We study the fundamental groups of algebraic stacks. We show that these fundamental groups carry an additional structure coming from the inertia groups. We use this additional structure to analyse geometric/topological properties of stacks. We give an explicit formula for the fundamental group of the coarse moduli space. As an application, we find an explicit formula for the fundamental group of the geometric quotient of an arbitrary algebraic group action. Also, we use these additional structures to give a necessary and sufficient for an algebraic stack to be uniformizable (i.e. quotient of an algebraic space by a finite group action).

AMS 2000 Mathematics subject classification: Primary 14A20. Secondary 14F35; 14L30

Segre proved that a smooth cubic surface over $Q$ is unirational if and only if it has a rational point. We prove that the result also holds for cubic hypersurfaces over any field, including finite fields.

AMS 2000 Mathematics subject classification: Primary 14G05; 14G15. Secondary 11G25; 11D25

Soient $\bm{G}$ un groupe réductif connexe sur un corps local non archimédien $F$ et $\bm{H}$ un groupe endoscopique de $\bm{G}$. On suppose $\bm{G}$ et $\bm{H}$ non ramifiés. Le {\og}lemme fondamental{\fg} affirme l’égalité de certaines sommes pondérées d’intégrales orbitales sur $\bm{G}(F)$ et sur $\bm{H}(F)$. On peut descendre cette assertion en un {\og}lemme fondamental pour les algèbres de Lie{\fg} affirmant l’égalité de sommes pondérées analogues sur $\bm{\mathfrak{g}}(F)$ et sur $\bm{\mathfrak{h}}(F)$, où $\bm{\mathfrak{g}}$ et $\bm{\mathfrak{h}}$ sont les algèbres de Lie de $\bm{G}$ et $\bm{H}$. Cette assertion est cruciale pour la théorie de l’endoscopie de Langlands. D’importants cas particuliers ont été prouvés récemment, sous l’hypothèse que la caractéristique de $F$ est positive. Nous donnons un sens précis à l’assertion suivante, et nous la prouvons : soient $F$ et $F'$ deux corps locaux de même corps résiduel $\mathbb{F}_{q}$ et de caractéristique résiduelle $p$ assez grande ; supposons le lemme fondamental (pour les algèbres de Lie) vrai sur le corps de base $F$ ; alors ce lemme est vrai sur le corps de base $F'$. Cela permet de relever ce lemme de la caractéristique positive à la caractéristique nulle. Une grande partie de l’article est consacrée à reformuler des théories bien connues (endoscopie, immeubles, réseaux de Moy–Prasad) de sorte que $F$ n’y intervienne que via le corps résiduel $\mathbb{F}_{q}$.

Let $\bm{G}$ be a connected reductive group over a non-archimedean local field $F$ and let $\bm{H}$ be an endoscopic group of $\bm{G}$. We suppose that $\bm{G}$ and $\bm{H}$ are unramified. The fundamental lemma asserts an equality between certain linear combinations of integral orbitals over $\bm{G}(F)$ and $\bm{H}(F)$. We can translate this assertion in a ‘fundamental lemma for Lie algebras’, that is, a conjectural equality between linear combinations of integral orbitals over $\bm{\mathfrak{g}}(F)$ and $\bm{\mathfrak{h}}(F)$, where $\bm{\mathfrak{g}}$ and $\bm{\mathfrak{h}}$ are the Lie algebras of $\bm{G}$ and $\bm{H}$. Important particular cases of this lemma were recently proved, assuming the characteristic of $F$ to be positive. We give a precise meaning to the following assertion and we prove it: let $F$ and $F'$ be two local fields with the same residue field $\mathbb{F}_{q}$ and with ‘big’ residual characteristic; suppose that the fundamental lemma (for Lie algebras) is true over $F$; then it is true over $F'$. In particular, we can lift the lemma to $0$-characteristic if it is known for positive characteristic. A large part of the article reformulates well-known constructions (endoscopy, buildings, Moy–Prasad filtrations) so that $F$ appears only via his residual field $\mathbb{F}_{q}$.

The purpose of this note is to show that the Betti realization of motives is compatible with Grothendieck's six operations and the nearby cycles functors, which in the motivic world, were previously studied by the author. We first review the construction of the Betti realization. Then, we establish some general criteria which, applied to the Betti realization, give the compatibilities we seek except for the one concerning the nearby cycles functors. The latter will be treated in a separate section.

A factorization formula for certain automorphisms of a Poisson algebra associated with a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing formulae for Donaldson–Thomas type invariants of Kontsevich and Soibelman.

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

A topological group $G$ is called extremely amenable if every continuous action of $G$ on a compact space has a fixed point. This concept is linked with geometry of high dimensions (concentration of measure). We show that a von Neumann algebra is approximately finite dimensional if and only if its unitary group with the strong topology is the product of an extremely amenable group with a compact group, which strengthens a result by de la Harpe. As a consequence, a $C^\ast$-algebra $A$ is nuclear if and only if the unitary group $U(A)$ with the relative weak topology is strongly amenable in the sense of Glasner. We prove that the group of automorphisms of a Lebesgue space with a non-atomic measure is extremely amenable with the weak topology and establish a similar result for groups of non-singular transformations. As a consequence, we prove extreme amenability of the groups of isometries of $L^p(0,1)$, $1\leq p<\infty$, extending a classical result of Gromov and Milman ($p=2$). We show that a measure class preserving equivalence relation $\mathcal{R}$ on a standard Borel space is amenable if and only if the full group $[\mathcal{R}]$, equipped with the uniform topology, is extremely amenable. Finally, we give natural examples of concentration to a non-trivial space in the sense of Gromov occurring in the automorphism groups of injective factors of type III.

We study basic properties of the category of smooth representations of a p-adic group G with coefficients in any commutative ring R in which p is invertible. Our main purpose is to prove that Hecke algebras are Noetherian whenever R is; this question arose naturally with Bernstein's fundamental work for R = ℂ, in which case he proved this Noetherian property. In a first step, we prove that Noetherianity would follow from a generalization of the so-called second adjointness property between parabolic functors, also due to Bernstein for complex representations. Then, to attack this second adjointness, we introduce and study ‘parahoric functors’ between representations of groups of integral points of smooth integral models of G and of their ‘Levi’ subgroups. Applying our general study to Bruhat-Tits parahoric models, we get second adjointness for minimal parabolic groups. For non-minimal parabolic subgroups, we have to restrict to classical and linear groups, and use smooth models associated with Bushnell-Kutzko and Stevens semi-simple characters. The same strategy should apply to ‘tame’ groups, using Yu's smooth models and generic characters.

For each group $G$ having an infinite normal subgroup with the relative property (T) (e.g. $G=H\times K$, with $H$ infinite with property (T) and $K$ arbitrary) and each countable abelian group $\varLambda$ we construct free ergodic measure-preserving actions $\sigma_\varLambda$ of $G$ on the probability space such that the first cohomology group of $\sigma_\varLambda$, $\ssm{H}^1(\sigma_\varLambda,G)$, is equal to $\text{Char}(G)\times\varLambda$. We deduce that $G$ has uncountably many non-stably orbit-equivalent actions. We also calculate 1-cohomology groups and show existence of ‘many’ non-stably orbit-equivalent actions for free products of groups as above.

We show that a continuous, odd, regular (non-exceptional), ordinary, irreducible, two-dimensional, $l$-adic representation of the absolute Galois group of the rational numbers is modular over some totally real field. We deduce that it occurs in the $l$-adic cohomology of some variety over the rationals and that its $L$-function has meromorphic continuation to the whole complex plane and satisfies the expected functional equation.

AMS 2000 Mathematics subject classification: Primary 11F80. Secondary 11G40

We give a survey on the Stokes structure of a good meromorphic flat bundle. We also show that a meromorphic flat bundle has the good formal structure if and only if it has a good lattice.

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.

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

We construct a relative de Rham–Witt complex $W\varOmega^{\cdot}_{X/S}$ for a scheme $X$ over a base scheme $S$. It coincides with the complex defined by Illusie (Annls Sci. Ec. Norm. Super.12 (1979), 501–661) if $S$ is a perfect scheme of characteristic $p>0$. The hypercohomology of $W\varOmega^{\cdot}_{X/S}$ is compared to the crystalline cohomology if $X$ is smooth over $S$ and $p$ is nilpotent on $S$. We obtain the structure of a $3n$-display on the first crystalline cohomology group if $X$ is proper and smooth over $S$.

AMS 2000 Mathematics subject classification: Primary 14F30; 14F40

For a matched pair of locally compact quantum groups, we construct the double crossed product as a locally compact quantum group. This construction generalizes Drinfeld’s quantum double construction. We study the modular theory and the $\mathrm{C}^*$-algebraic properties of these double crossed products, as well as several links between double crossed products and bicrossed products. In an appendix, we study the Radon–Nikodym derivative of a weight under a quantum group action (following Yamanouchi) and obtain, as a corollary, a new characterization of closed quantum subgroups.

AMS 2000 Mathematics subject classification: Primary 46L89. Secondary 46L65

We continue our study of the reduction of PEL Shimura varieties with parahoric level structure at primes p at which the group defining the Shimura variety ramifies. We describe ‘good’ p-adic integral models of these Shimura varieties and study their étale local structure. In the present paper we mainly concentrate on the case of unitary groups for a ramified quadratic extension. Some of our results are applications of the theory of twisted affine flag varieties that we developed in a previous paper.

We study operators of Kramers–Fokker–Planck type in the semiclassical limit, assuming that the exponent of the associated Maxwellian is a Morse function with a finite number n0 of local minima. Under suitable additional assumptions, we show that the first n0 eigenvalues are real and exponentially small, and establish the complete semiclassical asymptotics for these eigenvalues.

Let $G$ be a Lie groupoid over $M$ such that the target-source map from $G$ to $M\times M$ is proper. We show that, if $\mathcal{O}$ is an orbit of finite type (i.e. which admits a proper function with finitely many critical points), then the restriction $G|_\mathcal{U}$ of $G$ to some neighbourhood $\mathcal{U}$ of $\mathcal{O}$ in $M$ is isomorphic to a similar restriction of the action groupoid for the linear action of the transitive groupoid $G|_\mathcal{O}$ on the normal bundle $N\mathcal{O}$. The proof uses a deformation argument based on a cohomology vanishing theorem, along with a slice theorem which is derived from a new result on submersions with a fibre of finite type.

AMS 2000 Mathematics subject classification: Primary 58H05. Secondary 57R99

We describe explicitly the Voevodsky's triangulated category of motives (and give a ‘differential graded enhancement’ of it). This enables us to able to verify that DMgm ℚ is (anti)isomorphic to Hanamura's (k).

We obtain a description of all subcategories (including those of Tate motives) and of all localizations of . We construct a conservative weight complex functor ; t gives an isomorphism . A motif is mixed Tate whenever its weight complex is. Over finite fields the Beilinson–Parshin conjecture holds if and only if tℚ is an equivalence.

For a realization D of we construct a spectral sequence S (the spectral sequence of motivic descent) converging to the cohomology of an arbitrary motif X. S is ‘motivically functorial’; it gives a canonical functorial weight filtration on the cohomology of D(X). For the ‘standard’ realizations this filtration coincides with the usual one (up to a shift of indices). For the motivic cohomology this weight filtration is non-trivial and appears to be quite new.

We define the (rational) length of a motif M; modulo certain ‘standard’ conjectures this length coincides with the maximal length of the weight filtration of the singular cohomology of M.

Par une méthode entièrement nouvelle utilisant les déformations $p$-adiques de pentes positives de représentations automorphes pour $\mathrm{GSp}_{4/\mathbb{Q}}$, nous prouvons que le $p$-groupe de Selmer $H^1_f(\mathbb{Q},V_f(k))$ associé à une forme modulaire $f$ de poids $2k$ et ordinaire en $p$ est infini si l’ordre d’annulation à l’entier $k$ de la fonction $L$ de $f$ est impair.

By an entirely new method that makes use of $p$-adic deformations of automorphic representations of $\mathrm{GSp}_{4/\mathbb{Q}}$, we prove that the $p$-adic Selmer group $H^1_f(\mathbb{Q},V_f(k))$ associated to a modular form $f$ of weight $2k$ that is ordinary at $p$ is infinite if the order of vanishing at $k$ of the $L$-function of $f$ is odd.