On k-universal quadratic lattices over unramified dyadic local fields

Let k be a positive integer and let F be a finite unramified extension of Q with ring of integers O. An integral (resp. classic) quadratic form over O is called k-universal (resp. classic k-universal) if it represents all integral (resp. classic) quadratic forms of dimension k. In this paper, we provide a complete classification of k-universal and classic k-universal quadratic forms over O. The results are stated in terms of the fundamental invariants associated to Jordan splittings of quadratic lattices.