EUCLIDEAN VERSUS LORENTZIAN LATTICES

JACQUES MARTINET

Abstract. It is known that indefinite, odd, unimodular lattices are classified up to isometry by their signature (p, q), p, q > 0. We consider n-dimensional Euclidean lattices embedded into a Lorentzian lattice of signature (n, 1). We attach this way a weight to unimodular Euclidean lattices.

Foreword —Warning

In 2001 I wrote down AmSTeX-files accounting for work done during the academic year 1999-2001 on embedding a Euclidean, n-dimensional lattice into a Lorentzian lattice of signature (n, 1), with the help of Christian Batut, who wrote various PARI -programs. In particular I introduced the notion of a weight for a unimodular lattice, which can be viewed as a kind of measure of complexity.

Fifteen years later I decided to put on line this LaTeX-file which accounts for the essential of the work done in those days. This is not intended for publication. However I believe it might be of some use for those who are interested in the interface between Euclidean and odd Lorentzian lattice.

1. Introduction

Let p, q ≥ 0 be integers, and let V be a real vector space equipped with a quadratic form of signature (p, q) (thus n := dim V = p + q), with corresponding bilinear form denoted by (x, y) ↦ x · y (the “scalar product” on V). A lattice in V is a ℤ-submodule of V of rank dim V. Unless otherwise stated, the word “lattice” will denote a Euclidean lattice, that is a lattice in a space of signature (n, 0), When V has signature (p, q), we say lattice of signature (p, q), and a Lorentzian lattice is a lattice of signature (n, 1), n ≥ 1. We say that V, or its scalar product, is indefinite if both p and q are strictly positive. Negating x·y changes a lattice L of signature (p, q) into a lattice L^{−} of signature (q, p). We also set L^{+} = L.

2000 Mathematics Subject Classification. 11H55, 11H71.

Key words and phrases. Euclidean lattices, Lorentzian lattices, quadratic forms First version: May 28th, 2016 (correction: June 2nd, 2016).

We say that a lattice L (of any signature) is integral if the scalar product takes integral values on L, and that an integral L is even (resp. odd) if the norm x · x takes only even values on L (resp. takes odd values on some vectors of L). The determinant det(L) of L is the determinant of the Gram matrix of one of its bases. Its sign is that of (−1)^{q}, and det(L) is an integer when L is integral. We say that L is unimodular if its determinant is ±1, indeed (−1)^{q} if L has signature (p, q). An important reference for Lorentzian, unimodular lattices is Chapter 26 of [C-S].

Definition 1.1. Let ℤ^{+} (resp. ℤ^{−}) be the module ℤ equipped with the form +xy (resp. −xy). We denote by I_{p,q} the ℤ-module (ℤ^{+})^{p} ⊥ (ℤ^{−})^{q} (an orthogonal sum of p copies of ℤ^{+} and q copies of ℤ^{−}), viewed as a lattice in V = ℝ ⊗ I_{p,q}. If q = 0, this is a Euclidean lattice simply denoted by ℤ^{n}.

The basic result we shall use is the following theorem, a proof of which can be read in [Se], Chapter IV, or [M-H], Chapter II, Section 4):

Theorem 1.2. An odd, indefinite unimodular lattice of signature (p, q) is isometric to I_{p,q}. □

Let now Λ be a unimodular, n-dimensional lattice in a Euclidean space E, let V = E ⊥ (ℝ, −xy), and let L = Λ ⊥ ℤ^{−}. Then L is an odd unimodular Lorentzian lattice. Hence by Theorem 1.2, there exists an isometry

φ : Λ ⊥ ℤ^{−} → I_{n,1} .

This isometry is far from being canonical, and any other such isometry φ′ is of the form φ′ = φ ∘ u where u is an automorphism of I_{n,1}, that is, an element of the orthogonal group O_{n,1}(ℤ). We identify this way Λ with the orthogonal v^{⊥} of a vector v ∈ I_{n,1} of norm −1, and lattices up to isometry are in one-to-one correspondence with orbits of vectors of norm −1 in O_{n,1}(ℤ). Vectors x ∈ I_{n,1} will be written in the form

x = (x_{1}, …, x_{n}; x_{0})

and thus have norm N(x) = x_{1}^{2} + ⋯ + x_{n}^{2} − x_{0}^{2}. We write

v = (v_{1}, …, v_{n}; v_{0}) with v_{1}^{2} + ⋯ + v_{n}^{2} − v_{0}^{2} = −1 .

Without loss of generality, we may assume that the v_{i} are non-negative. Among all vectors v orthogonal to Λ, there exist finitely many for which v_{0} attains its minimum.

Definition 1.3. The smallest possible value of |v_{0}| for a vector v ∈ Λ^{⊥} is called the weight of Λ, denoted by wt(Λ).

Problem 1.4. To find good bounds for the the weight of n-dimensional unimodular lattices of minimum m.

In a given dimension the weight seems to be a measure of the complexity of the lattice. For example, among the even, 24-dimensional lattices, the weight has the value 145 for the Leech lattice which does not contain norm 2 vectors, 73 for the lattice which contains only 24 pairs of such vectors, and is ≤ 51 otherwise; see Section 7, Table 2.

Note that such a weight can also be attached to those lattices Λ of determinant d which can be embedded in I_{n,1}, and thus can then be realized as the orthogonal of a vector of norm −d.

2. Parity Classes

In this section Λ stands for an integral lattice of any signature (p, q). Integrality is equivalent to the inclusion

Λ ⊂ Λ^{∗} := {x ∈ V, ∀ y ∈ Λ, x · y ∈ ℤ} .

The map x ↦ N(x) mod 2 is a homomorphism of Λ into ℤ/2ℤ. Hence there exists e′ ∈ Λ^{∗}, such that N(x) ≡ e′ · x mod 2 for every x ∈ Λ, and e′ is uniquely defined modulo 2Λ^{∗}. We call the class of e′ modulo 2Λ^{∗} the parity class of Λ in Λ^{∗}.

Let d = det(Λ) ; this is the product of the elementary divisors of (Λ^{∗}, Λ). If d is odd, the map s ↦ dx induces an isomorphism pf Λ^{∗}/2Λ^{∗} onto Λ/2Λ. This shows that there exist parity vectors in Λ (e.g., e := de′), and that these parity vectors constitute a class modulo 2 in Λ. This is the parity class of Λ.

[In general there are 2^{k} parity classes in Λ, where k is the number of even elementary divisors of (Λ^{∗}, Λ).]

Note that Λ is even if and only if 0 is a parity class of Λ.

Proposition 2.1. (1) On a given parity class the norm is an in-

variant modulo 8.

(2) If Λ is unimodular this invariant is congruent to p−q modulo 8.

(3) The signature of an even, unimodular lattice satisfies the congruence q ≡ p mod 8.

Proof. (1) follows from the identity N(e+2x) = N(e)+4(e·x+N(x)), and (3) is a consequence of (2), observing that Λ is even if and only if 0 is a parity class of Λ.

(2) Let e be a parity vector of Λ. If Λ = I_{p,q}, we may chose for e the all ones vector, of norm p − q. This proves (2) for Λ = I_{p,q}. To prove (2) in general consider the lattices Λ ⊥ ℤ^{+} and Λ ⊥ ℤ^{−}. Both are odd, and at least one of them is indefinite, hence isometric to I_{p+1,q} or I_{p,q+1} by Theorem 1.2, which implies either N(e) + 1 ≡ (p + 1) − q or N(e) − 1 ≡ p − (q + 1), that is N(e) ≡ p − q mod 2. □

Let v = (v_{1}, …, v_{p+q}) ∈ I_{p,q} be a vector of odd norm d, and let H be the hyperplane orthogonal to v in V. Then Λ := I_{p,q} ∩ H is a lattice of determinant d in H, hence has a unique parity class.

Proposition 2.2. Let v = (v_{1}, …, v_{p+q}) ∈ I_{p,p}, of norm ±1, let H the hyperplane v^{⊥} ⊂ V, and let Λ = H ∪ I_{p,p}. Then the parity class of Λ is that of the vectors e with components e_{i} ≡ v_{i} + 1 mod 2.

Proof. Consider more generally the case of a vector v of norm d ≠ 0, assumed to be primitive (this is automatic if d = ±1), so that det(Λ) = (−1)^{q}d. The norm of x = (x_{i}) ∈ I_{p,p} is N(x) = ∑_{i} ε_{i}x_{i}^{2} where ε_{i} = ±1 and ∑ ε_{i} = p − q. Hence

N(x) ≡ e · x mod 2 ⟺ ∑_{i} (e_{i} − 1)x_{i} ≡ 0 mod 2 . Taking x = ε_{i}v_{j} − ε_{j}v_{i} ∈ Λ, we obtain

∀ i, ∀ j, (e_{i} − 1)v_{j} ≡ (e_{j} − 1)v_{i} mod 2 .

Pick i with v_{i} odd (such an i exists since v is primitive). We then see that we must have e_{j} ≡ 1 mod 2 if v_{j} is even, and e_{j} ≡ e_{i} mod 2 if v_{j} is odd. This shows that e belongs to one of the two classes C_{1}, C_{2} modulo 2 defined as follows:

v_{i} even: e_{i} ≡ 1 on C_{1} and C_{2} ;

v_{i} odd : e_{i} ≡ 0 on C_{1}, e_{i} ≡ 1 on C_{2}.

When d is even, there are two parity classes, necessarily C_{1} and C_{2}. When d is odd, we make use of the fact that for any integral lattice of odd determinant, the norm of parity vectors is congruent to the dimension modulo 2 (a consequence of Proposition 2.1 if d = ±1). Here we must have N(e) ≡ p+q −1 mod 2, whereas the all ones vector has norm p − q mod 2.

[Direct proof: e + v is a parity vector for Λ ⊥ ℤv, and because of the double inclusion dΛ ⊂ Λ ⊥ ℤv ⊂ Λ, e + v is a parity vector for Λ.] □

Corollary 2.3. The lattice Λ is even if and only if all components of v are odd. □

3. Small Norms and Small Weights

From now on, and unless otherwise stated, we study unimodular lattices inside V := ℝ ⊗ I_{n,1}, defined as the orthogonal of a vector v = (v_{1}, …, v_{n}; v_{0}) of norm −1, i.e., we have v_{1}^{2} + ⋯ + v_{n}^{2} − v_{0}^{2} = −1. In this section we essentially consider questions related to the existence of vectors of norm 1 or 2.

3.1. Lattices of Weight 1. This question is closely related to the existence of norm-1 vectors. In an integral lattice L (of any signature), vectors of norm ±1 have the following two properties:

(1) If N(x) = N(y) = ±1 and y ≠ ±x, then we have |x · y| < 1, hence x · y = 0;

(2) If N(e) = ±1, the identity x = (x ∓ (e · x)e) + (e · x)e shows that L splits as an orthogonal sum ℤe ⊥ L′. As a consequence the existence of k distinct pairs ±x of vectors of norm ±1 factors out from L an orthogonal sum of k lattices ℤ^{±}.

To state the theorem below, we need to identify a lattice defined by orthogonality with the celebrated lattice 𝔼_{8}, the definition of which we recall in Subsection 3.2 below. Suffices to know that min E_{8} = 2 and s(𝔼_{8}) = 120 (s > 64 suffices).

Proposition 3.1. The lattice orthogonal to (1^{8}; 3) is isometric to 𝔼_{8}.

Proof. Indeed this lattice is even and possesses 28 + 56 + 28 + 8 = 120 pairs of norm 2 vectors, obtained as permutations of the following four types: (1, −1, 0^{6}; 0), (1^{3}, 0^{5}; 1), (1^{6}, 0^{2}; 2) and (2, 1^{7}; 3), dividing its set S of minimal vectors into four subsets S_{1}, S_{2}, S_{3}, S_{4}; see the proof of Theorem 3.3. □

Theorem 3.2. Let Λ be an n-dimensional unimodular, Euclidean lattice. Then either wt(Λ) = 1, and Λ is isometric to ℤ^{n}, or wt(Λ) ≥ 3, and n ≥ 8; and if wt(Λ) = 3, Λ is isometric to 𝔼_{8} ⊥ ℤ^{n−8}.

Proof. Observe that if v_{i} = 0 for some i then the vector e_{i} with component 1 at i and 0 otherwise is a norm 1 vector, orthogonal to all vectors x with x_{i} = 0. Thus if v has k components equal to 0 then Λ is isometric to a direct sum Λ′ ⊥ ℤ^{k}. If v_{0} = 1, the vectors e_{i} constitute an orthogonal basis for Λ, which shows that Λ ≃ ℤ^{n}.

Assume now that v_{0} = 2. Then v extends the vector w := (1, 1, 1; 2) with v_{i} = 0 for i = 4, …, n. It thus suffices to consider w itself. Then the three vectors (1, 1, 0; −1), (1, 0, 1; −1) and (0, 1, 1; −1) have norm 1 and are pairwise orthogonal. This proves that weights never take the value 2.

Assume finally that v_{2} = 3. Then it suffices to consider three possible vectors v, namely (2, 2; 3) (n = 2), (2, 1^{4}; 3) (n = 5) and (1^{8}; 3) (n = 8), and to show that the first two define ℤ^{n}. This is achieved by considering the two vectors (2, 1; 2) and (1, 2; 2) in the first case, and the vectors e_{1} := (1, 1, 1, 1, 1; 2), and e_{i}, i = 2, 3, 4, 5 having components 1 at 0, 1, i and 0 otherwise in the second case. □

3.2. Roots. Let e be a vector of non-zero norm in an integral lattice L (of arbitrary signature). The symmetry σ_{e} along e (or with respect to the hyperplane e^{⊥}) is given by the formula

σ_{e}(x) = x − \frac{2(e · x)}{e · e} e ,

so that reflections defined by vectors of norm ±1 and ±2 are automorphisms of L. These vectors are usually called the roots of L, though we shall often discard norms ±1 (we have seen that they induce orthogonal decompositions),

[A general definition is that e is a root of L if e is primitive, and if σ_{e}(L) = L. Then N(e) must divide the annihilator of L^{∗}/L, so that when L is unimodular, roots are effectively vectors of norm dividing 2.]

In the Euclidean setting, a root lattice is an integral lattice generated by vectors of norm 2, its roots (even if some vectors of large norm may also be roots according to the general definition). A celebrated theorem is that root lattices are orthogonal sums of irreducible root lattices, which share out among two infinite series and three exceptional lattices.

Consider ℤ^{n+1} and ℤ^{n}, equipped with their canonical bases (ε_{i}), 0 ≤ i ≤ n and 1 ≤ i ≤ n, respectively. Then the infinite series are

𝔸_{n} = {x ∈ ℤ^{n+1} | ∑ x_{i} = 0} (n ≥ 1)

and

𝔻_{n} = {x ∈ ℤ^{n} | ∑ x_{i} ≡ 0 mod 2} (n ≥ 4) .

[We did not consider D_{2} ≃ 𝔸_{1} ⊥ 𝔸_{1} and 𝔻_{3} ≃ A_{3}.]

The Gram matrix of the basis (ε_{0} − ε_{i}) for 𝔸_{n} has entries 2 on the diagonal and 1 off the diagonal, which gives an easy way for characterizing 𝔸_{n} up to isomorphism. Replacing in this matrix 1 by 0 for one pair of entries (i, j), (j, i), we obtain a Gram matrix for 𝔻_{n}. (Consider the basis ε_{i} + ε_{j}, ε_{i} − ε_{k}, k ≠ i.)

The roots of 𝔸_{n} are the \frac{n(n+1)}{2} pairs ±(ε_{i} − ε_{j}), i < j and those of 𝔻_{n} are the n(n − 1) pairs ±(ε_{i} ± ε_{j}), i < j. The exceptional lattices are then 𝔼_{8} := 𝔻_{8}^{+} (see below, Coxeter lattices) and its successive densest cross-sections 𝔼_{7} and 𝔼_{6}. Their roots consist of 120, 63 and 36 pairs of norm-2 vectors, respectively. For the sets of roots of the irreducible lattices, we use the notation A A A_{n}, D D D_{n} and E E E_{n}.

3.3. Coxeter lattices. Given an integral lattice L, a necessary condition for a lattice M ⊃ L to be integral is that M should be contained in L^{∗}, which gives some importance to the study of lattices M such that L ⊂ M ⊂ L^{∗}. When L is an irreducible root lattice, lattices M as above are called Coxeter lattices.

When L = A_{n}, L^{∗}/L is cyclic of order det(L) = n + 1, so that there exists for each divisor r of n + 1 a unique Coxeter lattice containing 𝔸_{n} to index r, denoted by 𝔸_{n}^{r}. Its determinant is \frac{n+1}{r^{2}}, so that it can be integral only when r^{2} | n+1. This condition suffices, and in particular, 𝔸_{n}^{r} is unimodular if and only if n + 1 = r^{2}.

When L = 𝔻_{n}, we only find ℤ^{n} if n is odd, but two lattices distinct from ℤ^{n} if n is even, denoted by 𝔻_{n}^{+} and 𝔻_{n}^{−}, equal to 𝔻_{n} ∪ (𝔻_{n} + \frac{e^{±}}{2} ), where e^{+} = (1^{n}) and e^{−} = (1^{n−1}, −1). These two lattices are isometric, half-integral and exchanged by duality if n ≡ 2 mod 4, unimodular otherwise, odd if n ≡ 4 mod 8, even if n ≡ 0 mod 8. We have S(𝔻_{n}^{+}) = S(𝔻_{n}), except 𝔻_{4}^{+} ≃ ℤ^{4} and 𝔻_{8}^{+}, which is the lattice 𝔼_{8}. Similarly, one has min 𝔸_{n}^{r} = 2 except if r = n + 1, r = \frac{n+1}{2} and (n, r) = (5, 2) and then S(A_{n}^{r}) = S(𝔸_{n}) except for 𝔸_{7}^{2} ≃ 𝔼_{7} and 𝔸_{8}^{3} ≃ 𝔼_{8}.

We now construct the unimodular Coxeter lattices as orthogonal to a Lorentzian vector v = (v_{1}, …, v_{n}; v_{0}) of norm −1.

Theorem 3.3. (1) Let r ≥ 2, let n = r^{2} − 1 and let v = (1^{n}; r).

Then the orthogonal of v is isometric to 𝔸_{n}^{r}.

(2) Let n = 4m ≥ 2 and let v = (m − 1, 1^{n−1}; m + 1). Then the orthogonal of v is isometric to 𝔻_{n}^{+}.

Proof. Consider any vector v.

(i) If v_{i} = 1 for some i, then x with x_{i} = 2 and x_{j} = v_{j} if j ≠ i is a vector of norm 2. (ii) If v_{i_{1}} = ⋯ = v_{i_{k}} = a, say, for some k ≥ 2, the vectors having components a permutation of 1, −1, 0^{k−2} at i_{1}, …, i_{k} and 0 otherwise constitute a root system of type A A A_{k−1}. (iii) If a = 1, the vectors defined by (i) or (ii) constitute a system of type A A A_{k}. (iv) If v_{ℓ} = v_{0} − 2 for some ℓ, the vectors of (iii) together with vectors having components 1 at ℓ and two subscripts in i_{1}, …, i_{k}, −1 at 0, and 0 otherwise constitute a root system of type D D D_{k+1}.

[Proof: one constructs a Gram matrix with the convenient entries 2 or 1 for systems A A A_{?}, and one pair of entries 0 for systems D D D_{?}, starting with k − 1 vectors (1, −1, 0^{n−2}; 0) for (ii), and completing them first by one vector of the form (i), and finally one vector of the form (iv).]

The proof of (1) is now easy. By (iii) above, v^{⊥} is a unimodular lattice which contains 𝔸_{n}, hence isometric to 𝔸_{n}^{r} with r = \sqrt{n + 1}.

To prove (2), we extend using (iv) the root system of type A A A_{n−1} arising from v_{2}, …, v_{n} to a system of type D D D_{n}. This shows that v^{⊥} is a unimodular lattice containing 𝔻_{n}, hence isometric to 𝔻_{n}^{+} or ℤ^{n}, thus to 𝔻_{n}^{+} if m is even, and in particular if m = 2.

Let m ≥ 3. Then the vectors of norm 2 in both ℤ^{n} and 𝔻_{n}^{+} are exactly those of 𝔻_{n}. Vectors of ℤ^{n} are then of the form \frac{x+y}{2} for two orthogonal vectors of norm 2. It is easily checked that no sum x + y has even components. (Transitivity of the automorphism group on the roots shows that we may even fix x, for instance, x = e_{1}.)

[With the notation of Proposition 3.1, in the identification of 𝔼_{8} with 𝔸_{8}^{3} (resp. 𝔻_{8}^{+}), one has S(𝔸_{8}) = S_{1} ∪ S_{4} (resp. S(𝔻_{8}) = S_{2}).] □

Coxeter-like lattices can also be considered inside reducible root lattices. Here are two examples, with which we exhaust all lattices occurring in Table 1 of Section 7.

We have [𝔼_{7}^{∗} : 𝔼_{7}] = 2, and may write 𝔼_{7}^{∗} = 𝔼_{7} ∪ (𝔼_{7} + \frac{e}{2}) where e may be chosen of norm 6. The formula

L_{14} = (𝔼_{7} ⊥ 𝔼_{7}) ∪ (𝔼_{7} ⊥ 𝔼_{7} + (e, e))

defines an odd, unimodular lattice with minimum 3 off 𝔼_{7} ⊥ 𝔼_{7}.

Let p ≥ q ≥ 2. Write 𝔻_{4p}^{∗} = ⟨𝔻_{4p}, e, ε⟩, e as above, ε in the canonical basis for ℤ^{4p}, and similarly 𝔻_{4q}^{∗} = ⟨𝔻_{4q}, e′, ε′⟩. Then the formula

L_{4p,4q} = ⟨𝔻_{4p} ⊥ 𝔻_{4q}, e + ε′, e′ + ε⟩

defines a unimodular lattice, with minimum q + 1 off 𝔻_{4p} ⊥ 𝔻_{4q}, even if p and q are odd (e.g., the lattice with root system 2D D D_{12} of Table 2) and odd otherwise (e.g., the lattice with root system 2D D D_{8} of Table 1).

[Among low-dimensional lattices of small weight, we may also quote lattices with root systems A A A_{11} +E E E_{6} (n = 17, w = 6), A A A_{17} +A A A_{1} (n = 18, w = 5), and 2A A A_{9} (n = 18, w = 7).]

4. Orthogonal Sums

It is well-known and easy to prove that an isometric embedding L ↪ M of a unimodular lattice L into an integral lattice M identifies L with a direct factor of M, a remark we previously made in Section 3.1 when L = ℤ; and consideration of vectors of norm 1 shows the inequality wt(L ⊥ ℤ^{k}) ≤ wt(L), that we generalize in the theorem below and its corollary.

Theorem 4.1. Let

v = (v_{1}, …, v_{n}; v_{0}) ∈ I_{n,1} and v′ = (v′_{1}, …, v′_{n}; v′_{0}) ∈ I_{n′,1} be primitive vectors, let Λ = v^{⊥}, and let Λ′ = v′^{⊥}. Let

w = (v′_{0}v_{1}, …, v′_{0}v_{n}, v′_{1}, …, v′_{n},; v_{0}v′_{0}) ∈ I_{n+n′,1} .

Then, if Λ is unimodular, the lattice orthogonal to w in I_{n+n′,1} is isometric to the orthogonal sum Λ ⊥ Λ′.

Proof. Set M = w^{⊥}. The map

x = (x_{1}, …, x_{n}; x_{0}) ↦ (x_{1}, …, x_{n}, 0 …, 0; x_{0})

maps Λ isometrically onto a sublattice Λ_{1} of M. Since Λ is unimodular, M is the direct sum of Λ_{1} and of its orthogonal Λ′_{1} in M. Let V′ be the orthogonal of Λ_{1} in the rational vector space V = ℚ ⊗ I_{n+n′,1}. Identify v with the vector (v_{1}, …, v_{n}, 0 …, 0; v_{0}) ∈ V. Then, V′ is the space of vectors u ∈ V of the form

u = λv + (0, …, 0, μ_{1}, …, μ_{n′}; 0) = (λv_{1}, …, λv_{n}, μ_{1}, …, μ_{n′}; λv_{0})

with λ, μ_{1}, …, μ_{n′} ∈ ℚ. We have

w · u = λv′_{0}v_{1}^{2} + ⋯ + λv′_{0}v_{n}^{2} + μ_{1}v′_{1} + ⋯ + μ_{n′}v′_{n′} − λv′_{0}v_{0}^{2}
= −λv′_{0} + μ_{1}v′_{1} + ⋯ + μ_{n′}v′_{n′} .

Hence, Λ′_{1} is the set of elements of w which satisfy the following two conditions:

(1) μ_{1}v′_{1} + ⋯ + μ_{n′}v′_{n′} = λv′_{0} ;

(2) λv_{0}, λv_{1}, …, λv_{n}, μ_{1}, …, μ_{n} ∈ ℤ. Since v is primitive, the second condition means simply that λ, μ_{1}, …, μ_{n} are integral. The map

(μ_{1}, …, μ_{n′}; λ) ↦ λv + (0, …, 0, μ_{1}, …, μ_{n′}; 0) : Λ′ → V′

is then clearly an isomorphism of Λ′ onto Λ′_{1}. But we have

N((λv_{1}, …, λv_{n}, μ_{1}, …, μ_{n′}; λv_{0})) = λ^{2}(v_{1}^{2} + ⋯ + v_{n}^{2} − v_{0}^{2}) +μ_{1}^{2}+…+μ_{n′}^{2}

=μ_{1}^{2}+…+μ_{n}^{2}−λ^{2} =N((μ_{1}, …, μ_{n′}; λ)) .

Hence, the isomorphism above is an isometry. □

Corollary 4.2. If Λ and Λ′ are unimodular, then

wt(Λ ⊥ Λ′) ≤ wt(Λ) · wt(Λ′).

□

Question 4.3. Does equality always hold in the corollary above? In particular, is it true that wt(Λ ⊥ ℤ^{k}) = wt(Λ) whatever Λ and k?

[Equality in the second question holds if wt(Λ) ≤ 7.]

It should be noted that the construction of the vector w used Theorem 4.1 is not symmetric on direct sums. For instance, 𝔻_{16}^{+} ⊥ 𝔼_{8} and 𝔼_{8} ⊥ 𝔻_{16}^{+} give for the Niemeier lattice with root system D D D_{16} + E E E_{8} the distinct representations (9, 3^{15}, 1^{8}; 15) and (5^{8}, 3, 1^{15}, 15). The other occurrences of distinct representations for a same Niemeier lattice are listed in Section 7.

5. Lower bounds for the weight

In this section we consider unimodular lattices Λ of minimum ≥ 3 (lattices without roots). The theorem below gives lower bounds for the weight of lattices of minimum ≥ 3, first in general, then in the particular case of even lattices. Note that this theorem does not give specific lower bounds for odd lattices of minimum ≥ 4.

Theorem 5.1. Let Λ be an n-dimensional unimodular lattice.

(1) If min Λ ≥ 3, then wt(Λ)^{2} ≥ \frac{(n+1)(n+2)(2n+3)}{6}, and equality holds only for one unimodular lattice of dimension 23, the shorter Leech lattice O_{23}, of weight 70.

(2) If Λ is even and if min Λ ≥ 4, then wt(Λ)^{2} ≥ \frac{(n+1)(2n+1)(2n+3)}{3}. If n = 24, this implies wt(Λ) ≥ 145, and equality then holds if and only if Λ is isometric to the Leech lattice Λ_{24}.

Proof. We express Λ as the orthogonal in I_{n,1} of v = (v_{1}, …, v_{n}; v_{0}) where v_{0} = wt(Λ),

Set S_{2}(m) = 1^{2} + 2^{2} + ⋯ + m^{2} and denote by S_{2,odd}(m) the sum of odd squares up to m. We have

S_{2}(m) = \frac{m(m+1)(2m+1)}{6}

and

S_{2,odd}(2m + 1) = S_{2}(2m + 1) − 2^{2}S_{2}(m) = \frac{(m+1)(2m+1)(2m+3)}{3} . Assertions (i) and (ii) at the beginning of the proof of Theorem 3.3

show that if min Λ ≥ 3, the components v_{1}, …, v_{n} of v are distinct and at least 2, and Corollary 2.3 shows that they must be odd under condition (2). This proves the bounds v_{0} ≥ S_{2}(n + 1) under condition (1) and v_{0} ≥ S_{2,odd}(2n + 1) under condition (2), which are the lower bounds we need.

Putting n = 23 in (1), we obtain v_{0}^{2} ≥ \frac{24·25·49}{6} = 70^{2}. This shows that if equality holds in (1) for n = 23, then Λ is necessarily orthogonal to v = (24, 23, …, 5, 3; 70) (up to a permutation of v_{1}, …, v_{n})), but we cannot exclude a priori the existence of vectors of norm 1 or 2 in Λ. That min Λ is 3 was proved (Batut), using a program producing a Gram matrix for Λ and computing the minimum of Λ by standard Euclidean procedures. This identifies Λ with O_{23}, the only unimodular lattice of dimension ≤ 23 and minimum ≥ 3.

That n = 23 is the only possible choice is a consequence of the following theorem: the only integral points with strictly positive coordinates on the elliptic curve 6y^{2} = x(x + 1)(2x + 1) are (1, 1) and (23, 70).

[This is a theorem of G.N. Watson (1918); other proof by W. Ljunggren (1952); see [Mor], p. 258; nowadays one could use Baker’s method.]

Finally putting n = 24 in (2), we obtain v_{0}^{2} ≥ \frac{25·49·51}{3} = 20825, hence v_{0} ≥ 144.308 …, i.e. v_{0} ≥ 145. We observe that 145^{2} − 20825 = 200 = 51^{2}−49^{2}, which implies that if equality holds in (2) for n = 24, then Λ is necessarily orthogonal to v = (51, 47, 45, …, 5, 3; 145). A computation with a Gram matrix shows that Λ has no roots. This identifies Λ with the Leech lattice Λ_{24}, the only unimodular lattice of dimension ≤ 24 and minimum ≥ 4. (That we can obtain Leech by this construction has been known for some time; see Chapter 26 of [C-S].) □

It is somewhat puzzling that searching for lattices of minimum ≥ 3 among those having a small weight quickly forces the discovery of the remarkable lattices O_{23} and Λ_{24}. Up to dimension 26 there are four unimodular lattices of minimum ≥ 3, namely O_{23}, Λ_{24}, the odd Leech lattice O_{24}, and one lattice (discovered by Conway, uniqueness proved by Borcherds) that we shall denote by O_{26}. They are also easily found using Theorem 5.1.

For n = 24, we must have wt(Λ) ≥ \sqrt{5525} = 74.33 …, and since 75^{2} −5525 = 100 = 26^{2} −24^{2}, the minimal possible weight is ≥ 75, and 75 may occur only with v = (26, 25, 23, 22, 21, …, 7, 5, 3; 75). It turns out that v^{⊥} has minimum 3.

For n = 26, the lower bound is \sqrt{6930} = 83.24 . . .. We must go up to v_{0} = 89. Then we have 89^{2} − 6930 = 991 = 28^{2} + 29^{2} − 25^{2} − 3^{2}, and we obtain this way a lattice of minimum 3.

Such lattices have been classified in dimensions 27 and 28 by Bacher and Venkov ([Bc-Ve]). We content ourselves with one example with n = 27, which needs v_{0} ≥ \sqrt{7714} = 88. Using the equality 91^{2}−7714 = 36^{2} − 27^{2}, we construct one of the Bacher-Venkov lattices (s = 1332, |Aut| = 7680).

To finish this section we establish lower bounds for the weight of unimodular lattices with root system nA A A_{1}.

Theorem 5.2. Let Λ be an n-dimensional unimodular lattice with root system nA A A_{1}. Then n is even, say, n = 2m, and we have

wt(Λ) ≥ \frac{(m + 1)(2m^{2} + 4m + 3)}{3} ,

and even

wt(Λ)^{2} ≥ \frac{(2m + 1)(4m^{2} + 4m + 3)}{3} = \frac{(n + 1)(n^{2} + 2n + 3)}{3}

if Λ is even.

Proof. Let Λ_{0} = 𝔸_{1}^{⊥n}. We have [Λ : Λ_{0}]^{2} = det(Λ_{0}) = 2^{n}. Hence n is even. Set n = 2m, and write Λ = v^{⊥} for some v ∈ I_{n,1}. Minimizing the sum v_{1}^{2} + ⋯ + v_{n}^{2} under assertions (i) and (ii) at the beginning of the proof of Theorem 3.3, we see that the best we can do is to choose v with one component 1 and m − 1 distinct pairs (v_{j}, v_{j}) with v_{j} ≥ 2. This implies v_{0}^{2} ≥ 1 + 1^{2} + 2(2^{2} + ⋯ + m^{2}) + (m + 1)^{2}, which is the first inequality above. Similarly choosing all v_{i} odd implies v_{0}^{2} ≥ 1+1^{2} +2(3^{2} +⋯+(2m−1)^{2})+(2m+1)^{2}, from which the second inequality follows. □

Here are examples of each of the lower bounds above.

Applied with n = 22 the first inequality reads v_{0}^{2} ≥ 34^{2}. This defines a lattice of minimum 2, thus the known lattice with root system 22A A A_{1}, which is consequently of weight 34.

Applied with n = 24 the second inequality implies v_{0} ≥ \sqrt{5225} = 72.284 . . .. Since 73^{2} − 5225 = 27^{2} − 25^{2}, replacing in the proof of Theorem 5.2 the last term 25^{2} by 27^{2}, we construct a lattice of minimum 2 (because it it even), whose root system contains, hence is equal to 24A A A_{1}.

Similar arguments apply to Niemeier lattices with root systems 12A A A_{2} and 8A A A_{3}, giving for their weights the lower bounds 37, 49 from which we easily deduce the exact values wt = 39, 51 respectively.

6. More on Small Norms and Small Weights

We give here some complements to Section 3. We consider a vector v = (v_{1}, …, v_{n}; v_{0}) ∈ I_{n,1} of strictly negative norm −d, and assume that v_{1} ≥ v_{2} ≥ ⋯ ≥ v_{n} and v_{0} > 0.

6.1. Small Norms. Let x = (x_{1}, …, x_{n}; x_{0}) of norm t > 0 in E = v^{⊥}. Negating x if need be, we may moreover assume that x_{0} is also nonnegative. We have the following three relations between the v_{i}, x_{i} and t:

(1)  v_{1}^{2} + ⋯ + v_{n}^{2} = v_{0}^{2} − d
(2)  v_{1}x_{1} + ⋯ + v_{n}x_{n} = v_{0}x_{0}
(3)  x_{1}^{2} + ⋯ + x_{n}^{2} = x_{0}^{2} + t .

This implies

(v_{0}x_{0})^{2} = (v_{1}x_{1}+⋯+v_{n}x_{n})^{2} ≤ (v_{1}^{2}+⋯+v_{n}^{2}) (x_{1}^{2}+⋯+x_{n}^{2}) = (v_{0}^{2}−d) (x_{0}^{2}+t) ,

whence the crude estimates

x_{0}^{2} ≤ t(\frac{v_{0}^{2}}{d} − 1) and x_{1}^{2} + ⋯ + x_{n}^{2} = x_{0}^{2} + t ≤ \frac{tv_{0}^{2}}{d} , which bound the components of x.

The combination (1) − 2 × (2) + (3) yields the following identity involving the Lorentz distance between x_{0} and v_{0}:

(x_{1} − v_{1})^{2} + ⋯ + (x_{n} − v_{n})^{2} = (x_{0} − v_{0})^{2} + t − d ,

which shows the existence of a kind of symmetry exchanging the values 0 and v_{0} in x: setting y = x − v (thus, y_{i} = x_{i} − v_{i}) transforms the conditions “ x · v = 0 and N(x) = t ” into “ y · v = d and N(y) = t − d˙’’.

[More generally, using combinations λ^{2}(1) − 2λμ (2) + μ^{2}(3), we obtain (μx_{1} − λv_{1})^{2} + ⋯ + (μx_{n} − λv_{n})^{2} = (μx_{0} − λv_{0})^{2} + μ^{2}t − λ^{2}d .]

6.2. Reduction. Let us say that v is reduced if for every vector v′ in the orbit of v we have |v′_{0}| ≥ v_{0}. It would be interesting to find restrictive conditions on the v_{i} that satisfy reduced vectors v. Unfortunately I must content myself with the following two examples, proved using reflections along vectors e = (e_{1}, …, e_{n}; e_{0}) of norm ±2, that we assume to have a non-negative component e_{0}.

(1) If N(e) = +2 and e_{0} ≥ 1, then e_{1}v_{1} + ⋯ + e_{n}v_{n} ≤ e_{0}v_{0} ;

(2) If N(e) = −2 and e_{0} ≥ 2, then e_{1}v_{1} + ⋯ + e_{n}v_{n} ≤ (e_{0} − \frac{1}{e_{0}} )v_{0}.

Taking e = (1^{3}, 0^{n−3}, 1), (1) reads (1’) : v_{1} + v_{2} + v_{3} ≤ v_{0}. We shall use this condition to prove the well-known classification result, easily proved by more usual methods (even under the less restrictive condition n ≤ 11 ): A unimodular lattice Λ of dimension n ≤ 9 is isometric to ℤ^{n} or to 𝔼_{8} ⊥ ℤ^{n−8}.

Proof. Write Λ = v^{⊥} with v reduced. We have

v_{0}^{2} − 1 = v_{1}^{2} + ⋯ + v_{n}^{2} = (v_{1} + v_{2} + v_{3})^{2} − T

where

T = (v_{1}v_{2} − v_{4}^{2}) + (v_{1}v_{2} − v_{5}^{2}) + (v_{1}v_{3} − v_{6}^{2}) + (v_{1}v_{3} − v_{7}^{2}) + (v_{2}v_{3} − v_{8}^{2}) + (v_{2}v_{3} − v_{9}^{2})

is the sum of six non-negative terms. We have v_{1} + v_{2} + v_{3} ≤ v_{0}, If this inequality is strict, then we have T = (v_{1} + v_{2} + v_{3})^{2} − v_{0}^{2} + 1 ≤ (v_{0} − 1)^{2} − v_{0}^{2} + 1 = −2(v_{0} − 1) ≤ 0 , hence v_{0} = 1 and Λ ≃ ℤ^{n}. Otherwise, we have v_{1} + v_{2} + v_{3} = v_{0}, hence T = 1, which implies v_{3} = v_{1}. Writing T = ∑_{i=4}^{9}(v_{1}^{2} − v_{i}^{2}), we see that v_{1} = ⋯ = v_{8}, v_{9} < v_{8} and v_{8}^{2} − v_{9}^{2} = 1. Thus we must have v_{8} = 1, v_{9} = 0 and v_{0} = 3. □

7. Tables

I reproduce here tables of unimodular lattices, first odd or even up to dimension 16 (with no component ℤ), then even in dimension 24. These tables were computed by Christian Batut. His calculations do not provide any classification result; they were just intended to give the weight of all unimodular lattices either of dimension up to 16, or even and of dimension up to 24: indeed they make use of the known classification results, which can be read in [C-S], Chapter 16.

Table 1. Unimodular lattices of dimension n ≤ 16

dim. weight vector root system
0  1  ∅  ∅
8  3  1^{8}  E E E_{8}
12  4  2, 1^{11}  D D D_{12}
14  6  2^{7}, 1^{7}  2E E E_{7}
15  4  1^{15}  A A A_{15}
16  5  3, 1^{15}  D D D_{16}
9  3^{8}, 1^{8}  2E E E_{8}
6  3^{2}, 2, 1^{13}  2D D D_{8}

In practice, to construct Table 2, say, we consider increasing odd weights w = 3, 5, …, and partitions 24 = a_{1}+⋯+a_{k} with non-negative a_{i} such that a_{1}(w − 2)^{2} + a_{2}(w − 4)^{2} + ⋯ + a_{k}1^{2} = w^{2} − 1, construct a Gram matrix M for the orthogonal to the vector v = (v_{1}, …, v_{24}; w) with v_{i} = w − 2 for 1 ≤ i ≤ a_{1}, v_{i} = w − 4 for a_{1} + 1 ≤ i ≤ a_{1} + a_{2}, etc, and test M for equivalence with matrices which have been previously found. The process stops when we have found the right number (24) of pairwise non-equivalent matrices.

Thanks to the weight calculations done at the end of Section 5 which gave the weights of lattices with root systems ∅, 24A A A_{1}, 12A A A_{2}, 8A A A_{3}, it actually suffices to consider weights only up to w = 37.

The PARI-GP-code below (an updated version of a code due to Batut) produces a Gram matrix for a given vector v, which can then be reduced to shorten its identification. \underline{Input} : a vector v with integer components; \underline{Output}: an n × n matrix with n = length(v) − 1.

{gramlz(v) = local(m=#v,u,w); v[m]=-v[m];u=matkerint(Mat(v));w=u;w[m,]=-w[m,]; w_{}˜ ∗u;}

It was observed in Section 4 that the weight of the Niemeier lattice with root system D D D_{16} + E E E_{8} could be produced by two distinct representations. Here are other examples: 4A A A5 + D D D4: (9^{4}, 7^{5}, 5^{4}, 3^{6}, 1^{5}), (9^{4}, 7^{4}, 5^{6}, 3^{6}, 1^{4}), (9^{3}, 7^{6}, 5^{6}, 3^{4}, 1^{5}) ; D D D_{10} + 2E E E_{7}: (11, 5^{10}, 3^{7}, 1^{6}), (7^{7}, 5, 3^{7}, 1^{9}) ; 4E E E_{6}: (15, 9^{5}, 7^{6}, 5^{6}, 3, 1^{5}), (11^{6}, 9^{2}, 5^{6}, 3^{5}, 1^{5}), (11^{6}, 9, 7^{2}, 5^{5}, 3^{6}, 1^{4}) ; 8A A A_{3}: (13^{3}, 11^{4}, 9^{3}, 7^{3}, 5^{4}, 3^{4}, 1^{3}), (13^{3}, 11^{3}, 9^{4}, 7^{4}, 5^{4}, 3^{3}, 1^{3}) .

Table 2. The 24 Niemeier lattices

weight symbol  vector  Coxeter  root system
5  δ  1^{24}  h = 25  A A A_{24}
7  α  5, 1^{23}  h = 46  D D D_{24}
9  ζ  3^{7}, 1^{17}  h = 18 A A A_{17}+E E E_{7}
11  θ  5, 3^{9}, 1^{14}  h = 16 A A A_{15}+D D D_{9}
κ  3^{12}, 1^{12}  h = 13  2A A A_{12}
13  ϵ  7, 3^{12}, 1^{11}  h = 22  2D D D_{12}
15  β  9, 3^{15}, 1^{8}  h = 30 D D D_{16}+E E E_{8}
λ  5^{6}, 3^{7}, 1^{11}  h = 12 A A A_{11}+D D D_{7}+E E E_{6}
ν  5^{5}, 3^{10}, 1^{9}  h = 10 2A A A_{9}+D D D_{6}
17  o  5^{8}, 3^{9}, 1^{7}  h = 9  3A A A_{8}
19  ι  9, 5^{8}, 3^{8}, 1^{7}  h = 14  3D D D_{8}
21  η  11, 5^{10}, 3^{7}, 1^{6}  h = 18 D D D_{10}+2E E E_{7}
ρ  7^{4}, 5^{7}, 3^{7}, 1^{6}  h = 7  4A A A_{6}
π  7^{5}, 5^{5}, 3^{7}, 1^{7}  h = 8 2A A A_{7}+2D D D_{5}
25  η  11, 7^{6}, 6^{6}, 3^{6}, 1^{5}  h = 10  4D D D_{6}
27  γ  9^{8}, 3^{8}, 1^{8}  h = 30  3E E E_{8}
σ  9^{4}, 7^{5}, 5^{4}, 3^{6}, 1^{5}  h = 6  4A A A_{5}+D D D_{4}
31  υ  11^{2}, 9^{4}, 7^{5}, 5^{4}, 3^{5}, 1^{4}  h = 5  6A A A_{4}
33  μ  15, 9^{5}, 7^{6}, 5^{6}, 3, 1^{5}  h = 12  4E E E_{6}
37  τ  15, 11^{4}, 9^{4}, 7^{4}, 5^{4}, 3^{4}, 1^{3}  h = 6  6D D D_{4}
39  ϕ  13^{3}, 11^{4}, 9^{3}, 7^{3}, 5^{4}, 3^{4}, 1^{3}  h = 4  8A A A_{3}
51  χ  17^{3}, 15^{2}, 13^{3}, 11^{3}, 9^{2}, 7^{3}, 5^{3}, 3^{3}, 1^{2} h = 3  12A A A_{2}
73  ψ  27,23^{2},21^{2},19^{2},17^{2},15^{2},13^{2},11^{2},9^{2},7^{2},5^{2},3^{2},1 h = 2  24A A A_{1}
145  ω  51, 47, 45, 43, ... , 9, 7, 5, 3  h = 0  ∅

References

[Bc-Ve] R. Bacher, B. Venkov, Réseaux entiers unimodulaires sans racines en dimension 27 et 28, in Réseaux euclidiens, designs sphériques et formes modulaires, J. Martinet, ed., L’Ens. Math., Monographie 37, Genève (2001), 212–267.

[C-S] J.H. Conway, N.J.A. Sloane, Sphere Packings, Lattices and Groups, Grundlehren 290, Springer-Verlag, Heidelberg (1988); third edition: 1999.

[Mar] J. Martinet, Perfect Lattices in Euclidean Spaces, Grundlehren 327, Springer-Verlag, Heidelberg, (2003).

[M-H] J. Milnor, D. Husemoller, Symmetric Bilinear Forms, Springer-Verlag,

Ergebnisse 73, Heidelberg (1973).

[Mor] L.J. Mordell, Diophantine Equations, Academic Press, New-york (1969).

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

[Se] J.-P. Serre, Cours d’arithmétique, P.U.F., Paris (1970), translated into English as: A Course in Arithmetic, Graduate Texts in Math. 7, Springer-Verlag, Heidelberg (1973).

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

E-mail address: Jacques.Martinet@math.cnrs.fr