Construction of a Hermitian lattice without a basis of minimal vectors
It is known that infinitely many imaginary quadratic fields allow Hermitian lattices which are generated by minimal vectors but have no basis of minimal vectors. \textit{B. M. Kim} and the author found in [Proc. Am. Math. Soc. 136, No. 9, 3041--3044 (2008; Zbl 1165.11040)] two types of binary free Hermitian lattices which are generated by their minimal vectors, but are not generated by any 3 minimal vectors. It was a variant of Conway-Sloane's answer. However it has not been analyzed how the Hermitian lattices were constructed yet. In this article, the authors show the structure of those lattices and other lattices of the same property have been constructed. The main idea of this article is to convert a binary diagonal lattice with non-principal coefficient ideals into a binary free lattice. Then, four vectors generate the lattice, but three of them cannot generate it. The main results are as follows. Let \(m\) be a positive square-free integer such that \( m = a^{2}-b^{2}\) with \(a\) odd and \(b\) even satisfying \( 0< 2b <a.\) Choose two integers \(A\) and \(B\) such that \( aA +2bB =1\). Then the binary unimodular Hermitian lattice \(L\) with Gram matrix NEWLINE\[NEWLINE\left( \begin{matrix} a(1+B^{2}) & B+(1+B^{2})(-b+\sqrt{-m}) \\ B+(1+B^{2})(-b-\sqrt{-m}) & a(1+B^{2})+A \\ \end{matrix} \right)NEWLINE\]NEWLINE over \( Q(\sqrt{-m})\) has minimal vectors of squared length \(a\), is generated by its \(8\) minimal vectors, but is not generated by any 3 minimal vectors. Further, if \( m\) is a positive squarefree integer such that \(m = (2a)^{2}-(2b+1)^{2}\) with \( 0< 2b+1 <a,\) choose two integers \(A\) and \(B\) such that \( aA +(2b+1)B =1\). Then the binary unimodular Hermitian lattice \(L\) with Gram matrix NEWLINE\[NEWLINE\left( \begin{matrix} a(1+B^{2}) & B+(1+B^{2})(-b-1+\omega) \\ B+(1+B^{2})(-b-1+\overline{\omega}) & a(1+B^{2})+A \\ \end{matrix} \right)NEWLINE\]NEWLINE over \( Q(\sqrt{-m})\) has minimal vectors of squared length \(a\), is generated by its \(8\) minimal vectors, but is not generated by any 3 minimal vectors. It is remarked that the method can be applied to infinitely many imaginary quadratic fields, but not all such lattices are generated by this method. An example is given to illustrate this.
scientific article; zbMATH DE number 6071044