Construction of indecomposable positive definite unimodular Hermitian forms (Q1202840)

Let \(F=\mathbb{Q}(\sqrt{-m})\) \((m>0\) and square-free) be an imaginary quadratic field and \(D_ m\) the ring of algebraic integers in \(F\). Let \(V\) be a positive definite Hermitian space over \(F\) equipped with a semi- bilinear form \(\phi\) and the associated Hermitian form \(H\). Let \(L\) be a \(D_ m\)-lattice on \(V\). A lattice \(L\) is said to be an even (resp. odd) \(D_ m\)-lattice if \(H(x)\in 2\mathbb{Z}\) for all \(x\in L\) (resp. if \(H(x)\in\mathbb{Z}\) for all \(x\in L\) but \(H(x)\in\mathbb{Z}-2\mathbb{Z}\) for some \(x\in L)\). The main results of the present paper are (i) a construction of an \(n\)- ary non-decomposable unimodular Hermitian form over \(\mathbb{Q}(\sqrt{-m})\) for \(m\not\equiv 3\pmod 4\) and \(n\geq 2\), (ii) a construction of an \(n\)- ary indecomposable positive definite odd unimodular \(\mathbb{Z}[i]\)-lattice for \(n\geq 6\), and the non-existence of such type of lattice for \(n=2,3,4,5\), (iii) the ``even'' version of (ii), (iv) a construction of indecomposable positive definite odd unimodular lattices over \(\mathbb{Z}[\sqrt 2 i]\) of rank \(n\), except \(n=2\) and 3, and the non-existence of such type of lattice for \(n=2\) and 3, (v) the ``even'' version of (iv) for all \(n=2k\geq 2\).