We show that the Galois cohomology groups of $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ can be computed via the generalization of Herr’s complex to multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules. Using Tate duality and a pairing for multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules we extend this to analogues of the Iwasawa cohomology. We show that all $p$-adic representations of a direct power of $\operatorname{Gal}(\overline{\mathbb{Q}_{p}}/\mathbb{Q}_{p})$ are overconvergent and, moreover, passing to overconvergent multivariable $(\unicode[STIX]{x1D711},\unicode[STIX]{x1D6E4})$-modules is an equivalence of categories. Finally, we prove that the overconvergent Herr complex also computes the Galois cohomology groups.

Soit $F$ un corps local non archimédien de caractéristique ${\geqslant}0$, et soit $G=GL(N,F)$, $N\geqslant 1$. Un élément $\unicode[STIX]{x1D6FE}\in G$ est dit quasi régulier si le centralisateur de $\unicode[STIX]{x1D6FE}$ dans $M(N,F)$ est un produit d’extensions de $F$. Soit $G_{\text{qr}}$ l’ensemble des éléments quasi réguliers de $G$. Pour $\unicode[STIX]{x1D6FE}\in G_{\text{qr}}$, on note $O_{\unicode[STIX]{x1D6FE}}$ l’intégrale orbitale ordinaire sur $G$ associée à $\unicode[STIX]{x1D6FE}$. On remplace ici le discriminant de Weyl $|D_{G}|$ par un facteur de normalisation $\unicode[STIX]{x1D702}_{G}:G_{\text{qr}}\rightarrow \mathbb{R}_{{>}0}$ permettant d’obtenir les mêmes résultats que ceux prouvés par Harish-Chandra en caractéristique nulle: pour $f\in C_{\text{c}}^{\infty }(G)$, l’intégrale orbitale normalisée $I^{G}(\unicode[STIX]{x1D6FE},f)=\unicode[STIX]{x1D702}_{G}^{\frac{1}{2}}(\unicode[STIX]{x1D6FE})O_{\unicode[STIX]{x1D6FE}}(f)$ est bornée sur $G$, et pour $\unicode[STIX]{x1D716}>0$ tel que $N(N-1)\unicode[STIX]{x1D716}<1$, la fonction $\unicode[STIX]{x1D702}_{G}^{-\frac{1}{2}-\unicode[STIX]{x1D716}}$ est localement intégrable sur $G$.

The flow past an obstacle is a fundamental object in fluid mechanics. In 1967 Finn and Smith proved the unique existence of stationary solutions, called the physically reasonable solutions, to the Navier–Stokes equations in a two-dimensional exterior domain modeling this type of flows when the Reynolds number is sufficiently small. The asymptotic behavior of their solution at spatial infinity has been studied in detail and well understood by now, while its stability has remained open due to the difficulty specific to the two-dimensionality. In this paper, we prove that the physically reasonable solutions constructed by Finn and Smith are asymptotically stable with respect to small and well-localized initial perturbations.

We prove that every separable Banach space containing an isomorphic copy of $\ell _{1}$ can be equivalently renormed so that the new bidual norm is octahedral. This answers, in the separable case, a question in Godefroy [Metric characterization of first Baire class linear forms and octahedral norms, Studia Math. 95 (1989), 1–15]. As a direct consequence, we obtain that every dual Banach space, with a separable predual and failing to be strongly regular, can be equivalently renormed with a dual norm to satisfy the strong diameter two property.

We prove constancy of Newton polygons of all convergent $F$-isocrystals on Abelian varieties over finite fields. Applying the constancy, we prove the isotriviality of proper smooth families of curves over Abelian varieties. More generally, we prove the isotriviality over projective smooth varieties on which any convergent $F$-isocrystal has constant Newton polygons.

We determine the irregular Hodge filtration, as introduced by Sabbah, for the purely irregular hypergeometric ${\mathcal{D}}$-modules. We obtain, in particular, a formula for the irregular Hodge numbers of these systems. We use the reduction of hypergeometric systems from GKZ-systems as well as comparison results to Gauss–Manin systems of Laurent polynomials via Fourier–Laplace and Radon transformations.

Let $K$ be an algebraically closed field of prime characteristic $p$, let $X$ be a semiabelian variety defined over a finite subfield of $K$, let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$, let $V\subset X$ be a subvariety defined over $K$, and let $\unicode[STIX]{x1D6FC}\in X(K)$. The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$-sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$, some rational numbers $c_{i}$ and some non-negative integers $k_{i}$. We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$, or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$. We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.

We establish the first global results for groups definable in tame expansions of o-minimal structures. Let ${\mathcal{N}}$ be an expansion of an o-minimal structure ${\mathcal{M}}$ that admits a good dimension theory. The setting includes dense pairs of o-minimal structures, expansions of ${\mathcal{M}}$ by a Mann group, or by a subgroup of an elliptic curve, or a dense independent set. We prove: (1) a Weil’s group chunk theorem that guarantees a definable group with an o-minimal group chunk is o-minimal, (2) a full characterization of those definable groups that are o-minimal as those groups that have maximal dimension; namely, their dimension equals the dimension of their topological closure, (3) as an application, if ${\mathcal{N}}$ expands ${\mathcal{M}}$ by a dense independent set, then every definable group is o-minimal.