Unimodular Lattices Research Articles

An Exact Percolation Point of Surface Filling in a Four-Dimensional Hyper-Cubic Lattice

An exact percolation threshold in a four-dimensional hyper-cubic lattice is obtained as p c = 0.5 from geometrical consideration. In a similar way, a macroscopically percolated structure consisting of (2n - n)-dimensional structure is formed at the critical point p c = 0.5 in a 2n-dimensional hyper-cubic lattice, because of the existence of a self-dual lattice in 2n-dimensional space and the orthogonal relation of the (2n - n)-dimensional structures in the original and dual hyper-cubic lattices.

The mass of unimodular lattices

We derive an explicit formula for the mass of a unimodular Z -lattice of arbitrary signature using Bruhat–Tits theory.

Minkowski’s conjecture, well-rounded lattices and topological dimension

Let A ⊂ SL n ( R ) A \subset {\operatorname {SL}}_n({\mathbb R}) be the diagonal subgroup, and identify SL n ( R ) / SL n ( Z ) {\operatorname {SL}}_n({\mathbb R})/ {\operatorname {SL}}_n({\mathbb Z}) with the space of unimodular lattices in R n {\mathbb R}^n . In this paper we show that the closure of any bounded orbit A ⋅ L ⊂ SL n ( R ) / SL n ( Z ) \begin{equation*} A \cdot L \subset {\operatorname {SL}}_n({\mathbb R})/{\operatorname {SL}}_n({\mathbb Z}) \end{equation*} meets the set of well-rounded lattices. This assertion implies Minkowski’s conjecture for n = 6 n=6 and yields bounds for the density of algebraic integers in totally real sextic fields. The proof is based on the theory of topological dimension, as reflected in the combinatorics of open covers of R n {\mathbb R}^n and T n T^n .

Representations of integral quadratic forms by sums of squares

Let g[n] be the minimum number of squares whose sum represents all positive definite integral quadratic forms of rank n which are represented by sums of squares. In this article, we first discuss representations of integers by unimodular lattices. We then estimate the order of magnitude of the diameter of the 2-graph of unimodular lattices of rank n. Combining these results we prove g[n]=O(3 n /2n log n). We also provide a lower bound for g[n]. Finally, we discuss s-integrable lattices as an application of our method.

Irredundant self-dual bases for self-dual lattice varieties

For any finitely based self-dual variety of lattices, we determine the sizes of all equational bases that are both irredundant and self-dual. We make the same determination for {0, 1}-lattice varieties.

On some self-dual codes and unimodular lattices in dimension 48

In this paper, binary extremal self-dual codes of length 48 and extremal unimodular lattices in dimension 48 are studied through their shadows and neighbors. We relate an extremal singly even self-dual [48, 24, 10] code whose shadow has minimum weight 4 to an extremal doubly even self-dual [48, 24, 12] code. It is also shown that an extremal odd unimodular lattice in dimension 48 whose shadow has minimum norm 2 relates to an extremal even unimodular lattice. Extremal singly even self-dual [48, 24, 10] codes with shadows of minimum weight 8 and extremal odd unimodular lattice in dimension 48 with shadows of minimum norm 4 are investigated.

Extremal Self-Dual Codes over ℤ6, ℤ8 and ℤ10

In this paper, upper bounds on the minimum Euclidean weights of Type I codes over ℤ6, and self-dual codes over ℤ8 and ℤ10, are derived for modest lengths. The notion of extremality for Euclidean weights is also introduced. We construct new extremal self-dual codes over these rings. Most of these codes are obtained via the double circulant and quasi-twisted constructions. New extremal odd unimodular lattices are obtained from some of these codes by Construction A.

Pieces of [formula omitted]: existence and uniqueness for Barnes–Wall and Ypsilanti lattices

We give a new existence proof for the rank 2 d even lattices usually called the Barnes–Wall lattices, and establish new results on uniqueness, structure and transitivity of the automorphism group on certain kinds of sublattices. Our proofs are relatively free of calculations, matrix work and counting, due to the uniqueness viewpoint. We deduce the labeling of coordinates on which earlier constructions depend. Extending these ideas, we construct in dimensions 2 d , for d ≫ 0 , the Ypsilanti lattices, which are families of indecomposable even unimodular lattices which resemble the Barnes–Wall lattices. The number ϒ ( 2 d ) of isometry types here is large: log 2 ( ϒ ( 2 d ) ) has dominant term at least r 4 d 2 2 d , for any r ∈ [ 0 , 1 2 ) . Our lattices may be the first explicitly given families whose sizes are asymptotically comparable to the Siegel mass formula estimate ( log 2 ( mass ( n ) ) has dominant term 1 4 log 2 ( n ) n 2 ). This work continues our general uniqueness program for lattices, begun in Pieces of Eight (Adv. in Math. 148 (1999) 75). See also our new uniqueness proof for the E 8 -lattice (J. Number Theory 103 (2003) 77).

Characteristic Elements of Unimodular -lattices

Characteristic elements have been useful in the classification of unimodular lattices over the integers. This article gives an explicit formula for characteristic elements of a lattice in terms of a basis for the lattice and the dual of that basis.

Codes over ?2m and Jacobi forms over the Quaternions

We introduce codes over the ring **. We relate self-dual codes over this ring to quaternionic unimodular lattices and to self-dual codes over ** via a gray map. We study a connection between the complete weight enumerators of codes over the quaternionic ring Σ2m and Jacobi forms over the half-space of quaternions. This motivates us to construct an algebra homomorphism from a certain invariant polynomial ring, where the complete weight enumerators belong, to the ring of Jacobi forms over the quaternions. Higher genus modular forms over the quaternions are also constructed using joint weight enumerators of codes.

The automorphism group of the vertex operator algebra [formula omitted] for an even lattice L without roots

The automorphism group of the vertex operator algebra V L + is studied by using its action on isomorphism classes of irreducible V L + -modules. In particular, the shape of the automorphism group of V L + is determined when L is isomorphic to an even unimodular lattice without roots, 2 R for an irreducible root lattice R of type ADE and the Barnes–Wall lattice of rank 16.

Codes over rings, complex lattices and Hermitian modular forms

We introduce the finite ring S 2m= Z 2m+i Z 2m . We develop a theory of self-dual codes over this ring and relate self-dual codes over this ring to complex unimodular lattices. We describe a theory of shadows for these codes and lattices. We construct a gray map from this ring to the ring Z 2m and relate codes over these rings, giving special attention to the case when m=2. We construct various Hermitian modular forms from weight enumerators and give the correspondence between the invariant space, where the weight enumerators of codes reside, and the space of Hermitian modular forms.

Gaps in √n mod 1 and ergodic theory

Cut the unit circle $S^1=\mathbb{R}/\mathbb{Z}$ at the points $\{\sqrt{1}\}, \{\sqrt{2}\},\ldots,\{\sqrt{N}\}$, where $\{x\} = x \bmod 1$, and let $J_1, \ldots, J_N$ denote the complementary intervals, or \emph{gaps}, that remain. We show that, in contrast to the case of random points (whose gaps are exponentially distributed), the lengths $|J_i|/N$ are governed by an explicit piecewise real-analytic distribution $F(t) \,dt$ with phase transitions at $t=1/2$ and $t=2$. The gap distribution is related to the probability $p(t)$ that a random unimodular lattice translate $\Lambda \subset \mathbb{R}^2$ meets a fixed triangle $S_t$ of area $t$; in fact, $p''(t) = -F(t)$. The proof uses ergodic theory on the universal elliptic curve \[ E = \big(\mathrm{SL}_2(\mathbb{R}) \ltimes \mathbb{R}^2\big)/ \big(\mathrm{SL}_2(\mathbb{Z}) \ltimes \mathbb{Z}^2\big) \] and Ratner's theorem on unipotent invariant measures.

Holomorphic vertex operator algebras of small central charge

We provide a rigorous mathematical foundation to the study of strongly rational, holomorphic vertex operator algebras V of central charge c = 8, 16 and 24 initiated by Schellekens. If c = 8 or 16 we show that V is isomorphic to a lattice theory corresponding to a rank c even, self-dual lattice. If c = 24 we prove, among other things, that either V is isomorphic to a lattice theory corresponding to a Niemeier lattice or the Leech lattice, or else the Lie algebra on the weight one subspace V 1 is semisimple (possibly 0) of Lie rank less than 24.

Unimodular hermitian lattices in dimension 13

All indecomposable unimodular hermitian lattices in dimension 13 over the ring of integers in Q( −3 ) are determined.

Construction of new extremal unimodular lattices

In this paper we construct new extremal and optimal unimodular lattices in dimensions 36, 38, 42, 45, 52, 54, 60 and 68. We construct them in two ways: first in the case of dimensions congruent to 4 modulo 8 by construction B 3 followed by density doubling, generalizing the constructions of Sphere Packing Lattices and Groups (1988) 148 and Théorie des Nombres (1989) 772; and second by applying the well known Construction A to self-dual codes over GF(5) and to codes over the ring Z/25 Z. In particular the lattice in dimension 60, P 60 q , generalizes the construction of the lattice P 48 q . We also give the complete weight enumerator of the extended ternary quadratic residue code of length 60 and we provide a table of the best known unimodular lattices of dimensions up to 80.

Steinitz classes of unimodular lattices

For each prime l with l≡7( mod 8) , we define an action of the ring O= Z [ 1 2 (1+ −l )] on the unimodular lattice D l+1 + using a Paley matrix. We determine the isomorphism class of D l+1 + as an O -module. In particular we show that unless l=7, D l+1 + is not a free O -module. We note a consequence for the Leech lattice.

Four-manifold systoles and surjectivity of period map

P. Buser and P. Sarnak showed in 1994 that the maximum, over the moduli space of Riemann surfaces of genus $s$, of the least conformal length of a nonseparating loop, is logarithmic in $s$. We present an application of (polynomially) dense Euclidean packings, to estimates for an analogous 2-dimensional conformal systolic invariant of a 4-manifold $X$ with indefinite intersection form. The estimate turns out to be polynomial, rather than logarithmic, in $\chi(X)$, if the conjectured surjectivity of the period map is correct. Such surjectivity is targeted by the current work in gauge theory. The surjectivity allows one to insert suitable lattices with metric properties prescribed in advance, into the second de Rham cohomology group of $X$, as its integer lattice. The idea is to adapt the well-known Lorentzian construction of the Leech lattice, by replacing the Leech lattice by the Conway-Thompson unimodular lattices which define asymptotically dense packings. The final step can be described, in terms of the successive minima $\lambda\_i$ of a lattice, as deforming a $\lambda\_2$-bound into a $\lambda\_1$-bound, illustrated by Figure 1.

Positive definite lattices of rank at most 8

We give a short uniqueness proof for the E 8 root lattice, and in fact for all positive definite unimodular lattices of rank up to 8. Our proof is done with elementary arguments, mainly these: (1) invariant theory for integer matrices; (2) an upper bound for the minimum of nonzero norms (either of the elementary bounds of Hermite or Minkowski will do). We make no use of p-adic completions, mass formulas or modular forms.

Novel multisoliton solutions of some soliton equations

The novel multisoliton solutions for the nonlinear lumped self-dual network equations, Toda lattice and KP equation were obtained by using the Hirota direct method.