Some questions in Diophantine approximation: real and p-adics
Dipendra Prasad
TL;DR
The notes analyze the topological closures of finitely generated subgroups $L\subset G(\mathbb{Q})$ within $G(\mathbb{R})$ and $G(\mathbb{Q}_p)$, focusing on commutative cases (tori and abelian varieties) and proposing conjectures that such closures are governed by algebraic subgroups and rank data. They connect real and $p$-adic phenomena through Diophantine approximation tools, including the Four and Six Exponential conjectures, and introduce structural-rank based invariants that link $p$-adic closures to logarithmic data via the $p$-adic regulator. The work also discusses Mazur-type conjectures for abelian varieties, Leopoldt-type questions in the $p$-adic setting, and Madhav Nori's questions on the independence of closure dimensions across primes, tying these to broader rank and density phenomena. Finally, it outlines a non-commutative generalization in the spirit of Ratner and Borel density, proposing that orbit closures in quotients by lattices should arise from algebraic subgroups, thereby aiming for a unified Diophantine-approximation framework across algebraic groups.
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''.