Related papers: Small points and free abelian groups

This paper is devoted to the more elementary aspects of the contramodule story, and can be viewed as an extended introduction to the more technically complicated arXiv:1503.05523. Reduced cotorsion abelian groups form an abelian category,…

We explore novel gauge enhancements from abelian to non-simply-connected gauge groups in F-theory. To this end we consider complex structure deformations of elliptic fibrations with a Mordell--Weil group of rank one and identify the…

Let $E$ be an elliptic curve defined over a number field $K$ with fixed non-archimedean absolute value $v$ of split-multiplicative reduction, and let $f$ be an associated Latt\`es map. Baker proved in 2003 that the N\'eron-Tate height on…

Given an epimorphism between topological groups $f:G\to H$, when can a generating set of $H$ be lifted to a generating set of $G$? We show that for connected Lie groups the problem is fundamentally abelian: generators can be lifted if and…

A proof of the Borel completeness of torsion free abelian groups is presented. This proof differs considerably from the approach of Paolini-Shelah.

We show that if a torsion free nilpotent group $G$ has a balanced presentations and Hirsch length $h(G)>3$ then $\beta_1(G;\mathbb{Q})=2$. There is just one such group which is torsion-free and of Hirsch length $h=4$, and none with $h=5$.…

Let $F$ be a finitely generated non-abelian free group and $Q$ a finite quotient. Denote by $L_Q$ the language obtained by adding unary predicates $P_q$, $q\in Q$ to the language of groups. Using a slight generalization of some of the…

Let $m$ be a positive integer and let $E$ be an elliptic curve over $\mathbb{Q}$ with the property that $m\mid#E(\mathbb{F}_p)$ for a density $1$ set of primes $p$. Building upon work of Katz and Harron-Snowden, we study the probability…

To define the notion of a generic property of finite dimensional 2-step nilpotent Lie algebras we use standard correspondence between such Lie algebras and points of an appropriate algebraic variety, where a negligible set is one contained…

We prove that for a countable, commutative ring $R$, the class of countable $R$-modules either has only countably many isomorphism types, or else it is Borel complete. The machinery gives a succinct proof of the Borel completeness of TFAB,…

We deal with some pcf investigations mostly motivated by abelian group theory problems and deal their applications to test problems (we expect reasonably wide applications). We prove almost always the existence of aleph_omega-free abelian…

In this article we study automorphisms of Toeplitz subshifts. Such groups are abelian and any finitely generated torsion subgroup is finite and cyclic. When the complexity is non superlinear, we prove that the automorphism group is, modulo…

The class $A$ of anabelian groups is defined as the collection of finite groups without abelian composition factors. We prove that the commutator word $[x_1,x_2]$ and the power word $x_1^p$ have bounded width in $A$ when $p$ is an odd…

The main emphasis will be on height upper bounds in the algebraic torus G^{n}_{m}. By height we will mean the absolute logarithmic Weil height. Section 3.2 contains a precise definition of this and other more general height functions. The…

Using integral methods we recover and generalize some results by F\'{e}lix, Halperin and Thomas on the growth of the rational homology groups of free loop spaces, and obtain a new family of spaces whose $p$-torsion in homotopy groups grows…

It is shown in a local strongly $F$-regular ring there exits natural number $e_0$ so that if $M$ is any finitely generated maximal Cohen-Macaulay module then the pushforward of $M$ under the $e_0$th iterate of the Frobenius endomorphism…

Let $\alpha$ be a complex number. We show that there is a finite subset $F$ of the ring of the rational integers $\mathbb{Z}$, such that $F\left[ \alpha\right] =\mathbb{Z}\left[ \alpha\right]$, if and only if $\alpha$ is an algebraic number…

The Addition Theorem for the algebraic entropy of group endomorphisms of torsion abelian groups was proved by Dikranjan, Goldsmith, Salce and Zanardo. It was later extended by Shlossberg to torsion nilpotent groups of class 2. As our main…

We construct a local deformation problem for residual Galois representations $\bar{\rho}$ valued in an arbitrary reductive group $\hat{G}$ which we use to develop a variant of the Taylor-Wiles method. Our generalization allows Taylor-Wiles…

Two (strongly) zero-dimensional Lindel\"of topological groups whose product has positive covering dimension are constructed. An example of a Lindel\"of (strongly) zero-dimensional space whose free and free Abelian topological groups are not…