On Hasse's Unit Index

TL;DR

This paper investigates the distribution of Hasse's unit index in certain CM-fields, revealing that the count of fields with a specific index grows proportionally to X divided by the square root of log X.

Contribution

It provides a new asymptotic formula for the distribution of Hasse's unit index in CM-fields of the form Q(√d, √-1).

Findings

Number of d ≤ X with Q(L)=2 is proportional to X/√(log X).

Distribution pattern of Hasse's unit index in these fields is characterized.

Asymptotic behavior of the index distribution is established.

Abstract

We study the distribution of Hasse's unit index $Q(L)$ for the CM-fields $L = \mathbb{Q}(\sqrt{d}, \sqrt{-1})$ as $d$ varies among positive squarefree integers. We prove that the number of $d\leq X$ such that $Q(L) = 2$ is proportional to $X/\sqrt{\log X}$.