# 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.

## Key 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.

---
Source: https://tomesphere.com/paper/1906.07787