Nous démontrons une nouvelle minoration de la hauteur normalisée d’une sous-variété algébrique d’un tore multiplicatif (et par conséquent des petits points d’une telle sous-variété). Si dans le cas torique, une preuve effective de la conjecture de Bogomolov généralisée était déjà connue ainsi que des estimations “pluri-exponentielles” en le degré de la variété (Schmidt et Bombieri–Zannier), puis monomiales inverses (par le deuxième auteur et Philippon), notre approche qui est entièrement nouvelle, permet de démontrer à un $\varepsilon$-près les conjectures les plus précises (en fonction du degré) que l’on peut formuler dans ce cadre. On obtient ainsi pour ce problème l’exact analogue de ce que l’on sait obtenir dans le cadre du problème de Lehmer. Enfin, nous démontrons pour les sous-variétés de codimension au moins 2 une conjecture du deuxième auteur et Philippon.

An analytically irreducible hypersurface germ $(S,0)\subset(\bm{C}^{d+1},0)$ is quasi-ordinary if it can be defined by the vanishing of the minimal polynomial $f\in\bm{C}\{X\}[Y]$ of a fractional power series in the variables $X=(X_1,\dots,X_d)$ which has characteristic monomials, generalizing the classical Newton–Puiseux characteristic exponents of the plane-branch case ($d=1$). We prove that the set of vertices of Newton polyhedra of resultants of $f$ and $h$ with respect to the indeterminate $Y$, for those polynomials $h$ which are not divisible by $f$, is a semigroup of rank $d$, generalizing the classical semigroup appearing in the plane-branch case. We show that some of the approximate roots of the polynomial $f$ are irreducible quasi-ordinary polynomials and that, together with the coordinates $X_1,\dots,X_d$, provide a set of generators of the semigroup from which we can recover the characteristic monomials and vice versa. Finally, we prove that the semigroups corresponding to any two parametrizations of $(S,0)$ are isomorphic and that this semigroup is a complete invariant of the embedded topological type of the germ $(S,0)$ as characterized by the work of Gau and Lipman.

AMS 2000 Mathematics subject classification: Primary 14M25; 32S25

We show that a twisted Hilbert space with an unconditional basis is isomorphic to a Hilbert space.

AMS 2000 Mathematics subject classification: Primary 46B10. Secondary 46C15

Let $G=GL_n(F)$, where $F$ is a $p$-adic field of characteristic zero and residual characteristic $p$. Assuming that $p>2n$, we compare germs of characters of irreducible admissible representations of $G$ with germs of characters of unipotent representations of direct products of general linear groups over finite extensions of $F$. We show that the character of an irreducible admissible representation has an $s$-asymptotic germ expansion, for some semisimple $s$ in the Lie algebra of $G$. Furthermore, this expansion matches with the $0$-asymptotic expansion (that is, the local character expansion) of the character of a unipotent representation of the centralizer of $s$ in $G$.

AMS 2000 Mathematics subject classification: Primary 22E50; 22E35

We exhibit a smooth complex affine 5-fold whose Witt group of quadratic forms does not inject into the Witt group of its function field. The dimension 5 is the smallest possible. The example depends on relating Witt groups, mod 2 Chow groups, and Steenrod operations.

AMS 2000 Mathematics subject classification: Primary 19G12. Secondary 19E15