Smooth variety of lattices

Abstract

In the previous work of the authors, the foundations of the theory of smooth manifolds of number-theoretic lattices were laid. The case of arbitrary multidimensional lattices was considered.
This article considers the general case of shifted multidimensional lattices. Note that the geometry of the metric spaces of multidimensional lattices is much more complicated than the geometry of ordinary Euclidean space. This is evident from the paradox
of non-additivity of the length of a segment in the space of shifted one-dimensional lattices. From the presence of this paradox it follows that the problem of describing geodesic lines in spaces of multidimensional lattices remains open, as well as finding a formula for the length of arcs of lines in these spaces. Naturally, it would be interesting not only to describe these objects, but also to obtain a number-theoretic interpretation of these concepts. A further direction of research may be the study of the analytic continuation of the
hyperbolic zeta function on the spaces of shifted multidimensional lattices. As is known, the analytic continuation of the hyperbolic zeta functions of lattices are constructed for an arbitrary Cartesian lattice. Even the question of the continuity of these analytical continuations in the left half-plane on the space of lattices has not been studied. All of these, in our opinion, are relevant directions for further research.