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A characterization of the $$\mathbb Z^ n$$ lattice. (English) Zbl 0855.11032
A vector $$w$$ of an $$n$$-dimensional integral unimodular lattice $$L$$ is called characteristic if $$v\cdot w$$ is congruent to $$v\cdot v\bmod 2$$ for all $$v$$ in $$L$$. Such characteristic vectors always exist and form a certain coset of $$2L$$ called the “shadow” of $$L$$. Using a bit of modular form theory, the author proves that the standard lattice $$\mathbb Z^n$$ is characterized by the property of having no characteristic vector $$w$$ such that $$w\cdot w< n$$. Some related results concerning lattices of type $$\mathbb Z^{n-r} \oplus L_0$$ are announced.

MSC:
 11H06 Lattices and convex bodies (number-theoretic aspects)
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