Soit $G$ un groupe réductif connexe défini sur un corps $p$-adique $F$ et $\ggo$ son algèbre de Lie. Les intégrales orbitales pondérées sur $\ggo(F)$ sont des distributions $J_M(X,f)$---$f$ est une fonction test---indexées par les sous-groupes de Lévi $M$ de $G$ et les éléments semi-simples réguliers $X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales composantes du côté géométrique des formules des traces locale et globale d'Arthur. Si $M=G$, on retrouve les intégrales orbitales invariantes qui, vues comme fonction de $X$, sont bornées sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un résultat bien connu de Harish-Chandra. Si $M \subsetneq G$, les intégrales orbitales pondérées explosent au voisinage des éléments singuliers. Nous construisons dans cet article de nouvelles intégrales orbitales pondérées $J_M^b(X,f)$, égales à $J_M(X,f)$ à un terme correctif près, qui tout en conservant les principales propriétés des précédentes (comportement par conjugaison, développement en germes, {\it etc.}) restent bornées quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons également que les intégrales orbitales pondérées globales, associées à des éléments semi-simples réguliers, se décomposent en produits de ces nouvelles intégrales locales.

We study the cohomology of connected components of Shimura varieties $S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on the stabilization of the twisted trace formula, due to Kottwitz and Shelstad. More precisely, for each character $\olomega$ on the group of connected components of $S_{K^p}$ we define an operator $L(\omega)$ on the cohomology groups with compact supports $H^i_c (S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual trace of the composition of $L(\omega)$ with a Hecke operator $f$ away from $p$ and a sufficiently high power of a geometric Frobenius $\Phi^r_p$, can be expressed as a sum of $\omega$-{\em weighted} (twisted) orbital integrals (where $\omega$-{\em weighted} means that the orbital integrals and twisted orbital integrals occuring here each have a weighting factor coming from the character $\olomega$). As the crucial step, we define and study a new invariant $\alpha_1 (\gamma_0; \gamma, \delta)$ which is a refinement of the invariant $\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done by using a theorem of Reimann and Zink.

We prepare for a comparison of global trace formulas of general linear groups and their metaplectic coverings. In particular, we generalize the local metaplectic correspondence of Flicker and Kazhdan and describe the terms expected to appear in the invariant trace formulas of the above covering groups. The conjectural trace formulas are then placed into a form suitable for comparison.

We will construct a finite union of finite quotients of the affine building of the group $\GL_3$ over the field of $p$-adic numbers $\mathbb{Q}_p$. We will view this object as a hypergraph and estimate the spectrum of its underlying graph.

We construct an automorphic realization of the global minimal representation of quaternionic exceptional groups, using the theory of Eisenstein series, and use this for the study of theta correspondences.

We prove a relative trace formula that establishes the cubic base change for $\GL(2)$. One also gets a classification of the image of base change. The case when the field extension is nonnormal gives an example where a trace formula is used to prove lifting which is not endoscopic.

This paper starts by introducing a bi-periodic summation formula for automorphic forms on a group $G(E)$, with periods by a subgroup $G(F)$, where $E/F$ is a quadratic extension of number fields. The split case, where $E = F \oplus F$, is that of the standard trace formula. Then it introduces a notion of stable bi-conjugacy, and stabilizes the geometric side of the bi-period summation formula. Thus weighted sums in the stable bi-conjugacy class are expressed in terms of stable bi-orbital integrals. These stable integrals are on the same endoscopic groups $H$ which occur in the case of standard conjugacy. The spectral side of the bi-period summation formula involves periods, namely integrals over the group of $F$-adele points of $G$, of cusp forms on the group of $E$-adele points on the group $G$. Our stabilization suggests that such cusp forms---with non vanishing periods---and the resulting bi-period distributions associated to ``periodic'' automorphic forms, are related to analogous bi-period distributions associated to ``periodic'' automorphic forms on the endoscopic symmetric spaces $H(E)/H(F)$. This offers a sharpening of the theory of liftings, where periods play a key role. The stabilization depends on the ``fundamental lemma'', which conjectures that the unit elements of the Hecke algebras on $G$ and $H$ have matching orbital integrals. Even in stating this conjecture, one needs to introduce a ``transfer factor''. A generalization of the standard transfer factor to the bi-periodic case is introduced. The generalization depends on a new definition of the factors even in the standard case. Finally, the fundamental lemma is verified for $\SL(2)$.

We study the correspondence of representations arising by restricting the minimal representation of the linear group of type $E_7$ and relative rank $4$. The main tool is computations of the Jacquet modules of the minimal representation with respect to maximal parabolic subgroups of $G_2$ and $\PU_3(D)$.

Let $U(n,n)$ be the rank $n$ quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of $U(n,n)$ has at most simple poles at the integers or half integers in certain strip of the complex plane.

The fundamental lemma in the theory of automorphic forms is proven for the (quasi-split) unitary group $U(3)$ in three variables associated with a quadratic extension of $p$-adic fields, and its endoscopic group $U(2)$, by means of a new, elementary technique. This lemma is a prerequisite for an application of the trace formula to classify the automorphic and admissible representations of $U(3)$ in terms of those of $U(2)$ and base change to $\GL(3)$. It compares the (unstable) orbital integral of the characteristic function of the standard maximal compact subgroup $K$ of $U(3)$ at a regular element (whose centralizer $T$ is a torus), with an analogous (stable) orbital integral on the endoscopic group $U(2)$. The technique is based on computing the sum over the double coset space $T\bs G/K$ which describes the integral, by means of an intermediate double coset space $H\bs G/K$ for a subgroup $H$ of $G=U(3)$ containing $T$. Such an argument originates from Weissauer's work on the symplectic group. The lemma is proven for both ramified and unramified regular elements, for which endoscopy occurs (the stable conjugacy class is not a single orbit).