An isomorphism between the p-adic integers and a ring associated with a tiling of N-space by permutohedra

AbstractThe classical lattice A∗n, whose Voronoi cells tile Euclidean n-space by permutohedra, can be given the generalized balance ternary ring structure GBTn in a natural way as a quotient ring of Z[x]. The ring GBTn can also be considered as the set of all finite sequences s0 s1…sk, with si ∈ GBTn⧸αGBTn for all i, where α is an appropriately chosen element in GBTn. The extended generalized balance ternary (EGBTn) ring consists of all such infinite sequences. A primary goal of this paper is to prove that if 2n+1−1 and n+1 are relatively prime, then EGBTn is isomorphic as a ring to the (2n+1−1)-adic integers

Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki

These notes survey the main ideas, concepts and objects of the work by Shinichi Mochizuki on interuniversal Teichmüller theory [31], which might also be called arithmetic deformation theory, and its application to diophantine geometry. They provide an external perspective which complements the review texts [32] and [33]. Some important developments which preceded [31] are presented in the first section. Several important aspects of arithmetic deformation theory are discussed in the second section. Its main theorem gives an inequality–bound on the size of volume deformation associated to a certain log-theta-lattice. The application to several fundamental conjectures in number theory follows from a further direct computation of the right hand side of the inequality. The third section considers additional related topics, including practical hints on how to study the theory

(non)commutative f-un geometry

Stressing the role of dual coalgebras, we modify the definition of affine schemes over the 'field with one element'. This clarifies the appearance of Habiro-type rings in the commutative case, and, allows a natural noncommutative generalization, the study of representations of discrete groups and their profinite completions being our main motivation

Class Field Theoretic Methods in the Design of Lattice Signal Constellations

Siirretty Doriast