International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2013 / Article

Research Article | Open Access

Volume 2013 |Article ID 837080 | 4 pages | https://doi.org/10.1155/2013/837080

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

Academic Editor: Frank Werner
Received13 Nov 2012
Accepted28 Jan 2013
Published14 Mar 2013

Abstract

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, wherein 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 neighbor method. For this, we made use of the following facts (cf. [4]): Let be an integral -lattice and a prime ideal of that does not divide the discriminant of . An integral -lattice is called a -neighbor of if All -neighbors of a given -lattice can be constructed as where with (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 , and by randomly choosing admissible vectors from a set of representatives and constructing or all integral overlattices of of suitable index. For details of the construction, we refer to [4].

Corollary 2. There are exactly 83 isometry classes of -lattices of rank that yield even unimodular -lattices of rank .

Proof. Since is generated by its unit group , one may determine the structures over of an Eisenstein lattice as follows. Let be a third root of unity. If the -module structure of can be extended to a module structure, the -lattice needs to be isometric to its complex conjugate lattice . Let be such an isometry, so Let Then we need to find representatives of all conjugacy classes of elements such that 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 14 ordered by the number of roots. For the sake of completeness, we have included the results from [2] in rank 4, 8 and 12. denotes the root system of the corresponding even unimodular -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 . For lattices with a structure over the Hurwitz quaternions (note that , so all lattices with a structure over have a structure over ), the name of the corresponding Hurwitz lattice used in [5] is given in the last column.


no. Aut

1 1555201


no. Aut

12 4837294080022


no. Aut

13 2256887925964800023
24 84633297223681
36 2063912140801 ( )
412 1010163052801
5 26900729856001


no. Aut

14 1403964840984182784000034
24 + 13162170384226713601
36 + 320979616137216001 ( )
412 + 157100557971456001
54 + 4 27421188300472321
64 + 2 401224520171521
7 4183601507205120001
810 + 2 714093445324801
98 4438236667576322 ( )
104 + 3 + 313456656384
1113 + 11604018486528
126 8255648563201
136 + + 48977602560
144 + 4 154793410561
157 + 21427701120
1616 18513533767683
178 + 2 87071293441
184 + 3 1451188224
194 + 9795520512
204 825564856321 ( )
21 + 1277045637120
226 + 2 3023308802
239 + 1836660096
24 + 22448067840
254 + 2 1074954241
267 + 52907904
2710 4081466881
286 + 22674816
292 + 2 1343692801
305 + 8398080
318 4232632322
328 75582724
334 + 4478976
342 76441190401 ( )
352 6569164801
367 1530550080
377 2834352
383 + 113374080
393 + 2519424
406 16796161
416 6298562
422 + 1710720
435 139968
44 + 3265920
45 + 2426112
464 1612431362
474 680244481
484 41990402
494 13996801
504 314928
514 1399681
524 699843
53 660290641920
54 1813985280
55 87091200 ( )
56 1990656
573 58320
583 15552
592 606528
602 1866241
612 414721
622 25920
632 181442
642 181442
652 162004
66 2204496
67 108864
68 3888
69 2916
70 3032167219202 ,
71 155520005
72 92897283
73 16588801
74 3870723
75 293762
76 103681
77 80642
78 57604
79 46082
80 25923

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

Remark 3. We have the following.(a)The corresponding -lattices belong to mutually different -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]).

References

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Copyright © 2013 Andreas Henn et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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