Positive definite Hermitian forms over imaginary quadratic fields
The author asserts that he has generalized Kneser's neighbor-lattice method to imaginary quadratic fields and has developed a method to determine the class numbers and the representative forms of modular lattices over imaginary quadratic fields with class number 1. As consequences, he obtains the results (i) to determine all the positive unimodular Hermitian lattices over \({\mathbb{Q}}(\sqrt{-2})\) in dimensions \(<6\), (ii) to construct a class of positive definite unimodular indecomposable even lattices over some imaginary quadratic fields, (iii) to construct a class of positive definite unimodular indecomposable Hermitian odd lattices, (iv) to prove that there is one class in \(gen(I_ n)\) (the genus to which the form associated with the identity matrix \(I_ n\) of order n belongs) if and only if n satisfies an inequality which varies according to the ground field in treatment. The result (iv) is already given by \textit{G. Otremba} [J. Reine Angew. Math. 249, 1-19 (1971; Zbl 0221.12007)] by an analytical method. The proofs and the details are completely lacking.
scientific article; zbMATH DE number 4204437