Equidistribution of Algebraic Numbers of Norm One in Quadratic Number Fields

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 α/\barα for some α\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 β\mapsto \log| β| \bmod{\log | ε^2 |} where εis a fundamental unit of O_K.

19 pages, 2 figures, comments welcome.