Let $k$ be a field of characteristic $p\>0$. Let $D_m$ be a ${BT}_m$ over $k$ (i.e., an $m$-truncated Barsotti–Tate group over $k$). Let $S$ be a $k$-scheme and let $X$ be a ${BT}_m$ over $S$. Let $S_{D_m}\left(X\right)$ be the subscheme of $S$ which describes the locus where $X$ is locally for the fppf topology isomorphic to $D_m$. If $p\ge 5$, we show that $S_{D_m}\left(X\right)$ is pure in $S$, i.e. the immersion $S_{D_m}\left(X\right)\hookrightarrow S$ is affine. For $p\in \{2,3\}$, we prove purity if $D_m$ satisfies a certain technical property depending only on its $p$-torsion $D_m\left[p\right]$. For $p\ge 5$, we apply the developed techniques to show that all level $m$...

We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A₅ + A₁.

We introduce the concept of quadratic modular symbol and study how these symbols are related to quadratic$p$-adic $L$-functions. These objects were introduced in [3] in the case of modular curves. In this paper, we discuss a method to attach quadratic modular symbols and quadratic $p$-adic $L$-functions to more general Shimura curves.

We apply functional analytical and variational methods in order to study well-posedness and qualitative properties of evolution equations on product Hilbert spaces. To this aim we introduce an algebraic formalism for matrices of sesquilinear mappings. We apply our results to parabolic problems of different nature: a coupled diffusive system arising in neurobiology, a strongly damped wave equation, and a heat equation with dynamic boundary conditions.

In the present text we give a geometric interpretation of quasi-modular forms using moduli of elliptic curves with marked elements in their de Rham cohomologies. In this way differential equations of modular and quasi-modular forms are interpreted as vector fields on such moduli spaces and they can be calculated from the Gauss-Manin connection of the corresponding universal family of elliptic curves. For the full modular group such a differential equation is calculated and it turns out to be the...

On étudie différentes propriétés d’approximation pour des espaces homogènes $X$ (à stabilisateur fini) de ${\mathrm{GL}}_n$ sur un corps de nombres. On discute également du lien avec le problème de Galois inverse et on établit une formule pour le groupe de Brauer non ramifié de $X$.

La méthode de la descente a été introduite et développée par Colliot-Thélène et Sansuc. Elle permet d’étudier l’arithmétique de certaines variétés rationnelles. Dans ce texte on montre comment il en résulte que pour certaines familles $f:X\to Y$ de variétés rationnelles sur un corps local $k$ de caractéristique nulle le nombre des classes de $R$-équivalence de la fibre $X_y\left(k\right)$ est localement constant quand $y$ varie dans $Y\left(k\right)$.

We consider Thue equations of the form $a{x}^k+b{y}^k=1$, and assuming the truth of the abc-conjecture, we show that almost all locally soluble Thue equations of degree at least three violate the Hasse principle. A similar conclusion holds true for Fermat equations $a{x}^k+b{y}^k+c{z}^k=0$ of degree at least six.

On construit des courbes elliptiques sur $\mathbb{Q}\left(T\right)$ de rang au moins 3, avec un sous-groupe de torsion non trivial. Par spécialisation, des courbes elliptiques de rang 5 et 6 sur $\mathbb{Q}$ sont obtenues.