ON INTEGRAL LATTICES HAVING AN ODD MINIMUM

JACQUES MARTINET AND BORIS VENKOV

Abstract. We study the kissing number of integral lattices of odd minimum, with special emphasis on the case of minimum 3.

1. Introduction.

In this paper, we consider the problem of the kissing number for integral lattices having an odd minimum m, with special emphasis on the case where m = 3. Thus we try to give some precisions about the set of possible values for the number s of pairs of minimal vectors on such a lattice, and in particular on its maximal value s_{max} for given minimum m and dimension n. Our results include (see below):

• A list of upper and lower bounds for m = 3 and 1 ≤ n ≤ 24.

• The value of s_{max} for all odd m in the range 1 ≤ n ≤ 7 and moreover for m = 3 when n = 8, 9, 16, 22, 23.

• The complete list of possible s for m = 3 and n ≤ 8.

We may and most of the time shall restrict ourselves to well rounded lattices, i.e. lattices possessing n independent minimal vectors, and even to lattices which are generated by their minimal vectors.

Our results rely on the use of various techniques arising from the theories of spherical designs, root systems, Watson’s study ([W2]) of lattices which do not possess any hexagonal cross–section having the same minimum, classification according to the index of well–rounded sublattices, and estimations for the minimum of the dual lattice; we also used calculations with the PARI package. However, the reader may check that proofs of the bounds for s_{max} displayed in the table below, except in dimension 9, do not rely on heavy computations; more generally, handy-computational techniques suffice to obtain up to dimension 8 the classification of lattices of minimum 3 having a relatively large kissing number.

Key words and phrases. Euclidean Lattices, Kissing Number First author: Laboratoire A2X, UMR 5465 and FR 2254 The second author acknowledges support from the Swiss National Science Foundation.

Here are the most important results which we prove in this paper.

Theorem 1.1. Lower and upper bounds for the values of s on an integral lattice of minimum 3 are as displayed in the table below:

Table 1.1. Lower and upper bounds for s_{3}(n), n ≤ 24.

\begin{matrix}n & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ s_{3}(n) ≥ & 1 & 2 & 4 & 6 & 10 & 16 & 28 & 30 \\ s_{3}(n) ≤ & 1 & 2 & 4 & 6 & 10 & 16 & 28 & 30 \\ n & 9 & 10 & 11 & 12 & 13 & 14 & 15 & 16 \\ s_{3}(n) ≥ & 34 & 40 & 52 & 68 & 88 & 112 & 160 & 256 \\ s_{3}(n) ≤ & 34 & 63 & 81 & 103 & 129 & 162 & 203 & 256 \\ n & 17 & 18 & 19 & 20 & 21 & 22 & 23 & 24 \\ s_{3}(n) ≥ & 288 & 352 & 448 & 640 & 896 & 1408 & 2300 & 2301 \\ s_{3}(n) ≤ & 322 & 411 & 531 & 703 & 965 & 1408 & 2300 & 4991 \end{matrix}

Moreover, in dimensions 3–9, 16, 22, 23, for which the upper and lower bounds in the table above coincide, lattices on which these bounds are attained are unique.

The lower bounds are proved in Section 2 and the upper bounds in Section 3, except for dimensions 8 and 9, for which we need techniques to be developed later in Sections 8–10. We also prove in Section 3 the uniqueness assertions for n = 16, 22 and 23. A proof of those for low dimensions is given in Section 4, where we introduce root systems associated with suitable sublattices. (More subtle constructions of root systems will occur in Section 9.)

Section 5, which we shall not use in the sequel, is devoted to lattices having any odd minimum m ≥ 3 in the range 2 ≤ n ≤ 7. To state precisely our result, we first introduce some notation. We say that a lattice Λ is primitive if it is integral and the scalar products on Λ generate ℤ. Given integers m ≥ 1 and n ≥ 2, we denote by s_{m}(n) the maximal value of s on primitive n-dimensional lattices of minimum m. (This makes sense for all m and all n ≥ 2, but ℤ is the only primitive 1-dimensional lattice.)

Theorem 1.2. Let m ≥ 5 odd. We have s_{m}(n) = s_{3}(n) for 2 ≤ n ≤ 6 and s_{m}(7) = 27 < s_{3}(7) = 28.

Define the index of a well–rounded lattice Λ of dimension n to be the largest possible value for the index [Λ : L] where L is generated by n independent minimal vectors in Λ.

In section 6, we consider essentially lattices of index 1 and dimension n ≤ 6. We describe the minimal classes (in the sense of [M], Chapter 9) for lattices of odd minimum in dimensions n ≤ 5; general results under the less restrictive condition that the lattices do not possess hexagonal sections having the same minimum are displayed in Appendix 1. We obtain a complete description of the possible values of s for all odd m in dimensions n ≤ 5, as well as the analogous 6-dimensional results for m = 3 and m = 5. (Probably, the result we obtain for m = 5 holds for all m ≥ 5 odd.)

In Section 7, we calculate the maximal value of the index up to dimension 8 under various restrictions, notably under the hypothesis that Λ be similar to an integral lattice having an odd minimum. These results are then applied to lattices of dimension n ≤ 6. We also consider more precisely integral lattices of minimum 3 in dimensions up to 8; in particular, we characterize 7-dimensional integral lattices of index 3 (and minimum 3).

The remainder of the paper essentially deals with integral lattices of minimum 3.

In Section 8, we prove new upper bounds for the minimum of the dual lattice of an integral lattice of minimum m = 3 and dimension n ≤ 9. Our bounds in dimensions 8 and 9 (1 and \frac{4}{3}) are most certainly far from being sharp.

Section 9, in which we utilize the results of Section 8, is devoted to the construction of root systems that we can attach (under some severe restrictions) to integral lattices of minimum 3. These constructions are used to bound from above the difference s(Λ)−s(Λ_{0}) where Λ_{0} denotes a suitably chosen hyperplane cross–section. We also use them to prove the uniqueness assertions of Theorem 1.1 for 3 ≤ n ≤ 8.

More precise results for dimensions 7 and 8 are proved in Section 10, in which we establish in particular the list of all possible kissing numbers. Section 11 is devoted to dimension 9, and four appendices complete this paper.

The results we prove for dimension 9 give us an opportunity for making two comments.

(1) The bound s ≤ 34 that we shall prove in Section 11 significantly improves on the bound s ≤ 49 of [V], indeed valid for any system of norm 3 vectors having mutual scalar products ±1. However, under this weaker hypothesis, a system with s = 48 was found by Neumaier ([Neu]); see the final remark in Section 3.

(2) In the whole paper, we have focused on large values of s. Lattices having a small s, say, slightly larger than n, cannot be reasonably expected to be classified. Indeed, we meet here a problem in graph theory: consider the set Γ of graphs of valence 3 with n vertices; with such a graph, we associate its adjacency matrix A and then the matrix B = 3 I_{n} − A; this matrix is positive, semi-definite, and defines a relative lattice, which is integral of minimum ≤ 3 and has a small kissing number. To classify graphs of a given valency is considered as being a highly complicated problem. So the same conclusion should hold for lattices having a low kissing number.

2. Constructions of lattices.

In this section, we explain various constructions of lattices of minimum 3, which will in particular provide examples of lattices on which the lower bounds of Table 1.1 are attained.

Given a lattice Λ and a > 0, we denote by ^{a}Λ the lattice Λ equipped with the scalar product a (x · y).

Let m be a positive integer. In dimension 1, there exists for all m a unique lattice of minimum m, namely ^{m}ℤ; for n = 2, there are ⌊\frac{m+1}{2}⌋ well– rounded lattices, defined by a basis (e_{1}, e_{2}) such that e_{1} · e_{1} = e_{2} · e_{2} = m and e_{1} · e_{2} = i, 0 ≤ i ≤ ⌊\frac{m−1}{2}⌋, hence two lattices if m = 3.

In the remainder of this section, we restrict ourselves to the case where m = 3.

Our first construction is an adaptation of various lamination procedures to the problem of the kissing number. It produces lattices denoted by W_{n}, or maybe W_{n}^{a}, W_{n}^{b}, etc. when they are not unique. We set W_{1} = ^{3}ℤ, take for W_{2} the lattice for which e_{1} · e_{2} = 1, and construct the further W_{n} by induction: starting with a lattice W_{n} of minimum 3 having a basis (e_{1}, …, e_{n}) of minimal vectors, we consider up to isometry (n + 1)-dimensional lattices of minimum 3 which have a basis (e_{1}, …, e_{n}, e_{n+1}) extending the previous one and for which s is as large as possible. These are the lattices W_{n+1}.

Theorem 2.1. The W_{n} lattices are unique in the range 1 ≤ n ≤ 16, except for n = 10 and n = 11, where there are two of them. For n ≤ 14, s(W_{n}) is the lower bound for s displayed in Table 1.1; we have s(W_{15}) = 140 and s(W_{16}) = 156. [In dimensions 10 and 11, we distinguish W_{n}^{a} and W_{n}^{b} by the properties of their duals: the annihilator of Λ^{∗}/Λ is 4 for Λ = W_{10}^{a} and 16 for Λ = W_{10}^{b} (in both cases, s(Λ^{∗}) = 4); we have s(W_{11}^{a}^{∗}) = 19 and s(W_{11}^{b}^{∗}) = 27, (in both cases, the annihilator of Λ^{∗}/Λ is 4). We have the inclusions W_{10}^{a} ⊂ W_{11}^{a}, W_{10}^{a} ⊂ W_{11}^{b} and W_{10}^{b} ⊂ W_{11}^{a}.]

Sketch of proof. We have used a computer to list all possible extensions of a given W_{n} with n ≤ 15. In all cases, we found lattices with a determinant equal to the determinant found by Plesken and Pohst in [Pl-P] for their arithmetical laminations for minimum 3. We could in this way identify the W_{n} with some of the Plesken–Pohst lattices. □

The notation W_{n} has been chosen after Watson, who discovered them in [W2] up to dimension 7. We have W_{7} ≃ ^{2} 𝔼_{7}^{∗}, but W_{7} is better understood as the pull back in ℤ^{7} of the binary [7, 4, 3] Hamming code. The lattices W_{6}, W_{5}, … are then obtained as antilaminations of W_{7} (see the definition below).

We now briefly explain explicit constructions for dimensions up to 12.

Given a lattice Λ ⊂ ℤ^{n}, we easily construct lattices of dimension n + 1 and n + 2 by the following trick: adjoin to Λ the vectors (0^{n−1}, 1, 1, 1) ∈ ℤ^{n+2} and then (0^{n−2}, 1, 0, 0, 1, 1) ∈ ℤ^{n+3}, and consider the sections of these lattices first by the hyperplanes x_{n+1} + x_{n+2} = 0 in ℝ^{n+2}, then by the hyperplane x_{n+1} + x_{n+2} + x_{n+3} = 0 in ℝ^{n+3}. We obtain lattices in dimension n + 1 with s = s(Λ)+2, then in dimension n+2 with s = s(Λ)+6. Taking Λ = W_{7}, we obtain in this way W_{8} and W_{9}. Identifying two weight 3 words in the concatenation of two copies of the Hamming code, we obtain a [12, 8, 3] code possessing 17 weight 3 words. Its pull back in ℤ^{12} has s = 17 · 4 = 68; it is the lattice W_{12}. Its dual has two orbits of minimal vectors, with corresponding hyperplane sections W_{11}^{a} and W_{11}^{b}; repeating the antilamination procedure yields all the W_{n} with n ≤ 10.

Let Λ be a lattice, with minimum m. By the antilaminations of Λ, we mean the descending chain of successive hyperplane sections of Λ of the lattices having the highest Hermite invariant. The antilaminations of O_{23} (the unique 23-dimensional unimodular lattice of minimum 3) are described in [Bt-M], from which we can extract the following theorem:

Theorem 2.2. There is a unique “antilaminated” lattice in O_{23} for each of the dimensions n = 23 to n = 14, denoted by O_{n}. The values of s(O_{n}) are the lower bounds for s which are displayed in Table 1.1. We have s(O_{n}) > s(W_{n}) for n = 15, 16 and s(O_{14}) = s(W_{14}), but O_{14} and W_{14} are not isometric. □

Proof of the lower bounds in Theorem 1.1. Consider the lattices W_{n} for 1 ≤ n ≤ 14, O_{n} for 14 ≤ n ≤ 23, and O_{23} ⊥ W_{1} for n = 24. □

We now indicate a way of obtaining integral lattices of minimum 3 from even integral lattices of minimum 4, for which numerous examples are known.

Proposition 2.3. An integral lattice Λ of minimum 3 with s > n possesses norm 4 vectors.

Proof. Let S(Λ) = {±e_{1}, …, ±e_{s}}. If the e_{i} are pairwise orthogonal, then s ≤ n. Otherwise, there exists i, j such that e_{i} · e_{j} = ±1. Then N(e_{i} ∓ e_{j}) = 4. □

Theorem 2.4. An integral lattice of minimum 3 contains vectors of norm 4 if and only if there exists an even lattice L of minimum 4 and a vector e ∈ L which satisfies the following three conditions:

(1) N(e) = 12.

(2) e is not congruent modulo 2 to a shorter vector of L.

(3) e ∈ 2L^{∗}. [Note that, because of the identity N(x + 2z) − N(x) = 4(x.z + N(z)), y ≡ x mod 2L ⇒ N(y) ≡ N(x) mod 4, so that it suffices to consider vectors of norm 4 or 8 to test condition (2).]

Proof. If Λ exists, let L = Λ_{even}. Then, min L = 4. Let f ∈ S(Λ), and set e = 2f. Then, N(e) = 12, and e ∈ 2 L^{∗}, since all scalar products e·x = 2 f ·x are even on L, thus for all x ∈ Λ. Moreover, if e′ ≡ e mod 2Λ, then \frac{e′}{2} is a non-zero vector of Λ, which implies N(e′) ≥ 4 min Λ ≥ 12.

Conversely, given L of norm 4, let e ∈ L which satisfies the three conditions of the theorem. Set f = \frac{e}{2} and Λ = ⟨L, f⟩. Then, L = Λ ∪ (f + Λ). Thanks to condition (3), all scalar products f · x, x ∈ L are integral. Moreover, given x, y ∈ L, we have the equality

(f + x) · (f + y) = N(f) + f · (x + y) + x · y ,

which shows that Λ is integral. Finally, for x ∈ Λ, we have either x ∈ L, hence x = 0 or N(x) ≥ 4, or x = f + y , y ∈ L, hence 2x ≡ e mod 2 L, whence N(2x) ≥ 12. □

This construction transforms for 14 ≤ n ≤ 23 the laminated lattices Λ_{n} into O_{n} and similarly, one obtains W_{11}^{b}, W_{12}, W_{13} from Λ_{11}^{max}, Λ_{12}^{mid}, Λ_{13}^{max}. Also, two important examples show up in dimension 10. The lattices K′_{10} ([M], Chapter 8, Definition 5.8) and Q_{10} (a 4-modular lattice discovered by Souvignier) yield integral lattices of minimum 3, which have the same value of s (s = 40) as W_{10}^{a} and W_{10}^{b}. [Curiously, K′_{10} and Q_{10} play an important rôle in Nebe and Venkov’s classification of 10-dimensional strongly eutactic lattices; compare [N-V], Theorem 3.6.]

We now apply for further use the theorem above to root lattices (rescaled to minimum 4). We restrict ourselves to the cases where the lattices Λ of minimum 3 we obtain are well rounded. The results, the proofs of which we leave to the reader, are summarized in the table below, which contains an integer n, an n-dimensional root lattice L, the kissing number of the unique lattice Λ of minimum 3 coming from ^{2}L, and the name of Λ if any.

Table 2.5. Application of root lattices.

n 3  4  5 6 7  7  8  9  n ≥ 5
L 𝔸_{1}^{⊥3} 𝔸_{3} ⊥ 𝔸_{1} 𝔸_{5} 𝔻_{6} 𝔼_{7} 𝔼_{6} ⊥ 𝔸_{1} 𝔼_{7} ⊥ 𝔸_{1} 𝔼_{8} ⊥ 𝔸_{1} 𝔻_{n−1} ⊥ 𝔸_{1}
s 4  6  10 16 28  10  12  16  2(n − 1)
Λ W_{3}  W_{4} W_{5} W_{6} W_{7}

[The lattice constructed from 𝔻_{n−1} ⊥ 𝔸_{1} (indeed, also for n = 3 or 4) possesses a basis (e_{1}, …, e_{n}) of minimal vectors such that the other minimal vectors (up to sign) are e_{1} + e_{2} + e_{3}, …, e_{1} + e_{2} + e_{n}.]

We now describe briefly two constructions based on weight 6 codes.

For the first construction, we consider the lattice 𝔸_{1}^{⊥n} endowed with its canonical basis (ε_{1}, … ε_{n}), its “doubly even sublattice” L_{n} (the scaled copy of 𝔻_{n} defined by the congruence ∑_{i} x_{i} ≡ 0 mod 2), and a non-zero, selfdual code C (i.e., such that C ⊂ C^{⊥}), of length n and weight 6. Let Λ_{C} be the lattice generated by L_{n} and the vectors \frac{1}{2} (a_{1}ε_{1} + ⋯ + a_{n}ε_{n}) where (a_{1}, …, a_{n}) reduce modulo 2 to a word of C. Then Λ_{C} is an integral lattice of minimum 3, and s(Λ_{C}) = 16 t, where t denotes the number of weight 6 words in C. For instance, if n = 6 and C the one-word code, Λ_{C} is isometric to W_{6}. These codes have been studied by Bachoc and Gaborit in [B-G]. From [B-G], we can extract examples in dimensions 6, 10, 14, 15, 16, 19, 20, and 21 for which s(Λ_{C}) meets the lower bound displayed in Table 1.1.

For the second construction, we start with the lattice ℤ ⊥ 𝔸_{1}^{⊥(n−1)}, and consider a self-dual code C of length n−1 and weight 6. This time, we attach to C the lattice Λ′_{C} generated by L_{n−1}, the vectors \frac{1}{2} (a_{1}ε_{1} +⋯+a_{n−1}ε_{n−1}) as above, and the 2(n − 1) vectors ε + τ for τ a root of 𝔸_{1}^{⊥n−1}. Then Λ′_{C} is again an integral lattice of minimum 3, with s = 16 t+2 (n−1). For instance, if n = 7 and C is the one-word code, Λ′_{C} is isometric to W_{7}. From [B-G], we can extract examples in dimensions 7, 11, 13, and 17 for which s(Λ_{C}) meets the lower bound in Table 1.1.

3. Upper bounds from spherical designs theory.

We first state a theorem due to the second author:

Theorem 3.1. Let S ⊂ ℝ^{n} be a finite symmetric set whose elements are 2s norm 3 vectors with pairwise scalar products 0, ±1, ±3. Then, the following upper bounds for s hold:

(1) ∀ n ≤ 8, s ≤ \frac{8n}{9 − n}.

(2) ∀ n ≤ 24, s ≤ \frac{8n(n + 2)}{25 − n}.

Proof. See [V], Theorem 7.13. □

The proof given in [V] of the two assertions above consists in comparing two inequalities. In both cases, one of them is a Cauchy-Schwarz inequality [f, f] ≥ 0 for the scalar product on the set of degree 2 harmonic polynomials (see [V], Section 1), an inequality which properly belongs to the theory of spherical 2-designs. In case (2), a second inequality is obtained in the same way, using this time degree 4 polynomials; in case (1), one simply writes that the number of pairs of orthogonal vectors is bounded from below by zero. This implies:

Theorem 3.2. Equality holds in 3.1 (1) if and only if S is a spherical 3design and S does not contain any pair of orthogonal vectors; equality holds in 3.1 (2) if and only if S is a spherical 5-design. [We refer to [V], Section 3 for the definition of a spherical design.] □

The integral lattices of minimum 3 whose sets of minimal vectors constitute a spherical 5-design have been classified in [V] (Theorem 7.4). In our notation, this results reads:

Theorem 3.3. The set of minimal vectors of an integral lattice Λ of minimum 3 is a spherical 5-design if and only if Λ is isometric to W_{1}, W_{7}, O_{16}, O_{22}, or O_{23}. □

Using Theorem 3.1 and taking into account Theorem 3.3, which tells us when the inequality in 3.1 (2) is strict, we justify the upper bounds given in Theorem 1.1 except for n = 8 and n = 9; we also obtain the uniqueness assertions for n = 7, 16, 22 and 23.

To complete the proof of Theorem 1.1, there just remains to prove the upper bounds stated for dimensions 8 and 9 and the uniqueness assertions for dimensions 3–8. A quick proof of the uniqueness assertions for n = 3, 5, 6, 7 will be given in the next section; easy proofs for n = 3 and 4 can be read at the beginning of Section 6.

Remark 3.4. The set of 2s = 96 vectors in ℤ^{9} with mutual scalar products ± 1, constructed by Neumaier in [Neu], and referred to at the end of the introduction, is constructed in the following way: start with the affine plane P over 𝔽_{3}, and identify its nine elements with the canonical basis B = (ε_{1}, …, ε_{9}) of ℝ^{9}; a line in P contains three points, corresponding to 3 basis vectors ε_{i}, ε_{j}, ε_{k}, to which correspond the 8 vectors ε_{i} ± ε_{j} ± ε_{k} ∈ ℤ^{9}. The 12 affine lines in P thus define 48 × 2 = 96 vectors in ℤ^{9}.

4. A first construction of root systems.

We still denote by Λ an n-dimensional integral lattice of minimum 3.

Proposition 4.1. Let T be a non-empty set of minimal vectors in Λ, stable under the map x ↦ −x, and which does not contain any pair of orthogonal vectors. Set r = rk T and t = \frac{1}{2} |T |. Let e ∈ T, and set

T_{e}^{+} = {x ∈ T | e · x = +1} .

(T is the disjoint union of T_{e}^{+} and T_{e}^{−} = −T_{e}^{+}.)

(1) If all scalar products x · y for x, y ∈ T_{e}^{+}, y ≠ x are equal to +1, then t = r, and the vectors e − x, x ∈ T_{e}^{+} constitute a basis for a scaled copy to minimum 4 of the root lattice 𝔸_{r−1}.

(2) Otherwise, the lattice L generated by the vectors e − x, x ∈ T_{e}^{+}, and x + y, x, y ∈ T_{e}^{+}, x · y = −1 is a scaled copy to minimum 4 of a rank r root lattice.

Proof. For x, x′ ∈ T_{e}^{+}, we have N(x ± x′) = 6 ± x · x′, which shows that the vectors \frac{1}{\sqrt{2}} (e − x) and \frac{1}{\sqrt{2}} (x + y) when x · y = −1 have norm 2 and integral mutual scalar products. Hence they generate a root lattice.

Under hypothesis (1), let e_{1} = e, e_{2}, …, e_{t} be the vectors of T_{e}^{+}. The Gram matrix of the t − 1 vectors \frac{1}{\sqrt{2}} (e_{1} − e_{i}) has entries 2 on the diagonal and 1 outside. Thus it is a Gram matrix for 𝔸_{t−1}. In particular, it has rank t − 1, which shows that

rk(e_{1}, …, e_{t}) = rk(e_{1}, e_{1} − e_{2}, …, e_{i} − e_{t}) = t ,

hence that t = r.

Under hypothesis (2), let e_{1}, …, e_{r} be independent vectors in T_{e}^{+}. The vectors e_{1} − e_{k}, together with one vector e_{i} + e_{j}, constitute a rank r system. Choosing i, j with e_{i} · e_{j} = −1 proves the second assertion. □

Corollary 4.2. If Λ is well rounded, if S(Λ) contains no pair of orthogonal vectors, and if s(Λ) > n, then Λ can be constructed by the procedure of Theorem 2.4, using as lattice L the scaled copy to minimum 4 of a rank n root lattice.

Proof. Since s > n, hypothesis (2) of Proposition 4.2 is satisfied. We clearly have Λ = ⟨L, e_{1}⟩, and this construction of Λ from L is precisely that of Theorem 2.4. □

Proof of the uniqueness assertions for n=3–7. By Theorems 3.1 and 3.2, the scalar product of two minimal vectors in a lattice of the dimensions above on which s attains its maximum are non-zero. Hence we may apply the corollary above, and the conclusion follows from the results displayed in Table 2.5. □

5. A theorem of Watson.

In this section, we use a theorem of Watson to prove Theorem 1.2. We consider general lattices in the Euclidean space ℝ^{n} (our lattices are no more assumed to satisfy integrality conditions). To state Watson’s theorem referred to above, we recall the notion of a minimal class; see [M], Chapter 9 for an extensive study: these are the classes for the equivalence relation

ΛRΛ′ ⟺ ∃ u ∈ GL_{n}(ℝ) such that u(Λ) = Λ′ and u(S(Λ)) = S(Λ′) .

A class is clearly a union of similarity classes of lattices. We define on the set of minimal classes an ordering relation by

cl(Λ) ≺ cl(Λ′) ⟺ ∃ u ∈ GL_{n}(ℝ)
such that u(Λ) = Λ′ and u(S(Λ)) ⊂ S(Λ′) .

The perfection rank of a lattice Λ is the rank r in the space Sym_{n}(ℝ) of symmetric endomorphisms of the set of orthogonal projections onto the minimal vectors of Λ; we have 1 ≤ r ≤ \frac{n(n+1)}{2}; the difference r′ = \frac{n(n+1)}{2} − r is the perfection corank of Λ. These are invariants of the minimal class of Λ. The similarity classes of lattices belonging to a class of corank r′ are represented by the set of their Gram matrices scaled to a given (arbitrary) minimum; this is parametrized by the relative interior of a convex set in an affine space of dimension r′ whose extremal points are perfect classes (for which r′ = 0). A class with r′ = 0 is the similarity class of a perfect lattice. A class with r′ = 1 is represented by the interior of a Voronoi path t ∈ (0, 1) ↦ M(t) connecting the perfect matrices M(0) and M(1).

Theorem 5.1. (Watson.) Let Λ be a lattice of dimension n ≤ 7 which does not possess hexagonal sections having the same minimum and whose kissing number is maximum among all such lattices. Then the minimal class of Λ is that of W_{n}. In particular, for a lattice of dimension n = 1, 2, 3, 4, 5, 6 or 7 possessing no hexagonal section having the same minimum, we have s ≤ 1, 2, 4, 6, 10, 16 or 28, respectively.

Proof. See [W2]. □

Using Watson’s theorem, we immediately recover the values of s_{n}(3) for n ≤ 7 which we gave in Theorem 1.1, since two minimal vectors in an integral lattice having an odd minimum cannot generate a hexagonal lattice. We now proceed to the proof of Theorem 1.2, which concerns integral lattices having an odd minimum m ≥ 5:

Theorem 5.2. Let m ≥ 5 odd. We have s_{m}(n) = s_{3}(n) for 2 ≤ n ≤ 6 and s_{m}(7) = 27 < s_{3}(7) = 28.

Proof. The upper bounds for s in the range 1–6 follow immediately from Theorem 4.1. To prove the sharper bound s ≤ 27 for n = 7, we observe that W_{7} is perfect. This shows that a lattice with s = 28 having an odd minimum is similar to W_{7}, hence isometric to \sqrt{a} W_{7} for some integer a ≥ 1. But such a lattice is not primitive unless a = 1.

We now prove that the upper bound 27 is attained for all m ≥ 5. To this end, we consider a Voronoi path C connecting W_{7} to a neighbouring perfect lattice. For W_{7}, we have s = r. Hence the Voronoi paths having W_{7} as an end point are in one-to-one correspondence with the complementary sets of one pair ±x of minimal vectors (see [M], Section 7.5). Since Aut(W_{7}) ≃ W (𝔼_{7}) acts transitively on S(W_{7}^{∗}) ∼ S(𝔼_{7}), all these Voronoi paths are isometric to one of them, and connect scaled copies of 𝔼_{7}^{∗} and 𝔼_{7}. An explicit one-parameter family of matrices can be found in Jaquet’s thesis [J]. Scaled to minimum 6, this family is given by the matrices

M(t) = ( \begin{matrix}6 & t+3 & t+3 & t+3 & t+3 & t+3 & 2t \\ t+3 & 2t+6 & t+3 & t+3 & t+3 & t+3 & 0 \\ t+3 & t+3 & 2t+6 & t+3 & t+3 & t+3 & t+3 \\ t+3 & t+3 & t+3 & 2t+6 & t+3 & t+3 & t+3 \\ t+3 & t+3 & t+3 & t+3 & 2t+6 & t+3 & t+3 \\ t+3 & t+3 & t+3 & t+3 & t+3 & 2t+6 & t+3 \\ 2t & 0 & t+3 & t+3 & t+3 & t+3 & 6 \end{matrix} ) ,

which connect the perfect lattices ^{3} 𝔼_{7} and ^{2}W_{7}. Set t = \frac{3p}{m} where p, m are integers with 0 < 3p < m. The entries of \frac{m}{6} M(t) are m (its minimum), p + m, \frac{p+m}{2}, p, and 0. Choosing for p an odd integer strictly smaller than \frac{m}{3}, e.g. p = 1, we obtain an integral lattice of minimum m which belongs to C, hence has s = 27. Denote by L_{7}(m) any such lattice.

By orthogonality to one of the 63 directions of minimal vectors in 𝔼_{7}, we obtain 63 hyperplane cross-sections of W_{7} with s = 16, all having the same configuration of minimal vectors (i.e., the minimal vectors have the same system of components in a suitably chosen basis for W_{7}). Removing one pair of vectors from S(W_{7}) preserves the existence of such configurations (indeed, there remains 27 of them). This proves that L_{7}(m) contains hyperplane sections L_{6}(m) with s = 16.

Since the configuration of S(L_{6}(m)) is that of S(W_{6}), L_{6}(m) contains a hyperplane section L_{5}(m) with s = 10, which itself contains a hyperplane section with s = 6, etc. This completes the proof of Theorem 1.2. □

It is easy to verify that the lattices L_{7}(5) and L_{7}(7) are unique in the class C. However, this does not proves the uniqueness of these lattices as integral lattices with minimum 5 or 7, for Watson’s theorem does not tell us that C is the unique 7-dimensional, well–rounded minimal class with s = 27 containing integral lattices of odd minimum. Nevertheless, we believe that this is actually the case. [The lattice L_{7}(5) is 9-modular. This was first constructed by Conway and Sloane in [C-S2], and later interpreted by Bergé and the first author (unpublished) as the fixed point (up to similarity) of an involution on C which exchanges a lattice and its dual.]

6. Low-dimensional lattices of index 1.

Let Λ be an n-dimensional, well–rounded lattice. This means that there exist in Λ systems of n independent minimal vectors. These systems generate sublattices of finite index in Λ. The largest possible value for this index is called the index of Λ, and denoted by ı(Λ) = ı. The index solely depends on the minimal class of Λ, and is bounded from above by γ_{n}^{n/2}; see Section 7 below.

In this Section, we consider lattices of index 1 and dimension n ≤ 6; in particular, we give an explicit description of all well–rounded minimal classes of index 1 which contain an integral lattice having an odd minimum for dimensions n ≤ 5, using conditions (C1) and (C2) from the lemma below; the complete minimal classification of 5-dimensional lattices having no hexagonal section with the same minimum is displayed in Appendix 1.

We begin with an easy lemma:

Lemma 6.1. Let Λ be an integral lattice having an odd minimum. Then Λ satisfies the following two conditions: (C1) For any ℤ-linear relation a_{1}e_{1} + ⋯ + a_{r}e_{r} between minimal vectors e_{1}, …, e_{r}, the sum a_{1} + ⋯ + a_{r} is even. (C2) If e_{1}, …, e_{r} are independent minimal vectors and if Λ contains e = \frac{e_{1}+···+e_{r}}{2} , then r is even and ≥ 6.

Proof. If r = 4 in condition (C2), Λ contains a 4-dimensional centred cubic lattice, which itself contains hexagonal sections. The other assertions result immediately from norm calculations. □

In what follows, we restrict ourselves to minimal classes whose elements satisfy conditions (C1) and (C2).

The complete classification of minimal classes up to dimension 4 (first made by Štogrin) can be read in [M], Section 9.3 and 9.4. Minimal classes in dimension n ≤ 4 satisfying conditions (C1) and (C2) are as follows; we describe the minimal vectors in a basis of minimal vectors (e_{1}, …, e_{n}); the notation a_{s}, b_{s}, etc. is that of [M].

\underline{n = 2}, \underline{s = 2} (a_{2}). \underline{n = 3}, \underline{s = 3} (a_{3}) ; \underline{s = 4} : a_{3}, e_{4} = e_{1} + e_{2} + e_{3} (a_{4}). \underline{n = 4}, \underline{s = 4} (a_{4}) ; \underline{s = 5} : a_{4}, e_{5} = e_{1} + e_{2} + e_{3} (b_{5}) ; \underline{s = 6} : b_{5}, e_{6} = e_{1} + e_{2} + e_{4} (a_{6}).

Using the list of classes above, we easily verify that all these classes contain integral lattices of minimum m for all m ≥ 3 odd, and that such a lattice is unique if n = 3, s = 4 and m = 3 or 5, and if n = 4, s = 6 and m = 3; here are Gram matrices for the three lattices above (for n = 4 and m = 5, there are three lattices):

( \begin{matrix}3 & −1 & −1 \\ −1 & 3 & −1 \\ −1 & −1 & 3 \end{matrix} ) , ( \begin{matrix}5 & −2 & −2 \\ −2 & 5 & −1 \\ −2 & −1 & 5 \end{matrix} ) and ( \begin{matrix}3 & −1 & −1 & −1 \\ −1 & 3 & −1 & −1 \\ −1 & −1 & 3 & 1 \\ −1 & −1 & 1 & 3 \end{matrix} ) .

The general 5-dimensional classification is known (Batut, [Bt]). However, the result is very complicated, and it is difficult to extract from the published data the list of classes we are interested in. Following the method used in [M], we have classified directly the minimal classes having no hexagonal section with the same minimum, starting with the classes with s = 5, and increasing inductively the kissing number. We give below a detailed description of what concerns classes satisfying conditions (C1) and (C2), denoted by λ_{s} or λ′_{s}.

\underline{n = 5}, \underline{s = 5} : S = \{±e_{1}, …, ±e_{5}\} (λ_{5}). \underline{s = 6} : λ_{5}, and e_{6} = e_{1} + e_{2} + e_{3} or e_{1} + e_{2} + e_{3} + e_{4} + e_{5} (λ_{6}, λ′_{6}). s ≥ 7 : C ≻ λ_{6}. \underline{s = 7} : λ_{6}, and e_{7} = e_{1} + e_{2} + e_{4} or e_{1} + e_{4} + e_{5} (λ_{7}, λ′_{7}). s ≥ 8 : C ≻ λ_{7}. \underline{s = 8} : λ_{7}, and e_{8} = e_{3} − e_{4} + e_{5} or e_{1} + e_{2} + e_{5} (λ_{8}, λ′_{8}); s ≥ 8 and e_{8} = e_{1} + e_{2} + e_{5} ⟹ s = 8. \underline{s = 9} : λ_{8}, and e_{9} = e_{1} + e_{3} + e_{5} (λ_{9}). \underline{s = 10} : λ_{9}, and e_{10} = e_{2} + e_{4} − e_{5} (λ_{10}).

[A more symmetric definition for λ_{8}, λ_{9}, λ_{10} could be obtained by extending λ_{7} successively with e′_{8} = e_{1} + e_{3} + e_{5}, e′_{9} = e_{1} + e_{4} + e_{5} and e′_{10} = e_{1} + e_{2} + e_{3} + e_{4} + e_{5}.]

Sketch of proof. We shall not justify the details. Because the index is one, there exist bases of minimal vectors. As in [M]. we use the fact that the characteristic determinants (the determinants whose entries are components of systems of r ≤ n minimal vectors on r basis vectors) are 0 or ±1, which implies that

(1) all components are 0 or ±1 ;

(2) there do not exist systems of two vectors with two components (1, 1) and (1, −1) ;

(3) there do not exist systems of three vectors with three components (1, 1, 0), (1, 0, 1) and (0, 1, 1).

We also prove that if s ≥ 7 (resp. s ≥ 8), we can find a base change which puts in evidence a hyperplane section with s ≥ 5 (resp. s ≥ 6), so that we can use the results we know for dimension 4. □

[The class λ′_{8} deserves a special remark: it belongs to a series with s = 2(n − 1), defined for all n ≥ 3, with minimal vectors e_{1}, …, e_{n}, which constitute a basis, and e_{1} + e_{2} + e_{i}, i = 3, …, n. One may take e_{i} · e_{i} = m, e_{1} · e_{2} = −1, e_{1} · e_{i} = e_{2} · e_{i} = − \frac{m−1}{2} for 3 ≤ i ≤ n, and e_{i} · e_{j} = \frac{m−1}{2} for 3 ≤ i < j ≤ n; the decomposition

\begin{aligned}N(x) & = \frac{m − 1}{2} (x_{1} + x_{2} − x_{3} − ⋯ − x_{n})^{2} + \frac{m + 1}{2} (x_{1} − x_{2})^{2} \\ & + \frac{m + 1}{2} (x_{3}^{2} + ⋯ + x_{n}^{2}) \end{aligned}

of the norm form easily shows the existence of lattices which fulfil the properties. Moreover, such a lattice is unique if m = 3, as one easily sees, using the uniqueness of hyperplane sections with s = 6, and indeed coincides with the lattice associated in Table 2.5 with the root lattice 𝔻_{n−1} ⊥ 𝔸_{1}.]

Theorem 6.2. All classes λ_{s}, λ′_{s} contain integral lattices of any odd minimum m ≥ 3, except λ_{9} if m = 3. Lattices of minimum 3 in each of the classes λ′_{7}, λ_{8}, λ′_{8}, λ_{10} are unique, whereas λ_{7} contains three lattices. Similarly, λ_{10} contains a unique lattice of minimum 5. □

Proof. The construction of lattices having an odd minimum m has been done using the techniques given above in the case of λ′_{8}. Examples show up in in all classes except in λ_{9}. We shall consider in detail only lattices with m = 3 and s ≥ 7, the only cases we shall need later. (Note, however, that the existence of a lattice in λ_{10} with any minimum m ≥ 3 has been proved in Section 4.)

If C ≻ λ_{7}, because e_{1} + e_{2} + e_{3} and e_{1} + e_{2} + e_{4} are minimal, we have e_{i} · e_{j} = −1 for 1 ≤ i < j ≤ 4, except e_{3} · e_{4} = 1 (for the last condition, note that N(e_{1} + e_{2} + e_{3} + e_{4}) = 2 + 2 e_{3} · e_{4} ≥ 4). Using the conditions e_{i}·e_{5} ∈ {0, ±1}, N(e_{i}+e_{j} +e_{5}) > 3, N(e_{3}+e_{4}−e_{5}) > 3, and the invariance of C_{7} under the elements (1, 2), (3, 4) of S_{5}, we quickly find the three isometry classes of lattices belonging to C_{7}.

For a lattice Λ in a class containing λ_{8}, thanks to the condition N(e_{3} − e_{4} + e_{5}) = 3, we have e_{4} · e_{5} = −e_{3} · e_{5} = 1. Using the lower bounds N(e_{1}+e_{2}+e_{3}+e_{5}) ≥ 4 and N(e_{1}+e_{2}+e_{4}−e_{5}) ≥ 4, we obtain e_{1}·e_{5}+e_{2}·e_{5} = 0. Exchanging e_{1} and e_{2}, we may moreover assume that e_{1} · e_{5} ≤ e_{2} · e_{5}, and we are left with the two possibilities e_{1} · e_{5} = e_{2} · e_{5} = 0, and then Λ ∈ λ_{8}, and e_{1} · e_{5} = −1, e_{2} · e_{5} = +1, and then Λ ∈ λ_{10}.

The easier cases of λ_{7} and λ_{8} are dealt with in the same way. □

We now prove partial classification results for 6-dimensional lattices of index 1, which will show that the large values of s exist only for lattices having a larger index, a situation which we shall consider in the next section.

Lemma 6.3. If s(Λ) ≥ 7 (resp. s(Λ) ≥ 9), then Λ has a hyperplane section Λ_{0} with min Λ_{0} = min Λ and s(Λ_{0}) ≥ 6 (resp. s(Λ_{0}) ≥ 7).

Proof. Since ı(Λ) = 1, any system e_{1}, …, e_{6} of six independent minimal vectors of Λ constitute a basis for Λ. Since the other minimal vectors have three or five non-zero components on e_{1}, …, e_{6}, it is clear that if s ≥ 7, we can find five basis vectors which generate a sublattice with s = 7.

Suppose now that s ≥ 9. If two minimal vectors have five components, say, e = e_{1}+⋯+e_{4}+e_{5} and e′ = e_{1}+⋯+e_{4}+e_{6}, then e′ = e−e_{5}+e_{6} and we have a vector with three components on the basis (e, e_{2}, …, e_{6}). A similar argument shows that if three minimal vectors have five components, we can get rid of two of them. So we assume that S(Λ) contains to vectors e_{7} and e_{8} with three components. If they have a common component, we are done. Otherwise, we may assume that e_{7} = e_{1} +e_{2} +e_{3} and e_{8} = e_{4} +e_{5} +e_{6}. Then an extra vector with three (resp. five) components shares two (resp. three) components with e_{7} or e_{8}, which completes the proof of the lemma. □

Remark 6.4. By a detailed analysis of classes having a large kissing number, we could show that when s ≥ 11, Λ contains a hyperplane section Λ_{0} with min Λ_{0} = min Λ and s(Λ_{0}) ≥ 8.

By the lemma above, for studying the kissing number of integral 6-dimensional lattices having an odd minimum, we can restrict ourselves to lattices having a hyperplane section of type λ_{7}, λ′_{7}, λ_{8}, λ′_{8}, λ_{9}, or λ_{10}. Using PARI-GP and some extra programs due to Batut, we have made a computer search for lattices of minimum 3 containing as a hyperplane section one of the seven lattices quoted in Theorem 6.2, and tested for isometry the lattices we found. We state the result as a theorem:

Theorem 6.5. There are two (resp. five, resp. ten) well-rounded 6-dimensional lattices with s = 11 (resp. s = 10, resp. s = 9). □ [Remarks. (1) The two lattices with s = 11 belong to the same minimal class; both contain a hyperplane section isometric to W_{5}. (2) One of the five

lattices with s = 10 is the the strongly eutactic, 5-modular lattice ⋀^{2}𝔸_{4}.]

Using Remark 6.4, we could show that for a 6-dimensional lattice of index 1 having an odd minimum, we have s ≤ 12, (and also give a handycomputational proof for the bound s ≤ 11 when m = 3). We shall not give the nevertheless complicated details of the proof, and content ourselves with the following statement, which is easily verified using a computer:

Theorem 6.6. On the set of integral, well–rounded 6-dimensional lattices of index 1 and minimum m = 3 (resp. m = 5), the values taken by s constitute the interval [6, 11] (resp. [6, 12]). □

[Note that since s runs through the whole interval [5, 10] for n = 5 and all m ≥ 5 odd, the set of values of s certainly contains the interval [6, 11] for n = 6 and all m ≥ 5 odd. Probably, the set of values taken by s on lattices of index 1 is the interval [6, 12] for all m ≥ 5 odd.]

7. The index of a sublattice.

In the preceding section, we proved upper bounds for the kissing number of well–rounded 6-dimensional lattices of index 1. We now consider lattices having a larger index, and first study upper bounds of the index under various conditions for low-dimensional lattices. We shall give detailed proofs for n ≤ 7, leaving to the reader some verifications for n = 8. We then prove some precise results for n = 6, ı = 2 or 3, and n = 7, ı = 3.

In the following table, the first two lines give the exact upper bound for the index, first among all lattices, then among those which are not similar to a root lattice. The third line (NHS) gives this bound on the set of lattices having no hexagonal section with the same minimum, and the last one (OddMin) on the set of lattices which are integral when scaled to some odd minimum.

Theorem 7.1. Let Λ be a well rounded lattice of dimension n ≤ 8. Then we have for the index of Λ the following sharp upper bounds:

n  ≤ 3 4 5 6 7 8
i_{max}  1 2 2 4 8 16
i′_{max}  1 1 2 3 4 8
NHS  1 1 2 3 4 6
OddMin 1 1 1 3 3 5

Proof. The proof heavily relies on results of [M1], in particular Table 11.1.

• Calculation of ı_{max} and ı′_{max}. It is proved in [M1] that the easy bound ı_{max} ≤ γ_{n}^{n/2} is attained on suitably chosen root lattices, and that the better bound ı ≤ ı′_{max} holds except if Λ is similar to one of the lattices 𝔻_{4}, 𝔻_{6}, 𝔼_{7}, and 𝔼_{8}. (These results were indeed proved by Watson in [W1], except for the slightly weaker result ı′_{max} ≤ 9 if n = 8.)

From now on we only consider lattices satisfying condition (NHS).

• Dimension n ≤ 5. There is nothing to prove for n ≤ 4. If n = 5, consideration of the centred cubic lattice shows that index 2 may occur, and condition (C1) that integral lattices having an odd minimum have index 1.

• Dimension n = 6. It suffices to exhibit an integral lattice of index 3 having an odd minimum. A more precise result will be proved below (Proposition 7.6).

• 2-elementary quotients. Suppose that Λ contains a sublattice L generated by vectors e_{1}, …, e_{n} ∈ S and such that Λ/L is 2-elementary of dimension r ≥ 2 over 𝔽_{2}. Then Table 11.1 of [M1] shows that n ≥ 8, and that if n = 8, then r = 2 (and condition (C2) does not hold). Hence, for such a lattice, the index is smaller than the upper bound given in Theorem 2.1 under condition (OddMin). Moreover, a look at the three types listed in [M1] shows that extensions to index 8 with a quotient of type (4, 2) and no hexagonal sections are impossible. Since cyclic quotient of order 8 or 7 do not exist, we have proved the bound ı ≤ 6 for 8-dimensional lattices.

• Dimension n = 7. Table 11.1 of [M1] shows that lattices of index 4 and cyclic type satisfying condition (NHS) are generated by minimal vectors e_{1}, …, e_{7} together with e = \frac{e_{1}+e_{2}+e_{3}+e_{4}+e_{5}+2e_{6}+2e_{7}}{4} . The example given there, for which s = 7, obviously satisfies condition (NHS), but condition (C2) does not hold, since 2e ≡ \frac{e_{1}+e_{2}+e_{3}+e_{4}+e_{5}}{2} mod 4 Λ. The existence of integral lattices of index 3 having an odd minimum follows from Theorem 1.2.

• Dimension n = 8. The bound ı ≤ 6 has been proved above. Table 11.1 of [M1] lists six types of index 6. A close look at the conditions given there shows that only one type (denoted by (1, 1, 1, 2, 2, 2, 3, 3) in [M1]) may occur under condition (NHS), but the existence of components 1, 1, 1, 3, 3 shows that this type cannot be realized by an integral lattice of odd minimum; we leave the details to the reader, as well as the proof that an example of index 5 satisfying (OddMin) can be constructed using the type (1, 1, 1, 1, 1, 2, 2, 2).

□

To study lattices of index 2 or 3, we first recall an identity of Watson:

Lemma 7.2. (Watson) Let e_{1}, …, e_{r} ∈ E, let a_{1}, …, a_{r} ∈ ℝ, let d > 0, and let e = \frac{a_{1}e_{1}+···+a_{r}e_{r}}{d}. Denote by sgn(x) the sign of the real number x (0 if x = 0). Then:

((∑_{i=1}^{n}|a_{i}|) − 2d)N(e) = ∑_{i=1}^{n} |a_{i}|(N(e − sgn(a_{i}) e_{i}) − N(e_{i})) . □

From this identity, we easily deduce (see [M1], Section 2 for the details):

Proposition 7.3. (Watson) Let Λ be a lattice, let e_{1}, …, e_{r} be minimal vectors in Λ, and let a_{1}, …, a_{r} and d ≥ 2 be integers such that e = \frac{a_{1}e_{1}+···+a_{r}e_{r}}{d} belongs to Λ. Then we have

∑_{i=1}^{n}|a_{i}| ≥ 2d ,

and equality holds if and only if e − sgn(a_{i})e_{i} is minimal for every index i with a_{i} ≠ 0. Moreover, if equality holds and if a_{i} = \frac{d}{2} for some i, then e is also minimal. □

In practice, if L is a sublattice of Λ having a basis (e_{1}, …, e_{n}) of minimal vectors and if Λ/L is cyclic of order d, we may write Λ = ⟨L, e⟩ with e = \frac{a_{1}e_{1}+···+a_{r}e_{r}}{d}, where r and the a_{i} are integers such that r ≤ n and 1 ≤ a_{1} ≤ ⋯ ≤ a_{r} ≤ \frac{d}{2}.

Remark 7.4. If f = \frac{b_{1}e_{1}+···+b_{r}e_{r}}{d} is minimal, then for any index i, the lattice generated by f and the e_{j} with j ≠ i has index |b_{i}| in Λ. Hence the b_{i} are bounded by the index bound for dimension r.

Example 7.5. If n = 6 and Λ has index 3, then Λ is generated over a suitably chosen sublattice L having a basis of minimal vectors by a vector e = \frac{e_{1}+e_{2}+e_{3}+e_{4}+e_{5}+e_{6}}{3} . Vectors in e+L are of the form f = \frac{b_{1}e_{1}+···+b_{6}e_{6}}{3} with b_{i} ≡ 1 mod 3. If such a vector is minimal, then the b_{i} are equal to 1 or −2. Hence minimal vectors in Λ∖L (up to sign) are of the form e−e_{i_{1}} −⋯−e_{i_{k}}.

Proposition 7.6. Well–rounded, 6-dimensional lattices of index 3 satisfying condition NHS constitute a single minimal class, whose elements have s = 12. This class contains integral lattices having an odd minimum m for all m ≥ 7 (unique if m = 7), but none for m ≤ 5.

Proof. Write as above Λ = ⟨L, e⟩. We know that the vectors e_{i} and e′_{i} = e − e_{i} are minimal, and we must show that under condition (NHS), there are no other minimal vectors in Λ.

First, if f = a_{1}e_{1} + ⋯ + a_{k}e_{k}, say, belongs to S(L), with non-zero a_{i}, a sublattice of index 3a_{i} in Λ shows up. Thus we have a_{i} ∈ {±1}, and k ≥ 3. But if, say, f = e_{1}+e_{2}−e_{3} ∈ S(L), then we can write e = e_{3}+ \frac{f−e_{3}+e_{4}+e_{5}+e_{6}}{3} , in contradiction with Proposition 7.3.

Next suppose that f = e − e_{1} − ⋯ − e_{k}, say, is an extra minimal vector. Then k ≥ 3 (because e − e_{1} ∈ S), and we have \frac{f+e_{k+1}+···+e_{6}}{2} ∈ Λ, which implies 7 − k ≥ 5, i.e. k ≤ 2, again a contradiction. This proves that s = 12, and that the minimal class of Λ is well defined.

Lemma 7.2 applied to the minimal vector e − e_{i} reads

2(N(e) − m) + ∑_{j≠i} (N(e − e_{i} − e_{j}) − m) = N(e − e_{i}) .

Since the six terms in the left hand side are not zero, this identity implies m ≥ 7 (and even m ≥ 14 if m is even). If m = 7, we must have N(e) = 8, e·e_{i} = \frac{1}{2}N(e) = 4, and e_{i} ·e_{j} = 1 for j ≠ i. These conditions define a unique lattice Λ of minimum m = 7. Thus, the lower bound m ≥ 7 is optimal. Examples for m ≥ 7 odd are obtained by taking for instance e_{i} · e_{j} = \frac{m−5}{2} if i + j = 7 and e_{i} · e_{j} = 1 otherwise. □

An n-dimensional, well–rounded lattice Λ of index 3 is of the form Λ = L ∪ ±(e + L) where L has a basis (e_{1}, …, e_{n}) of minimal vectors and e = \frac{e_{1}+···+e_{r}}{3} for some integer r with 6 ≤ r ≤ n. For n = 7, we have r = 6 or 7. If Λ is integral with an odd minimum m, we have seen that the first case may occur if and only if m ≥ 7. We now consider the case where n = r = 7. We are going to prove a characterization of integral lattices of index 3 and minimum 3. However, before stating the result, we write down some identities valid for any lattice as above with r = n. The first two are Watson’s identity (7.2) relative to e and e − e_{i}:

∑_{i=1}^{n} (N(e − e_{i}) − m) = N(e) ;  (∗)

∀ i, 2 (N(e) − m) + ∑_{j≠i} (N(e − e_{i} − e_{j}) − m) = (n − 5) N(e − e_{i}) . (∗∗)

Taking the sum over i in (∗∗) and using (∗), we obtain the new identity

∑_{1≤i<j≤n} (N(e − e_{i} − e_{j}) − m) = \frac{n(n − 3)}{2} m − \frac{n + 5}{2} N(e) ,  (∗∗′)

which yields for the norm of e the upper bound N(e) ≤ \frac{n(n−3)}{n+5} m, where equality holds if and only if the \frac{n(n−1)}{2} vectors e − e_{i} − e_{j} are minimal.

Lemma 7.7. If Λ is integral and if both n and m are odd, then N(e) ≥ n.

Proof. Since 3e = ∑_{i} e_{i}, we have N(e) ≡ nm ≡ 1 mod 2. This implies that N(e − e_{i}) is even for all i, whence N(e − e_{i}) ≥ m + 1 and N(e) ≥ n by (∗). □

Theorem 7.8. An integral 7-dimensional lattice of minimum 3 and index 3 is isometric to W_{7} ≃ \sqrt{2} 𝔼_{7}^{∗}.

Proof. Since m = 3 < 7, Λ is of the form above. We have N(e) ≤ \frac{n(n−3)}{n+5} m = 7 and N(e) ≥ n = 7 by the previous lemma, hence N(e) = 7, then N(e−e_{i}) = 4 for all i by (∗), which implies e·e_{i} = 3, whence N(e − e_{i} − e_{j}) = 1 + 2 e_{i} · e_{j}, and finally e_{i} · e_{j} = 1, since N(e − e_{i} − e_{j}) = 3 by (∗∗′). This shows that the Gram matrix of the basis (e_{1}, …, e_{6}, e) of Λ is uniquely defined, hence that up to isometry, there exists at most one possible lattice, which indeed exists, since we know W_{7}. □

We display below Gram matrices first for (e_{1}, …, e_{7}), then for the basis (e − e_{6} − e_{7}, e_{2}, …, e_{7}) of Λ:

A_{28} = ( \begin{matrix}3 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 3 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 3 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 3 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 3 & 1 & 1 \\ 1 & 1 & 1 & 1 & 1 & 3 & 1 \\ 1 & 1 & 1 & 1 & 1 & 1 & 3 \end{matrix} ) ; C_{28} = ( \begin{matrix}3 & 1 & 1 & 1 & 1 & −1 & −1 \\ 1 & 3 & 1 & 1 & 1 & 1 & 1 \\ 1 & 1 & 3 & 1 & 1 & 1 & 1 \\ 1 & 1 & 1 & 3 & 1 & 1 & 1 \\ 1 & 1 & 1 & 1 & 3 & 1 & 1 \\ −1 & 1 & 1 & 1 & 1 & 3 & 1 \\ −1 & 1 & 1 & 1 & 1 & 1 & 3 \end{matrix} ) .

Lattices of minimum 5 and index 3 also are of the form above, and the upper bound which follows (∗∗′) shows that N(e) takes one of the values 7, 9, or 11. The corresponding lattices have not been classified.

We now come back to dimension 6 and consider the problem of classifying integral lattices of index 2 and minimum 3.

Theorem 7.9. Let Λ be an integral 6-dimensional lattice of minimum 3 and index 2. Then Λ is isometric to one of three lattices W_{6} (the Watson lattice), W′_{6}, W′′_{6}, with s = 16, 12, 10 respectively.

Proof. Because index 2 does not occur in lower dimensions, Λ is of the form Λ = L∪(e+L) where L possesses a basis (e_{1}, …, e_{6}) of minimal vectors and

e = \frac{e_{1} + e_{2} + e_{3} + e_{4} + e_{5} + e_{6}}{2} .

Since we may change any signs in e (because, say, e − e_{1} = \frac{−e_{1}+e_{2}+···+e_{6}}{2} ), we may assume that e is the shortest of the vectors \frac{e_{1}±e_{2}±e_{3}±e_{4}±e_{5}±e_{6}}{2} .

In our setting, Watson’s identity now reads

∑_{i} (N(e − e_{i}) − m) = 2 N(e) ,  (†)

Using (†) and the inequality N(e − e_{i}) ≥ N(e), we obtain 6 (N(e) − 3) ≤ 2 N(e), i.e. N(e) ≤ \frac{9}{2}, hence N(e) = 3 or 4. If N(e) = 4, then N(e − e_{i}) ≡ 1 mod 2, thus N(e − e_{i}) ≥ 5, and (†) implies 2N(e) ≥ 12, a contradiction. Hence N(e) = 3, and again by (†), for all i, N(e − e_{i}) = 4, i.e. e · e_{i} = 1, which also reads

∀ i, ∑_{j≠i} e_{i} · e_{j} = −1 .

Since |e_{i} · e_{j}| ≤ 1, the numbers of indices j ≠ i such that e_{i} · e_{j} = −1, 0, 1 are equal to (3, 0, 2), (2, 2, 1), or (1, 4, 0). Denoting by t_{1}, t_{2}, t_{3} the respective numbers of the systems above, we have s(Λ) = 7 + \frac{1}{2} (3t_{1} + 2t_{2} + t_{1}). We recover in this way the bound s ≤ 16, and the fact that this is attained if and only if t_{3} = 6.

Also, we have N(e − e_{i} − e_{j}) = 5 + 2 e_{i} · e_{j}, so that e − e_{i} − e_{j} is minimal if and only if e_{i}·e_{j} = −1, and the condition N(e−e_{i}−e_{j} −e_{k}) ≥ 4 (which holds because this vector has an even norm) amounts to e_{i}·e_{j} +e_{i}·e_{k} +e_{j} ·e_{k} ≥ −1.

Suppose first that a system (3, 0, 2) exists for some i. We may then assume that e_{1} · e_{2} = e_{1} · e_{3} = e_{1} · e_{4} = −1 and e_{1} · e_{5} = e_{1} · e_{6} = +1. This implies successively e_{2} · e_{3} = e_{2} · e_{4} = +1, then e_{2} · e_{5} = e_{2} · e_{6} = −1, and similarly e_{3} · e_{4} = +1 and e_{3} · e_{5} = e_{3} · e_{6} = −1, and finally e_{4} · e_{5} = e_{4} · e_{6} = −1 and e_{5} · e_{6} = +1. This proves that for all i, we have a system (3, 0, 2), that the corresponding lattice is unique up to isometry, hence isometric to W_{6}, and that we have s < 16 otherwise.

Suppose now that no system (3, 0, 2) exists, but that there is a system (2, 2, 0). We may then assume that e_{1} · e_{2} = e_{1} · e_{3} = −1, e_{1} · e_{4} = e_{1} · e_{5} = 0, and e_{1} · e_{6} = +1. Arguments like those we used in the first case then show that we have a unique choice for the other products e_{i} · e_{j}, and that for i = 1, 2, 3, 6 (resp. i = 4, 5), we have systems of type (2, 0, 1) (resp. (1, 4, 0)). This time, we have s = 12, again attained on a unique lattice (up to isometry) which we denote by W′_{6}, but we have s = 10 if all systems are of type (1, 4, 0). The case where s = 10 can be dealt with with similar arguments that we leave to the reader. □

Putting together Theorems 6.5 and 7.9, we indeed obtain the following theorem which proves for dimension 6 the results stated in the introduction:

Theorem 7.10. The set of values taken by s on 6-dimensional, well-rounded lattices of minimum 3 is [6, 12] ∪ {16}. □

A computer search easily produces lattices of minimum 5 and index 2 with s running through the interval [12, 16] This shows:

Theorem 7.11. The set of values taken by s on 6-dimensional, well-rounded lattices of minimum 5 is the interval [6, 16]. □

Probably, the same result holds for all m ≥ 7 odd.

Remark 7.12. We found only one lattice of minimum 5 for each of the values 13, 14, 15, 16 of s. Probably, these four lattices are unique. In contrast, we found many examples with a lower value of s, including one with index 2 and s = 6.

A lattice Λ of dimension 7 and index ı = 2 having an odd minimum extends a 6-dimensional lattice with the same property. This proves that when its minimum is 3, if ı ≥ 2, then Λ has a hyperplane section isometric to one of the three lattices W_{6}, W′_{6} or W′′_{6} (if ı = 3, then Λ ≃ W_{7} ⊃ W_{6}). However, new possibilities exist in dimension 8. We give below a partial result:

Proposition 7.13. Let Λ be an 8-dimensional lattice of minimum 3 and index ı ≥ 3. Then either Λ has a section isometric to W_{6}, W′_{6} or W′′_{6}, or s(Λ) ≤ 15.

Proof. We assume for the proof Λ has no 6-dimensional section of minimum 3 isometric to W_{6}, W′_{6} or W′′_{6}. Hence all sublattices of Λ of minimum 3 and dimension ≤ 7 have index 1.

(a) We first prove that index 4 is impossible. Indeed, we could write otherwise Λ = ⟨e_{1}, …, e_{8}, e⟩ with minimal e_{i} and e = \frac{e_{1}+···+e_{8}}{4} . By Watson’s identity, the vectors e−e_{i} are minimal. Applied to e−e_{i}, this identity reads (N(e) − 3) + ∑_{j≠1} (N(e − e_{1} − e_{j}) − 3) = 2N(e − e_{1}) = 6, a contradiction, since the left hand side is made of eight odd terms.

(b) Next we observe that in case of index 3 or 5, writing Λ as above, then for the lattice L generated by the e_{i}, we have s(L) = 8. For instance, is e = \frac{e_{1}+···+e_{8}}{3} and if, say, e′ = e_{1} + e_{2} − e_{3} were minimal, we could write e − e′ = \frac{e′−e_{3}+e_{4}+···+e_{8}}{3} , and an index 3 shows up in dimension 7; the same kind of arguments work in all cases of index 3 or 5.

(c) We now prove that s ≤ 8 if ı = 5. Write e = \frac{e_{1}+···+e_{r}+2e_{r+1}+···+2e_{8}}{5} . We have Λ = L ∪ ±(e + L) ∪ ±(2e + L), and replacing e by 2e amounts to exchanging r and 8 − r; we also have 2 ≤ r ≤ 6. The minimal vectors of e + L are of the form e − e_{i} − e_{j} − . . .. If i ≤ r, an index 4 occurs by (a); by Watson’s identity, this always occurs if r = 6. We may thus assume that 3 ≤ r ≤ 5. If some e − e_{i} − e_{j} − … is minimal, then we can construct a lattice of dimension ≤ 7 and index 3. If the minimal vectors are among e or the e − e_{i}, then there are at least two coefficients 2 in the denominator of e, and we may use them this time to construct a lattice of index 2 and dimension 7. Hence we have S(Λ) ⊂ S(L), whence s(Λ) ≤ s(L) = 8.

(d) Finally, let e = \frac{e_{1}+···+e_{8}}{3} . Then e has even norm. Hence the minimal vectors of Λ∖L are of the form e−e_{i}−e_{j} −… with 1, 3, … indices i, j, … If, say, some vector e′ = e−e_{1} −e_{2} −e_{3} −… is minimal, then e′ +e_{1} +e_{2} +e_{3} = \frac{e′+e_{4}+···+e_{8}}{2} , and an index 2 shows up in dimension 6. This shows that the minimal vectors in Λ∖L lie among the e − e_{i}. Since ∑_{i}(N(e − e_{i}) − 3) = 2N(e) > 0, there are at most 7 such vectors, whence the (actually, not optimal) bound s ≤ 15. □

Remark 7.14. The proposition above shows that an 8-dimensional lattice Λ of index ı ≥ 2 with no section W_{6}, W′_{6}, W′′_{6} and s ≥ 16, if any, may be written in the form Λ = ⟨e_{1}, …, e_{8}, e⟩ with e_{i} and e = \frac{e_{1}+···+e_{8}}{2} minimal. Then its other minimal vectors, up to sign, are of one of the forms e−e_{i} −e_{j} or e − e_{i} − e_{j} − e_{k} − e_{ℓ}.

8. The minimum of the dual lattice.

In the remainder of the paper, we only consider (unless otherwise explicitly stated) integral lattices Λ of minimum 3. To construct cross-sections with the same minimum and comparatively large values of s, we shall need to bound the minimum of the dual lattice Λ^{∗} of Λ. Some “critical” values (especially \frac{2}{3}, 1, \frac{4}{3}) will show up.

Recall that the Hermite invariant of a lattice L is γ(L) = \frac{min L}{det(L)} and

that the Hermite constant for dimension n is γ_{n} = sup_{dim L=n} γ(L). What we really need are their dual versions (the geometric means of their values for a lattice and its dual) γ′(L) and γ′_{n}: indeed, we have

γ′(L)^{2} = γ(L)γ(L^{∗}) = min L min L^{∗} ,

hence

min Λ^{∗} = \frac{γ′(Λ)^{2}}{3} ≤ \frac{γ′_{n}^{2}}{3} .  (∗)

However, our knowledge on γ′_{n} is very poor, and most of the time, we shall have to content ourselves with the trivial bound γ′_{n} ≤ γ_{n}. The values of γ_{n} and the corresponding critical lattices are known up to n = 8. Beyond dimension 8, upper bounds for γ_{n}, coming from estimations of Rogers, can be obtained from Table 1.2 of [C-S]; [C-S] uses the center density δ; we have γ_{n} = 4 δ_{n}^{2/n}. (Roger’s bounds have been recently improved by Cohn and Elkies in [C-E]; the new bounds could be used to shorten the proof of Theorem 8.2.)

We are going to prove three theorems for n ≤ 7, n = 8, and n = 9 respectively. Recall that Λ stands for an integral lattice of minimum 3.

Theorem 8.1. For n ≤ 7, we have min Λ^{∗} ≤ 1, and equality holds if and only if n = 7 and Λ ≃ W_{7}.

Proof. For n ≤ 6, the result follows from (∗), using the crude estimate γ_{n} < \sqrt{3}. Let now n = 7. Set d = det(Λ). The even part of Λ has determinant 4d and minimum m ≥ 4. We have γ(Λ_{even}) ≥ \frac{4}{(4d)^{1/7}} and

γ(Λ_{even}) ≤ γ_{7} = γ(𝔼_{7}) = 2^{6/7}, hence d ≥ 2^{6}. In the other direction, we have det(Λ^{∗}) = \frac{1}{d}. Thus if min Λ^{∗} ≥ 1, then γ(Λ^{∗}) ≥ d^{1/7}, and γ(Λ^{∗}) ≤ γ_{7} = 2^{6/7} implies d ≤ 2^{6}. Hence if min Λ^{∗} ≥ 1, then min Λ^{∗} = 1, γ(Λ^{∗}) = γ(𝔼_{7}), and since 𝔼_{7} is (up to scale) the unique critical 7-dimensional lattice, Λ^{∗} is isometric to \frac{1}{\sqrt{2}} 𝔼_{7}, and Λ to \sqrt{2} 𝔼_{7}^{∗} ≃ W_{7}. Conversely, W_{7}^{∗} obviously has minimum 1. □

The same kind of methods can be used to handle the case of dimension 9, but the result we obtain is probably far from being optimal.

Theorem 8.2. For n = 9, we have min Λ^{∗} < \frac{4}{3}.

Proof. We use Roger’s bound γ_{9} < 2.1411672 (Table 1.2 of [C-S] gives δ_{9} ≤ 0.06007).

Let d = det(Λ). We have as above det(Λ_{even}) = 4d and min Λ_{even} ≥ 4, which implies γ(Λ_{even}) ≥ \frac{4}{(4d)^{1/9}} , hence d ≥ 4^{8}γ_{9}^{−9} > 69.28 …; similarly, if

min Λ^{∗} ≥ \frac{4}{3}, we have γ(Λ^{∗}) ≥ \frac{4}{3} · d^{1/9}, hence d ≤ (\frac{3}{4})^{9} γ_{9} < 71.02 … We have thus proved that d must be equal to 70 or 71. Since these integers are square free, d is the annihilator of Λ^{∗}/Λ, which shows that min Λ^{∗} = \frac{p}{q} with q = d and (p, q) = 1. We then have p ≤ ⌊γ_{9} d^{1/9} q⌋, i.e. p ≤ 93 if d = 70 and p ≤ 94 if d = 71. But min Λ^{∗} = \frac{p}{q} ≥ \frac{4}{3} now reads p ≥ 93.33 … if d = 70 and p ≥ 94.66 … if d = 71, which contradicts the bounds above. □

To handle the case of dimension 8, we shall use different techniques, namely the possibility of embedding any integral lattice into a unimodular one:

Theorem 8.3. (Conway and Sloane.) An n-dimensional integral lattice can be embedded in an (odd) unimodular lattice of dimension n′ ≤ n + 3.

Proof. This is Corollary 8 of [C-S1]. □

First, we prove a general lemma.

Lemma 8.4. Let L be an odd n-dimensional sublattice of a unimodular lattice M of the form M_{0} ⊥ ℤ^{k} where M_{0} is even. Then min L^{∗} ≤ 1. Moreover, if equality holds, then either M_{0} ≠ 0 or L can be embedded in ℤ^{n}.

Proof. We may assume that k is minimal. Denote by (ε_{1}, …, ε_{k}) the canonical basis for ℤ^{k}. Since L is odd, k is non-zero. Since k is minimal, the ε_{i} do not lie in L^{⊥} (for otherwise, L could be embedded in M_{0} ⊥ ℤ^{k−1}). Hence the projections ε′_{i} of the ε_{i} onto the span of L are non-zero. For all x ∈ L, we have x · ε′_{i} = x · ε_{i} ∈ ℤ. Hence the vectors ε′_{i} are non-zero vectors in L^{∗}. We thus have

min L^{∗} ≤ min_{i} N(ε′_{i}) ≤ min_{i} N(ε_{i}) = 1 .

Moreover, if min L^{∗} = 1, all ε′_{i} have norm 1, hence coincide with the ε_{i}. Since the ε_{i} are independent, we then have dim L ≥ k, whence dim L = k if M_{0} = 0. □ [Actually, the strict inequality min L^{∗} < 1 holds for any non-zero sublattice of ℤ^{k} (even or odd) which is not isometric to a sublattice of ℤ^{n}.]

We now return to the case of a lattice Λ with min Λ = 3 and dim Λ = 8. Again, the results we are going to prove are probably far from being optimal. The lemma below is just a first step towards the proof of Theorem 8.6:

Lemma 8.5. For n = 8, we have min Λ^{∗} ≤ 1.

Proof. We consider more generally an integral lattice Λ of minimum 3 and dimension n ≤ 8. By Conway and Sloane’s theorem 8.3, Λ can be embedded in an (odd) unimodular lattice M of dimension n′ ≤ n + 3. We still assume that n′ is the smallest possible dimension. Since dim M ≤ 11, M is isomorphic to a lattice ℤ^{k}, k ≤ 11, or 𝔼_{8} ⊥ ℤ^{k}, k ≤ 3. (This is a theorem of Kneser; see [C-S], Table 16.7.) The lemma now follows immediately from Lemma 8.4. □

With respect to the bounds for s to be proved in the next section, the value 1 for min Λ^{∗} is critical: the results we shall obtain under the assumption min Λ^{∗} ≤ 1 are better when the strict inequality holds. The following partial result will suffice:

Theorem 8.6. An 8-dimensional lattice Λ which is generated by its minimal vectors satisfies the strict inequality min Λ^{∗} < 1.

Proof. We first prove the theorem in the case where Λ can be embedded in some ℤ^{k}. By Lemma 8.4, it suffices to consider the case where Λ is a sublattice of ℤ^{8}. We then denote by h the index [ℤ^{8} : Λ]. Suppose that min Λ^{∗} = 1, and let d = det(Λ). We have γ(Λ^{∗}) = d^{1/8} ≤ γ_{8} = 2, hence d ≤ 2^{8}, and indeed d < 2^{8}, since Λ is not similar to 𝔼_{8}. Calculating γ(Λ_{even}), we obtain the bound 4 (4d)^{−1/8} < 2, i.e. d > 2^{6}. Hence we have 8 < h < 16. Let (ε_{1}, …, ε_{8}) be the canonical basis for ℤ^{8}, set H = ℤ^{8}/Λ and h = |H|, and let φ : ℤ^{8} → H be the canonical surjection. Then the 16 vectors φ(±ε_{i}) are non-zero and distinct except perhaps for a pair ±ε_{i}, for otherwise, Λ would contain vectors of norm 1 or 2 (namely, some ε_{i} or some ε_{i} ± ε_{j}), and we would have h > 16, a contradiction. In case H contains a pair ±ε_{i}, h is even, hence equal to 10, 12 or 14. Since [Λ^{∗} : ℤ^{8}] = [ℤ^{8} : Λ] = h, there exists in Λ^{∗} a vector of the form e = \frac{a_{1}ε_{1}+···+a_{8}ε_{9}}{p} for p = 3, 5 or 7. Reducing e modulo ℤ^{8}, we may assume that |a_{i}| ≤ \frac{p−1}{2}. We now consider the three possible values for p. \underline{p = 3}. We have |a_{i}| ≤ 1, hence N(e) ≤ \frac{8}{9} < 1. \underline{p = 5}. Replacing e by 2e if need be, we may assume that there are at most ⌊\frac{8}{2}⌋ = 4 coefficients a_{i} equal to ±2. Hence N(e) ≤ \frac{4·2^{2}+4·1^{2}}{25} = \frac{4}{5} < 1. \underline{p = 7}. Replacing e by 2e or 3e if need be, we may assume that there are at most ⌊\frac{8}{3}⌋ = 2 coefficients a_{i} equal to ±3. Hence N(e) ≤ \frac{2·3^{2}+6·2^{2}}{49} = \frac{6}{7} < 1.

This completes the proof of Theorem 8.6 for lattices contained in a ℤ^{k}.

We are now left with the three cases where Λ ⊂ ℤ^{k} ⊥ 𝔼_{8}, k = 1, 2, or 3. We still denote by (ε_{i}) the canonical basis for ℤ^{k}. The minimal vectors of Λ are of the form either ε_{i} +τ, where τ is a root of 𝔼_{8}, or also ±(ε_{1} ±ε_{2} ±ε_{3}) if k = 3. We denote by R_{i} the set of roots τ ∈ 𝔼_{8} such that ε_{i} + τ is minimal in Λ.

Let L be the orthogonal projection of Λ on the span of 𝔼_{8}, This is a lattice of dimension 8 − k, generated by the roots τ ∈ 𝔼_{8} such that ε_{i} + τ ∈ S(Λ) for some index i. We have the inclusion Λ ⊂ ℤ^{k} ⊥ L. If L has a component of type 𝔸_{ℓ}, then min Λ^{∗} < 1, and there is nothing to prove. Otherwise, L is isometric either to 𝔻_{8−k} ⊂ ℤ^{8−k}, and L is again a sublattice of ℤ^{8} for which we have just proved the bound min Λ^{∗} < 1, or to 𝔼_{7} (resp. 𝔼_{6}), and then k = 1 (resp. 2).

First consider the case where Λ ⊂ ℤ ⊥ 𝔼_{7}. The set R must then span ℝ 𝔼_{7} and be of affine rank 7, hence must be equal to Ẽ E E_{7}. A Gram matrix for Λ is of the form J_{8}+M where M is a Gram matrix for Ẽ E E_{7} and J_{8} denotes the all ones 8 × 8 matrix. A direct calculation shows that min Λ^{∗} = \frac{43}{81} = 0.530... < 1.

Finally, we consider the more difficult case where Λ ⊂ ℤ^{2} ⊥ 𝔼_{6}. We suppose that min Λ^{∗} = 1.

Let H = (ℤ^{2} ⊥ 𝔼_{6})/Λ and let h = |H|. We first prove that H is elementary Abelian of order 9. We have

γ(Λ^{∗}) = det(Λ^{∗})^{−1/6} = det(Λ)^{1/6} = (3h^{2})^{1/6} ≤ γ_{8} = 2 ,

whence h^{2} ≤ \sqrt{\frac{256}{3}}, i.e. h ≤ 9. For each a ∈ H, denote by R_{a} the set of roots of 𝔼_{6} which map to a. If r_{1}, r_{2} are distinct roots in R_{a}, r_{1} − r_{2} is a non-zero element of Λ, which implies N(r_{1} − r_{2}) ≥ 3, hence r_{1} · r_{2} ≤ 0. Thus R_{a} is an affine root system. If 2a = 0, since r_{1} + r_{2} also belongs to Λ, we have r_{1} · r_{2} = 0, hence R_{a} is of type k𝔸_{1} with k ≤ 4 (𝔼_{6} has deficiency 2). We thus have |R_{a}| ≤ 8 in this case. Conversely, if R_{a} contains a component A A A_{1}, then 2a = 0, since r_{1} + (−r_{1}) = 0. Consequently, if 2a ≠ 0, R_{a} has no component A A A_{1}, which implies |R_{a}| ≤ 9, with equality if and only if R_{a} is of type 3Ã A A_{2}, the unique largest possible system of rank 6. Since R_{0} = ∅, we have ∑_{a∈H}|R_{a}| ≤ 9(h − 1) = 72, and since 𝔼_{6} contains exactly 72 roots, we must have h = 9, and indeed H must be elementary, since R_{a} is of type 3Ã A A_{2}. So E E E_{6} is the union of four systems {a, −a} of type 3A A A_{2} graded by H/{1, −1}.

Choose a system 3A A A_{2} as above in some class a ∈ H, and denote by r_{i}, 1 ≤ i ≤ 9 its elements, ordered in such a way that r_{i} + r_{i+1} + r_{i+2} = 0 for i = 1, 4, 7. Let r be any root in E E E_{6}. Then r · r_{1} + r · r_{2} + r · r_{3} = 0, so that if r ≠ r_{1}, r_{2}, r_{3}, the set (r·r_{1}, r·r_{2}, r·r_{3}) is (0, 0, 0) or a permutation of (1, 0, −1). Clearly, the first case occurs for any r ∈ {r_{4}, …, r_{9}} and the second one for any r ≠ ±r_{1}, …, ±r_{9} (write r as a combination of r_{1}, r_{2}, r_{4}, r_{5}, r_{7}, r_{8} with coefficients a_{i} ∈ ℤ not all zero, hence, say, a_{1} ≠ 0; then r · r_{1} = 2a_{1} − a_{2} must be ±1). Since Aut(𝔼_{6}) acts transitively on its sublattices of type 𝔸_{2}^{⊥3}, we may assume (up to isometry) first that L contains the lattice L_{7} with basis (ε_{1} + r_{i}), i = 1, 2, 4, 5, 7, 8, 3, whose minimal vectors are the 9 pairs ±(ε_{1}+r_{i}), i = 1, …, 9, then that Λ is obtained by extending L_{7} with a vector ε_{2} + r. Its minimal vectors are the 18 pairs ±(ε_{1} + r_{i}), ±(ε_{2} + r′_{i}), where {r′_{i}} is a system of type 3A A A_{2} containing r. Using automorphisms of 𝔼_{6}, we can perform all circular permutations inside the three systems (−1, 0, 1) occurring as scalar products r·r_{i}, and we can realize the transpositions using convenient sign changes of r, (r_{4}, r_{5}, r_{6}) and (r_{7}, r_{8}, r_{9}). This shows that the lattice Λ is unique up to isometry and that

A = ( \begin{matrix}3 & 0 & 1 & 1 & 1 & 1 & 0 & 1 \\ 0 & 3 & 1 & 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 3 & 0 & 1 & 1 & 1 & 1 \\ 1 & 1 & 0 & 3 & 1 & 1 & 1 & 0 \\ 1 & 1 & 1 & 1 & 3 & 0 & 1 & 1 \\ 1 & 1 & 1 & 1 & 0 & 3 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 & 1 & 3 & −1 \\ 1 & 0 & 1 & 0 & 1 & 0 & −1 & 3 \end{matrix} )

is a Gram matrix for Λ. Using this matrix, we have verified that min Λ^{∗} = \frac{7}{9} < 1. This completes the proof of Theorem 8.6. □ [Remarks: (1) for a convenient ordering of the r_{i}, the vector \frac{−ε_{1}+r_{1}+r_{4}+r_{7}}{3} belongs to Λ^{∗}; this proves “by hand” the bound min Λ^{∗} ≤ \frac{7}{9}.

(2) The lattice Λ above is a section of codimension 2 of the short Coxeter-Todd lattice. We have verified that it cannot be embedded into a strictly larger integral, 8-dimensional lattice of minimum 3.]

As for dimension 9, the methods used to prove Theorem 8.6 do not apply, because of the existence of the odd unimodular lattice 𝔻_{12}^{+}, which would only yield the weak bound min Λ^{∗} ≤ 2.

9. Bounds relative to a hyperplane section.

We still consider an integral lattice of minimum 3, whose dimension is denoted by n. Hyperplane cross–sections of Λ are in one-to-one correspondence with pairs ±e of primitive vectors in its dual Λ^{∗} (we associate with e the section of Λ by e^{⊥}). Under various assumptions about the norm of such a vector, we shall prove lower bounds for the kissing number of e^{⊥}, and use these bounds to prove the assertions of the introduction relative to dimensions n ≤ 8. Most of the proofs make use of root systems. However, our first results relies on the following well–known lemma, for which we shall nevertheless give a proof:

Lemma 9.1. Let V be a Euclidean space, and let T ⊂ V be a set of vectors with strictly negative mutual scalar products. Then |T | ≤ 1 + dim V.

Proof. If the vectors in T are independent, then |T | ≤ dim V. Otherwise, there exists a non-trivial dependence relation, that we may write in the form

∑_{x∈T_{1}} λ_{x} x = ∑_{y∈T_{2}} λ_{y} y ,

where T = T_{1} ∪ T_{2} is a partition of T and λ_{x} > 0 (resp. λ_{x} ≤ 0) if x ∈ T_{1} (resp. x ∈ T_{2}). Since it is a norm, the scalar product of the two sides of the equality above is non-negative. Hence ∑_{x∈T_{1},y∈T_{2}} λ_{x}λ_{y} (x · y) ≥ 0, which implies that all λ_{y}, y ∈ T_{2} are zero, or otherwise stated, that any linear relation on T, has non-negative (or non-positive) coefficients. It is then easy to see that this implies that such a relation is unique up to proportionality.

□

[An example of n+1 vectors as above in an n-dimensional lattice is provided by a suitably chosen half–system of minimal vectors in 𝔸_{n}^{∗}.]

We shall apply the lemma above to systems of vectors associated with a suitably chosen vector e ∈ Λ^{∗}. We say that a non-zero vector e ∈ Λ^{∗} is reduced if its norm is minimal in the coset e + Λ modulo Λ. This amounts to the condition

∀ x ∈ Λ , |e · x| ≤ \frac{N(x)}{2} ;

Λ^{∗} contains reduced vectors if and only if Λ is not unimodular.

Let e ∈ Λ^{∗} be reduced. We then have |e · x| ≤ 1 for all x ∈ S = S(Λ). Let

S_{i} = {x ∈ S | x · e = i} .

Then, S = S_{0} ∪ S_{1} ∪ S_{−1} is a partition of S (with S_{−1} = −S_{1}), and S_{0} is the set of norm 3 vectors in the lattice Λ_{0} = Λ ∩ (ℝe)^{⊥}, of dimension ≤ n − 1. Set s_{1} = |S_{1}|.

We first state and prove a lemma concerning norm 1 vectors in Λ^{∗}.

Lemma 9.2. Suppose that S_{1} is non-empty. Then for a reduced vector e ∈ Λ^{∗}, the following conditions are equivalent:

(1) 2e is a sum of two vectors in S_{1}.

(2) N(e) = 1 and 2e ∈ Λ.

Proof. Under hypothesis (1), we have 2N(e) = e · x + e · y = 2, and of course 2e ∈ Λ. Conversely, under hypothesis (2), given any x ∈ S_{1}, we have e · (2e − x) = 1 and N(2e − x) = 3, hence 2e = x + (2e − x). This proves that (1) is satisfied (and moreover, that x ↦ 2e − x is an involution on S_{1}). □

Lemma 9.3. Let e ∈ S(Λ^{∗}), set α = N(e) and β = 1/α, and let x, y be distinct vectors in S_{1}.

(1) If 2e = x + y, then α = 1 and x · y = −1.

(2) If α < \frac{4}{3} and either α ≠ 1 or α = 1 and e is not a sum x_{1} + x_{2},

x_{i} ∈ S_{1}, then x · y = 0 or 1.

(3) If α < \frac{2}{3}, then x · y = 1.

Proof. If 2e = x + y, then α = 1 by the lemma above, and N(x) + N(y) + 2x · y = N(x + y) = 4, hence x · y = −1.

Under the hypotheses of (2) or (3), 2e − x − y is non-zero. Since e is minimal in Λ^{∗}, this implies N(2e − x − y) = 4α − 2 + 2 x · y ≥ α, i.e.

x · y ≥ 1 − \frac{3}{2} α .

If α < \frac{4}{3} (resp. α < \frac{2}{3}), we then have x · y > −1 (resp. x · y > 0), which completes the proof of the lemma. □

Let \overline{S}_{1} be the image of S_{1} under the translation x ↦ x̄ = x − β e. Note that \overline{x} is the orthogonal projection of x onto (ℝ e)^{⊥}. In particular, \overline{S}_{1} is contained in (ℝ e)^{⊥}.

Lemma 9.4. Let e ∈ Λ^{∗} with N(e) < 1 and let Λ_{0} = Λ ∩ (ℝ e)^{⊥}. Then: (1) s_{1} ≤ n.

(2) If s_{1} = n, S_{1} generates a sublattice Λ_{1} of finite index in Λ.

(3) If moreover s > n, the index [Λ : Λ_{1}] is even, and Λ contains a vector of the form f = \frac{e_{1}+···+e_{p}−e_{p+1}−···−e_{2p}}{2} for some p with 6 ≤ 2p ≤ n.

Proof. (1) For x, y ∈ S_{1} we have

x̄ · ȳ = (x − β e) · (y − β e) = x · y − β ,

which shows that distinct vectors in S_{1} have a strictly negative scalar product, hence that s_{1} ≤ dim(ℝ e)^{⊥} + 1 = n by Lemma 9.1.

(2) Let ∑_{i} λ_{i} e_{i} be a non-trivial dependence relation between the e_{i}. Between the projections of the e_{i} onto (ℝ e)^{⊥}, we have the relation ∑_{i} λ_{i} ē_{i}. The proof of Lemma 9.1 shows that this is unique up to proportionality, and that changing the signs if need be, we may assume that it has non-negative coefficients. Then ∑_{i} λ_{i} = ∑_{i} λ_{i}(e_{i} · e) = 0, which implies that all λ_{i} are zero, a contradiction.

(3) The lattice Λ_{1} ∩ (ℝe)^{⊥} is generated by the differences e_{i} − e_{j}, hence is contained in the even part of Λ_{0}. Since s > n, Λ_{0} contains a norm 3 vector f_{0}. Let a be the smallest integer such that a f_{0} ∈ Λ_{1}. We have a f_{0} ∈ Λ_{1} ∩ (ℝ e)^{⊥} ⊂ Λ_{even}. Since f_{0} ∈ Λ∖Λ_{even}, a is even. Since e · e_{i} = 1 for all i, Λ∩(ℝ e)^{⊥} contains a vector of the form f = \frac{e_{1}+···+e_{p}−e_{p+1}−···−e_{2p}}{2} for some p with 2p ≤ n, and we have 2p ≥ 6 since integral lattices of dimensions n ≤ 5 having an odd minimum must have index 1. This completes the proof of the lemma. □

Corollary 9.5. Let Λ be a lattice with min Λ = 3 and min Λ^{∗} < 1. Then Λ has a hyperplane section Λ_{0} such that s(Λ) ≤ S(Λ_{0}) + n.

Proof. Just apply the proposition above to a vector e ∈ Λ^{∗} with norm N(e) < 1. □

Corollary 9.6. If n ≤ 8 and Λ is not isometric to W_{7}, then Λ has a hyperplane section Λ_{0} such that s(Λ) ≤ S(Λ_{0}) + n.

Proof. Apply the corollary above together with Theorems 8.1 and 8.6. □

By the results of Section 7, if n ≤ 5, every system of n independent minimal vectors in Λ generates Λ. Lemma 9.4 then shows that the corollary above is not optimal in these dimensions. It is for n = 6, since s(W_{6}) = 16 and s(W_{5}) = 10, and it allows us to recover the bound s ≤ 16 for n = 6 from the bound s ≤ 10 for n = 5. Note that in dimension 6, if Λ is not isometric to W_{6}, it has a hyperplane section Λ_{0} such that s(Λ) − s(Λ_{0}) ≤ 5: this is clear if s(Λ) ≤ 10, and the three lattices with 11 ≤ s < 16 all have a section W_{5}, with s = 10. The classification results to be proved in the next section will show that in dimension 7, every lattice Λ with min Λ^{∗} < 1 has a section Λ_{0} with s(Λ) − s(Λ_{0}) ≤ 6.

Applied to dimension 7, the results above imply interesting restrictions on the possible values for s. However, these results will be significantly improved in the next section, so that we state them as a mere proposition.

Proposition 9.7. Let Λ be an integral, 7-dimensional lattice of minimum 3. Then either s = 28 and Λ ≃ W_{7}, or Λ is isometric to a uniquely defined lattice W′_{7} with s = 18, or s(Λ) ≤ 17.

Proof. Let Λ_{0} be a hyperplane section of Λ on which s_{0} = s(Λ_{0}) attains its maximum. We know by Proposition 7.6 and Theorems 6.5 and 7.9 that either s_{0} ≤ 10, or Λ_{0} is one of the lattices W_{6} or W′_{6}, and then s_{0} = 16 or 12, or Λ_{0} is one of two lattices with s = 11. In particular, we have s_{0} ≤ 16, which immediately implies s ≤ 23. Classifying the extensions of W_{6} (which could be fairly easily done by hand, thanks to the rich structure of S(W_{6})), we see that besides W_{7}, there are exactly three extensions of W_{6}, namely a lattice W′_{7} of determinant 128 with s = 18 and two lattices with s = 17 (a lattice W′′_{7} of determinant 144 and W_{6} ⊥ W_{1}). This shows that either Λ is isometric to W_{7}, or s(Λ) ≤ s(W′_{6}) + 7 = 19. To complete the proof of the proposition, we have used a computer search to classify the extensions of the remaining three lattices, and again only W′_{7}, W′′_{7} and W_{6} ⊥ W_{1} showed up. □

Using the proposition above together with the bound s − s_{0} ≤ 8, we can now prove:

Proposition 9.8. Let Λ be an integral, 8-dimensional lattice of minimum 3. Then either s = 30 and Λ ≃ W_{8}, or s = 29 and Λ ≃ W_{7} ⊥ W_{1}, or s ≤ 25.

Proof. We first classify the extensions of W_{7} (which can be fairly easily done by hand, thanks to the rich structure related to the Hamming code of S(W_{7})). We find this way W_{8} and W_{7} ⊥ W_{1}. If Λ does not extend W_{7}, we have s_{0} ≤ 18, whence s ≤ 26, and even s ≤ 25 if Λ does not extend W′_{7}. In this last case, we have checked that extensions of W′_{7} except W_{8} and a uniquely defined lattice W′_{8} of determinant 192 with s = 22 have s ≤ 20. □

We now return to the case of an arbitrary dimension n. The lemma that we state and prove below will be used in Section 11 to bound the kissing number for 9-dimensional, integral lattices of dimension 9.

Lemma 9.9. With the notation of Lemma 9.3, suppose that N(e) ≤ 1, i.e. β ≥ 1. Let t = (β(β − 1))^{(1/2)}, and let \overline{\overline{S}}_{1} be the image of S_{1} under the translation

x ↦ \overline{\overline{x}} = t e + x̄ = (t − β) e + x .

(We have \overline{\overline{x}} = x̄ if and only if N(e) = 1.) If N(e) < 1, or if N(e) = 1 and e is not a sum x + y, x, y ∈ S_{1}, then \overline{\overline{S}}_{1} is the set of simple roots (of norm 2) in an affine root system of rank ≤ n. We have s_{1} ≤ n + 2.

Proof. We have

\overline{\overline{x}} · \overline{\overline{y}} = ((t − β) e + x) · ((t − β) e + y) = \frac{1}{β} (t − β)^{2} + 2 (t − β) + x · y = x · y − 1 .

This shows that \overline{\overline{x}} · \overline{\overline{y}} = −2, −1, 0 for x · y = −1−, 0, 1, that all vectors in \overline{\overline{S}}_{1} have norm 2, and that \overline{\overline{x}}·\overline{\overline{y}} = −2 is equivalent to x·y = −1 ⟺ N(x+y) = 4 on the one hand, and to x + y = −2 (t − β) e on the other hand. Comparing the norms, we obtain that

x · y = −1 ⟺ (t − β)^{2} β = 1 ⟺ β = 1 ,

and that when β = 1, then 2 e = x + y. Consequently, vectors in \overline{\overline{S}}_{1} have mutual scalar products 0 or −1. Thus the elements of \overline{\overline{S}}_{1} can be viewed as the vertices of an affine Dynkin diagram defined on an affine space of dimension n − 1.

Let S′ be an irreducible component of the Dynkin diagram and let x ∈ S′. The structure of the Dynkin diagram (see [C-S], Table 4.1) shows that the equality \overline{\overline{x}}·\overline{\overline{y}} = 0 holds on S_{1}∖S′ and for at least |S′|−3 elements in S′, hence for at least s_{1} −3 vectors in \overline{\overline{S}}_{1}. Now, for y, z ∈ S_{1} such that \overline{\overline{x}}·\overline{\overline{y}} = \overline{\overline{x}}·\overline{\overline{z}} = 0, we have x · y = x · z = 1, hence (x − y) · (x − z) = 2 and N(x − y) = 4. This shows that the set \{x − y, | \overline{\overline{x}} · \overline{\overline{y}} = 0\}, is a scaled copy of a root system of type 𝔸_{r} contained in ℝe^{⊥}, with r ≥ s_{1} − 3. We thus have s_{1} − 3 ≤ n − 1.

This completes the proof of the proposition. □

Remark 9.10. Except if S′ defines an affine system of type 𝔸̃_{r}, r ≥ 2, x can be chosen in such a way that |S′| − 2 or |S′| − 1 elements in S′ are orthogonal to \overline{\overline{x}}. Hence if the bound s_{1} = n + 2 is sharp, all components of the root system \overline{\overline{S}}_{1} must be of type Ã A A_{r}, r ≥ 2. Examples with s_{1} = n + 2 are known for n = 10, with systems 4Ã A A_{2} and 3Ã A A_{3}, obtained on the lattices with even parts K′_{10} and Q_{10} respectively that we mentioned in Sections 2 and 8.

10. Dimensions 7 and 8.

In this section, in which we keep the notation of the previous section, we shall make more precise the results of Theorems 9.7 and 9.8. To this end, we shall prove new bounds for the difference s(Λ) − s(Λ_{0}) relying on properties of the index. As above, Λ stands for an integral, n-dimensional lattices (n = 7 or 8) of minimum 3 and s_{0} = s(Λ_{0}), where Λ_{0} is a hyperplane section of Λ whose kissing number is maximal. Recall that 6-dimensional lattices of minimum 3 have index ı = 1 to within the three exceptions of W_{6}, W′_{6}, W′′_{6} for which ı = 2 (Theorem 6.5), that 7-dimensional lattices of minimum 3 have index 1 or 2, except W_{7} for which ı = 3, and that lattices with ı ≥ 2 are those which extend one of the lattices W_{6}, W′_{6}, W′′_{6}. (In dimension 8, it is easily verified that ı(W_{8}) = 4 and ı(W_{7} ⊥ W_{1}) = 3; a consequence of the results we are going to prove is that ı = 1 or 2 otherwise.)

Unless otherwise stated, all lattices are integral and have minimum 3.

Theorem 10.1. Let Λ be an integral, 7-dimensional lattice of minimum 3. Then s(Λ) ≤ 14 to within the following five exceptions: W_{7} (s = 28), W′_{7} (s = 18), W′′_{7} and W_{6} ⊥ W_{1} (s = 17), and one lattice with s = 16. Moreover, there are four (resp. nine, resp. 27) lattices with s = 14 (resp. s = 13, resp. s = 12).

Since lattices of the form Λ ⊥ W_{1} with dim Λ = 6 cover the interval [7, 13], we have:

Corollary 10.2. The set of values taken by s on the set of wellrounded, 7-dimensional lattices is [7, 14] ∪ [16, 18] ∪ {28}. □

To prove Theorem 10.1, we shall take into account the index of Λ, making use of three lemmas.

Lemma 10.3. Besides the five lattices with s ≥ 16 listed in Theorem 10.1, there are seventeen 7-dimensional lattices of index 2, namely four with s = 14, six with s = 13, three with s = 12 and four with s = 11.

Proof. Just list and test for isometry the extensions of W_{6}, W′_{6}, W′′_{6}. □

Lemma 10.4. The 7-dimensional lattices of index 1 all have s ≤ 13. There are 3 such lattices with s = 13 and 24 with s = 12.

Proof. We know by Theorem 6.5 that there are altogether seventeen 6-dimensional lattices of index 1 with s ≥ 9 (and indeed, s = 9, 10, or 11). Listing the extensions of these lattices, we find 24 lattices with s = 12, 3 with s = 13, and none with s ≥ 14. The problem is now to show that no new lattice shows up among the extensions of well-rounded, 6-dimensional lattices with s = 6, 7 or 8. This is a consequence of the following lemma, in which we more generally consider integral lattices of any odd minimum. □

Lemma 10.5. Let L be an integral, well-rounded, 7-dimensional lattice having an odd minimum m. If s(L) ≥ 12, then L possesses a hyperplane section of minimum m with s ≥ 9.

Proof. It involves many details, so that we only sketch it. We classify minimal classes of index 1 obtaining relatively easily the description of the 1 + 3 + 6 + 7 = 17 classes with s ≤ 10 which do not extend a 6-dimensional class with s = 9. The method is the one we sketched in Section 6 to deal with dimension 5, together with the following improvement: we have shown that the equivalence class of a Bacher matrix (defined in [Bt]) characterizes the corresponding minimal class; the proof is just a modification of the proof of a proposition due to Bergé (Proposition 2.9 in [Bt]). We prove that these classes can all be expressed using uniquely vectors with 3 or 5 components on a convenient basis B = (e_{1}, …, e_{7}) of minimal vectors (except for one obvious class with s = 7). We distinguish two classes with s = 9, namely (a_{9}) and (b_{9}), defined by

(a_{9}) : e_{8} = e_{1} + e_{2} + e_{3}, e_{9} = e_{1} + e_{2} + e_{4} and (b_{9}) : e_{8} = e_{1} + e_{2} + e_{3}, e_{9} = e_{1} + e_{4} + e_{5} ,

which we now use to list the seven classes with s = 10 which do not extend a 6-dimensional class with s = 9:

(a_{10}): (a_{9}), e_{5} + e_{6} + e_{7} ; (b_{10}) : (b_{9}), e_{1} + e_{6} + e_{7} ; (c_{10}) : (b_{9}), e_{2} + e_{6} + e_{7} ; (d_{10}) : (a_{9}), e_{1} + e_{2} + e_{5} + e_{6} + e_{7} ; (e_{10}) : (a_{9}), e_{1} + e_{3} + e_{5} + e_{6} + e_{7} ; (f_{10}) : (b_{9}), e_{1} + e_{2} + e_{4} + e_{6} + e_{7} ; (g_{10}) : (b_{9}), e_{2} + e_{3} − e_{4} + e_{6} + e_{7}.

We now consider extensions of these seven minimal classes to a class with s = 11. It is easily verified that adding a vector with seven non-zero components to one of these classes amounts to adding a vector with three or five non-zero components to another one (maybe the same). Then we have considered in detail the first three classes and proved that adding a vector with five components yield classes which could all be expressed with four vectors with only three components. It is easy to check that a minimal class defined on B by four extra minimal vectors having only three nonzero components always possesses a hyperplane section containing 9 pairs of minimal vectors. This proves Lemma 10.5 for classes with s ≥ 11 which extend (a_{10}), (b_{10}) or (c_{10}).

Since extending one of the four classes (d_{10}) — (g_{10}) with a vector having three non-zero components on B amounts to extending one of the classes (a_{10}), (b_{10}), (c_{10}), we are left with extensions of (d_{10}) — (g_{10}) with a vector with five components.

In the case of (d_{10}) or (e_{10}), any extension without a convenient hyperplane section must involve e_{5}, e_{6} and e_{7}. It turns out that a characteristic determinant equal to ±2 always shows up. In each of the cases (f_{10}), (g_{10}), we find an essentially unique extension, which both define the same class (a_{11}) with s = 11, and which cannot be extended to a class of index 1 with s ≥ 12. □ [Explicitly, (a_{11}) = (b_{9}), e_{1} + e_{2} + e_{4} + e_{6} + e_{7}, e_{2} + e_{3} + e_{5} + e_{6} + e_{7}.]

Proof of 10.1. Clear using the three lemmas above. □

We now consider 8-dimensional lattices.

Theorem 10.6. Let Λ be an integral, 8-dimensional lattice of minimum 3. Then s(Λ) ≤ 20 to within the following three exceptions: W_{8} (s = 30), W_{7} ⊥ W_{1} (s = 29), and W′_{8} (s = 22). Moreover, there are seven lattices with s = 20.

Proof. We know by Theorem 8.6 and Lemma 9.4, (1) that Λ possesses a hyperplane section Λ_{0} of minimum 3 with s(Λ_{0}) ≥ s(Λ) − 8. Since we have a complete list of small 7-dimensional lattices Λ_{0} with s(Λ_{0}) ≥ 12, we can list all 8-dimensional lattices Λ with s ≥ 20. □

Corollary 10.7. The set of values taken by s on the set of well-rounded, 8-dimensional lattices is [8, 20] ∪ {22} ∪ {29, 30}.

Proof. Using the transformation Λ ↦ Λ ⊥ W_{1} together with the results of Corollary10.2, we see that the set of values for s contains the union [8, 15] ∪ [17, 19]. Taking into account Theorem 10.6, it suffices to exhibit a lattice with s = 16. An easy example is provided by the pull-back of the binary code generated by the four vectors

(1, 1, 1, 0, 0, 0, 0, 0), (0, 0, 0, 1, 1, 1, 0, 0), (1, 0, 0, 0, 0, 1, 1, 0), (0, 1, 0, 0, 1, 0, 0, 1) ,

a 4-modular lattice of determinant 256. □

We could prove a more precise result, the proof of which we just sketch:

Theorem 10.8. There are exactly five integral, 8-dimensional lattices of minimum 3 with s = 19. □

To prove this theorem, since we know all Λ_{0} with s(Λ_{0}) ≥ 12, it suffices to prove the existence of hyperplane sections Λ_{0} such that s(Λ) − s(Λ_{0}) ≤ 7. We can produce two kind of proofs for this result. The first one consists in studying closely the possible root systems that we have constructed in Section 9. The second one relies on considerations about the index. Using Proposition 7.13 for lattices of index ı ≥ 3 and theorem 10.1 for lattices which extend a 7-dimensional lattice with s ≥ 12, we are left with lattices of index 2 of the form Λ = ⟨e_{1}, …, e_{8}, f⟩ where e_{1}, …, e_{8} are minimal vectors and f = \frac{e_{1}+e_{2}+e_{3}+e_{4}−e_{5}−e_{6}−e_{7}−e_{8}}{2} . (See Remark 7.14; f is orthogonal to a minimal vector in Λ^{∗}.) For these particular lattices, we can prove the existence of a hyperplane section Λ_{0} such that s(Λ) − s(Λ_{0}) ≤ 5.

11. Bounds for dimension 9.

Let Λ be a 9-dimensional lattice of minimum 3 and let e ∈ S(Λ^{∗}) be reduced. As above, let S_{1} = {x ∈ S(Λ) = S | e · x = 1}, s_{1} = |S_{1}|, α = N(e), and β = \frac{1}{α}.

Lemma 11.1. If N(e) ≠ 1, or if N(e) = 1 and 2e ∉ Λ, then s_{1} ≤ 13. [Note that if N(e) < 1 (resp. N(e) = 1), we have s_{1} ≤ 9 (resp. s_{1} ≤ 11) by Corollary 9.5 (resp. by Lemma 9.9) The proof will also show the following refinement: if β > 0.902, then s_{1} ≤ 12.]

Proof. Let \overline{S}_{1} and \overline{\overline{S}}_{1} be the images of S_{1} under the translations x ↦ x̄ = x − βe and x ↦ ¯ x̄ = x̄ + ζe where ζ is the (positive) square root of β(β − \frac{3}{4}). (This definition makes sense since we have proved the upper bound N(Λ^{∗}) < \frac{4}{3}.) Note that \overline{S}_{1} is contained in the set of norm 3 vectors in the n − 1-dimensional lattice Λ ∩ e^{⊥}, hence that \overline{\overline{S}}_{1} is contained in an affine hyperplane of ℝ^{9}.

Let x, y ∈ S_{1}. We have x · y ≥ 0 by Lemma 9.3. An easy calculation then shows the formulae

x̄ · ȳ = x · y − β ≥ −β

and

\overline{\overline{x}} · \overline{\overline{y}} = x̄ · ȳ + ζ^{2}α = x̄ · ȳ + β − \frac{3}{4} = x · y − \frac{3}{4} ≥ −\frac{3}{4} .

This proves that \overline{\overline{x}} · \overline{\overline{x}} = 3 − \frac{3}{4} = \frac{9}{4} and that \overline{\overline{x}} · \overline{\overline{y}} = \frac{1}{4} or − \frac{3}{4} if \overline{\overline{y}} ≠ \overline{\overline{x}}. To get rid of denominators, we now replace \overline{\overline{S}}_{1} by its scaled copy

Z = \{2z | z ∈ \overline{\overline{S}}_{1}\};

the formulae above now read z · z = 9 and z_{1} · z_{2} = 1 or − 3 if z_{2} ≠ z_{1}.

Denote by A, B, C the number of pairs (z_{1}, z_{2}) ∈ Z × Z such that z_{1} · z_{2} = 9, 1, −3 respectively. We have

A = |S_{1}| and A^{2} = A + B + C .

We shall use this to express |S_{1}| = A in terms of the single parameter C. Set Z = {z_{1}, …, z_{A}} , and let z = z_{1} + ⋯ + z_{A}. We have

N(z) = ∑_{i,j} z_{i} · z_{j} = 9A + B − 3C = A^{2} + 8A − 4C ≥ 0 ,

hence C ≤ \frac{1}{4} A^{2} + 2A.

To obtain a lower bound for C, we consider the harmonic polynomial F_{z}(t) = ∑_{i} P_{2}^{(t)}(z_{i}). (The notation is that of [V], § 3.) We have

\begin{aligned}[F_{z}, F_{z}] & = ∑_{i} ((z_{i} · z_{j})^{2} − 0) = 9^{2}A + B + 9C − 9A^{2} \\ & = 8(−A^{2} + 10A + C) ≥ 0 . \end{aligned}

Hence C ≥ A^{2} − 10A.

Comparing the two bounds above for C, we get

A^{2} − 10A ≤ \frac{1}{4} A^{2} + 2A ⟺ A ≤ 16 ,

which is not yet optimal.

In the proof above, we used the weak inequality v · v ≥ 0 valid for any vector v in any Euclidean vector space E, applied successively with E = ℝ^{9} and E = Harm_{2}(ℝ^{9}). To go further, we shall on the one hand make a better choice for v, and on the other hand use the Cauchy-Schwarz inequality in the form

v · v ≥ \frac{(v · u)^{2}}{u · u}

with a suitably chosen vector u.

First, we consider the vector t = z − ξe where we chose ξ in order that t be orthogonal to e. We have t · e = (∑_{i} z_{i} − ξe) · e = 2Aζα − ξα. Hence we take ξ = 2Aα. Then

t · t = ( ∑_{i} z_{i} − ξe) · ( ∑_{j} z_{j} − ξ) = ∑_{i,j} z_{i} · z_{j} − ξ(e · z)

= A^{2} + 8A − 4C − 4(β − \frac{3}{4}) A^{2}

= 4(1 − β) A^{2} + 8A − 4C ,

which implies the new bound

C ≤ (1 − β) A^{2} + 2A ,

better than the previous one, since we have replaced \frac{1}{4} by 1 − β < \frac{1}{4} in front of the A^{2} term.

To improve on the lower bound for C, we consider together with F_{z} another harmonic polynomial, namely F_{e}(t) = P_{2}^{(e)}(t) = (e · t)^{2} − \frac{α}{9} (t · t). We have

[F_{z}, F_{e}] = ∑_{i} ((z_{i} · e)^{2} − α) = 4ζ^{2}α^{2} A − α A = 4(β − 1)α A

and [F_{e}, F_{e}] = (e · e)^{2} − \frac{1}{9} α^{2} = \frac{8}{9} α^{2}.

Using the inequality [F_{z}, F_{z}] ≥ [F_{z}, F_{e}]^{2} / [F_{e}, F_{e}], we obtain

8(−A^{2} + 10A + C) ≥ \frac{16(β − 1)^{2}α^{2} A^{2}}{(8/9) α^{2}} = 18(β − 1)^{2} A^{2} ,

i.e. C ≥ (1 + \frac{9}{4} (β − 1)^{2}) A^{2} − 10A, better than the previous lower bound C ≥ A^{2} − 10A.

Comparing the last upper and lower bounds for C, we get

(1 + \frac{9}{4} (1 − β)^{2}) A − 10A ≤ (1 − β) A^{2} + 2A ⟺ A ≤ \frac{12}{\frac{9}{4} β^{2} − \frac{7}{2} β + \frac{9}{4}} .

The maximum of the right hand side is attained for β = \frac{7}{9}, and is then equal to \frac{27}{2} < 14. This proves the desired bound A ≤ 13 and the refinement A < 13 for β > (91 + 4\sqrt{13})/117 = 0.90104... □

Lemma 11.2. If N(Λ^{∗}) = 1, then there exists e ∈ S(Λ^{∗}) with 2e ∉ Λ.

Proof. It resembles that of Theorem 8.6, and involves many details. Since the proof of Theorem 11.4 only needs a weaker result which could be obtained by the methods of Appendix 4, we only sketch it.

Since e belongs to Λ^{∗} and N(e) is integral, the lattice M = ⟨Λ, e⟩ is integral. By Theorem 8.3, M can be embedded in a unimodular lattice M′ of dimension n′ ≤ 12. Since M contains a vector of norm 1, M′ is not isometric to 𝔻_{12}^{+}. Hence, we have M′ ≃ ℤ^{k} (k ≤ 12, and in fact k = 9 if we assume that k is minimal) or M′ ≃ 𝔼_{8} ⊥ ℤ^{k} (k ≤ 4). As usual, we denote by (ε_{i}) the canonical basis for ℤ^{k}.

If Λ ⊂ ℤ^{9}, all norm 1 vectors in ℤ^{9} must belong to Λ^{∗} (see the proof of Theorem 8.6). Hence we have 2ℤ^{9} ⊂ Λ ⊂ ℤ^{9}, which shows that Λ is the pull back of a code of length 9 and weight 3 (together with a vector ε_{i} + ε_{j} + ε_{k}, Λ contains the eight vectors ε_{i} ± ε_{j} ± ε_{k}). It can be verified that such codes do not exist.

Suppose now that Λ ⊂ 𝔼_{8} ⊥ ℤ^{k}. As in the proof of Theorem 8.6, consider the projection L of Λ on the span of 𝔼_{8}. This is a lattice of dimension 8 − k, generated by the roots τ ∈ 𝔼_{8} such that ε_{i} + τ ∈ S(Λ) for some index i.

We have the inclusion Λ ⊂ ℤ^{k} ⊥ L. If L has a component of type 𝔸_{ℓ}, then min Λ^{∗} < 1. If L ≃ 𝔻_{8−k}, then we may embed L into some ℤ^{ℓ}. Hence, it suffices to consider the cases where L ⊂ 𝔼_{7} ⊥ ℤ^{2} or L ⊂ 𝔼_{6} ⊥ ℤ^{3}. These cases are proved to be impossible by arguments like those we used in the proof of Theorem 8.6. □

Theorem 11.3. Let Λ be a 9-dimensional integral lattice of minimum 3. Then either Λ is isometric W_{9}, with s = 34, or s(Λ) ≤ 32.

Proof. By Theorem 10.6, a hyperplane section of Λ is isometric to W_{8}, W_{7} ⊥ W_{1}, W′_{8}, or to one of seven lattices with s = 20, or has s ≤ 19. By a direct computer computation, we have checked that the extensions of these ten lattices either are isometric to W_{9}, or have s ≤ 32. Lemmas 11.1 and 11.2 then prove that extensions of the other 7-dimensional lattices have s ≤ 19 + 13 = 32. □ [As a matter of fact, extensions of W_{8} and of W_{7} ⊥ W_{1} are easily dealt with by hand, but this time, such calculations do not suffice to prove the bound s ≤ 34.]

The same method, using Theorem 10.8 (that we have not proved in detail) instead of Theorem 10.6 immediately proves the following more precise result:

Theorem 11.4. Let Λ be a 9-dimensional integral lattice of minimum 3. Then either Λ is isometric W_{9}, with s = 34, or to one of three lattices with s = 32, or has s ≤ 31. □

We conjecture that on the set of 9-dimensional, integral lattices of minimum 3 which are generated by their minimal vectors, the bound N(Λ^{∗}) < \frac{4}{3} can be improved to N(Λ^{∗}) ≤ 1 (which would then be optimal). Moreover, it might well be that a hyperplane section Λ_{0} of minimum 3 with s(Λ) − s(Λ_{0}) ≤ n − 1 exist in dimension 9 as in dimension 8. We could then be able to list all lattices with s ≤ 27. We state the result as a conjecture:

Conjecture 11.5. There are 11 integral, well-rounded, 9-dimensional lattices of minimum 3 with s ≥ 27, one with s = 34, two with s = 31 and 30, and three with s = 32 and 28.

We have found well-rounded lattices for small values of s in the range 9 — 26. Thus a consequence of the conjecture above is that the set of possible values for s on the set of integral, well-rounded, 9-dimensional lattices should be

[9, 26] ∪ {28} ∪ [30, 32] ∪ {34} .

Things are less clear for dimension 10, where the largest known value for s, namely 40, is attained on four lattices, including two “non-Watson” lattices. Experimental data suggest that for n = 10, we have s = 40 or s ≤ 38.

We end this section with a few words about integral lattices of minimum 5. We know that up to n = 6, the set of possible values for s on a well–rounded lattice is the whole interval [n, s_{5}(n)] and that s_{5}(n) = s_{3}(n). (This probably also holds for all m ≥ 7 odd.) When n = 7, there is probably a gap: a computer search only found lattices with s ∈ [7, 21] ∪ {27}. Maybe, one could use methods à la Watson to prove the inequality s ≤ 21 on lattices having index ı ≤ 2, and then prove that there exists a unique lattice of index 3.

Probably, all 8-dimensional lattices have s ≤ 30. However, the bounds we proved or conjectured for lattices of minimum 3 in dimensions 9 or 10 do not extend to larger minima. Indeed, for r ≡ 3 mod 4, the unique sublattice of 𝔸_{r}^{∗} containing 𝔸_{r} to index \frac{r+1}{2} is proportional to an integral lattice of minimum m = \frac{r−1}{2} ≡ 1 mod 2, denoted by Cox_{r}. We have Cox_{7} ≃ W_{7}. For r ≥ 11, the lattice Cox_{r} has s = \frac{r(r+1)}{2}, and possesses sections with s = \frac{n(n−1)}{2} + 1 if n = r − 1 and s = \frac{n(n−1)}{2} if n ≤ r − 2. Hence, for all m ≥ 5 odd, lattices with s = 36 > 34 (resp. s ≥ 45 > 40) exist in dimension n = 9 (resp. 10), and we can even have s = 46 for m = 5 and n = 10. Note that for m = 5 (taking r = 11), we obtain the lattices with n = 6 and s = 15 of Theorem 7.11 and with n = 5 and s = 10 of Theorem 6.2. A.-M. Bergé ([Be]) has proved that the values of s on sections of Cox_{r} given above are the largest possible for all n ≥ 4.

Appendix 1: 5- and 6-dimensional lattices without a hexagonal section having the same minimum.

[figure: λ_{10}; λ_{9}; ν_{9}; λ′_{8}; λ_{8}; ν_{8}; λ′_{7}; λ_{7}; μ_{7}; ν_{7}; λ′_{6}; λ_{6}; μ_{6}; ν_{6}; λ_{5}; ν_{5}]

λ_{i}, λ′_{i}: classes of index 1 satisfying condition (C1) ; μ_{i}: other classes of index 1 ; ν_{i}: classes of index 2.

Figure App1.

Inclusion graphs for NHS minimal classes in dimension 5.

Minimum minimorum. We list below for each class as above the smallest possible minimum for an integral lattice belonging to this class. m = 1 : λ_{5} ; m = 3 : λ_{i}, i ≠ 1, 9 ; λ′_{i}; m = 4 : ν_{5}, μ_{6} ; m = 5 : λ_{9} ; m = 6 : μ_{7} ; m = 10 : ν_{6}, ν_{7}, ν_{8} ; m = 16 : ν_{9}. All classes λ_{i}, λ′_{i} contain integral lattices of any odd minimum m ≥ 5.

Correspondence with Batut’s classification. λ_{i}: a_{5}, b_{6}, e_{7}, m_{8}, p_{9}, w_{10} ; λ′_{i}: d_{6}, g_{7}, j_{8} ; μ_{i}: c_{6}, f_{7} ; ν_{i}: b_{5}, e_{6}, j_{7}, p_{8}, w_{9}. [Incidentally, the fifth column of the row w_{9} in [Bt], which reproduces that of the row u_{9}, is a slip. One should read [36, 88, 2].]

Dimension 6. We know that for an integral, well-rounded lattice having an odd minimum, of index ı = 1 (resp. 2, resp. 3), we have 6 ≤ s ≤ 12 (resp. 6 ≤ s ≤ 16, resp. s = 12). We have verified more generally that minimal classes having no hexagonal sections and such that s ≥ 14 are contained in the class defined by W_{6}. In particular, we recover this way Watson’s theorem for dimension 6. As for classes of index 1 containing an integral lattice, for s = 7 (resp. 8, 9, 10) there are 2 (resp. 4, 4, 5) classes.

Appendix 2: improved bounds in the range 2–24

The second statement in Theorem 3.1 can be made more precise. The proof, which relies on a complicated calculation, resembles that of Lemma 11.1. We state the result without giving its proof.

Theorem. Let Λ be an integral lattice of minimum 3 and dimension n in the range 2–24. Suppose that Λ is not unimodular. Let e ∈ S(Λ^{∗}), α = N(e),

s_{1} = |{x ∈ S(Λ) | e · x = 1}|,

and λ = \frac{s_{1}}{s}. Then we have

s(Λ) ≤ \frac{8n(n + 2)}{25 − n} − A_{1} − A_{2}

where

A_{1} = \frac{n^{2}(n + 2)(50 − n)}{9(n − 1)(n + 4)(25 − n)} × \frac{(λ − 3α/n)^{2}s}{α^{2}}

and

A_{2} = \frac{n}{9(n − 1)(n + 4)(25 − n)} × \frac{(3^{3}α^{2} − (18α − (n + 4))(n + 2)λ)^{2}s}{α^{4}} .

□

The restriction that Λ should not be unimodular is necessary: indeed, the proof makes use of the existence of a non-zero reduced vector e, a property which fails for unimodular lattices.

We know that on non-unimodular lattices, equality holds in Theorem 3.1 if and only if n = 7, 16 or 22, and in each case, we know the value of s. An easy calculation shows that A = B = 0 ⟹ α = \frac{n+2}{9} and \frac{s_{1}}{s} = \frac{n+2}{3}. This gives for n = 7, 16, 22 the following values for s and s_{0} = s − s_{1}:

n = 7: s = 28, s_{0} = 16;

n = 16: s = 256, s_{0} = 190;

n = 22: s = 1408, s_{0} = 896.

We recognize the values of s_{0}, namely those of s(Λ_{0}) for Λ_{0} = W_{6}, O_{15}, and O_{22} respectively. It should be noticed that our results imply that a lattice of dimension n_{0} ∈ {6, 15, 21}, if not isometric to one of the lattices Λ_{0} listed above, cannot be embedded in a Λ with dim Λ = n_{0} + 1 and s(Λ) maximal. The existence of such a lattice has been proved to be impossible if n_{0} = 6, and is very unlikely if n_{0} = 15 or 21.

Appendix 3: Large values of min Λ^{∗}.

We display the largest known values of min Λ^{∗} for integral lattices of minimum 3 which are generated by their minimal vectors in the range 1 ≤ n ≤ 24. We have verified that these bounds are optimal for 1 ≤ n ≤ 7; thanks to the results of Section 7, it then suffices to assume that Λ is well-rounded.

For n = 1, 2, 3, 7, the bounds are even optimal on the set of all lattices of minimum 3: for n = 1, 2, because Λ and Λ^{∗} are similar; for n = 3, because γ′_{3} is attained on 𝔸_{3}^{∗} ∼ W_{3} — cf. [M], Theorem 6.3.4; and for n = 7, by Theorem 8.1. The case n = 5 would follow from a proof of the conjectural value of γ′_{5}.

The lattice Λ ⊂ ℤ^{2} ⊥ 𝔼_{6} constructed in the proof of Theorem 8.6 attains the largest known value (indeed, \frac{7}{9}) for n = 8. However, there exists for n = 8 a well-rounded lattice (of index 2) for which min Λ^{∗} outdoes the value given in the table (\frac{4}{5} = 0.8 > \frac{7}{9} = 0.77 …). Note that the results of Section 7 imply that well-rounded lattices which are not generated by their minimal vectors do not exist in dimensions n ≤ 7.

Next, we find the value 1 for 9 ≤ n ≤ 13, and then larger values (2 is attained on O_{16}, 3 on O_{23} and O_{24}); for n = 10 examples are provided by the lattices related to K′_{10} or to Q_{10} which were constructed in Section 2; for n = 9 and 11 ≤ n ≤ 23, one may consider convenient sublattices of O_{23} or Plesken-Pohst lattices (denoted by plp_{∗} with suitable subscripts ∗), for which Gram matrices can be found in [Bt-M].

Large values of min Λ^{∗}

n  1 2 3 4 5 6 7 8
min Λ^{∗} \frac{1}{3} \frac{3}{8} \frac{1}{2} \frac{1}{2} \frac{2}{3} \frac{3}{4} 1 \frac{7}{9}
n  9 10 11 12 13 14 15 16
min Λ^{∗} 1 1 1 1 1 \frac{4}{3} \frac{3}{2} 2
n 17 18 19 20 21 22 23 24
min Λ^{∗} \frac{3}{2} \frac{3}{2} \frac{3}{2} 2 2 \frac{8}{3} 3 3

We list below the lattices which are known to us to meet the bound for dimensions which were not considered above.

\underline{n = 9} : O_{9a3}, O_{9b2} ; \underline{n = 11} : W_{11a}, W_{11b}, O_{11} ;

\underline{n = 12} : W_{12}, O_{12a}, O_{12b} ; \underline{n = 13}: W_{13}, O_{13a}, O_{13b}, plp_{13a} ;

\underline{n = 14, 15, 16} : O_{n} ; \underline{n = 17} : plp_{17c} ; \underline{n = 18} : plp_{18c}, plp_{18d} ; \underline{n = 19} : O_{19}, plp_{19d} ; \underline{n = 20 to 24} : O_{n}.

Appendix 4: On lattices with min Λ^{∗} = 1.

We shall now consider bounds for S_{1} for lattices Λ with min Λ^{∗} = 1 for which all norm 1 vectors in Λ^{∗} satisfy the conditions of Lemma 9.3. We assume that these lattices are generated by their minimal vectors. Proposition. Let e ∈ Λ^{∗}. If N(e) = 1 and 2e ∈ Λ, then \overline{S}_{1} is a root system of type κA A A_{1} for some κ ∈ [1, n − 1] . (Otherwise stated, S_{1} consists of κ ≤ n − 1 pairs ±z of norm 2 vectors.)

Proof. Let 2e = x+y, x, y ∈ S_{1}. For any z ≠ x, y in S_{1}, set t = x+y −z. Then t is an element of S such that t · e = 1, hence an element of S_{1}, and we have t · z = 2 e · z − z · z = −1, which shows that t̄= −z̄. Moreover we have x · z + y · z = 2 e · z = 2, hence x · z = y · z = 1 since both terms are bounded from above by 1, whence x̄ · z̄ = ȳ · z̄ = 0. Since N(z̄) = N(z) − 1 = 2 for every z, \overline{S}_{1} is a root system of type κA A A_{1} for some κ ≤ dim ℝe^{⊥} = n − 1, and κ is not zero because Λ is well rounded. □

Denote by r_{i} (i = 1, …, κ) the elements of \overline{S}_{1} (the roots of the system κA A A_{1} above), let P be the lattice they generate, let V be the span of P, and let P_{0} the lattice of index 2 in P (a scaled copy of 𝔻_{κ}) generated by the r_{i} ± r_{j}. We have P_{0} = Λ ∩ P = Λ_{even} ∩ P. Up to permutation and change of signs of the r_{i}, a norm 3 vector f ∈ Λ ∩ V is of the form f = \frac{r_{1}+···+r_{6}}{2} : indeed, 2f, of norm 12 is a sum 2r_{1} + r_{2} + r_{3} or r_{1} + ⋯ + r_{6}, and the first case must be excluded, since then (e + r_{1}) · f = 2. Together with f, Λ ∩ V contains all vectors which differ from f by an even number of sign changes, and the supports of such vectors f constitute a self-dual code of length κ and weight 6. When κ = n − 1, we recover the construction we gave at the end of Section 2.

We have seen in the previous sections that for n ≤ 9, N(e) = 1 and e ∈ S_{1} + S_{1} occurs only for Λ ≃ W_{7}. We could show that this is also impossible for n = 10. The discussion above proves immediately the weaker result κ ≤ n − 2 for n ≤ 10 to within the unique exception of W_{7}, which corresponds to the fact that every root in 𝔼_{7} is a sum x + y with x, y ∈ \frac{1}{2} S(𝔼_{7}^{∗}).

It is relatively easy to handle the cases where κ = n − 3, for which Λ contains the lattice Λ′ generated by the vectors e ± r_{i} together with some norm 3 vector e′ ∈ (ℚe)^{⊥}∖V. Using this device, we can obtain a short proof for n = 9 of the bound κ ≤ n − 3 = 6, which can be used in place of Lemma 11.2 to prove Theorem 11.4.

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Jacques Martinet, A2X Boris Venkov, POMI Inst. Math., Université Bordeaux 1 Fontanka 27 351, cours de la Libération 191011 St.Petersbourg 33405 Talence cedex, France Russia e-mail: martinet@math.u-bordeaux.fr bvenkov@pdmi.ras.ru