For a smooth variety $X$ over a perfect field $k$ in characteristic $p$, an equivalence of category is established between the category of crystals on $X/W(k)$ and the category of $p$-adically nilpotent, integrable de Rham–Witt connections.

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

This paper deals with the stabilization of the contribution of elliptic elements to the geometric side of the general twisted trace formula. We extend the results of Langlands, Kottwitz and Shelstad to all elliptic elements for the general twisted trace formula.

AMS 2000 Mathematics subject classification: Primary 11F72; 11R39; 11R34

Pour chaque entier $d=2,3,4$, il existe un corps $F$ de dimension cohomologique $1$ et une surface de del Pezzo de degré $d$ sur $F$ sans zéro-cycle de degré $1$, en particulier sans point rationnel. Les démonstrations utilisent le théorème de Merkur’ev et Suslin, le théorème de Riemann-Roch sur une surface et la formule du degré de Rost.

An initial boundary-value problem for the modified Korteweg–de Vries equation on the half-line, $0<x<\infty$, $t>0$, is analysed by expressing the solution $q(x,t)$ in terms of the solution of a matrix Riemann–Hilbert (RH) problem in the complex $k$-plane. This RH problem has explicit $(x,t)$ dependence and it involves certain functions of $k$ referred to as the spectral functions. Some of these functions are defined in terms of the initial condition $q(x,0)=q_0(x)$, while the remaining spectral functions are defined in terms of the boundary values $q(0,t)=g_0(t)$, $q_x(0,t)=g_1(t)$, and $q_{xx}(0,t)=g_2(t)$. The spectral functions satisfy an algebraic global relation which characterizes, say, $g_2(t)$ in terms of $\{q_0(x),g_0(t),g_1(t)\}$. It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.

AMS 2000 Mathematics subject classification: Primary 37K15; 35Q53. Secondary 35Q15; 34A55

Soit $G$ un groupe réductif $p$-adique connexe. Nous effectuons une décomposition spectrale sur $G$ à partir de la formule d’inversion de Fourier utilisée dans ‘Une formule de Plancherel pour l’algèbre de Hecke d’un groupe réductif $p$-adique’, V. Heiermann, Commun. Math. Helv.76 (2001), 388–415. Nous en déduisons essentiellement qu’une représentation cuspidale d’un sous-groupe de Levi M appartient au support cuspidal d’une représentation de carré intégrable de $G$ si et seulement si c’est un pôle de la fonction $\mu$ de Harish-Chandra d’ordre égal au rang parabolique de M. Ces pôles sont d’ordre maximal. Plus précisément, nous montrons que cette condition est nécessaire et que sa suffisance équivaut à une propriété combinatoire de la fonction $\mu$ de Harish-Chandra qui s’avère être une conséquence d’un résultat de E. Opdam. En outre, nous obtenons des identités entre des combinaisons linéaires de coefficients matriciels. Ces identités contiennent des informations sur le degré formel des représentations de carré intégrable ainsi que sur leur position dans la représentation induite.

Let $G$ be a reductive connected $p$-adic group. With help of the Fourier inversion formula used in ‘Une formule de Plancherel pour l’algèbre de Hecke d’un groupe réductif $p$-adique’, V. Heiermann, Commun. Math. Helv.76 (2001), 388–415, we give a spectral decomposition on $G$. In particular, we deduce from it essentially that a cuspidal representation of a Levi subgroup $M$ is in the cuspidal support of a square-integrable representation of $G$ if and only if it is a pole of Harish-Chandra’s $\mu$-function of order equal to the parabolic rank of $M$. These poles are of maximal order. In more explicit terms, we show that this condition is necessary and that its sufficiency is equivalent to a combinatorical property of Harish-Chandra’s $\mu$-function, which appears to be a consequence of a result of E. Opdam. We get also identities between some linear combinations of matrix coefficients. These identities contain information on the formal degree of square-integrable representations and on their position in the induced representation.

AMS 2000 Mathematics subject classification: Primary 22E50. Secondary 22E35; 11F70; 11F72; 11F85

We analyse the non-commutative space underlying the quantum group $\textrm{SU}_q(2)$ from the spectral point of view, which is the basis of non-commutative geometry, and show how the general theory developed in our joint work with Moscovici applies to the specific spectral triple defined by Chakraborty and Pal. This provides the pseudo-differential calculus, the Wodzciki-type residue, and the local cyclic co-cycle giving the index formula. The co-chain whose co-boundary is the difference between the original Chern character and the local one is given by the remainders in the rational approximation of the logarithmic derivative of the Dedekind eta function. This specific example allows us to illustrate the general notion of locality in non-commutative geometry. The formulae computing the residue are ‘local’. Locality by stripping all the expressions from irrelevant details makes them computable. The key feature of this spectral triple is its equivariance, i.e. the $\textrm{SU}_q(2)$-symmetry. We shall explain how this naturally leads to the general concept of invariant cyclic cohomology in the framework of quantum group symmetries.

AMS 2000 Mathematics subject classification: Primary 81R50; 19K33; 46L; 58B34

Given a lattice in an isocrystal, Mazur’s inequality states that the Newton point of the isocrystal is less than or equal to the invariant measuring the relative position of the lattice and its transform under Frobenius. Conversely, it is known that any potential invariant allowed by Mazur’s inequality actually arises from some lattice. These can be regarded as statements about the group $GL_n$. This paper proves an analogous converse theorem for all split classical groups.

AMS 2000 Mathematics subject classification: Primary 14L05. Secondary 11S25; 14F30; 20G25

An affine Hecke algebra $\mathcal{H}$ contains a large abelian subalgebra $\mathcal{A}$ spanned by the Bernstein–Zelevinski–Lusztig basis elements $\theta_x$, where $x$ runs over (an extension of) the root lattice. The centre $\mathcal{Z}$ of $\mathcal{H}$ is the subalgebra of Weyl group invariant elements in $\mathcal{A}$. The natural trace (‘evaluation at the identity’) of the affine Hecke algebra can be written as integral of a certain rational $n$-form (with values in the linear dual of $\mathcal{H}$) over a cycle in the algebraic torus $T=\textrm{Spec}(\mathcal{A})$. This cycle is homologous to a union of ‘local cycles’. We show that this gives rise to a decomposition of the trace as an integral of positive local traces against an explicit probability measure on the spectrum $W_0\setminus T$ of $\mathcal{Z}$. From this result we derive the Plancherel formula of the affine Hecke algebra.

AMS 2000 Mathematics subject classification: Primary 20C08; 22D25; 22E35; 43A32

In this paper, we shall study the Dirichlet problem for the minimal surfaces equation. We prove some results about the boundary behaviour of a solution of this problem. We describe the behaviour of a non-converging sequence of solutions in term of lines of divergence in the domain. Using this second result, we build some solutions of the Dirichlet problem on unbounded domain. We then give a new proof of the result of Cosín and Ros concerning the Plateau problem at infinity for horizontal ends.

AMS 2000 Mathematics subject classification: Primary 53A10

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

In this paper, we define and study the Weil–Petersson geometry. Under the framework of the Weil–Petersson geometry, we study the Weil–Petersson metric and the Hodge metric. Among the other results, we represent the Hodge metric in terms of the Weil–Petersson metric and the Ricci curvature of the Weil–Petersson metric for Calabi–Yau fourfold moduli. We also prove that the Hodge volume of the moduli space is finite. Finally, we proved that the curvature of the Hodge metric is bounded if the Hodge metric is complete and the dimension of the moduli space is 1.

AMS 2000 Mathematics subject classification: Primary 53A30. Secondary 32C16

Using a connected sum construction, we prove the existence of complete non-compact embedded constant-mean-curvature-1 surfaces in hyperbolic 3-space with finitely many ends and non-trivial topology. These surfaces are obtained by desingularizing finitely many mutually tangent horospheres.

AMS 2000 Mathematics subject classification: Primary 53A10; 53A35

Nous appelons polynôme quasi-ordinaire de Laurent un polynôme unitaire $f(Y)$ dont les coefficients sont des séries de Laurent à plusieurs variables et tel que son discriminant soit le produit d’un monôme de Laurent et d’une série entière de terme constant non-nul. Si la dérivée $\partial f/\partial Y$ rendue unitaire est encore quasi-ordinaire de Laurent—ce qui peut être toujours obtenu par changement de base—nous montrons que l’on peut mesurer le contact de ses facteurs avec ceux de $f$ en fonction d’invariants discrets de $f$ qui mesurent le contact entre ses racines, codés sous la forme de l’arbre d’Eggers–Wall. Tous les calculs sont faits en termes de chaînes et de cochaînes supportées par cet arbre. Ce travail constitue une généralisation de résultats connus pour les germes de courbes planes.

AMS 2000 Mathematics subject classification: Primary 32S25. Secondary 14M25

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

We study $l$-adic analogues of the classical polylogarithms. Using an explicit arithmetic formula we show that they satisfy the same distribution equations as classical polylogarithms. We also identify $l$-adic polylogarithms evaluated at roots of unity with Soulé classes.

AMS 2000 Mathematics subject classification: Primary 11G55; 11D88; 17B70