Project A1: The structure of (almost) lattices – algebra, analysis, and arithmetic

To each lattice in Euclidean space, one associates theta functions (counting lattice points in spherical shells by radius) and zeta functions (counting sublattices by index), both established objects of study with well-developed theories. Project A1 has three main objectives. First, we will connect zeta functions associated with various lattice enumeration problems to Ehrhart polynomials of polytopes and Hecke algebras of $p$-adic Lie groups. Second, we will investigate Euler factors of Dirichlet series counting various kinds of similar submodules (including cases related to exceptional root systems such as $E_8$ and to general root lattices) in particular with the objective to establish connections with rational results from $p$-adic integration theory for $p$-adic Lie groups as well as combinatorial Weyl group statistics. The third goal is to develop a theory of theta series for almost lattices. In known examples, the averaged counting problem in this context leads to very particular algebraic integers, and the aim is to uncover the underlying reasons.