Unimodular Lattices Research Articles

On the existence of frames of the Niemeier lattices and self-dual codes over [formula omitted

Motivated by the work of Montague in 1994, we demonstrate that every Niemeier lattice can be constructed as an even unimodular neighbor of the odd unimodular lattice obtained from some self-dual code over F p by Construction A for each prime number p with 5 ⩽ p ⩽ 23 . This implies that every Niemeier lattice contains a 2 k-frame for k = 2 , 3 , … , 28 .

CONGRUENCES BETWEEN EXTREMAL MODULAR FORMS AND THETA SERIES OF SPECIAL TYPES MODULO POWERS OF 2 AND 3

Heninger, Rains and Sloane proved that the nth root of the theta series of any extremal even unimodular lattice in Rn has integer coefficients if n is of the form 2i3j5k(i ≥ 3). Motivated by their discovery, we find the congruences between extremal modular forms and theta series of special types modulo powers of 2 and 3. This assertion enables us to prove that the 2nth root and the (3n/2)th root of the extremal modular form of weight n/2 have at least one non-integer coefficient.

Logarithm laws for unipotent flows, I

We prove analogs of the logarithm laws of Sullivan and Kleinbock--Margulis in the context of unipotent flows. In particular, we obtain results for one-parameter actions on the space of lattices SL(n, $\R$)/SL(n, $\Z$). The key lemma for our results says the measure of the set of unimodular lattices in $\R^n$ that does not intersect a 'large' volume subset of $\R^n$ is 'small'. This can be considered as a 'random' analog of the classical Minkowski Theorem in the geometry of numbers.

Classification of ternary extremal self-dual codes of length 28

All 28-dimensional unimodular lattices with minimum norm 3 are known. Using this classification, we give a classification of ternary extremal self-dual codes of length 28. Up to equivalence, there are 6,931 such codes.

Unimodular lattices in dimensions 14 and 15 over the Eisenstein integers

All indecomposable unimodular hermitian lattices in dimensions 14 and 15 over the ring of integers in $\mathbb{Q}(\sqrt{-3})$ are determined. Precisely one lattice in dimension 14 and two lattices in dimension 15 have minimal norm 3.

Hirzebruch-Mumford proportionality and locally symmetric varieties of orthogonal type

For many classical moduli spaces of orthogonal type there are results about the Kodaira dimension. But nothing is known in the case of dimension greater than 19 . In this paper we obtain the first results in this direction. In particular the modular variety defined by the orthogonal group of the even unimodular lattice of signature (2,8m+2) is of general type if m\ge 5 .

Integral Springer Theorem for Quaternionic Forms

AbstractJ. S. Hsia has conjectured an arithmetical version of Springer Theorem, which states that no two spinor genera in the same genus of integral quadratic forms become identified over an odd degree extension. In this paper we prove by examples that the corresponding result for quaternionic skew-hermitian forms does not hold in full generality. We prove that it does hold for unimodular skew-hermitian lattices under all extensions and for lattices whose discriminant is relatively prime to 2 under Galois extensions.

Resolution of matrix equations over arbitrary Brouwerian lattices

This paper studies the problem of solving a matrix equation over an arbitrary (not necessarily complete) Brouwerian lattice. A criterion for the solvability and a method for finding all solutions of the equation are obtained. Over an arbitrary self-dual Brouwerian lattice, an equivalent condition for the unique solution, a necessary and sufficient condition for the minimal solution and a procedure for constructing minimal solutions less than or equal to any given solution of the equation are presented. It is shown that the results of the paper can be applied to the determination of generalized inverses of a matrix over an arbitrary Brouwerian lattice.

Modular forms for the even modular lattice of signature (2,10)

We consider the principal congruence subgroup Γ [ 2 ] \Gamma [2] of level two inside the orthogonal group of an even and unimodular lattice of signature ( 2 , 10 ) (2,10) . Using Borcherds’ additive lifting construction, we construct a 715 715 -dimensional space of singular modular forms. This space is the direct sum of the one-dimensional trivial and a 714 714 -dimensional irreducible representation of the finite group O ⁡ ( F 2 12 ) \operatorname {O}(\mathbb {F}^{12}_2) (even type). It generates an algebra whose normalization is the ring of all modular forms. We define a certain ideal of quadratic relations. This system appears as a special member of a whole system of ideals of quadratic relations. At least some of them have geometric meaning. For example, we work out the relation to Kondo’s approach to the modular variety of Enriques surfaces.

S-extremal strongly modular lattices

S-extremal strongly modular lattices maximize the minimum of the lattice and its shadow simultaneously. They are a direct generalization of the s-extremal unimodular lattices defined in [6]. If the minimum of the lattice is even, then the dimension of an s-extremal lattices can be bounded by the theory of modular forms. This shows that such lattices are also extremal and that there are only finitely many s-extremal strongly modular lattices of even minimum.

Characteristic vectors of unimodular lattices which represent two

On améliore un majorant connu pour la dimension n d’un réseau unimodulaire indécomposable dont la longuer de l’ombre prend la troisième plus grande valeur possible, n-16.

Mixed state of a latticed-wave superconductor

We study the mixed state in an extreme type-II lattice d-wave superconductor in the regime of intermediate magnetic fields H_{c1} << H << H_{c2}. We analyze the low energy spectrum of the problem dominated by nodal Dirac-like quasiparticles with momenta near k_F=(\pm k_D,\pm k_D) and find that the spectrum exhibits characteristic oscillatory behavior with respect to the product of k_D and magnetic length l. The Simon-Lee scaling, predicted in this regime, is satisfied only on average, with the magnitude of the oscillatory part of the spectrum displaying the same 1/l dependence as its monotonous ``envelope'' part. The oscillatory behavior of the spectrum is due to the inter-nodal interference enhanced by the singular nature of the low energy eigenfunctions near vortices. We also study a separate problem of a single vortex piercing an isolated superconducting grain of size L by L. Here we find that the periodicity of the quasiparticle energy oscillations with respect to k_D L is doubled relative to the case where the field is zero and the vortex is absent, both such oscillatory behaviors being present at the leading order in 1/L. Finally, we review the overall features of the tunneling conductance experiments in YBCO and BSCCO, and suggest an interpretation of the peaks at 5-20 meV observed in the tunneling local density of states in these materials.

On the integrality of nth roots of generating functions

Motivated by the discovery that the eighth root of the theta series of the E 8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + x Z 〚 x 〛 ) can be written as f = g n for g ∈ R , n ⩾ 2 . Let P n : = { g n | g ∈ R } and let μ n : = n ∏ p | n p . We show among other things that (i) for f ∈ R , f ∈ P n ⇔ f ( mod μ n ) ∈ P n , and (ii) if f ∈ P n , there is a unique g ∈ P n with coefficients mod μ n / n such that f ≡ g n ( mod μ n ) . In particular, if f ≡ 1 ( mod μ n ) then f ∈ P n . The latter assertion implies that the theta series of any extremal even unimodular lattice in R n (e.g. E 8 in R 8 ) is in P n if n is of the form 2 i 3 j 5 k ( i ⩾ 3 ). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed–Muller code of length 2 m is in P 2 r (and similarly that the theta series of the Barnes–Wall lattice B W 2 m is in P 2 m ). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f ∈ P n ( n ⩾ 2 ) with coefficients restricted to the set { 1 , 2 , … , n } .

EISENSTEIN LATTICES, GALOIS RINGS AND QUATERNARY CODES

Self-dual codes over the Galois ring GR(4,2) are investigated. Of special interest are quadratic double circulant codes. Euclidean self-dual (Type II) codes yield self-dual (Type II) ℤ4-codes by projection on a trace orthogonal basis. Hermitian self-dual codes also give self-dual ℤ4-codes by the cubic construction, as well as Eisenstein lattices by Construction A. Applying a suitable Gray map to self-dual codes over the ring gives formally self-dual 𝔽4-codes, most notably in length 12 and 24. Extremal unimodular lattices in dimension 38, 42 and the first extremal 3-modular lattice in dimension 44 are constructed.

Transformations among large c conformal field theories

We show that there is a set of transformations that relates all of the 24 dimensional even self-dual (Niemeier) lattices, and also leads to non-lattice objects some of which can perhaps be interpreted as a basis for the construction of holomorphic conformal field theory. In the second part of this paper, we extend our observations to higher-dimensional conformal field theories build on extremal partition functions, where we generate c = 24 k theories. We argue that there exists generalizations of the c = 24 models based on Niemeier lattices and of the non-Niemeier spin-1 theories. The extremal cases have spectra decomposable into the irreducible representations of the Fischer–Griess Monster. This additional symmetry leads us to conjecture that these extremal theories, as well as the higher-dimensional analogs of the group lattice bases Niemeiers, will eventually yield to a full construction of their associated CFTs. We observe interesting periodicities in the coefficients of extremal partition functions and characters of the extremal vertex operator algebras.

Theta Series of Unimodular Lattices, Combinatorial Identities and Weighted Symmetric Polynomials

Hecke proved that the theta series of a positive definite even unimodular lattice is a polynomial of the well-known Essenstein series E4(z) and the Ramanujan series Δ24(z). A natural question is what kind of polynomials in E4(z) and Δ24(z) could be the theta series of positive definite even unimodular lattices. In this paper, we find two combinatorial identities on the theta series of the root lattices of the finite-dimensional simple Lie algebras of type D4n and the cosets in their integral duals, in terms of E4(z) and Δ24(z). Using these two identities, we prove that three families of weighted symmetric polynomials of two fixed families of polynomials of E4(z) and Δ24(z) are the theta series of certain positive definite even unimodular lattices, obtained by gluing finitely many copies of the root lattices of the finite-dimensional simple Lie algebras of type D2n. The results also show that the full permutation groups are the hidden symmetry of the theta series of certain unimodular lattices.

A note on odd unimodular Euclidean lattices

We prove that the theta series of any odd unimodular Euclidean lattice is not congruent to 1 modulo any odd prime p.

The automorphism groups of the vertex operator algebras V+L: general case

In this article, we give a method of calculating the automorphism groups of the vertex operator algebras V + L associated with even lattices L. For example, by using this method we determine the automorphism groups of V + L for even lattices of rank one, two and three, and even unimodular lattices.

Gitter und Anwendungen

Respecting the different backgrounds of the participants, most of the talks were aimed to a broad audience, among those nine survey talks invited by the organizers. This concept was very successful and opened many discussions and interdisciplinary interactions. It was also very fruitful for the many young participants. Also most of the non speakers took the opportunity to present their recent work via posters. The theory of lattices has many applications and interactions with various other mathematical and technical disciplines such as information technology, topology, algebraic geometry, representation theory, combinatorics, number theory and modular forms to name only the most prominent ones. One of the most classical problems is the construction of dense lattice sphere packings. The densest lattice sphere packings are only known in dimensions up to 8 and, due to a recent work by H. Cohn and A. Kumar, also in dimension 24. A. Kumar gave a very illuminating talk on the strategy of their proof of this great result which uses certain linear programming techniques based on the fact that the minimal vectors of the Leech lattice form a very good design. Design techniques have also been used by B. Venkov to define the notion of strongly perfect lattices, which realize certain local maxima of the density function and which are now classified up to dimension 12. The theory of modular forms is one well established tool in the investigation of lattices as shown in the nice introduction by Krieg. Bannai applies this theory to bound the strength of designs provided by layers of unimodular lattices. On the other hand, lattices provide one important tool for the construction of modular forms. The last talk by Böcherer showed that in many situations all modular forms are linear combinations of theta-series. Of increasing interest but computationally very difficult is the investigation of thin lattice sphere coverings. Vallentin and Schürmann developed new algorithms to find local optimal coverings and discovered lattice coverings better than the ones previously known. Nguyen presented new reduction algorithms for lattices in small dimensions motivated by practical applications to cryptosystems. For certain applications in information technology not only the density of the lattice but also other properties that minimize the fading error play a role. It turns out that lattices constructed from algebraic number fields yield good codes for Rayleigh fading channels. These ideal lattices are also used to investigate the ring of integers in algebraic number fields as shown in the talk by Schoof and also in the applications of ideal lattices introduced by Bayer-Fluckiger. Strongly related to this is the application to Arakelov theory as illustrated in the talks by Bost and Künnemann. Also Voronoi's classical theory of perfect forms (see Martinet's talk for an introduction) has new fruitful applications in number theory more precisely in the calculation of the homology of GL_n({\bf Z}) as presented by Elbaz-Vincent. Last but not least one should mention the talk by J.-P. Serre on BL-bases and unitary groups in characteristic 2.

SHELLS OF SELFDUAL LATTICES VIEWED AS SPHERICAL DESIGNS

We find out for which t shells of selfdual lattices and of their shadows are spherical t-designs. The method uses theta series of lattices, which are modular forms. We analyze fully cubic and Witt lattices, as well as all selfdual lattices of rank at most 24.