Norms in central simple algebras

Daniel Goldstein and Murray Schacher

Vol. 292 (2018), No. 2, 373–388

DOI: 10.2140/pjm.2018.292.373

Abstract

Let $A$ be a central simple algebra over a number field $K$. We study the question of which integers of $K$ are reduced norms of integers of $A$. We prove that if $K$ contains an integer that is the reduced norm of an element of $A$ but not the reduced norm of an integer of $A$, then $A$ is a totally definite quaternion algebra over a totally real field (i.e., $A$ fails the Eichler condition).

To Robert Steinberg, a cherished teacher, colleague, and friend

Keywords

central simple algebras, reduced norms, super singular elliptic curves

Mathematical Subject Classification 2010

Primary: 11S45

Secondary: 11S15

Milestones

Received: 22 January 2017

Revised: 7 July 2017

Accepted: 9 July 2017

Published: 17 October 2017