Extremal lattices. Gabriele Nebe. Oberwolfach August Lehrstuhl D für Mathematik

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1 Extremal lattices Gabriele Nebe Lehrstuhl D für Mathematik Oberwolfach August 22

2 Lattices and sphere packings Hexagonal Circle Packing θ = + 6q + 6q 3 + 6q 4 + 2q 7 + 6q

3 Even unimodular lattices Definition A lattice L in Euclidean n-space (R n, (, )) is the Z-span of an R-basis B = (b,..., b n ) of R n The dual lattice is L = b,..., b n Z = { n a i b i a i Z}. i= L # := {x R n (x, l) Z for all l L} L is called unimodular if L = L #. Q : R n R, Q(x) := 2 (x, x) associated quadratic form L is called even if Q(l) Z for all l L. min(l) := min{2q(l) l L} minimum of L. The sphere packing density of an even unimodular lattice is proportional to its minimum.

4 Theta-series of lattices Let (L, Q) be an even unimodular lattice of dimension n so a regular positive definite integral quadratic form Q : L Z. The theta series of L is θ L = l L q Q(l) = + where a k = {l L Q(l) = k}. k=min(l)/2 a k q k θ L defines a holomorphic function on the upper half plane by substituting q := exp(2πiz). Then θ L is a modular form of weight n 2 for the full modular group SL 2 (Z). n is a multiple of 8. θ L M n 2 (SL 2(Z)) = C[E 4, ] n 2 where E 4 := θ E8 = + 24q +... is the normalized Eisenstein series of weight 4 and = q 24q q 3 472q of weight 2

5 Extremal modular forms Basis of M 4k (SL 2 (Z)): E4 k = + 24kq+ q E4 k 3 = q+ q E4 k 6 2 = q E k 3m k 4 m k =... q m k +... where m k = n 24 = k 3. Definition This space contains a unique form f (k) := + q + q q m k + a(f (k) )q m k+ + b(f (k) )q m k f (k) is called the extremal modular form of weight 4k. f () = + 24q +... = θ E8, f (2) = + 48q +... = θ 2 E 8, f (3) = + 96, 56q = θ Λ24, f (6) = + 52, 46, q = θ P48p = θ P48q = θ P48n, f (9) = + 6, 28, 75, 6q = θ Γ72.

6 Extremal even unimodular lattices Theorem (Siegel) a(f (k) ) > for all k Corollary Let L be an n-dimensional even unimodular lattice. Then min(l) n 24 = 2 + 2m n/8. Lattices achieving this bound are called extremal. Extremal even unimodular lattices L R n n , 264 min(l) number of extremal lattices

7 Extremal even unimodular lattices Theorem (Siegel) a(f (k) ) > for all k and b(f (k) ) < for large k (k 248). Corollary Let L be an n-dimensional even unimodular lattice. Then min(l) n 24 = 2 + 2m n/8. Lattices achieving this bound are called extremal. Extremal even unimodular lattices L R n n , 264 min(l) number of extremal lattices

8 Extremal even unimodular lattices in jump dimensions Let L be an extremal even unimodular lattice of dimension 24m so min(l) = 2m + 2 All non-empty layers {l L Q(l) = a} form spherical -designs. The density of the associated sphere packing realises a local maximum of the density function on the space of all 24m-dimensional lattices. If m =, then L = Λ 24 is unique, Λ 24 is the Leech lattice. The 9656 minimal vectors of the Leech lattice form the unique tight spherical -design and realise the maximal kissing number in dimension 24. Λ 24 is the densest 24-dimensional lattice (Cohn, Kumar). For m = 2, 3 these lattices are the densest known lattices and realise the maximal known kissing number.

9 Extremal even unimodular lattices in jump dimensions The extremal theta series f (3) = + 96, 56q = θ Λ24. f (6) = + 52, 46, q = θ P48p = θ P48q = θ P48n. f (9) = + 6, 28, 75, 6q = θ Γ72. The automorphism groups Aut(Λ 24 ) = 2.Co order = Aut(P 48p ) = (SL 2 (23) S 3 ) : 2 order = Aut(P 48q ) = SL 2 (47) order 3776 = Aut(P 48n ) = (SL 2 (3)Y SL 2 (5)).2 2 order 5246 = Aut(Γ 72 ) = (SL 2 (25) PSL 2 (7)) : 2 order 5246 =

10 Construction of extremal lattices From codes. Let (e,..., e n ) be a p-frame, so (e i, e j ) = pδ ij. Z := e,..., e n Z = pz n, Z # = p Z. Z # /Z = F n p. Given C F n p the codelattice is Λ(C) := { p ci e i (c,..., c n ) C} Λ(C) # = Λ(C ). Λ(C) is even if p = 2 and C is doubly even. min(λ(c)) = min(p, d(c) p ). Aut(C) Aut(Λ(C)). Binary extremal codes. length d(c) extremal h 8 G , 47 QR 48? 4

11 Canonical constructions of lattices A canonical construction of a lattice is a construction that is respected by (a big subgroup of) its automorphism group. The Leech lattice has at least 23 constructions, none of them is really canonical: Leech as a neighbor of a code lattice Let G 24 F 24 2 be the binary Golay code (the extended quadratic residue code). Then d(g 24 ) = 8. Min(Λ(G 24 )) = {±e,..., ±e 24 }. Neighbor lattice: v = 2 (3e e 24 ) Λ 24 := Λ(G 24 ) (v),2 := {x Λ(G 24 ) (x, v) even }, v : M 24 Aut(Λ 24 ) = 2.Co.

12 Canonical constructions of the 48-dimensional lattices Two of the 48-dimensional extremal lattices have a canonical construction with codes: Theorem (Koch) Let C = C F 48 3 with d(c) = 5. Then Λ(C) (v),2 is an extremal even unimodular lattice, where v = 3 (e e 48 ). Theorem (N) Let C = C F 48 3 with d(c) = 5 such that Aut(C) is divisible by some prime p 5. Then C = Q 48 or C = P 48. We have Aut(Q 48 ) = SL 2 (47) and Aut(P 48 ) = (SL 2 (23) C 2 ) : 2. Remark Λ(Q 48 ) (v),2 = P 48q, Aut(P 48q ) = SL 2 (47) Λ(P 48 ) (v),2 = P 48p, Aut(P 48p ) = (SL 2 (23) S 3 ) : 2

13 How many 48-dimensional extremal lattices are there? Let L be an extremal even unimodular lattice of dimension 48 and p be a prime dividing Aut(L). Theorem p = 47, 23 or p 3. Let σ Aut(L) be of order p. Then the fixed lattice F := F ix(σ) := {v L σv = v} is as follows. p dim F det(f ) F example unique yes unique yes 3 unique yes 8 4 unique yes 7 unique yes A6 no Λ24 yes O24 yes

14 Hermitian lattices Definition Let K be an imaginary quadratic number field, Z K its ring of integers, (V, h) an n-dimensional Hermitian positiv definit K-vectorspace. A lattice P V is a finitely generated Z K -moduls that contains a basis of V. The minimum of P is min(p ) := min{h(l, l) l P }. The Hermitian Hermite function γ h (P ) := min(p ) measures the det(p ) /n density of P.. If P = b,..., b n ZK is a free Z K -module then det(p ) = det(h(b i, b j )) i,j. The Hermitian dual lattice is P := {v V h(v, l) Z K for all l P } We call P Hermitian unimodular, if P = P (then det(p ) = ).

15 Hermitian tensor products (Renaud Coulangeon) Minimal vectors in tensor products Let (L, h L ) and (M, h M ) be Hermitian Z K -lattices, n = dim ZK (L) m := dim ZK (M). Each v L M is the sum of at most n pure tensors v = r l i m i, such that r =: rk(v) minimal. i= Put A := (h L (l i, l j )) and B := (h M (m i, m j )), then h(v, v) = Trace AB r det(a) /r det(b) /r. so min(l M) min{rd r (L) /r d r (M) /r r =,..., n} where d r (L) = min{det(t ) T L, Rg(T ) = r}. In particular d r (L) /r min(l)/γ r (Z K ) where γ r is the Hermitian Hermite constant.

16 Trace lattices Trace lattices Any Hermitian Z K -lattice (P, h) is also a Z-lattice (L, Q) of dimension 2n, where L = P and Q(x) := h(x, x) R K = Q. Then the polar form of Q is (x, y) = Trace K/Q (h(x, y)) and (L, Q) is called the trace lattice of (P, h). min(l) = 2 min(p ), L # = Z # K P and det(l) = d n K det(p )2. Z # K = {x K Trace K/Q(xl) Z for all l Z K } d K = det(z K, Trace(xy) = Z # K /Z K

17 K = Q[ ], Z K = Z[η], η = ( + )/2 Then η 2 η + 3 =, Z [ η] has a Euclidean algorithm, for any x K there is some a Z[η] such that N(x a) 9. The densest 2-dimensional lattice ( 3/ ) Let L K have Gram matrix 3/ Then L K is the densest 2-dimensional Z[η]-lattice, the trace lattice is D 4. Let T be the 2-dimensional ( ) unimodular Hermitian Z K -lattice with 2 η Gram matrix. Then T is Hermitian unimodular, η 2 min(t ) = 2, Aut(T ) = ±S 3. The lattice P 48n as Hermitian tensor product Let (P, h) be some 2-dimensional Z K -lattice such that the trace lattice Trace(P ) is isometric to the Leech lattice. Then the Hermitian tensor product R := P ZK T has Hermitian minimum either 2 or 3. The minimum of R is 3, if and only if (P, h) does not represent one of the lattices L K or T.

18 A canonical construction for P 48n Some Z[η]-structure of Leech. Let P be the Z[η]-lattice with Aut(P ) = SL 2 (3).2 such that Trace(P ) = Λ 24. Then Trace(P T ) = P 48n. Canonical construction for P 48n As a trace lattice of a quaternion tensor product: SL 2 (3) GL 3 (Q 3,, ) and SL 2(5) GL (Q 5,, ) act on quaternionic lattices L 3 resp. L 5. Then P 48n = Trace(L 3 Q,2 L 5 )

19 The discovery of the 72-dimensional extremal even unimodular lattice. 967 Turyn: Constructed the Golay code G 24 from the Hamming code 78,82,84 Tits; Lepowsky, Meurman; Quebbemann: Construction of the Leech lattice Λ 24 from E Gross, Elkies: Λ 24 from Hermitian structure of E N.: Tried similar construction of extremal 72-dimensional lattices (Bordeaux). 998 Bachoc, N.: Two extremal 8-dimensional lattices using Quebbemann s generalization and the Hermitian structure of E 8 2 Griess: Reminds Lepowsky, Meurman construction of Leech. proposes to construct 72-dimensional lattices from Λ 24 2 N.: Used one of the nine Z[α = ] structures of Λ 24 to find extremal 72-dimensional lattice Γ 72 = L(αΛ 24, αλ 24 ) 2 Parker, N.: Check all other polarisations of Λ 24 to show that Γ 72 is the unique extremal lattice obtained from Λ 24 by Turyn s construction. Chance: : 6 to find extremely good polarisation.

20 K = Q[ 7], Z K = Z[α], α = ( + 7)/2 Then α 2 α + 2 =, β = α = α, αβ = 2 and Z[α] has a Euclidean algorithm, for any x K there is some a Z[α] such that N(x a) 4 7. The densest 2-dimensional lattice ( Denote by P a the Z[α]-lattice with Gram matrix 2/ ) 7 2/. Then min(p 7 a ) = and det(p a ) = 3/7. The Barnes-lattice P b = (β, β, ), (, β, β), (α, α, α) Z[α] 3 with Hermitian form h : P b P b Z[α], h((a, a 2, a 3 ), (b, b 2, b 3 )) = 3 2 i= a ib i is Hermitian unimodular, Aut Z[α] (P b ) = ± PSL 2 (7), 2 β γ h (P b ) = min(p b ) = 2. Gram matrix 2 β α α 3

21 Densest Z[α]-lattices E 8 as trace lattice P c := Z[α] 4 + (,,, 3), (,, 3, 2) K Then min(p c ) =, det(p c ) = (/7) 2, P c = 7P c, Trace(P c ) = E 8 Theorem P a, P b and P c are the densest Z[α]-lattices in dimension 2,3,4. γ 2 (Z[α]) = 7/3, γ 3 (Z[α]) = 2, γ 4 (Z[α]) = 7. Proof: Trace(P c ) is densest Z-lattice. For P a enough to apply reduction theory. For P b explicit application of Voronoi s algorithm.

22 Extremal lattices as Hermitian tensor products The Leech lattice Let P := P b Z[α] P c. Then min(p ) = 2 and Trace(P ) is an extremal even unimodular lattice of dimension 24, so Trace(P ) = Λ 24. Proof: Trace(P ) is even unimodular, since P b Hermitian unimodular and E 8 = Trace(P c ) even unimodular. Show that min(p ) 2: r 2 3 d r (P b ) 2 2 d r (P c ) 3/7 /8 rd r (P b ) /r d r (P c ) /r 2, 85, 5 min(l M) min{rd r (L) /r d r (M) /r r =,..., n}

23 Dimension 72. Theorem (R. Coulangeon, N) Let P be an Hermitian Z[α]-lattice with min(p ) = 2. Then min(p P b ) 3 and min(p P b ) > 3 if and only if P has no sublattice isometric to P b. Proof. r 2 3 d r (P b ) /r 2 2 d r (P ) /r 2 2 3/7 rd r (P b ) /r d r (P ) /r 4 3, 7 3 And d 3 (P ) > if P b is not a sublattice of P. Corollary Let P be some 2-dimensional Z[α]-lattice such that Trace(P ) = Λ 24. Then min(p P b ) 3 and min(p P b ) = 4 if P does not contain P b.

24 Hermitian structures of the Leech lattice Theorem (M. Hentschel, 29) There are exactly nine Z[α]-structures of the Leech lattice. group #P b P i } SL 2 (25) 2 2.A 6 D 8 2 2, 6 3 SL 2 (3) , 46 4 (SL 2 (5) A 5 ).2 2, 8 5 (SL 2 (5) A 5 ).2 2, , 48 7 ± PSL 2 (7) (C 7 : C 3 ) 2 36, PSL 2 (7) 2.A , 9 2.J 2.2 2, 29, 6 Theorem (R. Coulangeon, N) d 3 (P i ) = for i = 2,..., 9 and d 3 (P ) >, so min(p P b ) = 4.

25 Stehlé, Watkins proof of extremality Theorem (Stehlé, Watkins (2)) Let L be an even unimodular lattice of dimension 72 with min(l) 6. Then L is extremal, if and only if it contains at least 6, 28, 75, 6 vectors v with Q(v) = 4. Proof: L is an even unimodular lattice of minimum 6, so its theta series is θ L = + a 3 q 3 + a 4 q = f (9) + a 3 3. f (9) = + 6, 28, 75, 6q = q 3 72q So a 4 = 6, 28, 75, 6 72a 3 6, 28, 75, 6 if and only if a 3 =. Remark A similar proof works in all jump dimensions 24k (extremal minimum = 2k + 2) for lattices of minimum 2k. For dimensions 24k + 8 and lattices of minimum 2k one needs to count vectors v with Q(v) = k + 2.

26 The extremal 72-dimensional lattice Γ 72 Main result Γ 72 is an extremal even unimodular lattice of dimension 72. Γ 72 has a canonical construction as trace lattice of Hermitian tensor product. Aut(Γ 72 ) = U := (PSL 2 (7) SL 2 (25)) : 2. U is an absolutely irreducible subgroup of GL 72 (Q). All U-invariant lattices are similar to Γ 72. Γ 72 is an ideal lattice in the 9st cyclotomic number field. Γ 72 realises the densest known sphere packing and maximal known kissing number in dimension 72. Γ 72 is a Z[ ]-lattice. This gives (n 2 + 5n + 5)-modular lattices of minimum 8 + 4n (n N ).