Some questions in Diophantine approximation: real and p-adics

2025/05/21 by Prasad, Dipendra
#11J13 #11J95 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2505.15744

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 ⊂ 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''.

Citations

• An analogue of a conjecture of Mazur a question in Diophantine approximation on tori

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