We study characters of an $n$-fold cover $\widetilde{SL} (n,\mathbb{F})$ of $SL(n,\mathbb{F})$ over a non-Archimedean local field. We compute the character of an irreducible representation of $\widetilde{SL}(n,\mathbb{F})$ in terms of the character of an irreducible representation of a cover $\widetilde{GL}(n,\mathbb{F})$ of $GL(n,\mathbb{F})$. We define an analogue of L-packets for $\widetilde{SL}(n,\mathbb{F})$, such that the character of a linear combination of the representations in such a packet is computed in terms of the character of an irreducible representation of $PGL(n,\mathbb{F})$. This is analogous to stable endoscopic lifting for linear groups. We also prove an ‘inversion’ formula expressing the character of a genuine irreducible representation of $\widetilde{SL}(n,\mathbb{F})$ as a linear combination of virtual characters, each of which is obtained from $PGL(n,\mathbb{F})$.

AMS 2000 Mathematics subject classification: Primary 22E50. Secondary 11F70

This paper gives an extension of the classical Zariski–van Kampen theorem describing the fundamental groups of the complements of plane singular curves by generators and relations. It provides a procedure for computation of the first non-trivial higher-homotopy groups of the complements of singular projective hypersurfaces in terms of the homotopy variation operators introduced here.

AMS 2000 Mathematics subject classification: Primary 14F35; 14D05; 32S20; 32S50. Secondary 14J70; 32S25; 55P99

Nous démontrons une nouvelle minoration de la hauteur normalisée d’une sous-variété algébrique d’un tore multiplicatif (et par conséquent des petits points d’une telle sous-variété). Si dans le cas torique, une preuve effective de la conjecture de Bogomolov généralisée était déjà connue ainsi que des estimations “pluri-exponentielles” en le degré de la variété (Schmidt et Bombieri–Zannier), puis monomiales inverses (par le deuxième auteur et Philippon), notre approche qui est entièrement nouvelle, permet de démontrer à un $\varepsilon$-près les conjectures les plus précises (en fonction du degré) que l’on peut formuler dans ce cadre. On obtient ainsi pour ce problème l’exact analogue de ce que l’on sait obtenir dans le cadre du problème de Lehmer. Enfin, nous démontrons pour les sous-variétés de codimension au moins 2 une conjecture du deuxième auteur et Philippon.

In this paper we finish the topological classification of real algebraic surfaces of Kodaira dimension zero and we make a step towards the Enriques classification of real algebraic surfaces, by describing in detail the structure of the moduli space of real hyperelliptic surfaces.

Moreover, we point out the relevance in real geometry of the notion of the orbifold fundamental group of a real variety, and we discuss related questions on real varieties $(X,\sigma)$ whose underlying complex manifold $X$ is a $K(\pi,1)$.

Our first result is that if $(S,\sigma)$ is a real hyperelliptic surface, then the differentiable type of the pair $(S,\sigma)$ is completely determined by the orbifold fundamental group exact sequence.

This result allows us to determine all the possible topological types of $(S,\sigma)$, and to prove that they are exactly 78.

It follows also as a corollary that there are exactly eleven cases for the topological type of the real part of $S$.

Finally, we show that once we fix the topological type of $(S,\sigma)$ corresponding to a real hyperelliptic surface, the corresponding moduli space is irreducible (and connected).

We also give, through a series of tables, explicit analytic representations of the 78 components of the moduli space.

AMS 2000 Mathematics subject classification: Primary 14P99; 14P25; 14J15; 32Q57

We establish a transfer of (twisted) orbital integrals in the context of twisted endoscopy between a real reductive algebraic group $G$ and a reductive quasi-split real group $H_1$ associated to an endoscopic datum.

AMS 2000 Mathematics subject classification: Primary 22E45; 11F70

Nous conjecturons que la réduction modulo $p$ des représentations cristallines irréductibles de dimension 2 sur $\bar{\bm{Q}}_p$ de $\Gal(\bar{\bm{Q}}_p/\bm{Q}_p)$ peut être prédite par la réduction modulo $p$ de représentations $p$-adiques localement algébriques de $\GL_2(\bm{Q}_p)$. Nous explicitons quelques calculs de telles réductions confirmant cette conjecture. Cela suggère un lien arithmétique non trivial entre les deux types de représentations.

AMS 2000 Mathematics subject classification: Primary 11F33; 11F70; 11F80; 11F85

An analytically irreducible hypersurface germ $(S,0)\subset(\bm{C}^{d+1},0)$ is quasi-ordinary if it can be defined by the vanishing of the minimal polynomial $f\in\bm{C}\{X\}[Y]$ of a fractional power series in the variables $X=(X_1,\dots,X_d)$ which has characteristic monomials, generalizing the classical Newton–Puiseux characteristic exponents of the plane-branch case ($d=1$). We prove that the set of vertices of Newton polyhedra of resultants of $f$ and $h$ with respect to the indeterminate $Y$, for those polynomials $h$ which are not divisible by $f$, is a semigroup of rank $d$, generalizing the classical semigroup appearing in the plane-branch case. We show that some of the approximate roots of the polynomial $f$ are irreducible quasi-ordinary polynomials and that, together with the coordinates $X_1,\dots,X_d$, provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa. Finally, we prove that the semigroups corresponding to any two parametrizations of $(S,0)$ are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ $(S,0)$ as characterized by the work of Gau and Lipman.

AMS 2000 Mathematics subject classification: Primary 14M25; 32S25

Soit $G$ le groupe des points définis sur un corps $p$-adique d’un groupe réductif connexe. On définit l’espace des fonctions de Schwartz-Harish-Chandra sur $G$: ce sont des fonctions sur $G$, à valeurs complexes, qui vérifient des conditions de croissance et de lissité. La formule de Plancherel exprime les valeurs d’une telle fonction $f$ en termes des opérateurs $\pi(f)$, où $\pi$ parcourt l’ensemble des classes de représentations lisses irréductibles et tempérées de $G$. On démontre cette formule, ainsi que quelques résultats utiles d’analyse harmonique: l’existence du prolongement rationnel d’un opérateur d’entrelacement, la finitude (si $G$ est semi-simple) de l’ensemble des classes de représentations lisses irréductibles de carré intégrable de $G$ possédant un $K$-type donné. Tous ces résultats sont dus à Harish-Chandra, qui les a démontrés dans un manuscrit non publié. Le présent article est une rédaction de ce manuscrit.

AMS 2000 Mathematics subject classification: Primary 22E35; 22E50

We show that if an automorphism of a non-abelian free group $F_n$ is irreducible with irreducible powers, it acts on the boundary of Culler–Vogtmann’s outer space with north–south dynamics: there are two fixed points, one attracting and one repelling, and orbits accumulate only on these points. The main new tool we use is the equivariant assignment of a point $Q(X)$ to any end $X\in\partial F_n$, given an action of $F_n$ on an $\bm{R}$-tree $T$ with trivial arc stabilizers; this point $Q(X)$ may be in $T$, or in its metric completion, or in its boundary.

AMS 2000 Mathematics subject classification: Primary 20F65; 20E05; 20E08

We consider a transitive uniformly quasi-conformal Anosov diffeomorphism $f$ of a compact manifold $\mathcal{M}$. We prove that if the stable and unstable distributions have dimensions greater than two, then $f$ is $C^\infty$ conjugate to an affine Anosov automorphism of a finite factor of a torus. If the dimensions are at least two, the same conclusion holds under the additional assumption that $\mathcal{M}$ is an infranilmanifold. We also describe necessary and sufficient conditions for smoothness of conjugacy between such a diffeomorphism and a small perturbation.

AMS 2000 Mathematics subject classification: Primary 37C; 37D

We show that a twisted Hilbert space with an unconditional basis is isomorphic to a Hilbert space.

AMS 2000 Mathematics subject classification: Primary 46B10. Secondary 46C15

For a variety over a local field, we show that the alternating sum of the traces of the composition of the actions of an element of the Weil group and an algebraic correspondence on the $\ell$-adic etale cohomology is independent of $\ell$. We prove the independence by establishing basic properties of weight spectral sequences.

AMS 2000 Mathematics subject classification: Primary 14G20. Secondary 11G25

Let $G$ be a compact $p$-valued $p$-adic Lie group, and let $\varLambda(G)$ be its Iwasawa algebra. The present paper establishes results about the structure theory of finitely generated torsion $\varLambda(G)$-modules, up to pseudo-isomorphism, which are largely parallel to the classical theory when $G$ is abelian (except for basic differences which occur for those torsion modules which do not possess a non-zero global annihilator). We illustrate our general theory by concrete examples of such modules arising from the Iwasawa theory of elliptic curves without complex multiplication over the field generated by all of their $p$-power torsion points.

AMS 2000 Mathematics subject classification: Primary 11G05; 11R23; 16D70; 16E65; 16W70

Let $G=GL_n(F)$, where $F$ is a $p$-adic field of characteristic zero and residual characteristic $p$. Assuming that $p>2n$, we compare germs of characters of irreducible admissible representations of $G$ with germs of characters of unipotent representations of direct products of general linear groups over finite extensions of $F$. We show that the character of an irreducible admissible representation has an $s$-asymptotic germ expansion, for some semisimple $s$ in the Lie algebra of $G$. Furthermore, this expansion matches with the $0$-asymptotic expansion (that is, the local character expansion) of the character of a unipotent representation of the centralizer of $s$ in $G$.

AMS 2000 Mathematics subject classification: Primary 22E50; 22E35

We exhibit a smooth complex affine 5-fold whose Witt group of quadratic forms does not inject into the Witt group of its function field. The dimension 5 is the smallest possible. The example depends on relating Witt groups, mod 2 Chow groups, and Steenrod operations.

AMS 2000 Mathematics subject classification: Primary 19G12. Secondary 19E15

For a function $f\in L^p(\mathbb{R}d)$, $d\ge 2$, let $A_tf(x)$ be the mean of $f$ over the sphere of radius $t$ centred at $x$. Given a set $E\subset(0,\infty)$ of dilations we prove various endpoint bounds for the maximal operator $M_E$ defined by $M_E f(x)=\sup_{t\in E}|A_tf(x)|$, under some regularity assumptions on $E$.

AMS 2000 Mathematics subject classification: Primary 42B25. Secondary 42B20

In this note we are studying the Lie algebras associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus\{0,\mu_n,\infty\}$. Let $n$ be a prime number. We assume that for any $m\geq 1$ the numbers $Li_{m+1}(\xi_n^k)$ for $1\leq k\leq (n-1)/2$ are linearly independent over $\mathbb{Q}$ in $\mathbb{C}/(2\pi\ri)^{m+1}\mathbb{Q}$. Let $S=\{k_1,\cdots,k_q\}$ be a subset of $\{1,\dots,p-1\}$ such that if $k\in S$, then $p-k\in S$ and $(S+S)\cap S=\emptyset$ (the sum of two elements of $S$ is calculated $\mathrm{Mod}p$). Then we show that in the Lie algebra associated to non-abelian unipotent periods on $P^1_{\mathbb{Q}(\mu_n)}\setminus \{0,\mu_n,\infty\}$ there are derivations $D^{k_1}_{m+1},\dots,D^{k_q}_{m+1}$ in each degree $m+1$ and these derivations are free generators of a free Lie subalgebra of this Lie algebra.

AMS 2000 Mathematics subject classification: Primary 11G55