Construction of indecomposable definite Hermitian forms (Q5917694)

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 \(F\). The field \(F\) has a unique non-trivial involution (the complex conjugation). Let \(V\) be an \(n\)-dimensional non-degenerate Hermitian space over \(F\) equipped with a sesquilinear form \(\varphi\) on \(V\) with respect to the involution and Hermitian form \(H\) associated with \(\varphi\). Let \(L\) be a \(D_m\)-lattice on \(V\), i.e. a finitely generated \(D_m\)-module in \(V\) and \(FL= V\). \(L\) is called an even lattice if \(H(x)\in 2\mathbb{Z}\) for any \(x\in L\). Otherwise, it is called odd. \(L\) is indecomposable if it cannot be expressed as an orthogonal sum of two non-zero sublattices. The minimum of a Hermitian \(D_m\)-lattice \(L\) with respect to its associated Hermitian form \(h\) is \[ \min L= \min_h L= \{\min |h(x)|\mid 0\neq x\in L\}. \] In the present paper the author proves Theorem 2: For any given natural numbers \(n> 1\) and square-free \(m\), we can explicitly construct an odd indecomposable positive definite Hermitian \(D_m\)-lattice \(L\) of rank \(n\) if \(\min L> 1\). This result generalizes the result previously obtained by the same author in [Sci. China, Ser. A 35, 917-930 (1992; Zbl 0769.11021) and Acta Math. Sin., New Ser. 10, 113-120 (1994; Zbl 0807.11023)]. In those papers the results are obtained for \(m= 1, 2, 3, 7, 11\) and 15. The reviewer would like to make a remark that the paper does not touch the more difficult problem of constructing odd unimodular lattice having the assigned \(\min L> 1\).