  T. Ikeda and H. Katsurada have developed the theory of the Gross-Keating
invariant of a quadratic form in their recent papers [IK1] and [IK2]. In
particular, they prove that the local factor of the Fourier coefficients of the
Siegel-Eisenstein series is completely determined by the Gross-Keating
invariant with extra datum, called the extended GK datum, in [IK2]. On the
other hand, such local factor is a special case of the local densities for a
pair of two quadratic forms. Thus we propose a general question if the local
density can be determined by certain series of the Gross-Keating invariants and
the extended GK datums.
  In this paper, we prove that the answer to this question is affirmative, for
the local density of a single quadratic form defined over an unramified finite
extension of $\mathbb{Z}_2$. In the appendix, T. Ikeda and H. Katsurada compute
the local density formula of a single binary quadratic form defined over any
finite extension of $\mathbb{Z}_2$.
