Generation of Local Unitary Groups

TL;DR

This paper investigates the generation of local unitary groups over non-Archimedean fields, demonstrating that they are generated by symmetries and rescaled Eichler isometries, with specific conditions where symmetries alone suffice.

Contribution

It establishes the generators of unitary groups over two-dimensional étale algebras, extending understanding of their structure in non-Archimedean local fields.

Findings

Unitary groups are generated by symmetries and rescaled Eichler isometries.

Symmetries alone generate the group unless certain ramified dyadic conditions occur.

The paper provides explicit conditions for when symmetries suffice as generators.

Abstract

Let $E$ be a two-dimensional \'etale algebra over a non-Archimedean local field $K$ of characteristic zero. We show that the unitary group of a non-degenerate hermitian lattice over $E$ is generated by symmetries and rescaled Eichler isometries. In the appendix we show that unless $E/K$ is a ramified dyadic field extension and the residue field has two elements, symmetries suffice.