# On the Classification of Lattices Overℚ(-3) Which Are Even Unimodular ℤ-Lattices of Rank 32

Authors: Andreas Henn; Michael Hentschel; Aloys Krieg; Gabriele Nebe
Journal: International Journal of Mathematics and Mathematical Sciences
 (2013)

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## Abstract

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


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## Body

## 1. Introduction

Let𝒪=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):=traceK/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=dimℤ(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.Forn=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 ringR 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
(1)R=𝔐2,∞=ℤ+ℤi+ℤj+ℤ1+i+j+ij2,R=𝔐3,∞=ℤ+ℤ1+i32+ℤj+ℤj+ij32,
respectively, wherei2=j2=-1,ij=-ji,in the rational definite quaternion algebra of discriminant 22 and 32 respectively. For the Hurwitz order 𝔐2,∞, these lattices have been determined in [5], and the classification over 𝔐3,∞ is new (cf. [6]).


## 2. Statement of Results

Theorem 1.
The mass of the genus of Eisenstein lattices of rank16 is
(2)μ16=∑i=1h‍1|U(Λi)|=16519·3617·1847·809·691·419·47·13231·322·54·11·17~0.002.
There are exactly h=80 isometry classes [Λi] 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
(3)Λ/Γ∩Λ≅𝒪/𝔭andΓ/Γ∩Λ≅𝒪/𝔭¯.
All 𝔭-neighbors of a given 𝒪-lattice Γ can be constructed as
(4)Γ(𝔭,x):=𝔭-1x+Γx,Γx:={y∈Γ∣h(x,y)∈𝔭},
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 Γx of suitable index. For details of the construction, we refer to [4].Corollary 2.
There are exactly 83 isometry classes of𝔐3,∞-lattices of rank 8 that yield even unimodular ℤ-lattices of rank 32.Proof.
Since𝔐3,∞ is generated by its unit group 𝔐3,∞*≅C3:C4, one may determine the structures over 𝔐3,∞ 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 𝔐3,∞ module structure, the 𝒪-lattice Γ needs to be isometric to its complex conjugate lattice Γ¯. Let τ0 be such an isometry, so
(5)τ0∈GLℤ(Γ),τ0σ=σ-1τ0,h(τ0x,τ0y)=h(x,y)¯∀x,y∈Γ.
Let
(6)U′(Γ):=〈U(Γ),τ0〉≅U(Γ)·C2.
Then we need to find representatives of all conjugacy classes of elements τ∈U′(Γ) such that
(7)τ2=-1,τσ=-σ2τ.
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 Tables1–4 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 𝔐3,∞. For lattices with a structure over the Hurwitz quaternions 𝔐2,∞ (note that (i+j+ij)2=-3, so all lattices with a structure over 𝔐2,∞ 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.
no.
R
#Aut
𝔐
3
,
∞
𝔐
2
,
∞
1
E
8
155520
1
E
8Table 2
The lattice of rank 8.
no.
R
#Aut
𝔐
3
,
∞
𝔐
2
,
∞
1
2E8
48372940800
2
2E8Table 3
The lattices of rank 12.
no.
R
#Aut
𝔐
3
,
∞
𝔐
2
,
∞
1
3E8
22568879259648000
2
3E8
2
4E6
8463329722368
1
3
6D4
206391214080
1
L
6(𝔓6)
4
12A2
101016305280
1
5
∅
2690072985600
1
Λ
24Table 4
The lattices of rank 16.
no.
R
#Aut
𝔐
3
,
∞
𝔐
2
,
∞
1
4E8
14039648409841827840000
3
4E8
2
4E6 + E8
1316217038422671360
1
3
6D4 + E8
32097961613721600
1
E
8
⊥
L
6(𝔓6)
4
12A2 + E8
15710055797145600
1
5
4A2 + 4E6
2742118830047232
1
6
4D4 + 2E6
40122452017152
1
7
E
8
418360150720512000
1
E
8
⊥
Λ
24
8
10A2 + 2E6
71409344532480
1
9
8D4
443823666757632
2
L
8(𝔓8)
10
4A2 + 3D4 + E6
313456656384
11
13A2 + E6
11604018486528
12
6D4
825564856320
1
13
6A2 + D4 + E6
48977602560
14
4A2 + 4D4
15479341056
1
15
7A2 + E6
21427701120
16
16A2
1851353376768
3
17
8A2 + 2D4
8707129344
1
18
4A2 + 3D4
1451188224
19
4A2 + E6
9795520512
20
4D4
82556485632
1
L
8(𝔓4)
21
D
4 + E6
1277045637120
22
6A2 + 2D4
302330880
2
23
9A2 + D4
1836660096
24
A
2 + E6
22448067840
25
4A2 + 2D4
107495424
1
26
7A2 + D4
52907904
27
10A2
408146688
1
28
6A2 + D4
22674816
29
2A2 + 2D4
134369280
1
30
5A2 + D4
8398080
31
8A2
423263232
2
32
8A2
7558272
4
33
4A2 + D4
4478976
34
2D4
7644119040
1
L
8(𝔓2)
35
2D4
656916480
1
36
7A2
1530550080
37
7A2
2834352
38
3A2 + D4
113374080
39
3A2 + D4
2519424
40
6A2
1679616
1
41
6A2
629856
2
42
2A2 + D4
1710720
43
5A2
139968
44
A
2 + D4
3265920
45
A
2 + D4
2426112
46
4A2
161243136
2
47
4A2
68024448
1
48
4A2
4199040
2
49
4A2
1399680
1
50
4A2
314928
51
4A2
139968
1
52
4A2
69984
3
53
D
4
660290641920
54
D
4
1813985280
55
D
4
87091200
L
8(𝔓)
56
D
4
1990656
57
3A2
58320
58
3A2
15552
59
2A2
606528
60
2A2
186624
1
61
2A2
41472
1
62
2A2
25920
63
2A2
18144
2
64
2A2
18144
2
65
2A2
16200
4
66
A
2
2204496
67
A
2
108864
68
A
2
3888
69
A
2
2916
70
∅
303216721920
2
B
W
32, Λ32′′
71
∅
15552000
5
Λ
32
′
72
∅
9289728
3
Λ
32
73
∅
1658880
1
74
∅
387072
3
75
∅
29376
2
76
∅
10368
1
77
∅
8064
2
78
∅
5760
4
79
∅
4608
2
80
∅
2592
3A list of the Gram matrices of the lattices is given in [12].Remark 3.
We have the following.(a)
The80 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]).



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