<p>The paper discusses certain conjectures relating topological closure and Zariski closure of finitely generated subgroups <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathcal {L}\subset \mathcal {A}(\mathbb {Q})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">L</mi> <mo>⊂</mo> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">Q</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> inside <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\mathcal {A}(\mathbb {Q}_{p})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <msub> <mi mathvariant="double-struck">Q</mi> <mi>p</mi> </msub> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\mathcal {A}(\mathbb {R})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">A</mi> <mo stretchy="false">(</mo> <mi mathvariant="double-struck">R</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> where <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\mathcal {A}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">A</mi> </math></EquationSource> </InlineEquation> is either a torus or an abelian variety defined over <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {Q}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="double-struck">Q</mi> </math></EquationSource> </InlineEquation>, in the spirit of the Leopoldt conjecture for <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, and those due to Mazur [<CitationRef CitationID="CR1">1</CitationRef>, <CitationRef CitationID="CR2">2</CitationRef>], for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$p=\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>. We also consider a question formulated by Nori [<CitationRef CitationID="CR3">3</CitationRef>], of a similar nature for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p<\infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo><</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, but much more general than the Leopoldt conjecture.</p>

Some questions on diophantine approximation, real and p-adics

期刊论文
发表日期：

2026年4月21日

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Dipendra Prasad

摘要

The paper discusses certain conjectures relating topological closure and Zariski closure of finitely generated subgroups $\mathcal {L}\subset \mathcal {A}(\mathbb {Q})$ $L \subset A (Q)$ inside $\mathcal {A}(\mathbb {Q}_{p})$ $A (Q_{p})$ and $\mathcal {A}(\mathbb {R})$ $A (R)$ where $\mathcal {A}$ $A$ is either a torus or an abelian variety defined over $\mathbb {Q}$ $Q$ , in the spirit of the Leopoldt conjecture for $p<\infty $ $p < \infty$ , and those due to Mazur [1, 2], for $p=\infty $ $p = \infty$ . We also consider a question formulated by Nori [3], of a similar nature for $p<\infty $ $p < \infty$ , but much more general than the Leopoldt conjecture.