It is well known in Kähler geometry that the infinite-dimensional symmetric space of smooth Kähler metrics in a fixed Kähler class on a polarized Kähler manifold is well approximated by finite-dimensional submanifolds of Bergman metrics of height k. Then it is natural to ask whether geodesics in can be approximated by Bergman geodesics in . For any polarized Kähler manifold, the approximation is in the C0 topology. For some special varieties, one expects better convergence: Song and Zelditch proved the C2 convergence for the torus-invariant metrics over toric varieties. In this article, we show that some C∞ approximation exists as well as a complete asymptotic expansion for principally polarized abelian varieties.

Soit A une variété abélienne définie sur un corps de nombres K, le nombre de points de torsion définis sur une extension finie L est borné polynomialement en terme du degré [L : K]. Lorsque A est isogène à un produit de variétés abéliennes simples de type GSp, c'est-à-dire dont le groupe de Mumford–Tate est « générique » (isomorphe au groupe des similitudes symplectiques) et vérifiant la conjecture de Mumford–Tate, nous calculons l'exposant optimal dans cette borne, en terme de la dimension des sous-variétés abéliennes de A. Le résultat est inconditionnel pour un produit de variétés abéliennes simples dont l'anneau d'endomorphismes est ℤ et la dimension n'appartient pas à un ensemble exceptionnel explicite S = {4, 10, 16, 32, …}. Par ailleurs nous prouvons, suivant une stratégie de Serre, que si la conjecture de Mumford–Tate est vraie pour des variétés abéliennes de type GSp, alors la conjecture de Mumford–Tate est vraie pour un produit de telles variétés abéliennes.

Soit p un nombre premier et F un corps local non archimédien de caractéristique p. Dans cet article, à une représentation lisse irréductible de GL2(F) sur avec caractère central, nous associons un diagramme qui détermine la représentation de départ à isomorphisme près. Nous le déterminons également dans certains cas.

Let X be the variety obtained by the Weil transfer with respect to a quadratic separable field extension of a generalized Severi–Brauer variety. We study (and, in some cases, determine) the canonical dimension, incompressibility, and motivic indecomposability of X. We determine the canonical 2-dimension of X (in the general case).

Let be a lattice in the real simple Lie group L. If L is of rank at least 2 (respectively locally isomorphic to Sp(n, 1)) any unbounded morphism ρ : Γ → G into a simple real Lie group G essentially extends to a Lie morphism ρL : L → G (Margulis's superrigidity theorem, respectively Corlette's theorem). In particular any such morphism is infinitesimally, thus locally, rigid. On the other hand, for L = SU(n, 1) even morphisms of the form are not infinitesimally rigid in general. Almost nothing is known about their local rigidity. In this paper we prove that any cocompact lattice Γ in SU(n, 1) is essentially locally rigid (while in general not infinitesimally rigid) in the quaternionic groups Sp(n, 1), SU(2n, 2) or SO(4n, 4) (for the natural sequence of embeddings SU(n, 1) ⊂ Sp(n, 1) ⊂ SU(2n, 2) ⊂ SO(4n, 4)).

Let C be a proper smooth geometrically connected hyperbolic curve over a field of characteristic 0 and ℓ a prime number. We prove the injectivity of the homomorphism from the pro-ℓ mapping class group attached to the two dimensional configuration space of C to the one attached to C, induced by the natural projection. We also prove a certain graded Lie algebra version of this injectivity. Consequently, we show that the kernel of the outer Galois representation on the pro-ℓ pure braid group on C with n strings does not depend on n, even if n = 1. This extends a previous result by Ihara–Kaneko. By applying these results to the universal family over the moduli space of curves, we solve completely Oda's problem on the independency of certain towers of (infinite) algebraic number fields, which has been studied by Ihara, Matsumoto, Nakamura, Ueno and the author. Sequentially we obtain certain information of the image of this Galois representation and get obstructions to the surjectivity of the Johnson–Morita homomorphism at each sufficiently large even degree (as Oda predicts), for the first time for a proper curve.

We introduce C*-pseudo-multiplicative unitaries and concrete Hopf C*-bimodules for the study of quantum groupoids in the setting of C*-algebras. These unitaries and Hopf C*-bimodules generalize multiplicative unitaries and Hopf C*-algebras and are analogues of the pseudo-multiplicative unitaries and Hopf–von Neumann-bimodules studied by Enock, Lesieur and Vallin. To each C*-pseudo-multiplicative unitary, we associate two Fourier algebras with a duality pairing and in the regular case two Hopf C*-bimodules. The theory is illustrated by examples related to locally compact Hausdorff groupoids. In particular, we obtain a continuous Fourier algebra for a locally compact Hausdorff groupoid.