Well-rounded sublattices (Q2909028)

Let \(\Lambda\) be a lattice in \(\mathbb{R}^{n}\). It is well rounded if its elements of smallest ( non-zero) length generate the ambient vector space \(\mathbb{R}^{n}\). The set of well rounded lattices has dimension \(n(n-1)/2\) and has been made use of in papers of Ash and McConnell in explicitly computing parts of cohomology of some congruence subgroups of \(SL(3,\mathbb{Z})\). More recently McMullen has studied the dynamics of the diagonal subgroup of \(SL(n,\mathbb{R})\) acting on the set of all lattices of covolume 1 and showed that among other things that for every lattice \(\Lambda\) having cocompact stabilizer in the diagonal subgroup there is a well rounded sublattice in the orbit. In this paper well rounded sublattices of a given lattice have been studied. This type of question has first been studied by Fukshansky and others. Let \(\Lambda \subset \mathbb{R}^{d} \) be a lattice. For \(n\in\text{N}\), let \(a_{n}=a_{n}(\Lambda)\) be the number of well rounded sublattices of \(\Lambda\) of index \(n\). Consider the Dirichlet series \(\zeta_{wr}(\Lambda,s)=\sum_{n\in\text{N}}a_{n}n^{-s}\). If \(\Lambda\) is an arithmetic lattice in the plane. Then it is proved here that the abscissa of convergence of \(\zeta_{wr}(\Lambda,s)\) is one. It is also shown that there are infinitely many similarity classes of well-rounded sublattices in a plane lattice if there is at least one. This generalizes results about the rings of Gaussian and of Eisenstein integers of Fukshansky and his coauthors.