LATTICES MODULO 2 AND 3 (COMMENTS)

JACQUES MARTINET

Abstract. We write a few complements (and an erratum) to the two papers Reduction Modulo 2 and 3 of Euclidean Lattices, published in J. Algebra (2002) and Contemporary Math. (2004), cited [LM1] and [LM2] below, respectively.

1. Erratum

In an e-mail dated May 14th, 2012, R. Idel pointed out to me that in Theorem 1.3 of [LM1], one must assume that d = 2 or 3. Fortunately larger values of d are not considered in the paper. Also, in (1), “orthogonal” holds if and only if N(y) = N(x) (= 2m). Table III, line 1, read s_{4} = 45.

2. Tables of [LM1]

2.1. Perfect lattices in dimension 8. Since [LM1] was written, Dutour-Sikirić, Schürmann and Vallentin proved that the list of perfect lattices listed in my homepage is indeed the complete list. In my homepage, they are now organized as a collection of six lists p8di, i = 7, 6, 5, 4, 3, 2 where i stands for the dimension of a largest perfect cross-section having the same minimum; p8d7 is a re-ordering of the former list of Laïhem lattices lh(j), j = 1, ..., 1175.

Two of these lattices are listed in Table 2.11: 𝔼_{8} = p8d7[1] and L_{8}^{4} = p8d7[4]. The others are

p8d7[181] (= lh(179)), p8d5[126] (= batu(5)), and p8d5[128] (= nap(160)) .

They could have been added to Table 2.12. For the first one, we have s_{6} = 40, s_{8} = 45, s_{10} = 96, s_{12} = 158 , whereas for p8d5[126] (resp. p8d5[128]) we need the consideration of all even norms between the minimum m = 14 (resp. m = 12) and 2m.

2000 Mathematics Subject Classification. 11H55, 11H71.

Key words and phrases. Euclidean lattices, classes modulo 2 First on line: June 2016; last update: March 5th, 2022.

2.2. Dimensions 7 and 8(June 10th, 2016). There is not a lot to say on Table 2.10, since all lattices lying in a small enough neighbourhood of any of the lattices listed there could be added to it, including all lattices of the half-open Voronoi path (𝔼_{6}—𝔼_{6}^{∗}].

I have then considered the Voronoi paths in dimension 7 of the form L—L′ where at least one of the lattices L, L′ is one of the three perfect lattices for which norm 2m suffices, namely P_{7}^{5}, P_{7}^{20} and P_{7}^{23}, and tested whether 2m would suffice on a neighbourhood of the convenient end point(s). This occurred for 10 paths, namely 3 paths P_{7}^{1}—P_{7}^{5} out of 10, 5 paths P_{7}^{5}—P_{7}^{5} out of 6, and the unique paths P_{7}^{5}—P_{7}^{20} and P_{7}^{5}—P_{7}^{23}.

Actually I can prove more: the norms between m and 2m suffice for the whole last seven paths (those which only involve P_{7}^{5}, P_{7}^{20} and P_{7}^{23}); for the first three, this holds only on an interval [t_{0}, 1] with 0 < t_{0} < 1.

This shows that 2m suffices in dimension 7 on some subsets of dimension 1 (even possibly of a larger dimension), whereas only isolated examples had been found in [LM1] and [LM2].

I have then turned to dimension 8, restricting myself to the perfect lattices p8d5[126] (s = 36) and p8d5[128] (s = 37), for which the small value of s makes easy the construction of Voronoi neighbours. I have detected two Voronoi graphs of the form p8d5[126]–𝔼_{8} and p8d5[128]–𝔼_{8} on which all lattices possess representatives modulo 2 of norm ≤ 2m. I also found a path p8d5[128]–𝔼_{8} on which the bound 2m applies uniquely for t = 0 and \frac{1}{3} ≤ t ≤ 1. This suggests that for many of the numerous paths of the form 𝔼_{8}–L the bound 2m suffices near 𝔼_{8}. Another example will be given in Subsection 2.3.

I state the results above as a proposition.

Proposition 2.1. For all dimensions n ≤ 8 the space of lattices up to similarity contain subsets of dimension d ≥ 1 in which all lattices possess representatives modulo 2 of norm not larger than twice their minimum. □

Whether examples of dimension n(n+1)/2 exist beyond dimension 6 remains an open problem, as well as the existence of non-isolated examples beyond dimension 8.

2.3. Extension of the Tables of [LM1] (extended on June 12th, 2016). An interesting example for Table 2.11 has been missed, namely the 5-modular lattice L of dimension 8 and minimum 4, strongly eutactic of perfection co-rank 1, the lattice stg60 of the file strongly eutactic lattices in the catalogue of Perfect Lattices of my homepage. Thus the following line could have been added to Table 2.11: n = 8 ℓ = 4 Λ_{8,5−mod} s_{4} = 60 s_{6} = 120 s_{8} = 300 . Here are two other definitions for L.

(a) \underline{Algebraic}. L = (𝔐, Trd_{H/ℚ}(x\overline{y})), where H is the quaternion fields with center ℚ(\sqrt{5}) ramified exactly at the two real primes of its center, 𝔐 is a maximal order in H, and Trd is the reduced trace. Its automorphism group, of order 28800 = 2^{7} · 3^{2} · 5^{2}, acts transitively on each of his layers of norm 4, 6, 8.

(b) \underline{Craig}. The cross-section of Craig’s lattice 𝔸_{9}^{(2)} (alias Barnes’s P_{9}) by the hyperplane orthogonal to the unique pair of minimal vectors of its dual is isometric to L.

Since the perfection co-rank of L is equal to 1, L belongs to a Voronoi path L(t), indeed connecting two copies of 𝔼_{8}. A matrix representation for such a path is U(t) = A + t M, of minimum 2, where

A = ( \begin{matrix}2 & −1 & −1 & 1 & 0 & 0 & 0 & 0 \\ −1 & 2 & 0 & −1 & 0 & 0 & 0 & 0 \\ −1 & 0 & 2 & −1 & −1 & 1 & 0 & 0 \\ 1 & −1 & −1 & 2 & 0 & −1 & 0 & 0 \\ 0 & 0 & −1 & 0 & 2 & −1 & −1 & 1 \\ 0 & 0 & 1 & −1 & −1 & 2 & 0 & −1 \\ 0 & 0 & 0 & 0 & −1 & 0 & 2 & −1 \\ 0 & 0 & 0 & 0 & 1 & −1 & −1 & 2 \end{matrix} ) and M = ( \begin{matrix}0 & 1 & 0 & −1 & 0 & 0 & 0 & 1 \\ 1 & 0 & −1 & 0 & 1 & 0 & 0 & 0 \\ 0 & −1 & 0 & 1 & 0 & −1 & 0 & 0 \\ −1 & 0 & 1 & 0 & −1 & 0 & 1 & 0 \\ 0 & 1 & 0 & −1 & 0 & 1 & 0 & −1 \\ 0 & 0 & −1 & 0 & 1 & 0 & −1 & 0 \\ 0 & 0 & 0 & 1 & 0 & −1 & 0 & 1 \\ 1 & 0 & 0 & 0 & −1 & 0 & 1 & 0 \end{matrix} ).

It turns out that all lattices L(t), 0 ≤ t ≤ 1, have representatives modulo 2 in the interval [2, 4]. The lattice L is a scaled copy of L(\frac{1}{2}), and is well defined as the eutactic lattice of the path L(t). This family of lattices has moreover the following two properties:

(1) For every t ∈ [0, 1], L(1 − t) is isometric to L(t) ;

(2) For every t ∈ [0, 1], L(t)^{∗} is similar to L(t). This results from the two formulae

U(1 − t) = ^{t}P U(t)P and ^{t}Q((−t^{2} + t + 1)U(t)^{−1})Q = U(t) , where

P = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & 0 & −1 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & −1 & 0 & 0 \\ 0 & 0 & 0 & −1 & 1 & −1 & 0 & 1 \\ 0 & 0 & 0 & −1 & 0 & 0 & 0 & 0 \\ 0 & −1 & 1 & −1 & 0 & 0 & 0 & 1 \\ 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 1 & −1 & 0 & 0 & 0 & 0 & 0 & 1 \end{matrix} ) and Q = ( \begin{matrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & −1 \\ 0 & 0 & 0 & 0 & −1 & 1 & 1 & −1 \\ 0 & 0 & 0 & 0 & 0 & 0 & −1 & 1 \\ 0 & 0 & 0 & 0 & 1 & 0 & −1 & 0 \\ 0 & 1 & 0 & −1 & 0 & 0 & 1 & 0 \\ 0 & −1 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & −1 & 1 & 1 & −1 & 0 & 0 & 0 \\ 1 & 1 & −1 & 0 & 0 & 0 & 0 & 0 \end{matrix} ) .

This Voronoi path is an analogue of the Voronoi path 𝔻_{4}—𝔻_{4}, having as mid point the 4-modular, strongly eutactic lattice 𝔸_{2}⊗𝔸_{2}, all lattices of which also have representatives modulo 2 in the interval [2, 4].

3. Odd Lattices (Section 4 of [LM2])

Theorem 4.2 of [LM2] shows that in an integral lattice of minimum m all layers up to norm 2m+1 must be considered when listing shortest representatives of classes modulo 2. This theorem gives a precise description of classes of norm 2m + 1 when m is odd, and is then applied to 𝔼_{7}^{∗} scaled to minimum 3.

(February 21st,2022) An example with m even is provided by the lattice 𝔻_{10}^{+} scaled to minimum 4. One must then consider the layers of norms 4, 5, 8 and 9, and the weighted formula reads

90 + 256 + (\frac{10}{10} + \frac{1680}{4}) + \frac{2560}{10} = 2^{10} − 1 .

On the basis (ε_{i}) for ℤ^{10} rescaled to N(ε_{i}) = 2, vectors of norm 9 congruent modulo 2 are permutations of \frac{−3ε_{1}±ε_{2}±···±ε_{10}}{2} (with an even

number of minus signs), of norm 2 × \frac{3^{2} + 9 · 1^{2}}{4} = 9, and we have

\frac{−3ε_{1} + ε_{2} + ε_{3} + ⋯ + ε_{10}}{2} · \frac{ε_{1} − 3ε_{2} + ε_{3} + ⋯ + ε_{10}}{2} = 2 × \frac{−6 + 8}{4} = 1 ,

as predicted by Theorem 4.2 of [LM2].

4. On Lemma 3.2 of [LM2]

(Completed February 21st, 2022) This lemma states that an integral lattice of minimum 4 which is the even sublattices of an integral lattice of minimum 3 contains classes modulo 2 of minimal norm 12. The converse is not true: given a class of minimal norm 12 represented by a vector X and a Gram matrix A for Λ, \frac{1}{2}X defines an integral lattice of minimum 3 if and only if AX ≡ 0 mod 2. Such an example is provided by the lattice K′_{15}; see below.

Inside the Leech lattice, the series Λ_{n} has been dealt with in [LM2]. Representatives of norm 8 exist for n ≤ 6 and n = 8, 9, 10, 24 and for no other dimensions. These lattices are even sublattices of integral lattices of minimum 3 for Λ_{7} ∼ 𝔼_{7}, Λ_{11}^{max}, Λ_{12}^{mid}, Λ_{12}^{max}, and all Λ_{n} for 13 ≤ n ≤ 23. There remains the cases of Λ_{11}^{min} and Λ_{12}^{min}, for which the four norms 4, 6, 8, 10 are needed.

Besides the series Λ_{n}, two important other series exist inside the Leech lattice Λ_{24}, that we are going to investigate:

• the series K_{n} for 7 ≤ n ≤ 17 (K_{n} = Λ_{n} if n ≤ 6 or n ≥ 18);

• the series K′_{n} for 3 ≤ n ≤ 10 and 14 ≤ n ≤ 21 (K′_{n} = K_{n} if n = 11, 12, 13; K′_{n} = Λ_{n} if n ≤ 2 or n ≥ 22),

that we consider below for n ≤ 15.

We know from [LM1] that representatives of norm ≤ 8 exist for K_{7}, K_{8} and K_{12}. For K_{n}, n = 9, 10, 11, the smallest norms are 4, 6, 8, 10. Finally, K_{13}, K_{14}, K_{15} requer norms 4 to 12, and are the even sublattices of lattices L_{13}, L_{14}, L_{15} of minimum 3.

Norms 4 and 6 suffice for K′_{3}, K′_{4}, K′_{5}, K′_{6}, then norms up to 8 for K′_{8}, up to 10 for K′_{7} and K′_{14}, and norms up to 12 are needed for K′_{9}, K′_{10} and K′_{15}; K′_{9} and K′_{10} are even sublattices of integral lattices of minimum 3, but not K′_{15}, for which the numbers of vectors of each of these norms satisfy the formula

822 + 10116 + 15687 + 6140 + 2 = 32767 = 2^{15} − 1 , which shows that norm 12 is needed.

The Kneser neighbour of K_{12} with respect to a norm-4 vector has a decomposition ℤ ⊥ 3ℤ ⊥ K′_{10odd}, where K′_{10odd} is a 3-modular, strongly eutactic lattice of minimum 3, with s = 40; its even sublattice is isometric to K′_{10}. By cross-section, one obtains a lattice K′_{9odd} with even sublattice K′_{9}.

References

[D-S-V] M. Dutour Sikirić, A. Schürmann, F. Vallentin, Classification of eight dimensional perfect forms, Electron. Res. Announc. Amer. Math. Soc. 13 (2007), 21–32; preprint at arXiv:math/0609388.

[LM1] J. Martinet, Reduction Modulo 2 and 3 of Euclidean Lattices, J. Algebra 251 (2002), 864–887.

[LM2] J. Martinet, Reduction Modulo 2 and 3 of Euclidean Lattices, II (Acts of the Conference on Quadratic Forms, Talca, Chile, December 2002), Contemporary Math. 344 (2004), 265–273.

[PARI] PARI/GP, a computer algebra system designed for fast computations in number theory, url http://pari.math.u-bordeaux.fr/ .

Jacques Martinet, Villa 30, Hameau de Noailles, 33400 Talence, France

Email address: Jacques.Martinet@math.cnrs.fr