Even unimodular 12-dimensional quadratic forms over \({\mathbb{Q}}(\sqrt{5})\) (Q1090698)

The various techniques known for the classification of definite integral quadratic forms are combined in this carefully written paper to find the 12-dimensional totally positive even unimodular lattices over the ring of integers in \({\mathbb{Q}}(\sqrt{5})\). Such lattices give rise to 24- dimensional positive definite even unimodular \({\mathbb{Z}}\)-lattices, and the authors are influenced by the elegant approach of \textit{B. B. Venkov} [Tr. Mat. Inst. Steklova 148, 65-76 (1978; Zbl 0443.10021)] to Niemeier's list of these \({\mathbb{Z}}\)-lattices. In particular, they make use of the information about the possible \({\mathbb{Z}}\)-root lattices, and they follow Venkov in proving uniqueness of a \({\mathbb{Q}}(\sqrt{5})\)-version of the Leech lattice which had been constructed earlier by \textit{J. Tits} [J. Algebra 63, 56-75 (1980; Zbl 0436.20004)]. To prove completeness of their list of 14 other lattice classes (some of which coincide over \({\mathbb{Z}})\), the authors also employ the mass formula and Siegel's formula for representations of numbers, the latter being put into the context of Hilbert modular forms over \({\mathbb{Q}}(\sqrt{5})\).