On the Classification of Lattices Over Which Are Even Unimodular -Lattices of Rank 32

We classify the lattices of rank 16 over the Eisenstein integers which are even unimodular -lattices (of dimension 32). There are exactly 80 unitary isometry classes. 1. Introduction Let be the ring of integers in the imaginary quadratic field . An Eisenstein lattice is a positive definite Hermitian -lattice such that the trace lattice with is an even unimodular -lattice. The rank of the free -lattice is where . Eisenstein lattices (or the more general theta lattices introduced in [1]) are of interest in the theory of modular forms, as their theta series is a modular form of weight for the full Hermitian modular group with respect to (cf. [2]). The paper [2] contains a classification of the Eisenstein lattices for , , and . In these cases, one can use the classifications of even unimodular -lattices by Kneser and Niemeier and look for automorphisms with minimal polynomial . For , this approach does not work as there are more than isometry classes of even unimodular -lattices (cf. [3, Corollary 17]). In this case, we apply a generalisation of Kneser’s neighbor method (compare [4]) over to construct enough representatives of Eisenstein lattices and then use the mass formula developed in [2] (and in a more general setting in [1]) to check that the list of lattices is complete. Given some ring that contains , any -module is clearly also an -module. In particular, the classification of Eisenstein lattices can be used to obtain a classification of even unimodular -lattices that are -modules for the maximal order respectively, where in the rational definite quaternion algebra of discriminant and respectively. For the Hurwitz order , these lattices have been determined in [5], and the classification over is new (cf. [6]). 2. Statement of Results Theorem 1. The mass of the genus of Eisenstein lattices of rank is There are exactly isometry classes of Eisenstein lattices of rank . Proof. The mass was computed in [2]. The 80 Eisenstein lattices of rank 16 are listed in Table 4 with the order of their unitary automorphism group. These groups have been computed with MAGMA. We also checked that these lattices are pairwise not isometric. Using the mass formula, one verifies that the list is complete. To obtain the complete list of Eisenstein lattices of rank 16, we first constructed some lattices as orthogonal sums of Eisenstein lattices of rank 12 and 4 and from known 32-dimensional even unimodular lattices. We also applied coding constructions from ternary and quaternary codes in the same spirit as described in [7]. To this list of lattices, we applied Kneser’s