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

TL;DR

This paper investigates the entropy of Cohen-Lenstra measures on finite abelian p-groups, showing it is positive, finite, decreases with increasing rank, and tends to zero as the rank grows infinitely large.

Contribution

It establishes the entropy bounds and monotonicity of Cohen-Lenstra measures across different ranks, revealing entropy maximization properties.

Findings

Entropy of Cohen-Lenstra measures is positive and finite for all ranks.

Entropy decreases as the rank increases.

Entropy approaches zero as the rank tends to infinity.

Abstract

Let ${\rm \mathbf{H}}(\nu^{u}_{\rm CL})$ 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 $0 < {\rm \mathbf{H}}(\nu^{u}_{\rm CL}) < \infty$ for all $u$, ${\rm \mathbf{H}}(\nu^{u}_{\rm CL})$ is a strictly decreasing function of $u \ge 0$, and ${\rm \mathbf{H}}(\nu^{u}_{\rm CL}) \xrightarrow{u \to \infty} 0$. 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.