Lens spaces isospectral on forms but not on functions

TL;DR

This paper corrects a previous error regarding the spectral properties of lens spaces, clarifying that certain formulas only hold for prime cases and providing detailed recalculations for composite cases.

Contribution

It identifies and corrects an error in earlier work on lens spaces, specifically regarding spectral formulas for composite q, and offers revised calculations for these cases.

Findings

Formulas valid for prime q are not valid for composite q.

Detailed recalculations confirm the validity of previous results for prime q.

Corrected formulas account for complexities in composite q cases.

Abstract

This paper means to correct an error by the authors for the composite $q$ case in the paper "Lens Spaces, Isospectral on Forms but not on Functions", published in LMS J. Comput. Math.} 9 (2006), 270-286. All calculations and examples presented in \cite{GM} for prime $q$ remain valid, and we include detailed calculations below justifying this. Our original mistake was to conclude that Formula (3.11) \cite[p. 399]{Ikeda} remained true for all $q$ when in fact it is only valid if $q$ is prime. This means formulas (3) and (4) in \cite{GM} must be reworked to account for complications when $q$ is composite.