Let G be a finite group. The Cohen-Lenstra-Martinet Heuristics give a prediction of the distribution of <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\operatorname {Cl}_K[p^\infty ]$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mo>Cl</mo> <mi>K</mi> </msub> <mrow> <mo stretchy="false">[</mo> <msup> <mi>p</mi> <mi>∞</mi> </msup> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> when K runs over G-fields and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$p\not \mid |G|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∤</mo> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>. In this paper, we prove several results on the distribution of ideal class groups for some <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p\mid |G|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∣</mo> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>, and show that the behaviour is qualitatively different than what is predicted by the heuristics when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p\not \mid |G|$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∤</mo> <mo stretchy="false">|</mo> <mi>G</mi> <mo stretchy="false">|</mo> </mrow> </math></EquationSource> </InlineEquation>. We do this by using genus theory and the invariant part of the class group to investigate the algebraic structure of the class group, and show the infinite moments of class groups for abelian fields and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$D_4$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>D</mi> <mn>4</mn> </msub> </math></EquationSource> </InlineEquation>-fields.

Distribution of the bad part of class groups

期刊论文
发表日期：

2026年6月25日

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Weitong Wang

摘要

Let G be a finite group. The Cohen-Lenstra-Martinet Heuristics give a prediction of the distribution of $\operatorname {Cl}_K[p^\infty ]$ ${Cl}_{K} [p^{\infty}]$ when K runs over G-fields and $p\not \mid |G|$ $p ∤ | G |$ . In this paper, we prove several results on the distribution of ideal class groups for some $p\mid |G|$ $p ∣ | G |$ , and show that the behaviour is qualitatively different than what is predicted by the heuristics when $p\not \mid |G|$ $p ∤ | G |$ . We do this by using genus theory and the invariant part of the class group to investigate the algebraic structure of the class group, and show the infinite moments of class groups for abelian fields and $D_4$ $D_{4}$ -fields.