Signature ranks of units in cyclotomic extensions of abelian number fields

David S. Dummit, Evan P. Dummit and Hershy Kisilevsky

Vol. 298 (2019), No. 2, 285–298

DOI: 10.2140/pjm.2019.298.285

Abstract

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of $\mathbb{Q}{\left({\zeta }_m\right)}^{+}$ tends to infinity with $m$. We also show the signature rank of the units differs from its maximum possible value by a bounded amount for all the real subfields of the composite of an abelian field with finitely many odd prime-power cyclotomic towers. In particular, for any prime $p$ the signature rank of the units of $\mathbb{Q}{\left({\zeta }_{p^n}\right)}^{+}$ differs from $\varphi \left(p^n\right)∕2$ by an amount that is bounded independent of $n$. Finally, we show conditionally that for general cyclotomic fields the unit signature rank can differ from its maximum possible value by an arbitrarily large amount.

Keywords

signature rank of units, cyclotomic fields, abelian extensions, circular units

Mathematical Subject Classification 2010

Primary: 11R18

Secondary: 11R27

Milestones

Received: 31 May 2018

Revised: 20 August 2018

Accepted: 21 August 2018

Published: 8 March 2019