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Tensor products of hermitian lattices

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1. Introduction. The properties of euclidean lattices with respect to tensor product have been studied in a series of papers by Kitaoka ([K, Chapter 7], [K1]). A rather natural problem which was investigated there, among others, was the determination of the short vectors in the tensor product L οtimes M of two euclidean lattices L and M. It was shown for instance that up to dimension 43 these short vectors are split, as one might hope.
The present paper deals with a similar question for tensor products of hermitian lattices over imaginary quadratic fields or quaternion division algebras. The main motivation for this work is in connection with modular lattices, as defined by Quebbemann ([Q]), that is to say, even lattices that are similar to their dual. In [B-N] it is shown how tensor product over the ring of integers in an imaginary quadratic field can be used to shift from one level to another (the level of a modular lattice L is the square of the rate of the similarity mapping L* to L), and above all a construction of an 80-dimensional extremal unimodular lattice from a 20-dimensional 7-modular one by tensoring is given. It is thus of some interest to know a priori how short vectors behave under tensor product.
In Section 2 we give the basic definitions and properties concerning hermitian lattices that are needed in the sequel. We establish in Section 3 a splitness criterion for minimal vectors (Corollary 3.4) based on a general lower bound (Proposition 3.2). Finally, Section 4 is devoted to examples; among others, we give an alternate proof of the extremality of Bachoc-Nebe's 80-dimensional lattice, and we give a new construction of the Barnes-Wall lattices.
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