On the classification of lattices over \(\mathbb Q(\sqrt{-3})\) which are even unimodular \(\mathbb Z\)-lattices of rank 32
Let \(\mathcal{O}=\mathbb{Z}[(1+\sqrt{-3})/2]\) be the ring of Eisenstein integers. A Hermitian (with respect to complex conjugation) \(\mathcal{O}\)-lattice \((\Lambda,h)\) of \(\mathcal{O}\)-rank \(r\) is called an Eisenstein lattice if it is positive definite and if its quadratic trace lattice \((\Lambda,q)\) of \(\mathbb{Z}\)-rank \(n=2r\) with \(q(x,y)=h(x,y)+\overline{h(x,y)}\) is an even unimodular lattice, so in particular, \(n\) must be a multiple of \(8\). The three last named authors classified Eisenstein lattices for \(n=8,\;16,\;24\) in [Abh. Math. Semin. Univ. Hamb. 80, No. 2, 183--192 (2010; Zbl 1229.11062)] using the Kneser-Niemeier classification of positive definite even unimodular \(\mathbb{Z}\)-lattices for these ranks and the fact, that any such lattice that is a trace lattice of an Eisenstein lattice must contain an automorphism with minimal polynomial \(X^2-X+1\). To treat the case \(n=32\) in this paper, this method is not feasible because there are simply too many positive definite even unimodular \(\mathbb{Z}\)-lattices in that case. Instead, the authors apply a generalization of Kneser's neighbor method to Hermitian lattices over \(\mathcal{O}\), starting from a pre-constructed set of Eisenstein lattices including such lattices that are orthogonal sums of Eisenstein lattices of smaller rank. Using a computer, they thus construct a list of representatives of Eisenstein lattices and use the mass formula to check that the list is indeed exhaustive. It turns out that up to isometry, there are \(80\) Eisenstein lattices of \(\mathcal{O}\)-rank \(16\). The paper contains a list of these lattices and of the orders of their automorphism groups.
scientific article; zbMATH DE number 6165373