Automorphisms of even unimodular lattices over number fields
Let \( L \) be a lattice over a number field \( \mathbb{K} \), i.e., \( L \) is a finitely generated \( \mathcal{O}_\mathbb{K} \) module (\( \mathcal{O}_\mathbb{K} \) is the ring of algebraic integers in \( \mathbb{K} \)). Moreover, let \( L \) be equipped with a non-degenerate symmetric bilinear form \( B \). Recall that a lattice \( L \) is called even if \( B(x,x)\in 2 \mathcal{O}_\mathbb{K} \) for all \( x\in L \) and a lattice \( L \) is called unimodular, if \( L=\{ x\in \mathbb{K}L \;| \;B(x,y)\in \mathcal{O}_\mathbb{K} \;\text{ for all } y\in L \} \). In this paper the author describes the powers of irreducible polynomials occuring as characteristic polynomials of proper automorphisms of even unimodular lattices over number fields (Theorem A). The main theorem generalizes the result of \textit{E. Bayer-Fluckiger} and \textit{L. Taelman} [``Automorphisms of even unimodular lattices and equivariant Witt groups'', Preprint, \url{arxiv:1708.05540}]. The proof of the main theorem bases on the construction of a trace lattice of suitable hermitian lattice with prescribed local structure. Bayer-Fluckiger and Taelman [loc. cit.] proved the existence of such lattices over non-dyadic fields. The author extended this result for lattice over dyadic fields. The main theorem follows by applying the local-global principle for Brauer groups.
scientific article; zbMATH DE number 6994674