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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2505.15744 |
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| _version_ | 1866915296616382464 |
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| author | Prasad, Dipendra |
| author_facet | Prasad, Dipendra |
| contents | 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''. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_15744 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some questions in Diophantine approximation: real and p-adics Prasad, Dipendra Number Theory 11J95, 11J13 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''. |
| title | Some questions in Diophantine approximation: real and p-adics |
| topic | Number Theory 11J95, 11J13 |
| url | https://arxiv.org/abs/2505.15744 |