Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields

TL;DR

This paper studies the distribution of algebraic numbers of norm one in quadratic fields, showing they are equidistributed on the unit circle or in a logarithmic sense depending on whether the field is imaginary or real.

Contribution

It establishes equidistribution results for algebraic numbers of norm one in quadratic fields under various orderings, extending understanding of their distribution properties.

Findings

Equidistribution on the unit circle for imaginary quadratic fields.

Equidistribution in a logarithmic sense for real quadratic fields.

Results depend on the type of quadratic extension and ordering used.

Abstract

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in N can be written as \alpha/\bar{\alpha} for some \alpha \in O_K, which yields another ordering of \mathcal N given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that N is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that N is equidistributed with respect to norm, under the map \beta \mapsto \log| \beta | \bmod{\log | \epsilon^2 |} where \epsilon is a fundamental unit of O_K.