Entropy of Cohen-Lenstra measures: the $u$-aspect
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909457617780736 |
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| author | Siad, Artane |
| author_facet | Siad, Artane |
| contents | Let ${\rm \mathbf{H}}(ν^{u}_{\rm CL})$ be the entropy of the Cohen-Lenstra measure on finite abelian $p$-groups associated to an integral unit-rank $0 \le u \in \mathbb{N}$. In this note, we show that $0 < {\rm \mathbf{H}}(ν^{u}_{\rm CL}) < \infty$ for all $u$, ${\rm \mathbf{H}}(ν^{u}_{\rm CL})$ is a strictly decreasing function of $u \ge 0$, and ${\rm \mathbf{H}}(ν^{u}_{\rm CL}) \xrightarrow{u \to \infty} 0$. In particular, this shows that the groupoid measure is an entropy maximizer in the class of Cohen-Lenstra measures of varying integral unit-rank on finite abelian $p$-groups. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_09222 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Entropy of Cohen-Lenstra measures: the $u$-aspect Siad, Artane Number Theory Let ${\rm \mathbf{H}}(ν^{u}_{\rm CL})$ be the entropy of the Cohen-Lenstra measure on finite abelian $p$-groups associated to an integral unit-rank $0 \le u \in \mathbb{N}$. In this note, we show that $0 < {\rm \mathbf{H}}(ν^{u}_{\rm CL}) < \infty$ for all $u$, ${\rm \mathbf{H}}(ν^{u}_{\rm CL})$ is a strictly decreasing function of $u \ge 0$, and ${\rm \mathbf{H}}(ν^{u}_{\rm CL}) \xrightarrow{u \to \infty} 0$. In particular, this shows that the groupoid measure is an entropy maximizer in the class of Cohen-Lenstra measures of varying integral unit-rank on finite abelian $p$-groups. |
| title | Entropy of Cohen-Lenstra measures: the $u$-aspect |
| topic | Number Theory |
| url | https://arxiv.org/abs/2501.09222 |