Automorphisms with eigenvalues in $S^1$ of a $\mathbb Z$-lattice with cyclic finite monodromy
Volume 192 / 2020
Acta Arithmetica 192 (2020), 1-30
MSC: 15B36, 13F20, 15A27, 32S40.
DOI: 10.4064/aa180125-7-12
Published online: 7 October 2019
Abstract
For any finite set $M$ of positive integers, there is up to isomorphism a unique $\mathbb Z $-lattice $H_M$ with a cyclic automorphism $h_M:H_M\to H_M$ whose eigenvalues are roots of unity with orders in $M$ and have multiplicity 1. The paper studies the automorphisms of the pair $(H_M,h_M)$ which have eigenvalues in $S^1$. The main result gives necessary and sufficient conditions on $M$ for the only such automorphisms to be $\pm h_M^k$, $k\in \mathbb Z $. The proof uses resultants and cyclotomic polynomials. It is elementary, but involved. Special cases of the main result have already been applied to the study of automorphisms of Milnor lattices of isolated hypersurface singularities.
