research article on the classification of lattices …downloads.hindawi.com › journals › ijmms...

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2013 Article ID 837080 4 pageshttpdxdoiorg1011552013837080

Research ArticleOn the Classification of Lattices Over Q(radicminus3)Which Are EvenUnimodular Z-Lattices of Rank 32

Andreas Henn Michael Hentschel Aloys Krieg and Gabriele Nebe

Lehrstuhl A fur Mathematik RWTH Aachen University 52056 Aachen Germany

Correspondence should be addressed to Andreas Henn andreashennmatharwth-aachende

Received 13 November 2012 Accepted 28 January 2013

Academic Editor Frank Werner

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

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

1 Introduction

Let O = Z[(1 + radicminus3)2] be the ring of integers in theimaginary quadratic field 119870 = Q[radicminus3] An Eisenstein latticeis a positive definite Hermitian O-lattice (Λ ℎ) such that thetrace lattice (Λ 119902) with 119902(119909 119910) = trace

119870Qℎ(119909 119910) = ℎ(119909 119910) +

ℎ(119909 119910) is an even unimodular Z-lattice The rank of the freeO-lattice Λ is 119903 = 1198992 where 119899 = dimZ(Λ) Eisenstein lattices(or the more general theta lattices introduced in [1]) are ofinterest in the theory of modular forms as their theta seriesis a modular form of weight 119903 for the full Hermitian modulargroup with respect to O (cf [2]) The paper [2] contains aclassification of the Eisenstein lattices for 119899 = 8 16 and24 In these cases one can use the classifications of evenunimodular Z-lattices by Kneser and Niemeier and look forautomorphisms with minimal polynomial 1198832 minus 119883 + 1

For 119899 = 32 this approach does not work as there aremorethan 10

9 isometry classes of even unimodular Z-lattices (cf[3 Corollary 17]) In this case we apply a generalisation ofKneserrsquos neighbormethod (compare [4]) overZ[(1+radicminus3)2]to construct enough representatives of Eisenstein lattices andthen use the mass formula developed in [2] (and in a moregeneral setting in [1]) to check that the list of lattices iscomplete

Given some ring 119877 that contains O any 119877-module isclearly also an O-module In particular the classification

of Eisenstein lattices can be used to obtain a classificationof even unimodular Z-lattices that are 119877-modules for themaximal order

119877 = M2infin

= Z + Z119894 + Z119895 + Z1 + 119894 + 119895 + 119894119895

2

119877 = M3infin

= Z + Z1 + 119894radic3

2

+ Z119895 + Z119895 + 119894119895radic3

2

(1)

respectively where 1198942 = 119895

2= minus1 119894119895 = minus119895119894 in the rational

definite quaternion algebra of discriminant 22 and 32 respec-tively For the Hurwitz order M

2infin these lattices have been

determined in [5] and the classification overM3infin

is new (cf[6])

2 Statement of Results

Theorem 1 Themass of the genus of Eisenstein lattices of rank16 is

12058316=

ℎ

sum

119894=1

1

1003816100381610038161003816

119880 (Λ119894)

1003816100381610038161003816

=

16519 sdot 3617 sdot 1847 sdot 809 sdot 691 sdot 419 sdot 47 sdot 13

2

31sdot 3

22sdot 5

4sdot 11 sdot 17

sim 0002

(2)

2 International Journal of Mathematics and Mathematical Sciences

Table 1 The lattice of rank 4

no 119877 Aut M3infin

M2infin

1 1198648

155520 1 1198648



M2infin

1 21198648

48372940800 2 21198648

Table 3 The lattices of rank 12


M2infin

1 31198648

22568879259648000 2 31198648

2 41198646

8463329722368 13 6119863

4206391214080 1 119871

6(P6)

4 121198602

101016305280 15 0 2690072985600 1 Λ

24

There are exactly ℎ = 80 isometry classes [Λ119894] of Eisenstein

lattices of rank 16

Proof The mass was computed in [2] The 80 Eisensteinlattices of rank 16 are listed in Table 4 with the order oftheir unitary automorphism group These groups have beencomputed with MAGMAWe also checked that these latticesare pairwise not isometric Using the mass formula oneverifies that the list is complete

To obtain the complete list of Eisenstein lattices of rank16 we first constructed some lattices as orthogonal sumsof Eisenstein lattices of rank 12 and 4 and from known32-dimensional even unimodular lattices We also appliedcoding constructions from ternary and quaternary codes inthe same spirit as described in [7] To this list of lattices weapplied Kneserrsquos neighbor method For this we made use ofthe following facts (cf [4]) Let Γ be an integral O-lattice andp a prime ideal of O that does not divide the discriminant ofΓ An integral O-lattice Λ is called a p-neighbor of Γ if

ΛΓ cap Λ cong Op and ΓΓ cap Λ cong Op (3)

All p-neighbors of a given O-lattice Γ can be constructed as

Γ (p 119909) = pminus1119909 + Γ119909 Γ

119909= 119910 isin Γ | ℎ (119909 119910) isin p (4)

where 119909 isin Γ pΓ with ℎ(119909 119909) isin pp (such a vector iscalled admissible) We computed (almost random) neighbors(after rescaling the already computed lattices to make themintegral) for the prime elements 2 2 minus radicminus3 and 4 minus radicminus3 byrandomly choosing admissible vectors 119909 from a set of repre-sentatives and constructing Γ(p 119909) or all integral overlatticesof Γ119909of suitable index For details of the construction we refer

to [4]

Corollary 2 There are exactly 83 isometry classes of M3infin

-lattices of rank 8 that yield even unimodular Z-lattices of rank32



M2infin

1 41198648

14039648409841827840000 3 41198648

2 41198646+ 1198648

1316217038422671360 13 6119863

4+ 1198648

32097961613721600 1 1198648perp 1198716(P6)

4 121198602+ 1198648

15710055797145600 15 4119860

2+ 41198646

2742118830047232 16 4119863

4+ 21198646

40122452017152 17 119864

8418360150720512000 1 119864

8perp Λ24

8 101198602+ 21198646

71409344532480 19 8119863

4443823666757632 2 119871

8(P8)

10 41198602+ 31198634

+ 1198646

313456656384

11 131198602+ 1198646

1160401848652812 6119863

4825564856320 1

13 61198602+1198634

+ 1198646

48977602560

14 41198602+ 41198634

15479341056 115 7119860

2+ 1198646

2142770112016 16119860

21851353376768 3

17 81198602+ 21198634

8707129344 118 4119860

2+ 31198634

145118822419 4119860

2+ 1198646

979552051220 4119863

482556485632 1 119871

8(P4)

21 1198634+ 1198646

127704563712022 6119860

2+ 21198634

302330880 223 9119860

2+1198634

183666009624 119860

2+ 1198646

2244806784025 4119860

2+ 21198634

107495424 126 7119860

2+1198634

5290790427 10119860

2408146688 1

28 61198602+1198634

2267481629 2119860

2+ 21198634

134369280 130 5119860

2+1198634

839808031 8119860

2423263232 2

32 81198602

7558272 433 4119860

2+1198634

447897634 2119863

47644119040 1 119871

8(P2)

35 21198634

656916480 136 7119860

21530550080

37 71198602

283435238 3119860

2+1198634

11337408039 3119860

2+1198634

251942440 6119860

21679616 1

41 61198602

629856 242 2119860

2+1198634

171072043 5119860

2139968

44 1198602+1198634

326592045 119860

2+1198634

242611246 4119860

2161243136 2

47 41198602

68024448 1

International Journal of Mathematics and Mathematical Sciences 3

Table 4 Continued


M2infin

48 41198602

4199040 249 4119860

21399680 1

50 41198602

31492851 4119860

2139968 1

52 41198602

69984 353 119863

4660290641920

54 1198634

181398528055 119863

487091200 119871

8(P)

56 1198634

199065657 3119860

258320

58 31198602

1555259 2119860

2606528

60 21198602

186624 161 2119860

241472 1

62 21198602

2592063 2119860

218144 2

64 21198602

18144 265 2119860

216200 4

66 1198602

220449667 119860

2108864

68 1198602

388869 119860

22916

70 0 303216721920 2 11986111988232 Λ1015840101584032

71 0 15552000 5 Λ

1015840

32

72 0 9289728 3 Λ32

73 0 1658880 174 0 387072 375 0 29376 276 0 10368 177 0 8064 278 0 5760 479 0 4608 280 0 2592 3

Proof Since M3infin

is generated by its unit group Mlowast3infin

cong

1198623 1198624 one may determine the structures over M

3infinof an

Eisenstein lattice Γ as follows Let (minus1 + radicminus3)2 = 120590 isin 119880(Γ)

be a third root of unity If the O-module structure of Γ can beextended to a M

3infinmodule structure the O-lattice Γ needs

to be isometric to its complex conjugate lattice Γ Let 1205910be

such an isometry so

1205910isin GLZ (Γ) 120591

0120590 = 120590

minus11205910 ℎ (120591

0119909 1205910119910) = ℎ (119909 119910)

forall119909 119910 isin Γ

(5)

Let

119880

1015840(Γ) = ⟨119880 (Γ) 1205910

⟩ cong 119880 (Γ) sdot 1198622 (6)

Then we need to find representatives of all conjugacy classesof elements 120591 isin 119880

1015840(Γ) such that

120591

2= minus1 120591120590 = minus120590

2120591 (7)

This can be shown as in [8] in the case of the Gaussianintegers

Alternatively one can classify these lattices directly usingthe neighbor method and a mass formula which can bederived from the mass formula in [9] as in [5] The resultsare contained in [6] For details on the neighbor method in aquaternionic setting we refer to [10]

The Eisenstein lattices of rank up to 16 are listed inTables 1ndash4 ordered by the number of roots For the sake ofcompleteness we have included the results from [2] in rank4 8 and 12 119877 denotes the root system of the correspondingeven unimodularZ-lattice (cf [11 Chapter 4]) In the columnAut the order of the unitary automorphism group is givenThe next column contains the number of structures of thelattice over M

3infin For lattices with a structure over the

Hurwitz quaternionsM2infin

(note that (119894 + 119895 + 119894119895)2 = minus3 so alllattices with a structure overM

2infinhave a structure over O)

the name of the corresponding Hurwitz lattice used in [5] isgiven in the last column

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 mutuallydifferent Z-isometry classes

(b) Each of the lattices listed previously is isometric toits conjugate Hence the associated Hermitian thetaseries are symmetric Hermitian modular forms (cf[1])

References

[1] M Hentschel A Krieg and G Nebe ldquoOn the classification ofeven unimodular lattices with a complex structurerdquo Interna-tional Journal of NumberTheory vol 8 no 4 pp 983ndash992 2012

[2] M Hentschel A Krieg and G Nebe ldquoOn the classification oflattices over Q(radicminus3) which are even unimodular Z-latticesrdquoAbhandlungen aus dem Mathematischen Seminar der Univer-sitat Hamburg vol 80 no 2 pp 183ndash192 2010

[3] O D King ldquoA mass formula for unimodular lattices with norootsrdquo Mathematics of Computation vol 72 no 242 pp 839ndash863 2003

[4] A Schiemann ldquoClassification of Hermitian forms with theneighbour methodrdquo Journal of Symbolic Computation vol 26no 4 pp 487ndash508 1998

[5] C Bachoc and G Nebe ldquoClassification of two Genera of32-dimensional Lattices of Rank 8 over the Hurwitz OrderrdquoExperimental Mathematics vol 6 no 2 pp 151ndash162 1997

[6] A Henn Klassifikation gerader unimodularer Gitter mit quater-nionischer Struktur [PhD thesis] RWTH Aachen University

[7] C Bachoc ldquoApplications of coding theory to the constructionof modular latticesrdquo Journal of Combinatorial Theory A vol 78no 1 pp 92ndash119 1997


[8] M Kitazume and A Munemasa ldquoEven unimodular Gaussianlattices of rank 12rdquo Journal of Number Theory vol 95 no 1 pp77ndash94 2002

[9] K Hashimoto ldquoOn Brandtmatrices associated with the positivedefinite quaternion Hermitian formsrdquo Journal of the Faculty ofScience vol 27 no 1 pp 227ndash245 1980

[10] R Coulangeon ldquoReseaux unimodulaires quaternioniens endimension le 32rdquoActa Arithmetica vol 70 no 1 pp 9ndash24 1995

[11] J H Conway and N J A Sloane Sphere Packings Lattices andGroups vol 290 Springer New York NY USA 3rd edition1999

[12] M Hentschel ldquoDie 120599-Gitter im MAGMA-Kode als Gram-Matrizenrdquo httpwwwmatharwth-aachendedemitarbeiterhentschel

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M2infin

1 1198648

155520 1 1198648



M2infin

1 21198648

48372940800 2 21198648



M2infin

1 31198648

22568879259648000 2 31198648

2 41198646

8463329722368 13 6119863

4206391214080 1 119871

6(P6)

4 121198602

101016305280 15 0 2690072985600 1 Λ

24

There are exactly ℎ = 80 isometry classes [Λ119894] of Eisenstein

lattices of rank 16

Proof The mass was computed in [2] The 80 Eisensteinlattices of rank 16 are listed in Table 4 with the order oftheir unitary automorphism group These groups have beencomputed with MAGMAWe also checked that these latticesare pairwise not isometric Using the mass formula oneverifies that the list is complete

To obtain the complete list of Eisenstein lattices of rank16 we first constructed some lattices as orthogonal sumsof Eisenstein lattices of rank 12 and 4 and from known32-dimensional even unimodular lattices We also appliedcoding constructions from ternary and quaternary codes inthe same spirit as described in [7] To this list of lattices weapplied Kneserrsquos neighbor method For this we made use ofthe following facts (cf [4]) Let Γ be an integral O-lattice andp a prime ideal of O that does not divide the discriminant ofΓ An integral O-lattice Λ is called a p-neighbor of Γ if

ΛΓ cap Λ cong Op and ΓΓ cap Λ cong Op (3)

All p-neighbors of a given O-lattice Γ can be constructed as

Γ (p 119909) = pminus1119909 + Γ119909 Γ

119909= 119910 isin Γ | ℎ (119909 119910) isin p (4)

where 119909 isin Γ pΓ with ℎ(119909 119909) isin pp (such a vector iscalled admissible) We computed (almost random) neighbors(after rescaling the already computed lattices to make themintegral) for the prime elements 2 2 minus radicminus3 and 4 minus radicminus3 byrandomly choosing admissible vectors 119909 from a set of repre-sentatives and constructing Γ(p 119909) or all integral overlatticesof Γ119909of suitable index For details of the construction we refer

to [4]

Corollary 2 There are exactly 83 isometry classes of M3infin

-lattices of rank 8 that yield even unimodular Z-lattices of rank32



M2infin

1 41198648

14039648409841827840000 3 41198648

2 41198646+ 1198648

1316217038422671360 13 6119863

4+ 1198648

32097961613721600 1 1198648perp 1198716(P6)

4 121198602+ 1198648

15710055797145600 15 4119860

2+ 41198646

2742118830047232 16 4119863

4+ 21198646

40122452017152 17 119864

8418360150720512000 1 119864

8perp Λ24

8 101198602+ 21198646

71409344532480 19 8119863

4443823666757632 2 119871

8(P8)

10 41198602+ 31198634

+ 1198646

313456656384

11 131198602+ 1198646

1160401848652812 6119863

4825564856320 1

13 61198602+1198634

+ 1198646

48977602560

14 41198602+ 41198634

15479341056 115 7119860

2+ 1198646

2142770112016 16119860

21851353376768 3

17 81198602+ 21198634

8707129344 118 4119860

2+ 31198634

145118822419 4119860

2+ 1198646

979552051220 4119863

482556485632 1 119871

8(P4)

21 1198634+ 1198646

127704563712022 6119860

2+ 21198634

302330880 223 9119860

2+1198634

183666009624 119860

2+ 1198646

2244806784025 4119860

2+ 21198634

107495424 126 7119860

2+1198634

5290790427 10119860

2408146688 1

28 61198602+1198634

2267481629 2119860

2+ 21198634

134369280 130 5119860

2+1198634

839808031 8119860

2423263232 2

32 81198602

7558272 433 4119860

2+1198634

447897634 2119863

47644119040 1 119871

8(P2)

35 21198634

656916480 136 7119860

21530550080

37 71198602

283435238 3119860

2+1198634

11337408039 3119860

2+1198634

251942440 6119860

21679616 1

41 61198602

629856 242 2119860

2+1198634

171072043 5119860

2139968

44 1198602+1198634

326592045 119860

2+1198634

242611246 4119860

2161243136 2

47 41198602

68024448 1


Table 4 Continued


M2infin

48 41198602

4199040 249 4119860

21399680 1

50 41198602

31492851 4119860

2139968 1

52 41198602

69984 353 119863

4660290641920

54 1198634

181398528055 119863

487091200 119871

8(P)

56 1198634

199065657 3119860

258320

58 31198602

1555259 2119860

2606528

60 21198602

186624 161 2119860

241472 1

62 21198602

2592063 2119860

218144 2

64 21198602

18144 265 2119860

216200 4

66 1198602

220449667 119860

2108864

68 1198602

388869 119860

22916

70 0 303216721920 2 11986111988232 Λ1015840101584032

71 0 15552000 5 Λ

1015840

32

72 0 9289728 3 Λ32

73 0 1658880 174 0 387072 375 0 29376 276 0 10368 177 0 8064 278 0 5760 479 0 4608 280 0 2592 3

Proof Since M3infin


cong


3infinof an





such an isometry so

1205910isin GLZ (Γ) 120591

0120590 = 120590

minus11205910 ℎ (120591

0119909 1205910119910) = ℎ (119909 119910)


(5)

Let

119880

1015840(Γ) = ⟨119880 (Γ) 1205910

⟩ cong 119880 (Γ) sdot 1198622 (6)



120591

2= minus1 120591120590 = minus120590

2120591 (7)













References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 4 Continued


M2infin

48 41198602

4199040 249 4119860

21399680 1

50 41198602

31492851 4119860

2139968 1

52 41198602

69984 353 119863

4660290641920

54 1198634

181398528055 119863

487091200 119871

8(P)

56 1198634

199065657 3119860

258320

58 31198602

1555259 2119860

2606528

60 21198602

186624 161 2119860

241472 1

62 21198602

2592063 2119860

218144 2

64 21198602

18144 265 2119860

216200 4

66 1198602

220449667 119860

2108864

68 1198602

388869 119860

22916

70 0 303216721920 2 11986111988232 Λ1015840101584032

71 0 15552000 5 Λ

1015840

32

72 0 9289728 3 Λ32

73 0 1658880 174 0 387072 375 0 29376 276 0 10368 177 0 8064 278 0 5760 479 0 4608 280 0 2592 3

Proof Since M3infin


cong


3infinof an





such an isometry so

1205910isin GLZ (Γ) 120591

0120590 = 120590

minus11205910 ℎ (120591

0119909 1205910119910) = ℎ (119909 119910)


(5)

Let

119880

1015840(Γ) = ⟨119880 (Γ) 1205910

⟩ cong 119880 (Γ) sdot 1198622 (6)



120591

2= minus1 120591120590 = minus120590

2120591 (7)













References





















Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article on the classification of lattices …downloads.hindawi.com › journals › ijmms...

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