Norms in Central Simple Algebras

TL;DR

This paper investigates the properties and distribution of outliers in central simple algebras over number fields, and concludes with a structure theorem related to products of supersingular elliptic curves over finite fields.

Contribution

It introduces a detailed study of outliers in central simple algebras and presents a structure theorem for products of supersingular elliptic curves over GF(p).

Findings

Characterization of outliers in central simple algebras

Distribution patterns of outliers over number fields

A structure theorem for supersingular elliptic curve products

Abstract

Let A be a central simple algebra central over a number field K whose ring of integers is R. An outlier is an element r of R so that: r is a reduced norm of an element of A, but not the norm of an algebraic integer in A. We study properties and distribution of outliers. We end with a structure theorem about products of super singular elliptic curves over GF(p).