Signature Ranks of Units in Cyclotomic Extensions of Abelian Number Fields

TL;DR

This paper investigates the behavior of signature ranks of units in cyclotomic fields, showing they tend to infinity or stay close to maximum in certain cases, with some results depending on unproven hypotheses.

Contribution

It establishes new bounds and asymptotic behaviors for the signature ranks of units in cyclotomic extensions, including both unconditional and conditional results.

Findings

Signature rank of units in ${ m Q}(zeta_m)^+$ tends to infinity with m.

Signature rank differs from maximum by a bounded amount in certain abelian subfields.

Conditional results show possible arbitrarily large differences in general cyclotomic fields.

Abstract

We prove the rank of the group of signatures of the circular units (hence also the full group of units) of ${\mathbb Q}( \zeta_m)^+$ 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 ${\mathbb Q}( \zeta_{p^n})^+$ differs from $\varphi(p^n)/2$ 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.