Abstract
Given a fixed quadratic extension K of , we consider the distribution of elements in K of norm one (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in 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-α for some αOK, which yields another ordering of 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 → log|β|(mod log|2|) where is a fundamental unit of OK.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1841-1861 |
| Number of pages | 21 |
| Journal | International Journal of Number Theory |
| Volume | 7 |
| Issue number | 7 |
| DOIs | |
| State | Published - Nov 2011 |
| Externally published | Yes |
Bibliographical note
Funding Information:The authors would like to thank Ted Chinburg whose questions motivated this work, and Ram Murty who supplied the idea behind the proof of Theorem 2.7. The second author was supported in part by the National Science Foundation (DMS-0801243).
Keywords
- Hecke L-function
- Weil height
- equidistribution
- quadratic algebraic numbers
- visible points