An important problem in the theory of quadratic and hermitian forms is to find good generators for their orthogonal, resp. unitary groups. For example, one wants to show that such a group is generated by symmetries, i.e., isometries that fix elementwise some hyperplane. In the present paper, the authors study this problem in the following setting. Let \(K\) be a local field of characteristic \(0\) with ring of integers \({\mathcal O}_K\), let \(E\) be a two-dimensional étale \(K\)-algebra and \({\mathcal O}_E\) be the integral closure of \({\mathcal O}_K\) in \(E\), i.e., \(E\) is a quadratic field extension of \(K\) with ring of integers \({\mathcal O}_E\), or \(E=K\times K\) and \({\mathcal O}_E={\mathcal O}_K\times {\mathcal O}_K\). A hermitian \({\mathcal O}_E\)-lattice is a finitely generated \({\mathcal O}_E\)-module \(L\) equipped with a hermitian form with respect to the standard involution \(\alpha\mapsto\overline{\alpha}\) on \(E\) which is just the Galois involution in the case of a field extension, and which corresponds to swapping components in the case \(E=K\times K\). We always assume the hermitian form to be nondegenerate. Let \(U(L)\) denote the unitary group of the hermitian lattice \(L\). Two types of elements in \(U(L)\) are of particular interest: symmetries and so-called (rescaled) Eichler isometries that are of a more technical nature.
If \(E/K\) is a non-dyadic field extension, then it was shown by S. Böge [J. Reine Angew. Math. 221, 85–112 (1966; Zbl 0199.09202)] that \(U(L)\) is generated by symmetries. If \(E/K\) is a ramified dyadic field extension, K. Hayakawa [J. Fac. Sci., Univ. Tokyo, Sect. I 15, 1–11 (1968; Zbl 0244.20045)] showed that symmetries also suffice provided \(2\) is a prime element in \(K\) and \(K\neq {\mathbb Q}_2\), but that Eichler isometries will be needed in the case \(K={\mathbb Q}_2\). In the present paper, the authors treat all remaining cases. They show that \(U(L)\) is generated by symmetries in the split case \(E=K\times K\) as well as in the case of an unramified field extension \(E/K\), and that in the case of a general ramified field extension \(E/K\), then \(U(L)\) will always be generated by symmetries and rescaled Eichler isometries. In the Appendix, a refined version of the latter result is shown, namely that in the ramified case, \(U(L)\) can be generated by symmetries if and only if the residue field of \(E\) has more than \(2\) elements. While the split and unramified case can be dealt with by adapting some of the classical techniques developed by Böge, the ramified case requires a new approach involving Jordan splittings of lattices.
The authors point out that their results provide an alternative proof for the computation of the determinant groups of unitary lattices due to M. Kirschmer [Arch. Math. 113, No. 4, 337–347 (2019; Zbl 1461.11059)].