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<p>Let <i>K</i> be an imaginary quadratic field in which the odd prime <i>p</i> does not split. When the <i>p</i>-part of the class group of <i>K</i> is cyclic, the main result of this paper is to describe the possible structures for the <i>p</i>-part of the class group of the first level of the cyclotomic <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb Z_p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">Z</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation>-extension of <i>K</i>. As a consequence, we can also show the compatibility of the heuristics of Cohen–Lenstra–Martinet for class groups with the heuristics of Ellenberg–Jain–Venkatesh for how often the cyclotomic Iwasawa invariant <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation> equals 1.</p>

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The first level of \(\mathbb Z_p\)-extensions and compatibility of heuristics

  • 期刊论文

  • 发表日期：

    2025年9月8日

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  • Debanjana Kundu,
  • Lawrence C. Washington

摘要

Let K be an imaginary quadratic field in which the odd prime p does not split. When the p-part of the class group of K is cyclic, the main result of this paper is to describe the possible structures for the p-part of the class group of the first level of the cyclotomic \(\mathbb Z_p\) Z p -extension of K. As a consequence, we can also show the compatibility of the heuristics of Cohen–Lenstra–Martinet for class groups with the heuristics of Ellenberg–Jain–Venkatesh for how often the cyclotomic Iwasawa invariant \(\lambda \) λ equals 1.