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<p>We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11139_2025_1108_Article_IEq1.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(100\%\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>100</mn> <mo>%</mo> </mrow> </math></EquationSource> </InlineEquation> of curves in these families have, in a precise sense, many such primes.</p>

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Normal distribution of bad reduction

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年5月22日

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  • Robert J. Lemke Oliver,
  • Daniel Loughran,
  • Ari Shnidman

摘要

We prove normal distribution laws for primes of bad semistable reduction in families of curves. As a consequence, we deduce that when ordered by height, \(100\%\) 100 % of curves in these families have, in a precise sense, many such primes.