On the Classification of Lattices Over ℚ(√−3) Which Are Even Unimodular ℤ-Lattices of Rank 32

Abstract

We classify the lattices of rank 16 over the Eisenstein integers which are even unimodular ℤ-lattices (of dimension 32). There are exactly 80 unitary isometry classes.

1. Introduction

Let 𝒪=ℤ[(1+√−3)/2] be the ring of integers in the imaginary quadratic field 𝐾=ℚ[√−3]. An Eisenstein lattice is a positive definite Hermitian 𝒪-lattice (Λ,ℎ) such that the trace lattice (Λ,𝑞) with 𝑞(𝑥,𝑦)∶=trace_𝐾/ℚℎ(𝑥,𝑦)=ℎ(𝑥,𝑦)+ℎ(𝑥,𝑦) is an even unimodular ℤ-lattice. The rank of the free 𝒪-lattice Λ is 𝑟=𝑛/2 where 𝑛=dim_ℤ(Λ). 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 𝑟 for the full Hermitian modular group with respect to 𝒪 (cf. [2]). The paper [2] contains a classification of the Eisenstein lattices for 𝑛=8, 16, and 24. In these cases, one can use the classifications of even unimodular ℤ-lattices by Kneser and Niemeier and look for automorphisms with minimal polynomial 𝑋²−𝑋+1.

For 𝑛=32, this approach does not work as there are more than 10⁹ isometry classes of even unimodular ℤ-lattices (cf. [3, Corollary 17]). In this case, we apply a generalisation of Kneser’s neighbor method (compare [4]) over ℤ[(1+√−3)/2] 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 𝑅 that contains 𝒪, any 𝑅-module is clearly also an 𝒪-module. In particular, the classification of Eisenstein lattices can be used to obtain a classification of even unimodular ℤ-lattices that are 𝑅-modules for the maximal order 𝑅=𝔐_2,∞=ℤ+ℤ𝑖+ℤ𝑗+ℤ1+𝑖+𝑗+𝑖𝑗2,𝑅=𝔐_3,∞√=ℤ+ℤ1+𝑖32√+ℤ𝑗+ℤ𝑗+𝑖𝑗32, (1) respectively, where𝑖²=𝑗²=−1,𝑖𝑗=−𝑗𝑖,in the rational definite quaternion algebra of discriminant 2² and 3² respectively. For the Hurwitz order 𝔐_2,∞, these lattices have been determined in [5], and the classification over 𝔐_3,∞ is new (cf. [6]).

2. Statement of Results

Theorem 1. The mass of the genus of Eisenstein lattices of rank 16 is 𝜇₁₆=^ℎ_𝑖=11𝑈Λ_𝑖=16519⋅3617⋅1847⋅809⋅691⋅419⋅47⋅132³¹⋅3²²⋅5⁴⋅11⋅17∼0.002. (2) There are exactly ℎ=80 isometry classes [Λ_𝑖] of Eisenstein lattices of rank 16.

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 neighbor method. For this, we made use of the following facts (cf. [4]): Let Γ be an integral 𝒪-lattice and 𝔭 a prime ideal of 𝒪 that does not divide the discriminant of Γ. An integral 𝒪-lattice Λ is called a 𝔭-neighbor of Γ if Λ/Γ∩Λ≅𝒪/𝔭andΓ/Γ∩Λ≅𝒪/𝔭. (3) All 𝔭-neighbors of a given 𝒪-lattice Γ can be constructed as Γ(𝔭,𝑥)∶=𝔭⁻¹𝑥+Γ_𝑥,Γ_𝑥∶=𝑦∈Γ∣ℎ𝑥,𝑦∈𝔭, (4) where 𝑥∈Γ⧵𝔭Γ with ℎ(𝑥,𝑥)∈𝔭𝔭 (such a vector is called admissible). We computed (almost random) neighbors (after rescaling the already computed lattices to make them integral) for the prime elements 2,2−√−3, and 4−√−3 by randomly choosing admissible vectors 𝑥 from a set of representatives and constructing Γ(𝔭,𝑥) or all integral overlattices of Γ_𝑥 of suitable index. For details of the construction, we refer to [4].

Corollary 2. There are exactly 83 isometry classes of 𝔐_3,∞-lattices of rank 8 that yield even unimodular ℤ-lattices of rank 32.

Proof. Since 𝔐_3,∞ is generated by its unit group 𝔐^∗_3,∞≅𝐶₃∶𝐶₄, one may determine the structures over 𝔐_3,∞ of an Eisenstein lattice Γ as follows. Let (−1+√−3)/2=∶𝜎∈𝑈(Γ) be a third root of unity. If the 𝒪-module structure of Γ can be extended to a 𝔐_3,∞ module structure, the 𝒪-lattice Γ needs to be isometric to its complex conjugate lattice Γ. Let 𝜏₀ be such an isometry, so 𝜏₀∈GL_ℤ(Γ),𝜏₀𝜎=𝜎⁻¹𝜏₀,ℎ𝜏₀𝑥,𝜏₀𝑦=ℎ𝑥,𝑦∀𝑥,𝑦∈Γ. (5) Let 𝑈^′(Γ)∶=𝑈(Γ),𝜏₀≅𝑈(Γ)⋅𝐶₂. (6) Then we need to find representatives of all conjugacy classes of elements 𝜏∈𝑈^′(Γ) such that 𝜏²=−1,𝜏𝜎=−𝜎²𝜏. (7) This can be shown as in [8] in the case of the Gaussian integers.

Alternatively, one can classify these lattices directly using the neighbor method and a mass formula, which can be derived from the mass formula in [9] as in [5]. The results are contained in [6]. For details on the neighbor method in a quaternionic setting, we refer to [10].

The Eisenstein lattices of rank up to 16 are listed in Tables 1–4 ordered by the number of roots. For the sake of completeness, we have included the results from [2] in rank 4, 8 and 12. 𝑅 denotes the root system of the corresponding even unimodular ℤ-lattice (cf. [11, Chapter 4]). In the column #Aut, the order of the unitary automorphism group is given. The next column contains the number of structures of the lattice over 𝔐_3,∞. For lattices with a structure over the Hurwitz quaternions 𝔐_2,∞ (note that (𝑖+𝑗+𝑖𝑗)²=−3, so all lattices with a structure over 𝔐_2,∞ have a structure over 𝒪), the name of the corresponding Hurwitz lattice used in [5] is given in the last column.

Table 1 

The lattice of rank 4.

Table 2 

The lattice of rank 8.

Table 3 

The lattices of rank 12.

Table 4 

The lattices of rank 16.

A list of the Gram matrices of the lattices is given in [12].

Remark 3. We have the following.(a)The 80 corresponding ℤ-lattices belong to mutually different ℤ-isometry classes.(b)Each of the lattices listed previously is isometric to its conjugate. Hence the associated Hermitian theta series are symmetric Hermitian modular forms (cf. [1]).