The p-adic closure of a subgroup of rational points on a commutative algebraic group

TL;DR

This paper investigates the p-adic closure of subgroups of rational points on commutative algebraic groups, linking its dimension to Leopoldt's conjecture for certain tori.

Contribution

It formulates a conjecture for the dimension of the p-adic closure and establishes its equivalence to Leopoldt's conjecture in specific cases.

Findings

Proposes a dimension conjecture for p-adic closures of rational point subgroups.

Shows the conjecture's validity for certain tori is equivalent to Leopoldt's conjecture.

Connects algebraic group theory with deep conjectures in number theory.

Abstract

Let G be a commutative algebraic group over Q. Let Gamma be a subgroup of G(Q) contained in the union of the compact subgroups of G(Q_p). We formulate a guess for the dimension of the closure of Gamma in G(Q_p), and show that its correctness for certain tori is equivalent to Leopoldt's conjecture.