This Note gives an extension of Mahler's theorem on lattices in ${\mathbb{R}}^n$ to simply connected nilpotent groups with a $Q$-structure. From this one gets an application to groups of Heisenberg type and a generalization of Hermite's inequality.

We show that the unimodular lattice associated to the rank 20 quaternionic matrix group ${\mathbf{SL}}_2\left({\mathbf{F}}_{41}\right)\otimes {\tilde{S}}_3\subset {\mathbf{GL}}_{80}\left(\mathbf{Z}\right)$ is a fourth example of an 80-dimensional extremal lattice. Our method is to use the positivity of the $\Theta$-series in conjunction with an enumeration of all the norm 10 vectors. The use of Aschbacher’s theorem on subgroups of finite classical groups (reliant on the classification of finite simple groups) provides one proof that this lattice is distinct from the previous three, while computing the inner product...

Some interesting lattices can be constructed using association schemes. We illustrate this by a unimodular lattice without roots of dimension 28 which admits $\mathrm{Sp}(6,{𝔽}_3)·2$ as its automorphism group.

Using the geometry of the projective plane over the finite field ${𝔽}_q$, we construct a Hermitian Lorentzian lattice $L_q$ of dimension $(q^2+q+2)$ defined over a certain number ring $𝒪$ that depends on $q$. We show that infinitely many of these lattices are $p$-modular, that is, $p{L}_q^{\text{'}}=L_q$, where $p$ is some prime in $𝒪$ such that ${\left|p\right|}^2=q$.The Lorentzian lattices $L_q$ sometimes lead to construction of interesting positive definite lattices. In particular, if $q\equiv 3mod4$ is a rational prime such that $(q^2+q+1)$ is norm of some element in $\mathbb{Q}\left[\sqrt{-q}\right]$, then we find a $2q(q+1)$ dimensional...

For algebraic number fields $K$ with $s\>0$ real and $2t\>0$ complex embeddings and “admissible” subgroups $U$ of the multiplicative group of integer units of $K$ we construct and investigate certain $(s+t)$-dimensional compact complex manifolds $X(K,U)$. We show among other things that such manifolds are non-Kähler but admit locally conformally Kähler metrics when $t=1$. In particular we disprove a conjecture of I. Vaisman.

On étudie ici du point de vue de la dualité les réseaux de dimension $5$ ayant un automorphisme d’ordre $5$. On y rencontre en particulier le premier exemple irrationnel de couple de réseaux duaux extrême pour le produit de leurs constantes d’Hermite, et l’on donne une réponse partielle à un problème de Conway et Sloane sur les réseaux isoduaux.