Related papers: Duality for higher local fields after Kato and Suz…

Let $k$ be a global field of characteristic $p>0$. Denote $\Omega_k$ the set of places of $k$ and let $S$ be a non-empty subset of $\Omega_k$. We consider a scheme $\mathscr{X} \rightarrow Spec(\mathcal{O}_S)$ smooth, separated, of finite…

Suppose that $K$ is an infinite field which is large (in the sense of Pop) and whose first order theory is simple. We show that $K$ is {\em bounded}, namely has only finitely many separable extensions of any given finite degree. We also…

We prove new modularity lifting theorems for p-adic Galois representations in situations where the methods of Wiles and Taylor--Wiles do not apply. Previous generalizations of these methods have been restricted to situations where the…

Let A be an excellent integral henselian local noetherian ring, k its residue field of characteristic p>0 and K its fraction field. Using an algebraization technique introduced by the first named author, and the one-dimension case already…

We develop a quantum duality principle for subgroups of a Poisson group and its dual, in two formulations. Namely, in the first one we provide functorial recipes to produce quantum coisotropic subgroups in the dual Poisson group out of any…

Given a number field extension $K/k$ with an intermediate field $K^+$ fixed by a central element of the corresponding Galois group of prime order $p$, we build an algebraic torus over $k$ whose rational points are elements of $K^\times$…

Let K be a number field containing the group of n-th roots of unity and S a set of primes of K including all those dividing n and all real archimedean places. We consider the cup product on the first Galois cohomology group of the maximal…

Using the theory of pro-p groups and relative Poincar\'{e} duality, we define a type of cobordism category well suited to arithmetic topology. We completely classify topological quantum field theories on these two-dimensional versions of…

C*-algebras generalizing Cuntz-Krieger algebras can be associated to hyperbolic homeomorphisms of compact metric spaces. They satisfy a non-commutative form of Spanier-Whitehead duality with respect to K-theory. We prove this for the case…

Let F be a finite field and l a prime not equal to the characteristic of F. Let K be the function field of a surface over F. Assume that K contains a primitive lth root of unity. In the paper we prove a certain local-global principle for…

For Shimura varieties of Hodge type, we show that there are natural isomorphisms between locally analytic complete cohomology groups and cohomology groups for flag varieties with coefficient which is given by their perfectoid covers. This…

Over any smooth algebraic variety over a $p$-adic local field $k$, we construct the de Rham comparison isomorphisms for the \'etale cohomology with partial compact support of de Rham $\mathbb Z_p$-local systems, and show that they are…

Let $X$ be a smooth projective geometrically connected variety defined over a number field $K$. We prove that the geometric \'etale cohomology of $X$ with $\mathbb{Q}/\mathbb{Z}$-coefficients has finitely many classes invariant under the…

For an arbitrary field $K$ and $K$-variety $V$, we introduce the \'etale-open topology on the set $V(K)$ of $K$-points of $V$. This topology agrees with the Zariski topology, Euclidean topology, or valuation topology when $K$ is separably…

Let $K$ be a field complete with respect to a discrete valuation $v$ of residue characteristic $p$. Let $f(z) \in K[z]$ be a separable polynomial of the form $z^\ell-c.$ Given $a \in K$, we examine the Galois groups and ramification groups…

We review the recently constructed `double field theory' which introduces in addition to the conventional coordinates associated to momentum modes coordinates associated to winding modes. Thereby, T-duality becomes a global symmetry of the…

We give a concrete characterization of the rational conjugacy classes of maximal tori in groups of type G2, focusing on the case of number fields and p-adic fields. In the same context we characterize the rational conjugacy classes of A2…

We study the geometry and the cohomology of the tamely ramified cover of Drinfeld's $p$-adic symmetric space. For this tame level, we prove, in a purely local way, most of a conjecture of Harris on the form of the $\ell$-adic cohomologies,…

Generalizing Duality Theorem of V. V. Fedorchuk, we prove Stone-type duality theorems for the following four categories: all of them have as objects the locally compact Hausdorff spaces, and their morphisms are, respectively, the continuous…

Let $K$ be the function field of a $p$-adic curve, $G$ a semisimple simply connected group over $K$ and $X$ a $G$-torsor over $K$. A conjecture of Colliot-Th\'el\`ene, Parimala and Suresh predicts that if for every discrete valuation $v$ of…