A characterization of the \(\mathbb Z^ n\) lattice (Q1902225)

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.