Unimodular Lattices Research Articles

Lattice vertex algebras on general even, self-dual lattices

In this note we analyse the Lie algebras of physical states stemming from lattice constructions on general even, self-dual lattices Gamma^{p,q} with p greater or equal to q. It is known that if the lattice is at most Lorentzian, the resulting Lie algebra is of generalized Kac-Moody type (or has a quotient that is). We show that this is not true as soon as q is larger than 1. By studying a certain sublattice in the case q>1 we obtain results that lead to the conjecture that the resulting non-GKM Lie algebra cannot be described conveniently in terms of generators and relations and belongs to a new and qualitatively different class of Lie algebras.

New asymptotic bounds for self-dual codes and lattices

We give an independent proof of the Krasikov-Litsyn bound d/n/spl lsim/(1-5/sup -1/4/)/2 on doubly-even self-dual binary codes. The technique used (a refinement of the Mallows-Odlyzko-Sloane approach) extends easily to other families of self-dual codes, modular lattices, and quantum codes; in particular, we show that the Krasikov-Litsyn bound applies to singly-even binary codes, and obtain an analogous bound for unimodular lattices. We also show that in each case, our bound differs from the true optimum by an amount growing faster than O(/spl radic/n).

Splitting the shadow

We derive formulae for the theta series of the two translates of the even sublattice L 0 of an odd unimodular lattice L that constitute the shadow of L. The proof rests on special evaluations of the Jacobi theta series attached to L and to a certain vector. We produce an analogous theorem for codes. Additionally, we construct non-linear formally self-dual codes and relate them to lattices.

Extremal odd unimodular lattices in dimensions 44, 46 and 47

In this note, extremal odd unimodular lattices in dimensions 44, 46 and 47 are constructed from self-dual codes over Z 4 and Z 6 by Construction A. The lattices in dimensions 46 and 47 seem to be the first explicit examples for such lattices.

Virasoro frames and their stabilizers for the E8 lattice type vertex operator algebra

The concept of a framed vertex operator algebra (FVOA) is new (cf. [DGH]). This article contributes to this theory with a full analysis of all Virasoro frame stabilizers in V , the important example of the E8 level 1 affine Kac-Moody VOA, which is isomorphic to the lattice VOA for the root lattice of E8(C). We analyze the frame stabilizers, both as abstract groups and as subgroups of Aut(V ) ∼ E8(C). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V ), but is not usually a maximal finite subgroup of this normalizer. In particular, we prove that there are exactly five orbits for the action of Aut(V ) on the set of Virasoro frames, thus settling an open question about V in Section 5 of [DGH]. The results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories, so this article contributes also to this theory. There are two main viewpoints in our analysis. The first is the theory of codes, lattices, markings and the resulting groups of automorphisms. The second is the theory of finite subgroups of Lie groups. We expect our methods to be applicable to the study of other FVOAs and their frame stabilizers. Appendices present aspects of the theory of automorphism groups of VOAs. In particular, there is a general result of independent interest, on embedding lattices into unimodular lattices so as to respect automorphism groups and definiteness.

Unimodular lattices with long shadow

Let L be an odd unimodular lattice of dimension n with shadow n−16. If min( L)⩾3 then dim( L)⩽46 and there is a unique such lattice in dimension 46 and no lattices in dimensions 44 and 45. To prove this, a shadow theory for theta series with spherical coefficients is developed.

Self-dual lattice identities

Given a finitely based self-dual variety of lattices, is it definable, modulo lattice theory, by a single self-dual lattice identity? There are infinitely many examples with “yes” as the answer and infinitely many with “no.”.

Automorphisms of even unimodular lattices and unramified Salem numbers

In this paper we study the characteristic polynomials S ( x )=det( xI − F |II p , q ) of automorphisms of even unimodular lattices with signature ( p , q ). In particular, we show that any Salem polynomial of degree 2 n satisfying S (−1) S (1)=(−1) n arises from an automorphism of an indefinite lattice, a result with applications to K3 surfaces.

Even Unimodular Gaussian Lattices of Rank 12

We classify even unimodular Gaussian lattices of rank 12, that is, even unimodular integral lattices of rank 12 over the ring of Gaussian integers. This is equivalent to the classification of the automorphisms τ with τ2=−1 in the automorphism groups of all the Niemeier lattices, which are even unimodular (real) integral lattices of rank 24. There are 28 even unimodular Gaussian lattices of rank 12 up to equivalence.

A mass formula for unimodular lattices with no roots

We derive a mass formula fornn-dimensional unimodular lattices having any prescribed root system. We use Katsurada’s formula for the Fourier coefficients of Siegel Eisenstein series to compute these masses for all root systems of even unimodular 32-dimensional lattices and odd unimodular lattices of dimensionn≤30n\leq 30. In particular, we find the mass of even unimodular 32-dimensional lattices with no roots, and the mass of odd unimodular lattices with no roots in dimensionn≤30n\leq 30, verifying Bacher and Venkov’s enumerations in dimensions 27 and 28. We also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the Minkowski-Siegel mass constants.

Birational automorphisms of quartic Hessian surfaces

We find generators of the group of birational automorphisms of the Hessian surface of a general cubic surface. Its nonsingular minimal model is a K3 surface with the Picard lattice of rank 16 which embeds naturally in the even unimodular lattice II 1,25 of rank 26 and signature (1,25). The generators are related to reflections with respect to some Leech roots. A similar observation was made first in the case of quartic Kummer surfaces in the work of Kondo. We shall explain how our generators are related to the generators of the group of birational automorphisms of a general quartic Kummer surface which is birationally isomorphic to a special Hessian surface.

Self-dual [formula omitted]-codes and Hadamard matrices

In this note, we investigate Type I codes over Z 4 constructed from Hadamard matrices. As an application, we construct a Type I Z 4 -code with minimum Euclidean weight 16 of length 40. This code is the first example of such a Type I Z 4 -code. This code also gives an example of a 40-dimensional extremal odd unimodular lattice with minimum norm 4.

Ternary Code Construction of Unimodular Lattices and Self-Dual Codes over \Bbb Z 6

We revisit the construction method of even unimodular lattices using ternary self-dual codes given by the third author (M. Ozeki, in Théorie des nombres, J.-M. De Koninck and C. Levesque (Eds.) (Quebec, PQ, 1987), de Gruyter, Berlin, 1989, pp. 772–784), in order to apply the method to odd unimodular lattices and give some extremal (even and odd) unimodular lattices explicitly. In passing we correct an error on the condition for the minimum norm of the lattices of dimension a multiple of 12. As the results of our present research, extremal odd unimodular lattices in dimensions 44, 60 and 68 are constructed for the first time. It is shown that the unimodular lattices obtained by the method can be constructed from some self-dual ℤ6-codes. Then extremal self-dual ℤ6-codes of lengths 44, 48, 56, 60, 64 and 68 are constructed.

Odd unimodular lattices of minimum 4

Odd unimodular lattices of minimum 4

Conference Matrices and Unimodular Lattices

Conference matrices are used to define complex structures on real vector spaces. Certain lattices in these spaces become modules for rings of quadratic integers. Multiplication of these lattices by nonprincipal ideals yields simple constructions of further lattices including the Leech lattice.

Minima of Odd Unimodular Lattices in Dimension 24m

Rains and Sloane established that the minimum of a unimodular Z-lattice in dimension 24m is bounded above by 2m+2. They conjectured that only even lattices could attain this bound. In this paper, I prove their conjecture.

Type II Codes OverF4 + uF4

A special class of self dual codes over an alphabet of size 16 which contains bothF4andF2+uF2is studied. Applications to codes over these two alphabets and unimodular lattices (Gaussian, golden and integer) are given.

Cubic rings and the exceptional Jordan algebra

In a previous paper [EG] we described an integral structure (J, E) on the exceptional Jordan algebra of Hermitian 3×3 matrices over the Cayley octonions. Here we use modular forms and Niemeier's classification of even unimodular lattices of rank 24 to further investigate J and the integral, even lattice J0=(ZE)⊥ in J. Specifically, we study ring embeddings of totally real cubic rings A into J which send the identity of A to E, and we give a new proof of R. Borcherds's result that J0 is characterized as a Euclidean lattice by its rank, type, discriminant, and minimal norm.

An extremal ternary self-dual [28,14,9] code with a trivial automorphism group

In this paper, we show that the smallest length for which there is an extremal ternary self-dual code with a trivial automorphism group is 28 by constructing such a code of length 28. This code is the first example of extremal ternary self-dual codes with trivial automorphism groups. An extremal ternary self-dual [32,16,9] code with a trivial automorphism group is also constructed. A 32-dimensional odd unimodular lattice with minimum norm 3 and a trivial automorphism group is obtained from the code by Construction A.

Local Borcherds products

The local Picard group at a generic point of the one-dimensional Baily-Borel boundary of a Hermitean symmetric quotient of type O(2,n) is computed. The main ingredient is a local version of Borcherds’ automorphic products. The local obstructions for a Heegner divisor to be principal are given by certain theta series with harmonic coefficients. Sometimes they generate Borcherds’ space of global obstructions. In these particular cases we obtain a simple proof of a result due to the first author: Suppose that Γ⊂O(2,n) is the orthogonal group attached to an even unimodular lattice. Then every meromorphic modular form for Γ, whose zeros and poles lie on Heegner divisors, is given by a Borcherds product.