On the construction of positive definite indecomposable unimodular even Hermitian forms
Let \(F=\mathbb{Q}(\sqrt{-m})\), \(m>0\) and square free, be an imaginary quadratic field and \(D_m\) its ring of integers. The field \(F\) has a unique non-trivial involution whose fixed field is \(\mathbb{Q}\). Let \(V\) be an \(n\)-ary positive definite Hermitian space over \(F\) equipped with a nondegenerate sesquilinear form \(\phi\) on \(V\) with respect to the involution and \(H\) the Hermitian form associated to \(\phi\). Let \(L\) be a \(D_m\)-lattice on \(V\). The set of all positive definite unimodular \(D_m\)-lattices on \(V\) consists of at most two genera with respect to the unitary group \(U(V,H)=\{\sigma\in\text{GL}_n(V)/H(\sigma x,\sigma y)= H(x,y)\) for all \(x\), \(y\) in \(V\}\). One, which always exists, is called the genus of odd unimodular \(D_m\)-lattices. The other is called the genus of even unimodular \(D_m\)-lattices, and its existence, which depends on the values of \(m\) and \(n\), was proved by \textit{F. Zhu} [Chin. Ann. Math., Ser. A 5, 313-320 (1984; Zbl 0534.10016)]. A \(D_m\)-lattices \(L\) is called indecomposable if \(L\) cannot be expressed as the orthogonal sum of two nonzero lattices. For any given natural numbers \(m\) and \(n\), the author in two earlier papers gave methods for constructing indecomposable positive definite odd unimodular Hermitian \(D_m\)-lattices of rank \(n\). In this paper, the problem of construction of indecomposable positive definite even unimodular Hermitian \(D_m\)-lattices is studied, and the main result obtained is that for any given natural numbers \(n\) and squarefree \(m\) satisfying the condition \(m\equiv 1\pmod 4\) and \(n\equiv 0\pmod 4\) or \(m\equiv 2\pmod 4\) and \(n\equiv 0\pmod 2\), one can construct explicitly indecomposable positive definite even unimodular Hermitian \(D_m\)-lattices of rank \(n\).
indecomposable positive definite even unimodular Hermitian lattices
scientific article; zbMATH DE number 729082