We prove that a random function in the Hardy space $H^2$ is a non-cyclic vector for the backward shift operator almost surely. The question of existence of a local pseudocontinuation for a random analytic function is also studied.

Nous donnons une nouvelle démonstration d’un théorème de Vogan qui classifie les représentations cohomologiques isolées dans le dual unitaire d’un groupe de Lie semisimple réel. Nous étudions la question analogue dans le dual automorphe et obtenons des résultats partiels. Nous appliquons finalement ces résultats (et conjectures) à l’étude de la cohomologie des variétés localement symétriques arithmétiques.

We give a new proof of a theorem of Vogan which classifies the cohomological representations of a real semisimple Lie group $G$ which are isolated in the unitary dual of $G$. We investigate the same question in the automorphic dual, and obtain partial results. We finally derive applications of all these results (and conjectures) to the study of the cohomology of arithmetic locally symmetric spaces.

We study product theorems for matrix spaces. In particular, we prove the following theorems.

Theorem 1. For all $\varepsilon>0$, there is $\delta>0$ such that if $A\subset\mathrm{SL}_3(\mathbb{Z})$ is a finite set, then either $A$ intersects a coset of a nilpotent subgroup in a set of size at least $|A|^{1-\varepsilon}$, or $|A^3|>|A|^{1+\delta}$.

Theorem 2. Let $A$ be a finite subset of $\mathrm{SL}_2(\mathbb{C})$. Then either $A$ is contained in a virtually abelian subgroup, or $|A^3|>c|A|^{1+\delta}$ for some absolute constant $\delta>0$.

Here $A^3=\{a_1a_2a_3:a_i\in A,\ i=1,2,3\}$ is the $3$-fold product set of $A$.

We explain three methods for showing that the $p$-adic monodromy of a modular family of abelian varieties is ‘as large as possible', and illustrate them in the case of the ordinary locus of the moduli space of $g$-dimensional principally polarized abelian varieties over a field of characteristic $p$. The first method originated from Ribet's proof of the irreducibility of the Igusa tower for Hilbert modular varieties. The second and third methods both exploit Hecke correspondences near a hypersymmetric point, but in slightly different ways. The third method was inspired by work of Hida, plus a group theoretic argument for the maximality of $\ell$-adic monodromy with $\ell\neq p$.

Dichotomies in various conjectures from algebraic geometry are in fact occurrences of the dichotomy among Zariski structures. This is what Hrushovski showed and which enabled him to solve, positively, the geometric Mordell–Lang conjecture in positive characteristic. Are we able now to avoid this use of Zariski structures? Pillay and Ziegler have given a direct proof that works for semi-abelian varieties they called ‘very thin’, which include the ordinary abelian varieties. But it does not apply in all generality: we describe here an abelian variety which is not very thin. More generally, we consider from a model-theoretical point of view several questions about the fields of definition of semi-abelian varieties.

We study the intersection theory on the moduli spaces of maps of $n$-pointed curves $f:(C,s_1,\dots,s_n)\to V$ which are stable with respect to the weight data $(a_1,\dots,a_n)$, $0\le a_i\le1$. After describing the structure of these moduli spaces, we prove a formula describing the way descendant invariants change under a wall crossing. As a corollary, we compute the weighted descendants in terms of the usual ones, i.e. for the weight data $(1,\dots,1)$, and vice versa.

One of the main obstacles in applying converse theorems to prove new cases of functoriality is that of stability of $\gamma$-factors for a certain class of $L$-functions obtained from the ‘Langlands–Shahidi’ method, where the $\gamma$-factors are defined inductively by means of ‘local coefficients’. The problem then becomes that of stability of local coefficients upon twisting the representation by a highly ramified character. In this paper we first establish that the inverses of certain local coefficients are, up to an abelian $\gamma$-factor, genuine Mellin transforms of partial Bessel functions of the type we analysed in our previous paper. The second main result is then the resulting stability of the local coefficients in this situation, which include all the cases of interest for functoriality. Hopefully, the analysis given here will open the door to a proof of the general stability and the equality of $\gamma$-factors obtained from different methods through integration over certain quotient spaces whose generic fibres are closed. They do not seem to have been studied before in any generality.

Let $H$ be a torsion-free compact $p$-adic analytic group whose Lie algebra is split semisimple. We show that the quotient skewfield of fractions of the Iwasawa algebra $\varLambda_H$ of $H$ has trivial centre and use this result to classify the prime $c$-ideals in the Iwasawa algebra $\varLambda_G$ of $G:=H\times\mathbb{Z}_p$. We also show that a finitely generated torsion $\varLambda_G$-module having no non-zero pseudo-null submodule is completely faithful if and only if it is has no central torsion. This has an application to the study of Selmer groups of elliptic curves.

For a linear group $G$ acting on an absolutely irreducible variety $X$ over $\mathbb{Q}$, we describe the orbits of $X(\mathbb{Q}_p)$ under $G(\mathbb{Q}_p)$ and of $X(\mathbb{F}_p((t)))$ under $G(\mathbb{F}_p((t)))$ for $p$ big enough. This allows us to show that the degree of a wide class of orbital integrals over $\mathbb{Q}_p$ or $\mathbb{F}_p((t))$ is less than or equal to $0$ for $p$ big enough, and similarly for all finite field extensions of $\mathbb{Q}_p$ and $\mathbb{F}_p((t))$.

Nous démontrons des théorèmes d'immersion holomorphe dans un espace projectif complexe pour des variétés kählériennes non compactes et des laminations par variétés complexes qui admettent un fibré en droites holomorphe strictement positif. En particulier, nous montrons que sur une lamination compacte par surfaces de Riemann, les fonctions méromorphes séparent les points si et seulement si aucun cycle feuilleté n'est homologue à $0$.

We prove holomorphic immersion theorems in a finite dimensional complex projective space for kählerian non-compact manifolds and for laminations by complex manifolds that carry a line bundle of positive curvature. In particular, we prove that on a Riemann surfaces lamination of a compact space, the space of meromorphic functions separates points if and only if every foliation cycle is non homologous to $0$.

This paper is a brief survey of recent results and some open problems related to linear groups of finite Morley rank, an area of research where Bruno Poizat's impact is very prominent. As a sign of respect to his strongly expressed views that mathematics has to be done, written and pulished only in the native tongue of the immediate author–the scribe, in effect–of the text, I insist on writing my paper in Russian, even if the results presented belong to a small but multilingual community of researchers of American, British, French, German, Kazakh, Russian, Turkish origin. To emphasise even further the linguistic subtleties involved, I use British spelling in the English fragments of my text.

A diffeomorphism $f$ has a heterodimensional cycle if there are (transitive) hyperbolic sets $\varLambda$ and $\varSigma$ having different indices (dimension of the unstable bundle) such that the unstable manifold of $\varLambda$ meets the stable one of $\varSigma$ and vice versa. This cycle has co-index $1$ if $\mathop{\mathrm{index}}(\varLambda)=\mathop{\mathrm{index}}(\varSigma)\pm1$. This cycle is robust if, for every $g$ close to $f$, the continuations of $\varLambda$ and $\varSigma$ for $g$ have a heterodimensional cycle.

We prove that any co-index $1$ heterodimensional cycle associated with a pair of hyperbolic saddles generates $C^1$-robust heterodimensioal cycles. Therefore, in dimension three, every heterodimensional cycle generates robust cycles.

We also derive some consequences from this result for $C^1$-generic dynamics (in any dimension). Two of such consequences are the following. For tame diffeomorphisms (generic diffeomorphisms with finitely many chain recurrence classes) there is the following dichotomy: either the system is hyperbolic or it has a robust heterodimensional cycle. Moreover, any chain recurrence class containing saddles having different indices has a robust cycle.

The pseudovariety $\mathbf{DG}$ of all finite monoids all of whose regular $\mathcal{D}$-classes are subgroups is shown to be local, that is, it is verified that the pseudovariety $g\mathbf{DG}$ of finite categories generated by $\mathbf{DG}$ coincides with the pseudovariety $\ell\mathbf{DG}$ of all finite categories whose local monoids all belong to $\mathbf{DG}$. Yet more general statements of this kind are deduced, yielding results such as that, for every prime number $p$, the pseudovariety $\mathbf{DG}_p$ of all finite monoids all of whose regular $\mathcal{D}$-classes are $p$-groups is local, or that the pseudovarieties $\mathbf{DG}_{\mathrm{sol}}$ and $\mathbf{DG}_{\mathrm{nil}}$ of all finite monoids all of whose regular \mbox{$\mathcal{D}$-classes} are, respectively, solvable groups and nilpotent groups are local.

The Main Conjecture of Iwasawa theory for an elliptic curve $E$ over $\mathbb{Q}$ and the anticyclotomic $\mathbb{Z}_p$-extension of an imaginary quadratic field $K$ was studied in \cite{bertolini_darmon}, in the case where $p$ is a prime of ordinary reduction for $E$. Analogous results are formulated, and proved, in the case where $p$ is a prime of supersingular reduction. The foundational study of supersingular main conjectures carried out by Perrin-Riou, Pollack, Kurihara, Kobayashi and Iovita and Pollack are required to handle this case in which many of the simplifying features of the ordinary setting break down.

We draw a connection between the model-theoretic notions of modularity (or one-basedness), orthogonality and internality, as applied to difference fields, and questions of descent in in algebraic dynamics. In particular we prove in any dimension a strong dynamical version of Northcott's theorem for function fields, answering a question of Szpiro and Tucker and generalizing a theorem of Baker's for the projective line.

The paper comes in three parts. This first part contains an exposition some of the main results of the model theory of difference fields, and their immediate connection to questions of descent in algebraic dynamics. We present the model-theoretic notion of internality in a context that does not require a universal domain with quantifier-elimination. We also note a version of canonical heights that applies well beyond polarized algebraic dynamics. Part II sharpens the structure theory to arbitrary base fields and constructible maps where in part I we emphasize finite base change and correspondences. Part III will include precise structure theorems related to the Galois theory considered here, and will enable a sharpening of the descent results for non-modular dynamics.

Nous présentons une expression explicite pour la hauteur normalisée d’une variété torique projective définie sur $\overline{\mathbb{Q}}$. Cette expression se décompose comme somme de contributions locales, chaque terme étant l’intégrale d’une certaine fonction concave et affine par morceaux, définie sur le polytope $Q_{\mathcal{A}}$ classiquement associé à l’action du tore. Plus généralement, nous obtenons une expression explicite pour la multihauteur normalisée d’un tore relative à plusieurs plongements monomiaux.

L’ensemble de fonctions introduit se comporte comme un analogue arithmétique du polytope $Q_{\mathcal{A}}$. En plus des formules pour les hauteur et multihauteurs, nous montrons que cet objet se comporte de manière naturelle par rapport à plusieurs constructions standard : décomposition en orbites, formation de joints, produits de Segre et plongements de Veronese.

La démonstration suit une démarche indirecte : à la place de la définition de la hauteur normalisée, on s’appuie sur le calcul d’une fonction de Hilbert arithmétique appropriée.

We present an explicit expression for the normalized height of a projective toric variety defined over $\overline{\mathbb{Q}}$. This expression decomposes as a sum of local contributions, each term being the integral of a certain function, concave and piecewise linear-affine, defined on the polytope $Q_{\mathcal{A}}$ classically associated with the torus action. More generally, we obtain an explicit expression for the normalized multiheight of a torus with respect to several monomial embeddings.

The set of functions introduced behaves as an arithmetic analogue of the polytope $Q_{\mathcal{A}}$. Besides the formulae for the height and multiheight, we show that this object behaves in a natural way with respect to several standard constructions: decomposition into orbits, joins, Segre products and Veronese embeddings.

The proof follows an indirect path: instead of the definition of the normalized height, we rely on the computation of an appropriate arithmetic Hilbert function.

Nous proposons une façon simple de compter le nombre de $G$-chtoucas de Drinfeld avec modifications arbitraires, inspirée par les travaux de Kottwitz sur les variétés de Shimura.

We propose a simple method to count the number of Drinfeld $G$-shtukas with arbitrary modifications which is inspired by works by Kottwitz on Shimura varieties.

Soit $\mathrm{F}$ un corps commutatif localement compact non archimédien, et soit $\mathrm{D}$ une algèbre à division de centre $\mathrm{F}$. Nous prouvons que toute représentation irréductible supercuspidale du groupe $\mathrm{GL}_m(\mathrm{D})$, de niveau non nul, est l'induite compacte d'une représentation d'un sous-groupe ouvert compact modulo le centre de $\mathrm{GL}_m(\mathrm{D})$. Plus précisément, nous prouvons que de telles représentations contiennent un type simple maximal au sens de Bushnell et Kutzko.

Let $\mathrm{F}$ be a non-Archimedean locally compact field and let $\mathrm{D}$ be a central $\mathrm{F}$-division algebra. We prove that any positive level supercuspidal irreducible representation of the group $\mathrm{GL}_m(\mathrm{D})$ is compactly induced from a representation of a compact mod centre open subgroup of $\mathrm{GL}_m(\mathrm{D})$. More precisely, we prove that such representations contain a maximal simple type in the sense of Bushnell and Kutzko.

This second part of the paper strengthens the descent theory described in the first part to rational maps and arbitrary base fields. We obtain in particular a decomposition of any difference field extension into a tower of finite, field-internal and one-based difference field extensions. This is needed in order to obtain the ‘dynamical Northcott’ Theorem 1.11 of Part I in sharp form.