Embedding $p$-elementary lattices

We investigate the ramification of embeddings of local lattices or quadratic forms over the ring $\mathbb Z_p$ of $p$-adic numbers. It is proved that every primitive embedding decomposes uniquely into an orthogonal sum of minimal indecomposable embeddings, and all such embeddings are constructed for $p$-elementary lattices. Ramification theory enables us to find the number of orbits of representations for forms and, in particular, for numbers by other quadratic forms over $\mathbb Z_p$, and to calculate the local multipliers in the weight formula for representations of a form by a genus of quadratic forms.