results due to Aleksandrov and Peller. 2000 Mathematics Subject Classification: 47B35, 46E15, 46E25.