Geodesic
Embedding $p$-elementary lattices
Izvestiya. Mathematics , Tome 63 (1999) no. 1, pp. 73-102

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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. 
@article{IM2_1999_63_1_a3,
     author = {V. G. Zhuravlev},
     title = {Embedding $p$-elementary lattices},
     journal = {Izvestiya. Mathematics },
     pages = {73--102},
     publisher = {mathdoc},
     volume = {63},
     number = {1},
     year = {1999},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_1_a3/}
}
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V. G. Zhuravlev. Embedding $p$-elementary lattices. Izvestiya. Mathematics , Tome 63 (1999) no. 1, pp. 73-102. http://geodesic.mathdoc.fr/item/IM2_1999_63_1_a3/