Some questions in Diophantine approximation: real and p-adics

TL;DR

This paper investigates the topological closure of finitely generated subgroups in algebraic groups over rationals, real numbers, and p-adic fields, focusing on tori, abelian varieties, and semisimple groups, with insights into Diophantine approximation.

Contribution

It extends the understanding of closures of algebraic points in various algebraic groups, especially for tori and abelian varieties, and explores variants for semisimple groups.

Findings

Closure of finitely generated subgroups often forms open normal subgroups.

Results align with the principle of Occam's razor for algebraic groups.

Provides new insights into topological properties in Diophantine approximation contexts.

Abstract

The Weak approximation theorem describes the closure of $G(Q)$ inside $G(Q_p)$ as well as inside $G(R)$ for $G$ an algebraic group over $Q$; the closure is always an open normal subgroup with finite abelian quotient, and is well understood in a certain sense even if precise results are not always available (such as for tori!). In this paper, for a finitely generated subgroup $ L \subset G(Q)$ we consider the topological closure of $ L$ inside $G(Q_p)$ and $G(R)$. The paper is written mostly for $G$ a torus or an abelian variety, but eventually considers a variant of the question for $G$ a semisimple group. The paper is written with the wishful thinking that when dealing with questions on topological closure of algebraic points in an algebraic group defined over a number field, the simplest answers hold, a well-known principle known as ``Occum's razor''.