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On primitively 2-universal quadratic forms. (English. Russian original) Zbl 1263.11048
St. Petersbg. Math. J. 23, No. 3, 435-458 (2012); translation from Algebra Anal. 23, No. 3, 31-62 (2011).
A positive definite integral quadratic form or corresponding quadratic $$\mathbb Z$$-lattice is said to be (primitively) $$m$$-universal if it (primitively) represents all positive definite integral quadratic forms of rank $$m$$. The well-known Conway-Schneeberger $$15$$-Theorem gives an effective criterion to determine whether a positive definite classically integral quadratic form is $$1$$-universal. Various generalizations of this result have been investigated. In particular, an analogous $$2$$-universality criterion has been obtained by B. M. Kim, M.-H. Kim and B.-K. Oh [Contemp. Math. 249, 51–62 (1999; Zbl 0955.11011)], and a partial result for primitive $$1$$-universality appears in a previous paper by the present author [Lith. Math. J. 50, No. 2, 140–163 (2010; Zbl 1247.11047)].
The paper under review contains a detailed study of the binary quadratic forms that are primitively represented by a quadratic form $$Q$$ of rank at least $$5$$ over each of the rings $$\mathbb Z_p$$ of $$p$$-adic integers, under restrictions on the structure of $$Q$$. These local results lead to a criterion for a positive definite, classically integral quadratic form of rank at least $$5$$ having odd square-free determinant to be locally primitively $$2$$-universal. A form of class number $$1$$ satisfying this criterion is therefore primitively $$2$$-universal.
##### MSC:
 1.1e+09 Quadratic forms over local rings and fields 1.1e+13 Quadratic forms over global rings and fields
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##### References:
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