This paper contains a complete proof of Fukaya and Kato’s $ϵ$-isomorphism conjecture for invertible $\Lambda$-modules (the case of $V=V_0\left(r\right)$, where $V_0$ is unramified of dimension $1$). Our results rely heavily on Kato’s proof, in an unpublished set of lecture notes, of (commutative) $ϵ$-isomorphisms for one-dimensional representations of $G_{{\mathbb{Q}}_p}$, but apart from fixing some sign ambiguities in Kato’s notes, we use the theory of $\left(\varphi ,\Gamma \right)$-modules instead of syntomic cohomology. Also, for the convenience of the reader we give a slight modification or rather reformulation of it in the language of Fukuya and Kato and extend it to the (slightly noncommutative) semiglobal setting. Finally we discuss some direct applications concerning the Iwasawa theory of CM elliptic curves, in particular the local Iwasawa Main Conjecture for CM elliptic curves $E$ over the extension of ${\mathbb{Q}}_p$ which trivialises the $p$-power division points $E\left(p\right)$ of $E$. In this sense the paper is complimentary to our work with Bouganis (Asian J. Math. 14:3 (2010), 385–416) on noncommutative Main Conjectures for CM elliptic curves.

In this paper we offer a combinatorial view on the adjunction theory of toric varieties. Inspired by classical adjunction theory of polarized algebraic varieties we explore two convex-geometric notions: the $\mathbb{Q}$-codegree and the nef value of a rational polytope $P$. We prove a structure theorem for lattice polytopes $P$ with large $\mathbb{Q}$-codegree. For this, we define the adjoint polytope $P^{\left(s\right)}$ as the set of those points in $P$ whose lattice distance to every facet of $P$ is at least $s$. It follows from our main result that if $P^{\left(s\right)}$ is empty for some $s<2∕\left(dimP+2\right)$, then the lattice polytope $P$ has lattice width one. This has consequences in Ehrhart theory and on polarized toric varieties with dual defect. Moreover, we illustrate how classification results in adjunction theory can be translated into new classification results for lattice polytopes.

Adams, Vogan, and D. Prasad have given conjectural formulas for the behavior of the local Langlands correspondence with respect to taking the contragredient of a representation. We prove these conjectures for tempered representations of quasisplit real $K$-groups and quasisplit $p$-adic classical groups (in the sense of Arthur). We also prove a formula for the behavior of the local Langlands correspondence for these groups with respect to changes of the Whittaker data.

We consider projective bundles (or Brauer–Severi varieties) over an abelian variety which are homogeneous, that is, invariant under translation. We describe the structure of these bundles in terms of projective representations of commutative group schemes; the irreducible bundles correspond to Heisenberg groups and their standard representations. Our results extend those of Mukai on semihomogeneous vector bundles, and yield a geometric view of the Brauer group of abelian varieties.

We prove finiteness results for Tate–Shafarevich groups in degree $2$ associated with $1$-motives. We give a number-theoretic interpretation of these groups, relate them to Leopoldt’s conjecture, and present an example of a semiabelian variety with an infinite Tate–Shafarevich group in degree $2$. We also establish an arithmetic duality theorem for $1$-motives over number fields, which complements earlier results of Harari and Szamuely.

The theory of $\left({\phi }_q,\Gamma \right)$-modules is a generalization of Fontaine’s theory of $\left(\phi ,\Gamma \right)$-modules, which classifies $G_F$-representations on ${\mathcal{O}}_F$-modules and $F$-vector spaces for any finite extension $F$ of ${\mathbb{Q}}_p$. In this paper following Colmez’s method we classify triangulable ${\mathcal{O}}_F$-analytic $\left({\phi }_q,\Gamma \right)$-modules of rank $2$. In the process we establish two kinds of cohomology theories for ${\mathcal{O}}_F$-analytic $\left({\phi }_q,\Gamma \right)$-modules. Using them, we show that if $D$ is an étale ${\mathcal{O}}_F$-analytic $\left({\phi }_q,\Gamma \right)$-module such that $D^{{\phi }_q=1,\Gamma =1}=0$ (i.e., $V^{G_F}=0$, where $V$ is the Galois representation attached to $D$), then any overconvergent extension of the trivial representation of $G_F$ by $V$ is ${\mathcal{O}}_F$-analytic. In particular, contrary to the case of $F={\mathbb{Q}}_p$, there are representations of $G_F$ that are not overconvergent.