Geodesic
Embedding $p$-elementary lattices
Izvestiya. Mathematics, Tome 63 (1999) no. 1, pp. 73-102 
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/
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     author = {V. G. Zhuravlev},
     title = {Embedding $p$-elementary lattices},
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     url = {http://geodesic.mathdoc.fr/item/IM2_1999_63_1_a3/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

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.

[1] Artin E., Geometricheskaya algebra, Nauka, M., 1969 | MR | Zbl

[2] Zhuravlev V. G., “Predstavlenie formy rodom kvadratichnykh form”, Algebra i analiz, 8:1 (1996), 21–112 | MR

[3] Kassels Dzh., Ratsionalnye kvadratichnye formy, Mir, M., 1982 | MR

[4] Konvei Dzh., Sloen N., Upakovki sharov, reshetki i gruppy, Mir, M., 1990

[5] Nikulin V. V., “Tselochislennye simmetricheskie bilineinye formy i nekotorye ikh geometricheskie prilozheniya”, Izv. AN SSSR. Ser. matem., 43:1 (1979), 111–177 | MR | Zbl

[6] Siegel C. L., “Über die analytische Theorie der quadratischen Formen”, Ann. of Math., 36:2 (1935), 527–606 | DOI | MR

[7] Zhuravlev V. G., “Orbity predstavlenii chisel lokalnymi kvadratichnymi formami”, Tr. MIRAN, 218 (1997), 151–164 | MR | Zbl