Densities of primes and realization of local extensions

In this paper we introduce new densities on the set of primes of a number field. If $K/K_0$ is a Galois extension of number fields, we associate to any element $x \in {\rm Gal}_{K/K_0}$ a density $\delta_{K/K_0,x}$ on primes of $K$. In particular, the density associated to $x = 1$ is the usual Dirichlet density on $K$. After establishing some properties of these densities, we use them to show that the maximal solvable extension of a number field unramified outside an almost Chebotarev set realize the maximal local extension at each prime lying outside this set.Comment: 19 pages; the main theorem appears now in a slightly generalized version compared to v1. A section recalling stable sets is added. Several small corrections are mad

Reconstructing decomposition subgroups in arithmetic fundamental groups using regulators

Our main goal in the present article is to explain how one can reconstruct the decomposition subgroups and norms of points on an arithmetic curve inside its fundamental group if the following data are given: the fundamental group, a part of the cyclotomic character and the family of the regulators of the fields corresponding to the generic points of all \'{e}tale covers of the given curve. The approach is inspired by that of Tamagawa for curves over finite fields but uses Tsfasman-Vl\u{a}du\c{t} theorem instead of Lefschetz trace formula. To the authors' knowledge, this is a new technique in the anabelian geometry of arithmetic curves. It is conditional and depends on still unknown properties of arithmetic fundamental groups. We also give a new approach via Iwasawa theory to the local correspondence at the boundary.Comment: 17 pages; this is a second version of the article. Certain significant changes were made. The two most important are: a section on local correspondence on the boundary via Greenberg conjecture in Iwasawa theory was added and, following a suggestion of a referee, the conditions in the main theorem were weakene

On a generalization of the Neukirch-Uchida theorem

In this paper we generalize a part of Neukirch-Uchida theorem for number fields from the birational case to the case of curves \Spec \caO_{K,S} with $S$ a stable set of primes of a number field $K$. In particular, such sets can have arbitrarily small (positive) Dirichlet density. The proof consists of two parts: first one establishes a local correspondence at the boundary $S$, which works as in the original proof of Neukirch. But then, in contrast to Neukirchs proof, a direct conclusion via Chebotarev density theorem is not possible, since stable sets are in general too small, and one has to use further arguments.Comment: 13 page

On a mechanical lens

In this paper, we consider the dynamics of a heavy homogeneous ball moving under the influence of dry friction on a fixed horizontal plane. We assume the ball to slide without rolling. We demonstrate that the plane may be divided into two regions, each characterized by a distinct coefficient of friction, so that balls with equal initial linear and angular velocity will converge upon the same point from different initial locations along a certain segment. We construct the boundary between the two regions explicitly and discuss possible applications to real physical systems