It is still an open question whether a compact embedded hypersurface in the Euclidean space $\mathbb{R}^{n+1}$ with constant mean curvature and spherical boundary is necessarily a hyperplanar ball or a spherical cap, even in the simplest case of surfaces in $\mathbb{R}^3$. In a recent paper, Alías and Malacarne (Rev. Mat. Iberoamericana18 (2002), 431–442) have shown that this is true for the case of hypersurfaces in $\mathbb{R}^{n+1}$ with constant scalar curvature, and more generally, hypersurfaces with constant higher-order $r$-mean curvature, when $r\geq2$. In this paper we deal with some aspects of the classical problem above, by considering it in a more general context. Specifically, our starting general ambient space is an orientable Riemannian manifold $\bar{M}$, where we will consider a general geometric configuration consisting of an immersed hypersurface into $\bar{M}$ with boundary on an oriented hypersurface $P$ of $\bar{M}$. For such a geometric configuration, we study the relationship between the geometry of the hypersurface along its boundary and the geometry of its boundary as a hypersurface of $P$, as well as the geometry of $P$ as a hypersurface of $\bar{M}$. Our approach allows us to derive, among others, interesting results for the case where the ambient space has constant curvature (the Euclidean space $\mathbb{R}^{n+1}$, the hyperbolic space $\mathbb{H}^{n+1}$, and the sphere $\mathbb{S}^{n+1}$). In particular, we are able to extend the symmetry results given in the recent paper mentioned above to the case of hypersurfaces with constant higher-order $r$-mean curvature in the hyperbolic space and in the sphere.

We formulate and study a new coarse (co-)assembly map. It involves a modification of the Higson corona construction and produces a map dual in an appropriate sense to the standard coarse assembly map. The new assembly map is shown to be an isomorphism in many cases. For the underlying metric space of a group, the coarse co-assembly map is closely related to the existence of a dual Dirac morphism and thus to the Dirac dual Dirac method of attacking the Novikov conjecture.

Two integral structures on the $\mathbb{Q}$-vector space of modular forms of weight two on $X_0(N)$ are compared at primes $p$ dividing $N$ at most once. When $p=2$ and $N$ is divisible by a prime that is $3$ mod $4$, this comparison leads to an algorithm for computing the space of weight one forms mod $2$ on $X_0(N/2)$. For $p$ arbitrary and $N>4$ prime to $p$, a way to compute the Hecke algebra of mod $p$ modular forms of weight one on $\varGamma_1(N)$ is presented, using forms of weight $p$, and, for $p=2$, parabolic group cohomology with mod $2$ coefficients. Appendix A is a letter of October 1987 from Mestre to Serre in which he reports on computations of weight one forms mod $2$ of prime level. Appendix B, by Wiese, reports on an implementation for $p=2$ in Magma, using Stein’s modular symbols package, with which Mestre’s computations are redone and slightly extended.

A method of choice for realizing finite groups as regular Galois groups over $\mathbb{Q}(T)$ is to find $\mathbb{Q}$-rational points on Hurwitz moduli spaces of covers. In another direction, the use of the so-called patching techniques has led to the realization of all finite groups over $\mathbb{Q}_p(T)$. Our main result shows that, under some conditions, these $p$-adic realizations lie on some special irreducible components of Hurwitz spaces (the so-called Harbater–Mumford components), thus connecting the two main branches of the area. As an application, we construct, for every projective system $(G_n)_{n\geq0}$ of finite groups, a tower of corresponding Hurwitz spaces $(\mathcal{H}_{G_n})_{n\geq0}$, geometrically irreducible and defined over some cyclotomic extension of $\mathbb{Q}$, which admits projective systems of $\mathbb{Q}_p^{\mathrm{ur}}$-rational points for all primes $p$ not dividing the orders $|G_n|$ ($n\geq0$).

In this paper we study a Haagerup inequality in the general case of discrete groupoids. We develop two geometrical tools, pinching and tetrahedral change of faces, based on deformation of triangles, to prove it. We show how to use these tools to find all the already known results just by manipulating triangles. We use these tools for groups acting freely and by isometries on the set of vertices of any affine building and give a first reduction of this inequality to its verification on some special triangles and prove the inequality when the building is of type $\tilde{A}_{k_1}\times\cdots\times\tilde{A}_{k_n}$, where $k_i\in\{1,2\}$, $i=1,\dots,n$.

We study the first occurrence of a cuspidal representation $\pi$ of $\mathrm{Sp}_{2n}(\mathbb{A})$ in the tower of split orthogonal groups under symplectic-orthogonal theta lift. We give upper and lower bounds for the first occurrence in terms of the wave front of $\pi$. Besides we describe a class of small cuspidal representations whose first occurrence is the maximal possible.

For a finite extension $F/\mathbb{Q}_p$ we construct and study a class of locally analytic representations of GL$_2(F)$ and related groups such as the quaternion algebra over $F$. The construction is based on inducing a locally analytic character of a maximal torus. We show that for a generic character the resulting representation is topologically irreducible, and not isomorphic to a locally analytic principal series, when the torus is non-split.

Une conjecture de Grothendieck prédit que le transport parallèle de toute classe de cycle algébrique invariante par monodromie est encore une classe de cycle algébrique. Nous prouvons qu'il s'agit d'une conséquence des conjectures standard.

Dans un travail antérieur, nous avions démontré, en caractéristique nulle, une variante de cette conjecture de déformation, en remplaçant la notion de cycle algébrique par celle de cycle motivé (adjonction formelle des inverses des isomorphismes de Lefschetz attachés à des polarisations). Nous étendons ici ce résultat en caractéristique arbitraire, sous diverses hypothèses (par exemple si la base est ‘presque complète’).

Nous définissons inconditionnellement des groupes de Galois motiviques dans ce contexte, et étudions leur variation en famille.

A conjecture due to Grothendieck predicts that the parallel transport of any monodromy-invariant algebraic cycle class remains an algebraic cycle class. We prove that this follows from the standard conjectures.

We have shown in a previous work a variant of this deformation conjecture in characteristic zero, in which algebraic cycles are replaced by motivated cycles (formal adjunction of inverses of Lefschetz isomorphisms attached to polarizations). We extend this result in any characteristic under various assumptions, for instance if the base is ‘almost complete’.

We give an unconditional construction of motivic Galois groups in this context, and study their variation in a family of projective smooth varieties.

Nous étudions la nature arithmétique de $q$-analogues des valeurs $\zeta(s)$ de la fonction zêta de Riemann, notamment des valeurs des fonctions $\zeta_q(s)=\sum_{k=1}^{\infty}q^k\sum_{d\mid k}d^{s-1}$, $s=1,2,\dots$, où $q$ est un nombre complexe, $|q|<1$ (ces fonctions sont intimement liées au monde automorphe). Le théorème principal de cet article montre que, si $1/q$ est un nombre entier différent de $\pm1$ et si $M$ est un nombre impair suffisamment grand, alors la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta_q(3),\zeta_q(5),\dots,\zeta_q(M)$ est au moins $c_1\sqrt{M}$, avec $c_1=0,3358$. Ce résultat peut être considéré comme un $q$-analogue du résultat de Rivoal et de Ball et Rivoal, qui affirme que la dimension de l’espace vectoriel engendré sur $\mathbb{Q}$ par $1,\zeta(3),\zeta(5),\dots,\zeta(M)$ est au moins $c_2\log M$, avec $c_2=0,5906$. Pour les mêmes valeurs de $q$, une minoration similaire pour les valeurs $\zeta_q(s)$ aux entiers $s$ pairs nous permet de redémontrer un cas particulier d’un résultat de Bertrand qui affirme la transcendance sur $\mathbb{Q}$ de l’une des deux séries d’Eisenstein $E_4(q)$ et $E_6(q)$ pour tout nombre complexe $q$ tel que $0<|q|<1$.

In this paper we study the reduction to characteristic $p$ of the Shimura variety associated to a unitary group which has signature $(n-1,1)$ at its real place. We describe the Newton polygon, the Ekedahl–Oort, and the final stratification. In addition we examine the moduli space of $p$-isogenies using a variant of the local model for Shimura varieties. We apply our results to obtain a proof of the Eichler–Shimura congruence relation for the case that $n$ is even.

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 $\mathbb{F}_{q}$.

La conjecture de Tate pour les variétés abéliennes sur les corps finis a été prouvée par Tate lui-même dans le cas d'un $H^2$, mais reste ouverte en degré supérieur. Nous démontrons ici une variante affaiblie de cette conjecture (en tout degré), qui décrit néanmoins tout cycle de Tate ‘en termes de’ cycles algébriques. Cette variante s'applique aussi aux variétés abéliennes $A$ sur un corps de type fini sur un corps fini, au moins dans le cas ‘de bonne réduction’, i.e. dans le cas où $A$ est fibre générique d'un schéma abélien sur une base projective normale sur un corps fini.

Tate's conjecture for abelian varieties over finite fields has been proven by Tate himself for $H^2$ but remains open in higher degree. We prove a weaker version of the conjecture in any degree, which describes every Tate cycle `in terms of' algebraic cycles. This variant also applies to abelian varieties $A$ over finitely generated fields of characteristic $p$, at least in the `good reduction' case, i.e. when $A$ is the generic fibre of an abelian scheme on a normal projective variety defined over a finite field.

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.

The Christoffel problem, in its classical formulation, asks for a characterization of real functions defined on the unit sphere $S^{n-1}\subset\mathbb{R}^n$ which occur as the mean curvature radius, expressed in terms of the Gauss unit normal, of a closed convex hypersurface, i.e. the boundary of a convex body in $\mathbb{R}^n$. In this work we consider the related problem in Lorentz space $\mathbb{L}^n$ and present necessary and sufficient conditions for a $C^1$ function defined in the hyperbolic space $H^{n-1}\subset\mathbb{L}^n$ to be the mean curvature radius of a spacelike embedding $\bm{M}\hookrightarrow\mathbb{L}^n$.

The relative trace formula is a tool in the theory of automorphic forms which was invented by Jacquet in order to study period integrals and relate them to Langlands functoriality. In this paper we give an analogue of Arthur’s spectral expansion of the trace formula to the relative setup in the context of $\text{GL}_n$. This is an important step toward application of the relative trace formula and it extends earlier work by several authors to higher rank. Our method is new and based on complex analysis and majorization of Eisenstein series. To that end we use recent lower bounds of Brumley for Rankin–Selberg $L$-functions at the edge of the critical strip.

Ellipticity of operators on a manifold with edges can be treated in the framework of a calculus of $2\times2$-block matrix operators with trace and potential operators on the edges. The picture is similar to the pseudodifferential analysis of boundary-value problems. The extra conditions satisfy an analogue of the Shapiro–Lopatinskij condition, provided a topological obstruction for the elliptic edge-degenerate operator in the upper left corner vanishes; this is an analogue of a condition of Atiyah and Bott in boundary-value problems. In general, however, we need global projection data, similarly to global boundary conditions, known for Dirac operators or other geometric operators. The present paper develops a new calculus with global projection data for operators on manifolds with edges. In particular, we show the Fredholm property in a suitable scale of spaces and construct parametrices within the calculus.

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.

Soient $p$ un nombre premier et $\mathcal{C}$ la courbe de Hecke $p$-adique de $\mathrm{GL}_2$ de niveau moderé $1$ introduite par Coleman et Mazur. Nous montrons que $\mathcal{C}$ est lisse en ses points Eisenstein critiques et nous donnons une condition nécessaire et suffisante pour que l’application poids soit étale en un tel point en terme de certaines valeurs de la fonction $\zeta$$p$-adique. Un point important est la détermination en ces points du lieu de réductibilité schématique du pseudo-caractère Galoisien porté par $\mathcal{C}$, et ce restreint à un groupe de décomposition en $p$. L’assertion de lissité découle alors de ce que les fonctions $L$ de Dirichlet ont au plus des zéros simples aux entiers.

Let $p$ be a prime number and $\mathcal{C}$ be the $p$-adic tame level $1$ eigencurve introduced by Coleman and Mazur. We prove that $\mathcal{C}$ is smooth at the evil Eisenstein points and we give necessary and sufficient conditions for etaleness of the map to the weight space at these points in terms of $p$-adic zeta values. A key step is the determination at these points of the schematic reducibility locus of the pseudo-character carried by $\mathcal{C}$ restricted to a decomposition group at $p$. Then, the smoothness appears to be a consequence of the fact that the Dirichlet $L$-functions only have simple zeros at integers.

On calcule les restrictions aux points fixes de deux bases de la cohomologie équivariante des tours de Bott, et on précise la structure multiplicative de ces algèbres de cohomologie. En s’aidant du travail de Duan, on en déduit une méthode de calcul des constantes de structure de la cohomologie équivariante des variétés de drapeaux (calcul de Schubert équivariant).

‘Equivariant cohomology of Bott towers and equivariant Schubert calculus’. We calculate the restrictions to fixed points of two bases of the equivariant cohomology of Bott towers, and we describe the multiplicative structure of these cohomology algebras. We build on the work of Duan to give a method to calculate the structure constants of the equivariant cohomology of the flag varieties (equivariant Schubert calculus).