Entropy of Cohen-Lenstra measures: the $u$-aspect

TL;DR

This work analyzes the entropy properties of Cohen-Lenstra measures on finite abelian $p$-groups as the integral unit rank $u$ varies. It derives that the entropy $H( u^{u}_{CL})$ is finite for all $u>-1$, strictly decreases with $u$, and tends to zero as $u$ grows, identifying the associated groupoid measure as an entropy maximizer within this family. A key contribution is the explicit relative-entropy formula between measures with different unit ranks, expressed in terms of the normalizing constants $F_{u}$ and a $p$-adic zeta function framework. The approach leverages Cohen-Lenstra zeta functions to relate entropy to zeta function derivatives, establishing a principled maximum-entropy perspective for Cohen-Lenstra measures and linking information-theoretic quantities to number-theoretic structures.

Abstract

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.

Theorems & Definitions (8)

• Theorem 1
• Remark 1
• Theorem 2
• Remark 2
• Lemma 1
• proof
• Theorem 3
• proof