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

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

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

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

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