Vol. 292, No. 2, 2018

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Norms in central simple algebras

Daniel Goldstein and Murray Schacher

Vol. 292 (2018), No. 2, 373–388

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

central simple algebras, reduced norms, super singular elliptic curves
Mathematical Subject Classification 2010
Primary: 11S45
Secondary: 11S15
Received: 22 January 2017
Revised: 7 July 2017
Accepted: 9 July 2017
Published: 17 October 2017
Daniel Goldstein
Center for Communications Research
San Diego, CA
United States
Murray Schacher
Center for Communications Research
San Diego, CA
United States