Scholarlink

<p>Let <i>X</i> be a projective variety over a number field <i>K</i> endowed with a height function associated to an ample line bundle on <i>X</i>. Given an algebraic extension <i>F</i> of <i>K</i> with a sufficiently big Northcott number, we can show that there are finitely many cycles in <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(X_{\bar{\mathbb {Q}}}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>X</mi> <mover accent="true"> <mrow> <mi mathvariant="double-struck">Q</mi> </mrow> <mrow> <mo stretchy="false">¯</mo> </mrow> </mover> </msub> </math></EquationSource> </InlineEquation> of bounded degree and small height defined over <i>F</i>. Fields <i>F</i> with the required properties were explicitly constructed in [<CitationRef CitationID="CR18">18</CitationRef>] and [<CitationRef CitationID="CR16">16</CitationRef>], motivating our investigation. We point out explicit specializations to canonical heights associated to abelian varieties and selfmaps of <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {P}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">P</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation>. We apply similar methods to the study of CM-points. As a crucial tool, we introduce a refinement of Northcott’s theorem.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Fields with few small points

  • 期刊论文

  • 发表日期：

    2025年11月21日

查看全文
  • Nuno Hultberg

摘要

Let X be a projective variety over a number field K endowed with a height function associated to an ample line bundle on X. Given an algebraic extension F of K with a sufficiently big Northcott number, we can show that there are finitely many cycles in \(X_{\bar{\mathbb {Q}}}\) X Q ¯ of bounded degree and small height defined over F. Fields F with the required properties were explicitly constructed in [18] and [16], motivating our investigation. We point out explicit specializations to canonical heights associated to abelian varieties and selfmaps of \(\mathbb {P}^n\) P n . We apply similar methods to the study of CM-points. As a crucial tool, we introduce a refinement of Northcott’s theorem.