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Minimal Vectors in the Second Exterior Power of a Lattice
194, 467]476 Ž1997. JA977039
JOURNAL OF ALGEBRA ARTICLE NO.
Minimal Vectors in the Second Exterior Power of a Lattice Renaud Coulangeon Dept. Uni¨ ersite´ Bordeaux I, 351 cours de la Liberation, ´ de Mathematiques, ´ ´ 33405 Talence Cedex, France Communicated by E¨ a Bayer-Fluckiger Received September 26, 1996
1. INTRODUCTION In wCx we studied the Rankin k-invariants which were defined by Rankin in wRx as a natural generalization of the classical Hermite invariant of a lattice in the following way: let L be a lattice of maximal rank in an n-dimensional Euclidean space E. We denote by d k Ž L. the minimal determinant of k-dimensional sublattices of L and put g k Ž L. s d k Ž L.rŽdet L. k r n. The upper bound of g k Ž L. for all lattices L in E is denoted by gn, k . It is readily verified that the k-dimensional sublattice spanned by a k-tuple of linearly independent vectors x 1 , . . . , x k in L depends only on its image x 1 n ??? n x k in the kth exterior power Hk L. Moreover, Hk E inherits a canonical Euclidean structure from the original scalar product on E, which is defined for split Žmulti-. vectors Žthat is, vectors which can be written as x 1 n ??? n x k . by:
Ž x 1 n ??? n x k . ? Ž y 1 n ??? n y k . [ det Ž x i ? y j . 1Fi , jFk . Thus, the norm Žor squared length. of a split non-zero vector x 1 n ??? n x k with respect to this scalar product coincides with the determinant of the sublattice spanned by x 1 , . . . , x k . In other words, one has:
d k Ž L . s min N Ž v . ¬ v g Hk L split 4 . 467 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
468
RENAUD COULANGEON
Consequently, a natural question, which already arose in wCx, is to ask whether the usual minimal norm of Hk L, as a lattice in Hk E, is attained on split elements. If so, one has g k Ž L. s g ŽHk L., where g ŽHk L. denotes the Hermite number of Hk L. As usual, the vectors of minimal length of a lattice will be referred to as minimal vectors. We give in Section 2 a partial answer to that problem, in the case k s 2, using arguments similar to those used by Y. Kitaoka for the analoguous problem for the tensor product Žcf. wK, Chap. 7x.. This is essentially based on estimates for the Ranking constant gn, 2 . This nevertheless does not seem to apply to higher exterior powers. Conversely, we show in Section 3 that asymptotically, the answer to the above question is negative Žfor any k ., and for k s 2, we give explicit counterexamples in dimension 24 and 48. Our counterexamples Žexplicit or not., arise from the theory of symplectic lattices initiated by Buser and Sarnak in wB-Sx. The analogy with Steinberg’s result concerning the tensor product Žsee wM-H, II, Theorem 9.6x. is worth noticing.
2. SHORT VECTORS IN THE SECOND EXTERIOR POWER In the following, the property for a lattice L that the minimal vectors of L n L Žresp. Hk L. are split will be referred to as property Ž S2 . Žresp. Sk .. In the following statement, the notation ? x @ stands for the greatest integer less or equal to x. THEOREM 2.1. Let r 0 s sup r G 2 ¬ g 2 s, 2 - s for all 2 F s F r 4 . Then Ž S2 . holds for any lattice L with ?dim Lr2@ F r 0 . Proof. Let v be a minimal vector of L n L. Using well-known properties of bilinear alternating forms over principal rings, we contend that v can be written in the form r
vs
Ý x i n yi , is1
where x 1 , . . . , x r , y 1 , . . . , yr are 2 r linearly independent vectors in L Ž2 r F n. Žcf. wB, Sect. 5, No. 1, Theoreme ´ ` 1.. Such a decomposition is essentially canonical, in the sense that the Z-module spanned by x 1 , . . . , x r , y 1 , . . . , yr and a fortiori the integer 2 r Žthe rank of v ., depends only on v . Then the
469
MINIMAL VECTORS OF LATTICES
norm of v is given by r
NŽ v . s s
ž
r
Ý x i n yi is1
?
/ž
Ý x i n yi is1
/
Ý Ž x i n yi ? x j n y j . i, j
s
Ý Ž x i ? x j . Ž yi ? y j . y Ž x i ? y j . Ž yi ? x j . i, j
s Tr Ž AB y C 2 . , where A s ŽŽ x i ? x j .1 F i, jF r , B s ŽŽ yi ? y j ..1 F i, jF r , C s ŽŽ x i ? y j ..1 F i, jF r . Thus, denoting by L2 r the sublattice spanned by x 1 , . . . , x r , y 1 , . . . , yr , by G the corresponding Gram matrix G s Ž tAC CB ., and by J the 2 r = 2 r matrix Ž yI0 r I0r . we obtain NŽ v . s
1 2
Tr Ž G t JGJ . G
2r 2
Ž det G .
1r2 r
Ž det t JGJ .
1r2 r
s r Ž det L2 r .
1rr
,
where we used the inequality TrŽ MN . G mŽdet M det N .1r m , valid for any two symmetric positive definite matrices of degree m Žcf. wK, Chap. 7, Lemma 7.1.3x.. Consequently we have
d 2 Ž L2 r . G N Ž v . G r Ž det L2 r .
1rr
,
Ž 2.1.
so that r F g 2 Ž L2 r . F g 2 r, 2 . Therefore, if ?dim Lr2@ F r 0 , then r F r 0 , and the condition r F g 2 r, 2 implies r s 1. LEMMA 2.2. g 2 r, 2 - r for 2 F r F 5. Proof. It is based on known estimates of g 2 r, 2 for these values of r: r s 2: g4, 2 s 3r2 - 2 ŽRankin, cf. wRx.. r s 3: we use the general inequality established by Rankin, gn, k F g h, k Žgn, h . k r h for 1 F k F h F n, applied to k s 2, h s 4, n s 6. We obtain v v
g6, 2 F g4, 2 Ž g6, 4 .
1r2
s g4, 2 Ž g6, 2 .
1r2
2
whence g6, 2 F Ž g4, 2 . s 9r4 - 3.
r s 4: we obtain in the same way: g 8, 2 F Žg6, 2 . 3r2 F Ž3r2. 3 - 4. A better estimate for g 8, 2 is obtained using: g 8, 2 F g 7, 2 Ž g 8 . 2r7 F g6, 2 Žg 7 .1r3 Žg 8 . 2r7 F 9r2 10r7. v
470
RENAUD COULANGEON v
r s 5: using the above upper bound, we obtain: g 10, 2 F Žg 8, 2 . 4r3 -
5. COROLLARY 2.3.
If dim L F 11, L satisfies Ž S2 ..
Proof. Straightforward using Theorem 2.1 and Lemma 2.2. Remark. If one assumes that g 8, 2 s g 2 ŽE 8 . s 3, which is likely to be the case, it follows that g 12, 2 F 3 5r3 , but it does not allow us to extend our result to dimension 12, since 3 5r3 ) 6. If we restrict ourselves to the case of integral lattices, i.e., lattices for which the scalar product takes integral values, we can obtain results with no condition on the dimension: PROPOSITION 2.4. L satisfies Ž S2 ..
Let L be an integral lattice such that d 2 Ž L. F 8. Then
Proof. Setting v s Ý ris1 x i n yi for a minimal vector v of L n L, as in the proof of Theorem 1, and L2 r the 2 r-dimensional sublattice spanned by x 1 , . . . , x r , y 1 , . . . , yr , we have 8 G d 2 Ž L . G N Ž v . G r Ž det L2 r .
1rr
G r,
Ž 2.2.
since det L2 r is integral. If r F 5, Lemma 2.2 shows that v , as a minimal vector of L2 r n L2 r , has to be split, so we can assume that r s 6, 7, or 8. In the first case we see, applying Ž2.2. with r s 6, that the sublattice L12 associated to v is an integral 12-dimensional lattice with detŽ L12 . F ?Ž 86 . 6 @ s 5. Such lattices are classified Žcf. wC-S2x., and are easily seen to satisfy d 2 Ž L12 . F 3. Consequently, 3 G d 2 Ž L. G r Žby Ž2.2.., whence a contradiction. In the same way, if r s 7, then L14 is a 14-dimensional integral lattice of determinant F ?Ž 87 . 7 @ s 2, whence, using the classification of wC-S2x, d 2 Ž L14 . F 3, which again leads to a contradiction. Finally, for r s 8, L16 is a 16-dimensional unimodular lattice, so that r F d 2 Ž L. F d 2 Ž L16 . s 3, whence again a contradiction. This corollary applies in particular to the root-lattices A n , Dn , E 6 , E 7 , E 8 , and to Dq n. To conclude this section, we investigate a wide class of lattices, including the laminated lattices L n , as well as the sequence K n Žcf. wC-S, Chap. 6x for a definition., for which the following lemma holds: LEMMA 2.5. Assume that L contains a 2-dimensional hexagonal section with the same minimal norm, and that dim L - 20. Then L satisifes Ž S2 ..
MINIMAL VECTORS OF LATTICES
471
Proof. The inequality d 2 Ž L. G 34 N Ž L. 2 holds for any L ŽHermite’s inequality; cf. wM, chap. 2, Theoreme ´ ` 2.1x. and becomes an equality under the assumption of the lemma. Let v be a minimal vector of L n L. As in the previous proofs, we associate to it a sublattice L2 r of L, with r F ? nr2@. Assuming that v is not split, we can furthermore restrict ourselves to r G 6. From the inequality r Ždet L2 r .1r r F d 2 Ž L., joined to the fact that det L2 r G d 2 r Ž L., we obtain that
d2 r Ž L. F
ž
d2 Ž L. r
r
/
.
Ž 2.3.
Let M be a 2 r-dimensional minimal section; we have
gŽ M. G
N Ž L.
Ž d2 r Ž L. .
1r2 r
2
G
'3
'r ,
so that g 2 r G Ž2r '3 .'r , which implies, using classical upper bounds for gn Žnamely the actual value of gn for n s 1, . . . , 8, and Blichfeldt’s inequality gn F Ž2rp . G Ž2 q nr2. 2r n for n ) 8; cf. wK, Chap. 2, Theorem 2.2.1x., that r G 10 whence the conclusion. PROPOSITION 2.6. K n satisfy Ž S2 ..
For n F 23, the laminated lattices L n and the lattices
Proof. Let L n denote either L n or K n . We first notice that L n satisfies the assumption of Lemma 2.5 for any n, which gives the conclusion for n - 20. For 20 F n F 23, we let v be a non-split minimal vector of L n n L n , and denote by 2 r the rank of the associated sublattice Ž L n . 2 r Ž r F ? nr2@.. Assuming that v is not split, we can restrict ourselves to r G 6. From the proof of Lemma 2.5, it follows that
d 2 r Ž Ln . F
ž
d 2 Ž Ln . r
r
/
,
Ž 2.4.
and also g 2 r G Ž2r '3 .'r so that r G 10. Then we notice that d 2 r Ž L n . G d 2 r Ž L24 .. But the actual values of d k Ž L24 ., for k F 8 and for k G 16, are known Žcf. wC-S, Chap. 7, Corollary 2.7x or wL-Sx.: one has d 20 Ž L24 . s 64 and d 22 Ž L24 . s 12, which in both cases are strictly greater than the bound given by Ž2.4., whence the conclusion. Remark. It will be shown in the next section that L 24 does not satisfy Ž S2 ..
472
RENAUD COULANGEON
3. COUNTEREXAMPLES We now use the notion of symplectic lattices to exhibit counterexamples to the above property. Recall Žcf. wB-Sx and its Appendix. that a lattice L is symplectic if there exists an isometry s from L onto its dual lattice LU and such that s 2 s yId. Such a lattice has even dimension 2 r and determinant 1. PROPOSITION 3.1. Let L be a symplectic lattice of dimension 2 r. Then there exists in L n L a non-split ¨ ector with squared length r. Accordingly, NŽ L n L. F r. Proof. Let s be a symplectic isometry from L onto LU , and let e1 , . . . , e r , e rq1 , . . . , e2 r be a symplectic basis of L with respect to s , that is:
ž
Ž ei ? s e j . s J s
0 yIr
Ir 0
/
Ž such a basis exists since det L s 1 . .
In other words, denoting by eUi 4 the dual basis of e i 4 , we have, for 1 F i F r:
s Ž e i . s yeUrq1 ,
s Ž e rqi . s eUi ,
s Ž eUi . s ye rqi ,
s Ž eUrqi . s e i .
It follows that the element v s e1 n e rq1 q ??? qe r n e2 r has the required property, namely, NŽ v . s
Ý
Ž e i n e rqi . ? Ž e j n e rqj .
1Fi , jFr
s
Ý
Ž e i ? e j . Ž e rqi ? e rqj . y Ž e i ? e rqj . Ž e rqi ? e j .
1Fi , jFr
s
Ý
Ž e i ? e j . Ž eUi ? eUj . y Ž e i ? e rqj . Ž yeUi ? eUrqj .
1Fi , jFr
s Tr Ž AAU q C t CU . , where A s Ž Ž e i ? e j . . 1Fi , jFr ,
B s Ž Ž e rqi ? e rqj . . 1Fi , jFr ,
C s Ž Ž e i ? e rqj . . 1Fi , jFr ,
473
MINIMAL VECTORS OF LATTICES
and AU , BU , CU are the corresponding blocks, replacing e i by eUi . From the equality
ž
A t C
C B
/ž
AU t U C
CU s I2 r , BU
/
we deduce that NŽ v . s TrŽ Ir . s r. From this we derive the following asymptotic behaviour for property Ž S2 .: COROLLARY 3.2. In any e¨ en dimension 2 r G 180, there exist lattices L which do not satisfy Ž S2 .. Proof. Define m 2 r s max L g S 2 r g Ž L. s max L g S 2 rNŽ L., where S 2 r denotes the set of Žisometry classes . of symplectic lattices in dimension 2 r. From Buser and Sarnak’s inequality ŽŽ1.12. in wB-Sx.
m2 r G
s2 r
y1 rr
ž / 2
,
where sn s
p n r2 G Ž nr2 q 1 .
,
and Hermite’s inequality d 2 Ž L. G 34 N Ž L. 2 , we conclude that for any r there exists a 2 r-dimensional symplectic lattice M2 r with
d 2 Ž M2 r . G
3 s2 r 4
ž / 2
y2 rr
.
It is then easy to check that this lower bound is strictly greater than r for r G 90, whence the conclusion, applying the previous proposition. From this, we can deduce the following more general statement about property Ž Sk .: COROLLARY 3.3. For any k G 2, there exists an integer n k such that n-dimensional lattices which do not satisfy Ž Sk . exist for any n G n k . Proof. For k s 2 this is the previous proposition, with n 2 F 180. For k ) 2, we consider lattices of the type L e s L H Z e, where H stands for the orthogonal sum. One easily checks that Hk Ž L H Z e . s Hk L H Hky 1 L n Z e , and N Ž Hk Ž L H Z e . . s min N Ž Hk L . , N Ž e . N Ž Hky 1 L . 4 .
474
RENAUD COULANGEON
We can choose e in such a way that NŽ e . - NŽHk L.rNŽHky 1 L., so that N Ž Hk Ž L H Z e . . s N Ž e . N Ž Hky 1 L .
and
S Ž Hk Ž L H Z e . . s S Ž Hky 1 L . n e, where S Ž M . stands for the set of minimal vectors of the lattice M. Then we can assume by induction that L does not satisfy Ž Sky 1 ., which is possible if the rank of L is greater than n ky 1 , and then conclude by the above property that Hk Ž L H Z e . itself does not satisfy Ž Sk .. In case k s 2, Corollary 3.2 also provides explicit counterexamples in lower dimensions. In dimension 24, we will consider the Leech lattice L 24 , which is symplectic Žsee for instance the Appendix to wB-Sx by J. Conway and N. Sloane where a symplectic basis is explicitly given.. In dimension 48 at least three even unimodular lattices of norm 6 are known, usually denoted by P48 p and P48 q for the first two ones Žcf. wC-S, p. 195x., and a third one N48 recently found by G. Nebe wNx. Furthermore P48 p is symplectic Žits automorphism group is isomorphic to 2 ? L2 Ž23. = S3 , and therefore contains elements of square yId Žcf. wATLASx.., and so does N48 wNx. Applying 3.1 to these lattices, we obtain: PROPOSITION 3.4. Ž1. NŽ L 24 n L 24 . s d 2 Ž L 24 . s 12 but L 24 n L 24 contains s1 s 2 6 ? 3 2 ? 5 3 ? 7 ? 13 ? 23 pairs of split minimal ¨ ectors and s2 s 2 6 ? 3 6 ? 5 2 ? 7 ? 11 ? 23 pairs of non-split ones which are in 1]1 correspondence with the symplectic automorphisms of L 24 . Ž2. Let L stand for P48 p or N48 . Then L n L contains only non-split minimal ¨ ectors. Proof. As we previously noticed ŽLemma 2.5., d 2 Ž L 24 . s 12, and it is attained exactly on the hexagonal planes spanned by two minimal vectors. Next we contend that the sublattice L s Ž L 24 . 2 r of L 24 associated to a non-split minimal vector v of L 24 n L 24 , if it exists, has dimension 24, i.e., r s 12 Žthe lower values of r are excluded by the arguments of Lemma 2.5 and Proposition 2.6.. Applying Ž2.1., we see that 12 s d 2 Ž L 24 . G N Ž v . G 12 Ž det L .
1r12
,
whence
w L 24 : L x s 'det L s 1. Therefore, L s L 24 and NŽ L 24 n L 24 . s NŽ v . s 12 s d 2 Ž L 24 .. By Proposition 3.1 such non-split vectors of rank 24 and squared length 12 in L 24 n L 24 exist. Moreover, setting v s Ý12 is1 x i n yi and denoting by G
475
MINIMAL VECTORS OF LATTICES
the Gram matrix of the vectors x 1 , . . . , x 12 , y 1 , . . . , y 12 , we have, using the notation of Theorem 2.1: N Ž v . s 12 Tr Ž G t JGJ . G 12 Ž det G .
1r12
G 12.
One easily checks that equality can hold if and only if t JGJ s Gy1 , that is to say, x 1 , . . . , x 12 , y 1 , . . . , y 12 is a symplectic basis of L 24 with respect to a symplectic automorphism s of L 24 . Then the map which to a symplectic automorphism s associates the element Ý12 is1 x i n yi , where x 1 , . . . , x 12 , y 1 , . . . , y 12 is Žany. symplectic basis of L 24 with respect to s , is clearly one to one, so that the number of non-split minimal vectors of L 24 n L 24 is equal to the number of symplectic automorphisms of L 24 , which can be easily computed using wATLASx. Ž2. Hermite’s inequality d 2 Ž L. G 34 NŽ L. 2 shows that d 2 Ž P48 p . G 27 ) 24, and from Proposition 3.1 we conclude that NŽ P48 p n P48 p . F 24 d 2 Ž P48 p . so that the minimal vectors of P48 p n P48 p are non-split. The same holds for N48 . Concluding Remarks. 1. In view of what was shown in Section 2, and of the various examples and counterexamples that have been investigated, we suggest the following conjecture: Conjecture. Ž S2 . holds for any lattice in dimension - 24. 2. One way to improve the results of Section 2 would be to get better estimates for gn, 2 than the known ones, which are certainly far from being optimal, even in low dimensions.
REFERENCES wATLASx J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘Atlas of Finite Groups,’’ Oxford Univ. Press, Oxford, 1985. wBx N. Bourbaki, ‘‘Algebre,’’ Chap. IX, Hermann, Paris, 1959. ` wB-Sx P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus, In¨ ent. Math. 117 Ž1994., 27]56. wWith an Appendix by J. H. Conway and N. J. A. Sloanx wC-Sx J. H. Conway and N. J. A. Sloane, ‘‘Sphere Packings, Lattices and Groups,’’ Grundlehren, Vol. 290, Springer-Verlag, Heidelberg, 1993. wC-S2x J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. I. Quadratic forms of small determinant, Proc. Roy. Soc. London Ser. A 418 Ž1988., 17]41. wCx R. Coulangeon, Reseaux k-extremes, Proc. London Math. Soc. Ž 3 ., 73 Ž1996., no. ´ ˆ 3, 555]574. wKx Y. Kitaoka, ‘‘Arithmetic of Quadratic Forms,’’ Cambridge Tracts in Math. Ž1993.. wL-Sx J. Leech and N. J. A. Sloane, Sphere packing and error-correcting codes, Canad. J. Math. 23 Ž1971., 718]745.
476 wMx wM-Hx wNx wRx
RENAUD COULANGEON
J. Martinet, ‘‘Les reseaux parfaits des espaces euclidiens,’’ Masson, Paris, 1996. ´ J. Milnor and D. Husemoller, ‘‘Symmetric Bilinear Forms,’’ Grundlehren, Vol. 73, Springer-Verlag, Heidelberg, 1973. G. Nebe, Some cyclo-quaternionic lattices, J. Algebra, in press. R. A. Rankin, On positive definite quadratic forms, J. London Math. Soc. Ž 2 . 28 Ž1953., 309]314.
JOURNAL OF ALGEBRA ARTICLE NO.
Minimal Vectors in the Second Exterior Power of a Lattice Renaud Coulangeon Dept. Uni¨ ersite´ Bordeaux I, 351 cours de la Liberation, ´ de Mathematiques, ´ ´ 33405 Talence Cedex, France Communicated by E¨ a Bayer-Fluckiger Received September 26, 1996
1. INTRODUCTION In wCx we studied the Rankin k-invariants which were defined by Rankin in wRx as a natural generalization of the classical Hermite invariant of a lattice in the following way: let L be a lattice of maximal rank in an n-dimensional Euclidean space E. We denote by d k Ž L. the minimal determinant of k-dimensional sublattices of L and put g k Ž L. s d k Ž L.rŽdet L. k r n. The upper bound of g k Ž L. for all lattices L in E is denoted by gn, k . It is readily verified that the k-dimensional sublattice spanned by a k-tuple of linearly independent vectors x 1 , . . . , x k in L depends only on its image x 1 n ??? n x k in the kth exterior power Hk L. Moreover, Hk E inherits a canonical Euclidean structure from the original scalar product on E, which is defined for split Žmulti-. vectors Žthat is, vectors which can be written as x 1 n ??? n x k . by:
Ž x 1 n ??? n x k . ? Ž y 1 n ??? n y k . [ det Ž x i ? y j . 1Fi , jFk . Thus, the norm Žor squared length. of a split non-zero vector x 1 n ??? n x k with respect to this scalar product coincides with the determinant of the sublattice spanned by x 1 , . . . , x k . In other words, one has:
d k Ž L . s min N Ž v . ¬ v g Hk L split 4 . 467 0021-8693r97 $25.00 Copyright Q 1997 by Academic Press All rights of reproduction in any form reserved.
468
RENAUD COULANGEON
Consequently, a natural question, which already arose in wCx, is to ask whether the usual minimal norm of Hk L, as a lattice in Hk E, is attained on split elements. If so, one has g k Ž L. s g ŽHk L., where g ŽHk L. denotes the Hermite number of Hk L. As usual, the vectors of minimal length of a lattice will be referred to as minimal vectors. We give in Section 2 a partial answer to that problem, in the case k s 2, using arguments similar to those used by Y. Kitaoka for the analoguous problem for the tensor product Žcf. wK, Chap. 7x.. This is essentially based on estimates for the Ranking constant gn, 2 . This nevertheless does not seem to apply to higher exterior powers. Conversely, we show in Section 3 that asymptotically, the answer to the above question is negative Žfor any k ., and for k s 2, we give explicit counterexamples in dimension 24 and 48. Our counterexamples Žexplicit or not., arise from the theory of symplectic lattices initiated by Buser and Sarnak in wB-Sx. The analogy with Steinberg’s result concerning the tensor product Žsee wM-H, II, Theorem 9.6x. is worth noticing.
2. SHORT VECTORS IN THE SECOND EXTERIOR POWER In the following, the property for a lattice L that the minimal vectors of L n L Žresp. Hk L. are split will be referred to as property Ž S2 . Žresp. Sk .. In the following statement, the notation ? x @ stands for the greatest integer less or equal to x. THEOREM 2.1. Let r 0 s sup r G 2 ¬ g 2 s, 2 - s for all 2 F s F r 4 . Then Ž S2 . holds for any lattice L with ?dim Lr2@ F r 0 . Proof. Let v be a minimal vector of L n L. Using well-known properties of bilinear alternating forms over principal rings, we contend that v can be written in the form r
vs
Ý x i n yi , is1
where x 1 , . . . , x r , y 1 , . . . , yr are 2 r linearly independent vectors in L Ž2 r F n. Žcf. wB, Sect. 5, No. 1, Theoreme ´ ` 1.. Such a decomposition is essentially canonical, in the sense that the Z-module spanned by x 1 , . . . , x r , y 1 , . . . , yr and a fortiori the integer 2 r Žthe rank of v ., depends only on v . Then the
469
MINIMAL VECTORS OF LATTICES
norm of v is given by r
NŽ v . s s
ž
r
Ý x i n yi is1
?
/ž
Ý x i n yi is1
/
Ý Ž x i n yi ? x j n y j . i, j
s
Ý Ž x i ? x j . Ž yi ? y j . y Ž x i ? y j . Ž yi ? x j . i, j
s Tr Ž AB y C 2 . , where A s ŽŽ x i ? x j .1 F i, jF r , B s ŽŽ yi ? y j ..1 F i, jF r , C s ŽŽ x i ? y j ..1 F i, jF r . Thus, denoting by L2 r the sublattice spanned by x 1 , . . . , x r , y 1 , . . . , yr , by G the corresponding Gram matrix G s Ž tAC CB ., and by J the 2 r = 2 r matrix Ž yI0 r I0r . we obtain NŽ v . s
1 2
Tr Ž G t JGJ . G
2r 2
Ž det G .
1r2 r
Ž det t JGJ .
1r2 r
s r Ž det L2 r .
1rr
,
where we used the inequality TrŽ MN . G mŽdet M det N .1r m , valid for any two symmetric positive definite matrices of degree m Žcf. wK, Chap. 7, Lemma 7.1.3x.. Consequently we have
d 2 Ž L2 r . G N Ž v . G r Ž det L2 r .
1rr
,
Ž 2.1.
so that r F g 2 Ž L2 r . F g 2 r, 2 . Therefore, if ?dim Lr2@ F r 0 , then r F r 0 , and the condition r F g 2 r, 2 implies r s 1. LEMMA 2.2. g 2 r, 2 - r for 2 F r F 5. Proof. It is based on known estimates of g 2 r, 2 for these values of r: r s 2: g4, 2 s 3r2 - 2 ŽRankin, cf. wRx.. r s 3: we use the general inequality established by Rankin, gn, k F g h, k Žgn, h . k r h for 1 F k F h F n, applied to k s 2, h s 4, n s 6. We obtain v v
g6, 2 F g4, 2 Ž g6, 4 .
1r2
s g4, 2 Ž g6, 2 .
1r2
2
whence g6, 2 F Ž g4, 2 . s 9r4 - 3.
r s 4: we obtain in the same way: g 8, 2 F Žg6, 2 . 3r2 F Ž3r2. 3 - 4. A better estimate for g 8, 2 is obtained using: g 8, 2 F g 7, 2 Ž g 8 . 2r7 F g6, 2 Žg 7 .1r3 Žg 8 . 2r7 F 9r2 10r7. v
470
RENAUD COULANGEON v
r s 5: using the above upper bound, we obtain: g 10, 2 F Žg 8, 2 . 4r3 -
5. COROLLARY 2.3.
If dim L F 11, L satisfies Ž S2 ..
Proof. Straightforward using Theorem 2.1 and Lemma 2.2. Remark. If one assumes that g 8, 2 s g 2 ŽE 8 . s 3, which is likely to be the case, it follows that g 12, 2 F 3 5r3 , but it does not allow us to extend our result to dimension 12, since 3 5r3 ) 6. If we restrict ourselves to the case of integral lattices, i.e., lattices for which the scalar product takes integral values, we can obtain results with no condition on the dimension: PROPOSITION 2.4. L satisfies Ž S2 ..
Let L be an integral lattice such that d 2 Ž L. F 8. Then
Proof. Setting v s Ý ris1 x i n yi for a minimal vector v of L n L, as in the proof of Theorem 1, and L2 r the 2 r-dimensional sublattice spanned by x 1 , . . . , x r , y 1 , . . . , yr , we have 8 G d 2 Ž L . G N Ž v . G r Ž det L2 r .
1rr
G r,
Ž 2.2.
since det L2 r is integral. If r F 5, Lemma 2.2 shows that v , as a minimal vector of L2 r n L2 r , has to be split, so we can assume that r s 6, 7, or 8. In the first case we see, applying Ž2.2. with r s 6, that the sublattice L12 associated to v is an integral 12-dimensional lattice with detŽ L12 . F ?Ž 86 . 6 @ s 5. Such lattices are classified Žcf. wC-S2x., and are easily seen to satisfy d 2 Ž L12 . F 3. Consequently, 3 G d 2 Ž L. G r Žby Ž2.2.., whence a contradiction. In the same way, if r s 7, then L14 is a 14-dimensional integral lattice of determinant F ?Ž 87 . 7 @ s 2, whence, using the classification of wC-S2x, d 2 Ž L14 . F 3, which again leads to a contradiction. Finally, for r s 8, L16 is a 16-dimensional unimodular lattice, so that r F d 2 Ž L. F d 2 Ž L16 . s 3, whence again a contradiction. This corollary applies in particular to the root-lattices A n , Dn , E 6 , E 7 , E 8 , and to Dq n. To conclude this section, we investigate a wide class of lattices, including the laminated lattices L n , as well as the sequence K n Žcf. wC-S, Chap. 6x for a definition., for which the following lemma holds: LEMMA 2.5. Assume that L contains a 2-dimensional hexagonal section with the same minimal norm, and that dim L - 20. Then L satisifes Ž S2 ..
MINIMAL VECTORS OF LATTICES
471
Proof. The inequality d 2 Ž L. G 34 N Ž L. 2 holds for any L ŽHermite’s inequality; cf. wM, chap. 2, Theoreme ´ ` 2.1x. and becomes an equality under the assumption of the lemma. Let v be a minimal vector of L n L. As in the previous proofs, we associate to it a sublattice L2 r of L, with r F ? nr2@. Assuming that v is not split, we can furthermore restrict ourselves to r G 6. From the inequality r Ždet L2 r .1r r F d 2 Ž L., joined to the fact that det L2 r G d 2 r Ž L., we obtain that
d2 r Ž L. F
ž
d2 Ž L. r
r
/
.
Ž 2.3.
Let M be a 2 r-dimensional minimal section; we have
gŽ M. G
N Ž L.
Ž d2 r Ž L. .
1r2 r
2
G
'3
'r ,
so that g 2 r G Ž2r '3 .'r , which implies, using classical upper bounds for gn Žnamely the actual value of gn for n s 1, . . . , 8, and Blichfeldt’s inequality gn F Ž2rp . G Ž2 q nr2. 2r n for n ) 8; cf. wK, Chap. 2, Theorem 2.2.1x., that r G 10 whence the conclusion. PROPOSITION 2.6. K n satisfy Ž S2 ..
For n F 23, the laminated lattices L n and the lattices
Proof. Let L n denote either L n or K n . We first notice that L n satisfies the assumption of Lemma 2.5 for any n, which gives the conclusion for n - 20. For 20 F n F 23, we let v be a non-split minimal vector of L n n L n , and denote by 2 r the rank of the associated sublattice Ž L n . 2 r Ž r F ? nr2@.. Assuming that v is not split, we can restrict ourselves to r G 6. From the proof of Lemma 2.5, it follows that
d 2 r Ž Ln . F
ž
d 2 Ž Ln . r
r
/
,
Ž 2.4.
and also g 2 r G Ž2r '3 .'r so that r G 10. Then we notice that d 2 r Ž L n . G d 2 r Ž L24 .. But the actual values of d k Ž L24 ., for k F 8 and for k G 16, are known Žcf. wC-S, Chap. 7, Corollary 2.7x or wL-Sx.: one has d 20 Ž L24 . s 64 and d 22 Ž L24 . s 12, which in both cases are strictly greater than the bound given by Ž2.4., whence the conclusion. Remark. It will be shown in the next section that L 24 does not satisfy Ž S2 ..
472
RENAUD COULANGEON
3. COUNTEREXAMPLES We now use the notion of symplectic lattices to exhibit counterexamples to the above property. Recall Žcf. wB-Sx and its Appendix. that a lattice L is symplectic if there exists an isometry s from L onto its dual lattice LU and such that s 2 s yId. Such a lattice has even dimension 2 r and determinant 1. PROPOSITION 3.1. Let L be a symplectic lattice of dimension 2 r. Then there exists in L n L a non-split ¨ ector with squared length r. Accordingly, NŽ L n L. F r. Proof. Let s be a symplectic isometry from L onto LU , and let e1 , . . . , e r , e rq1 , . . . , e2 r be a symplectic basis of L with respect to s , that is:
ž
Ž ei ? s e j . s J s
0 yIr
Ir 0
/
Ž such a basis exists since det L s 1 . .
In other words, denoting by eUi 4 the dual basis of e i 4 , we have, for 1 F i F r:
s Ž e i . s yeUrq1 ,
s Ž e rqi . s eUi ,
s Ž eUi . s ye rqi ,
s Ž eUrqi . s e i .
It follows that the element v s e1 n e rq1 q ??? qe r n e2 r has the required property, namely, NŽ v . s
Ý
Ž e i n e rqi . ? Ž e j n e rqj .
1Fi , jFr
s
Ý
Ž e i ? e j . Ž e rqi ? e rqj . y Ž e i ? e rqj . Ž e rqi ? e j .
1Fi , jFr
s
Ý
Ž e i ? e j . Ž eUi ? eUj . y Ž e i ? e rqj . Ž yeUi ? eUrqj .
1Fi , jFr
s Tr Ž AAU q C t CU . , where A s Ž Ž e i ? e j . . 1Fi , jFr ,
B s Ž Ž e rqi ? e rqj . . 1Fi , jFr ,
C s Ž Ž e i ? e rqj . . 1Fi , jFr ,
473
MINIMAL VECTORS OF LATTICES
and AU , BU , CU are the corresponding blocks, replacing e i by eUi . From the equality
ž
A t C
C B
/ž
AU t U C
CU s I2 r , BU
/
we deduce that NŽ v . s TrŽ Ir . s r. From this we derive the following asymptotic behaviour for property Ž S2 .: COROLLARY 3.2. In any e¨ en dimension 2 r G 180, there exist lattices L which do not satisfy Ž S2 .. Proof. Define m 2 r s max L g S 2 r g Ž L. s max L g S 2 rNŽ L., where S 2 r denotes the set of Žisometry classes . of symplectic lattices in dimension 2 r. From Buser and Sarnak’s inequality ŽŽ1.12. in wB-Sx.
m2 r G
s2 r
y1 rr
ž / 2
,
where sn s
p n r2 G Ž nr2 q 1 .
,
and Hermite’s inequality d 2 Ž L. G 34 N Ž L. 2 , we conclude that for any r there exists a 2 r-dimensional symplectic lattice M2 r with
d 2 Ž M2 r . G
3 s2 r 4
ž / 2
y2 rr
.
It is then easy to check that this lower bound is strictly greater than r for r G 90, whence the conclusion, applying the previous proposition. From this, we can deduce the following more general statement about property Ž Sk .: COROLLARY 3.3. For any k G 2, there exists an integer n k such that n-dimensional lattices which do not satisfy Ž Sk . exist for any n G n k . Proof. For k s 2 this is the previous proposition, with n 2 F 180. For k ) 2, we consider lattices of the type L e s L H Z e, where H stands for the orthogonal sum. One easily checks that Hk Ž L H Z e . s Hk L H Hky 1 L n Z e , and N Ž Hk Ž L H Z e . . s min N Ž Hk L . , N Ž e . N Ž Hky 1 L . 4 .
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RENAUD COULANGEON
We can choose e in such a way that NŽ e . - NŽHk L.rNŽHky 1 L., so that N Ž Hk Ž L H Z e . . s N Ž e . N Ž Hky 1 L .
and
S Ž Hk Ž L H Z e . . s S Ž Hky 1 L . n e, where S Ž M . stands for the set of minimal vectors of the lattice M. Then we can assume by induction that L does not satisfy Ž Sky 1 ., which is possible if the rank of L is greater than n ky 1 , and then conclude by the above property that Hk Ž L H Z e . itself does not satisfy Ž Sk .. In case k s 2, Corollary 3.2 also provides explicit counterexamples in lower dimensions. In dimension 24, we will consider the Leech lattice L 24 , which is symplectic Žsee for instance the Appendix to wB-Sx by J. Conway and N. Sloane where a symplectic basis is explicitly given.. In dimension 48 at least three even unimodular lattices of norm 6 are known, usually denoted by P48 p and P48 q for the first two ones Žcf. wC-S, p. 195x., and a third one N48 recently found by G. Nebe wNx. Furthermore P48 p is symplectic Žits automorphism group is isomorphic to 2 ? L2 Ž23. = S3 , and therefore contains elements of square yId Žcf. wATLASx.., and so does N48 wNx. Applying 3.1 to these lattices, we obtain: PROPOSITION 3.4. Ž1. NŽ L 24 n L 24 . s d 2 Ž L 24 . s 12 but L 24 n L 24 contains s1 s 2 6 ? 3 2 ? 5 3 ? 7 ? 13 ? 23 pairs of split minimal ¨ ectors and s2 s 2 6 ? 3 6 ? 5 2 ? 7 ? 11 ? 23 pairs of non-split ones which are in 1]1 correspondence with the symplectic automorphisms of L 24 . Ž2. Let L stand for P48 p or N48 . Then L n L contains only non-split minimal ¨ ectors. Proof. As we previously noticed ŽLemma 2.5., d 2 Ž L 24 . s 12, and it is attained exactly on the hexagonal planes spanned by two minimal vectors. Next we contend that the sublattice L s Ž L 24 . 2 r of L 24 associated to a non-split minimal vector v of L 24 n L 24 , if it exists, has dimension 24, i.e., r s 12 Žthe lower values of r are excluded by the arguments of Lemma 2.5 and Proposition 2.6.. Applying Ž2.1., we see that 12 s d 2 Ž L 24 . G N Ž v . G 12 Ž det L .
1r12
,
whence
w L 24 : L x s 'det L s 1. Therefore, L s L 24 and NŽ L 24 n L 24 . s NŽ v . s 12 s d 2 Ž L 24 .. By Proposition 3.1 such non-split vectors of rank 24 and squared length 12 in L 24 n L 24 exist. Moreover, setting v s Ý12 is1 x i n yi and denoting by G
475
MINIMAL VECTORS OF LATTICES
the Gram matrix of the vectors x 1 , . . . , x 12 , y 1 , . . . , y 12 , we have, using the notation of Theorem 2.1: N Ž v . s 12 Tr Ž G t JGJ . G 12 Ž det G .
1r12
G 12.
One easily checks that equality can hold if and only if t JGJ s Gy1 , that is to say, x 1 , . . . , x 12 , y 1 , . . . , y 12 is a symplectic basis of L 24 with respect to a symplectic automorphism s of L 24 . Then the map which to a symplectic automorphism s associates the element Ý12 is1 x i n yi , where x 1 , . . . , x 12 , y 1 , . . . , y 12 is Žany. symplectic basis of L 24 with respect to s , is clearly one to one, so that the number of non-split minimal vectors of L 24 n L 24 is equal to the number of symplectic automorphisms of L 24 , which can be easily computed using wATLASx. Ž2. Hermite’s inequality d 2 Ž L. G 34 NŽ L. 2 shows that d 2 Ž P48 p . G 27 ) 24, and from Proposition 3.1 we conclude that NŽ P48 p n P48 p . F 24 d 2 Ž P48 p . so that the minimal vectors of P48 p n P48 p are non-split. The same holds for N48 . Concluding Remarks. 1. In view of what was shown in Section 2, and of the various examples and counterexamples that have been investigated, we suggest the following conjecture: Conjecture. Ž S2 . holds for any lattice in dimension - 24. 2. One way to improve the results of Section 2 would be to get better estimates for gn, 2 than the known ones, which are certainly far from being optimal, even in low dimensions.
REFERENCES wATLASx J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker, and R. A. Wilson, ‘‘Atlas of Finite Groups,’’ Oxford Univ. Press, Oxford, 1985. wBx N. Bourbaki, ‘‘Algebre,’’ Chap. IX, Hermann, Paris, 1959. ` wB-Sx P. Buser and P. Sarnak, On the period matrix of a Riemann surface of large genus, In¨ ent. Math. 117 Ž1994., 27]56. wWith an Appendix by J. H. Conway and N. J. A. Sloanx wC-Sx J. H. Conway and N. J. A. Sloane, ‘‘Sphere Packings, Lattices and Groups,’’ Grundlehren, Vol. 290, Springer-Verlag, Heidelberg, 1993. wC-S2x J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. I. Quadratic forms of small determinant, Proc. Roy. Soc. London Ser. A 418 Ž1988., 17]41. wCx R. Coulangeon, Reseaux k-extremes, Proc. London Math. Soc. Ž 3 ., 73 Ž1996., no. ´ ˆ 3, 555]574. wKx Y. Kitaoka, ‘‘Arithmetic of Quadratic Forms,’’ Cambridge Tracts in Math. Ž1993.. wL-Sx J. Leech and N. J. A. Sloane, Sphere packing and error-correcting codes, Canad. J. Math. 23 Ž1971., 718]745.
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J. Martinet, ‘‘Les reseaux parfaits des espaces euclidiens,’’ Masson, Paris, 1996. ´ J. Milnor and D. Husemoller, ‘‘Symmetric Bilinear Forms,’’ Grundlehren, Vol. 73, Springer-Verlag, Heidelberg, 1973. G. Nebe, Some cyclo-quaternionic lattices, J. Algebra, in press. R. A. Rankin, On positive definite quadratic forms, J. London Math. Soc. Ž 2 . 28 Ž1953., 309]314.