Uniqueness of the Leech lattice

Abstract

We give Conway’s proof for the uniqueness of the Leech lattice.

1 Uniqueness of the Leech lattice

Theorem 1.1 There is a unique even unimodular lattice Λ in R^{24} without vectors of squared length 2. It is known as the Leech lattice. The group .0 of automorphisms fixing the origin has order 8315553613086720000.

Let Λ be an even unimodular lattice in R^{n} where n < 32.

(‘Unimodular’ is the same as ‘self-dual’, and says that the volume of the fundamental domain is 1. In other words, the lattice has one point per unit volume. ‘Even’ means that the squared length |x|^{2} = (x, x) is an even integer for each x ∈ Λ. It follows that all inner products (x, y) with x, y ∈ Λ are integers: (x, y) = \frac{1}{2}(|x + y|^{2} − |x|^{2} − |y|^{2}).)

The theta function θ_{Γ} of a lattice Γ is defined by θ_{Γ}(z) = ∑_{x∈Γ} q^{\frac{1}{2} (x,x)}, where q = e^{2πiz}.

A code has a weight enumerator, and the MacWilliams relation describes the relation between a weight enumerator of a linear code and its dual. If the code is self-dual, this yields an invariance property for the weight enumerator. For lattices similar things are true: there is a relation between the theta function of a lattice and the theta function of its dual, and if the lattice is self-dual its theta function has an invariance proterty. We quote Hecke’s theorem.

Theorem 1.2 Let Γ be an even unimodular lattice in R^{n}. Then (i) n ≡ 0 (mod 8), and (ii) θ_{Γ} is a modular form of weight \frac{1}{2}n.

Let M_{k} be the vector space of modular forms of weight 2k, and let M_{k}^{0} be the subspace consisting of cusp forms, that is, of modular forms that vanish at i∞. The following theorem says that there are not too many modular forms, so that one has very strong information when something is a modular form of low weight.

Theorem 1.3 (i) M_{k} = 0 for k < 0 and k = 1.

(ii) For k = 0, 2, 3, 4, 5 we have dim M_{k} = 1 and dim M_{k}^{0} = 0.

(iii) M_{k−6} ≃ M_{k}^{0}.

(iv) For k ≥ 2 we have dim M_{k} = dim M_{k}^{0} + 1.

(The proof is easy, but belongs elsewhere.)

Look at our even unimodular lattice Λ in R^{n} with n < 32. We have 8|n and θ_{Γ} ∈ M_{n/4}. Let N_{m} be the number of vectors of squared length m, so that N_{0} = 1 and N_{2} = 0, and θ_{Γ}(z) = ∑_{m} N_{m}q^{m/2}.

If n = 8 then θ_{Γ} is uniquely determined by N_{0} = 1, since dim M_{2} = 1. We already know an even unimodular lattice in R^{8}, namely E_{8}. It is a root lattice, that is, is generated by vectors of squared length 2, so certainly N_{2} ≠ 0. (In fact N_{2} = 240.)

If n = 16 then again θ_{Γ} is uniquely determined by N_{0} = 1, since dim M_{4} = 1. And the lattice E_{8} ⊕ E_{8} shows that N_{2} ≠ 0 also here.

So n = 24. Here dim M_{6} = 2, and the two conditions N_{0} = 1, N_{2} = 0 determine the function uniquely. Computing the coefficients one finds N_{4} = 196560, N_{6} = 16773120, N_{8} = 398034000.

(In fact, N_{2m} = \frac{65520}{691}(σ_{11}(m) − τ(m)) for m > 0, where σ_{h}(m) = ∑_{d|m} d^{h} and τ(m) is Ramanujan’s function, defined by q ∏_{m=1}^{∞}(1 − q^{m})^{24} = ∑_{m=1}^{∞} τ(m)q^{m}.)

Call x ∈ Λ short if (x, x) ≤ 8.

Claim Each coset of 2Λ inside Λ contains a short vector. The classes that contain more than a single pair ±x of short vectors are precisely the classes that contain vectors of length \sqrt{8}, and these contain 48 short vectors, namely 24 mutually orthogonal pairs ±x of vectors of length \sqrt{8}.

Indeed, let x, y be short vectors with y ≠ ±x and y − x ∈ 2Λ. We may suppose (x, y) ≥ 0. (Otherwise, replace y by −y.) Now |y − x|^{2} ≤ |y|^{2} + |x|^{2} ≤ 16, but for nonzero vectors u ∈ 2Λ we know |u|^{2} ≥ 16, so equality must hold everywhere, and x and y are orthogonal. In R^{24} we can have at most 24 mutually orthogonal pairs of vectors ±x. The number of cosets that contain short vectors is at least \frac{N_{0}}{1} + \frac{N_{4}}{2} + \frac{N_{6}}{2} + \frac{N_{8}}{48} = 2^{24}, but since this is the total number of cosets, we must have equality everywhere. This proves the claim.

Fix one set of 24 mutually orthogonal pairs of vectors of length \sqrt{8} to define a basis of R^{24}. Then Λ contains the vectors \frac{1}{\sqrt{8}}(±8, 0^{23}) and their halved differences \frac{1}{\sqrt{8}}((±4)^{2}, 0^{22}) (where this notation means that there are 2 places with ±4 and 22 places with 0, in any order). These vectors generate a sublattice Λ_{0} of Λ, and Λ_{0} = \{\frac{1}{\sqrt{8}}(x_{1}, …, x_{24}) | 4|x_{i} for all i and 8| ∑ x_{i}\}.

Consider an arbitrary vector x = \frac{1}{\sqrt{8}}(x_{1}, …, x_{24}) ∈ Λ. Since the inner products with vectors in Λ_{0} must be integers, it follows that the x_{i} must be integers, all of the same parity (all even or all odd). If there is such a vector with all x_{i} odd, then pick one and make sure that for that one x_{i} ≡ 1 (mod 4) for all i, by changing the sign of some coordinates, if necessary.

The next step identifies the extended binary Golay code inside the lattice. Consider the sublattice Λ_{1} of Λ consisting of the vectors for which all x_{i} are even, and let C be the image of Λ_{1} in {0, 1}^{24} under the map defined coordinatewise by sending 0 (mod 4) to 0 and 2 (mod 4) to 1. Note that if x ↦ c then also x+x_{0} ↦ c for any x_{0} ∈ Λ_{0} (since such x_{0} has all coordinates divisible by 4).

Claim C is the extended binary Golay code.

There is a unique linear code with word length 24, dimension 12 and minimum distance 8. Clearly, C is a linear code with word length 24. Suppose c ∈ C, c ≠ 0. Then c is the image of some x ∈ Λ, and by subtracting a vector in Λ_{0} we may assume that x_{i} ∈ {0, 2} for all i except one, and x_{i} ∈ \{−2, 2\} for the last i. Now 4 ≤ (x, x) = \frac{4}{8}wt(c), so that wt(c) ≥ 8. This shows that C has minimum distance at least 8.

Look at the supports of code words of weight 8. No 5-set can be covered twice, otherwise the minimum distance would be smaller than 8. Since each 8-set covers \binom{8}{5} 5-sets, and there are \binom{24}{5} 5-sets altogether, there cannot be more than \binom{24}{5}/\binom{8}{5} = 759 words of weight 8 in C.

The balls of radius 4 around code words cover the words at distance less than 4 to C precisely once, and the words at distance 4 to C at most six times: if a word w has distance 4 to distinct code words c_{1} and c_{2}, then c_{1} − w and c_{2} − w are vectors of weight 4 with disjoint supports (since c_{1} and c_{2} have distance 8), and there are at most six disjoint 4-sets in a 24-set. Since 1 + 24 + \binom{24}{2} + \binom{24}{3} + \frac{1}{6} \binom{24}{4} = 2^{12}, there can be at most 2^{24}/2^{12} = 2^{12} code words, i.e. C has dimension at most 12.

Now count vectors of length 2. If (x, x) = 4, then ∑ x_{i}^{2} = 32, and x must have one of the shapes \frac{1}{\sqrt{8}}((±4)^{2}, 0^{22}) or \frac{1}{\sqrt{8}}((±2)^{8}, 0^{16}) or \frac{1}{\sqrt{8}}(∓3, (±1)^{23}). The number of vectors of these shapes is at most \binom{24}{2}.2^{2} + 759.2^{7} + 24.2^{12} = 196560. But this is N_{4}, so we must have equality everywhere.

(For the vectors of shape \frac{1}{\sqrt{8}}((±2)^{8}, 0^{16}) there are 2^{8} choices for the signs, but we can use only half of these, since two choices that differ in only one place would differ by a vector of squared length 2.

For the vectors of shape \frac{1}{\sqrt{8}}(∓3, (±1)^{23}), subtract a vector with all coordinates 1 (mod 4) to get a vector with all x_{i} even. The fact that C has dimension at most 12 means that at most 2^{12} choices for the signs are possible.)

This shows that C has dimension 12, and therefore is the extended binary Golay code.

Now we can describe the Leech lattice.

Theorem 1.4 x = \frac{1}{\sqrt{8}}(x_{1}, ..., x_{24}) ∈ Λ if and only if (i) x_{i} ∈ Z, all x_{i} have the same parity; and (ii) if all x_{i} are even, then ∑ x_{i} ≡ 0 (mod 8); if all x_{i} are odd, then then

∑ x_{i} ≡ 4 (mod 8); and

(iii) {i | x_{i} ≡ a (mod 4)} ∈ S for a = 0, 1, 2, 3, where S is the collection of supports of vectors in C.

Proof: Exercise. □

About the group of automorphisms: starting from any of the N_{8}/48 sets of 24 mutually orthogonal pairs of vectors ±x of length \sqrt{8}, we arrived at a unique description of Λ. That means that its group of automorphisms is transitive on these N_{8}/48 coordinate frames. If we fix a frame, then the possible automorphisms consist of sign changes and permutations of the coordinates, and correspond to automorphisms of the extended binary Golay code, which has group 2^{12}.M_{24}. Altogether we find a group of order (N_{8}/48).2^{12}.|M_{24}| = 8292375.4096.24.23.22.21.20.48 = 8315553613086720000 = 2^{22}.3^{9}.5^{4}.7^{2}.11.13.23.

2 Related sporadic simple groups

Co.1 The group of the Leech lattice stabilizing the origin 0 is called .0 and was found above to have order 8315553613086720000. It has a center of order 2 (the map x ↦ −x) and the quotient is a simple group called .1 or Co_{1} of order \frac{1}{2}|.0| = 2^{21}.3^{9}.5^{4}.7^{2}.11.13.23.

Co.2 The subgroup of .0 fixing a vector of squared length 4 is Co_{2} of order |.0|/N_{4} = 2^{18}.3^{6}.5^{3}.7.11.23. This is the group of automorphisms of a strongly regular graph on 2300 points.

Co.3 The subgroup of .0 fixing a vector of squared length 6 is Co_{3} of order |.0|/N_{6} = 2^{10}.3^{7}.5^{3}.7.11.23. This is the group of automorphisms of a regular 2-graph on 276 points.

McL The subgroup of .0 fixing a triangle with sides of squared lengths 4, 4, 6 is McL of order 2^{7}.3^{6}.5^{3}.7.11. The group McL.2 is the point stabilizer of Co_{3} in its action on 276 points. It is the group of automorphisms of a strongly regular graph on 275 points.

HS The subgroup of .0 fixing a triangle with sides of squared lengths 4, 6, 6 is HS of order 2^{9}.3^{2}.5^{3}.7.11. The group HS.2 is the group of automorphisms of a strongly regular graph on 100 points.

M_{24} The group M_{24} of order 2^{10}.3^{3}.5.7.11.23 is 5-transitive on 24 points (and the subgroup fixing 5 points has order 48, so |M_{24}| = 24.23.22.21.20.48). It is the automorphism group of the Steiner system S(5, 8, 24). The automorphism group of the extended binary Golay code (both translations and coordinate permutations) is 2^{12}.M_{24}.

M_{23} A point stabilizer in M_{24} is M_{23} of order \frac{1}{24}|M_{24}| = 2^{7}.3^{2}.5.7.11.23. It is the automorphism group of the Steiner system S(4, 7, 23). The automorphism group of the perfect binary Golay code is 2^{12}.M_{23}.

M_{22} A point stabilizer in M_{23} is M_{22} of order \frac{1}{23}|M_{23}| = 2^{7}.3^{2}.5.7.11. It is the automorphism group of the Steiner system S(3, 6, 22). It is the group of automorphisms of a strongly regular graph on 77 points.

M_{12} The group M_{12} of order 12.11.10.9.8 = 2^{6}.3^{3}.5.11 is sharply 5transitive on 12 points. It is the stabilizer of a word of weight 12 in the action of M_{24} on the extended binary Golay code. It is the automorphism group of the Steiner system S(5, 6, 12). The automorphism group of the extended ternary Golay code is 3^{6}.2.M_{12}.

M_{11} A point stabilizer in M_{12} is M_{11} of order \frac{1}{12}|M_{12}| = 2^{4}.3^{2}.5.11. It is the automorphism group of the Steiner system S(4, 5, 11).

The letters here abbreviate Co: Conway, McL: McLaughlin, HS or HiS: Higman-Sims, M: Mathieu.

3 Theta functions

Let us give some more detail for the sentence above that said ‘Computing the coefficients one finds N_{4} = 196560, N_{6} = 16773120, N_{8} = 398034000’.

The theta function for the E_{8} lattice is

1 + 240 ∑_{m=1}^{∞} σ_{3}(m)q^{m} = 1 + 240(q + 9q^{2} + 28q^{3} + 73q^{4} + ⋯).

The theta function for the two nonisomorphic even unimodular lattices in R^{16} is

1 + 480 ∑_{m=1}^{∞} σ_{7}(m)q^{m} = 1 + 480(q + 129q^{2} + 2188q^{3} + 16513q^{4} + ⋯).

Since one of these lattices is E_{8} ⊕ E_{8} with theta function

(1 + 240 ∑_{m=1}^{∞} σ_{3}(m)q^{m})^{2},

we find the identity 1 + 480 ∑_{m=1}^{∞} σ_{7}(m)q^{m} = (1 + 240 ∑_{m=1}^{∞} σ_{3}(m)q^{m})^{2}.

A basis for the 2-dimensional space M_{6} of modular forms of weight 12 is given by f = 1 + \frac{65520}{691} ∑_{m=1}^{∞} σ_{11}(m)q^{m} and g = q ∏_{m=1}^{∞}(1 − q^{m})^{24} = ∑_{m=1}^{∞} τ(m)q^{m}. Since

f = 1 + \frac{65520}{691}(q + 2049q^{2} + 177148q^{3} + 4196353q^{4} + ⋯)

and

g = q − 24q^{2} + 252q^{3} − 1472q^{4} + ⋯,

a linear combination af + bg = 1 + 0q + ⋯ must have a = 1 and b = −\frac{65520}{691}. Therefore, the theta function of the Leech lattice is

θ_{Λ}(z) = 1 + \frac{65520}{691} ∑_{m=1}^{∞}(σ_{11}(m) − τ(m))q^{m}
= 1 + \frac{65520}{691}(2073q^{2} + 176896q^{3} + 4197825q^{4} + ⋯)
= 1 + 196560q^{2} + 16773120q^{3} + 398034000q^{4} + ⋯

as desired.