Higher Aspects of Counting Points on Schemes

Counting points on varieties over finite fields is a famous and well studied problem. Works of Igusa and Denef generalized this problem to counting points over the rings Z/N. We propose a research on higher versions of this problem: (1) Replacing the rings Z/N by finite rings with higher-dimensional Zariski tangent spaces. For a scheme X, we propose to study the relation between the size of a ring R and the number of R-points of X, as R ranges over all rings with a fixed Zariski tangent space. (2) Replacing schemes by stacks. For a stack X and a finite ring R, we want to relate the number of isomorphism classes of the groupoid X(R) and the size of R. (3) More generally, replacing schemes by higher stacks. For a higher stack X and a finite ring R, one gets a higher groupoid X(R), which can be represented by a homotopical type. π0(X(R)) is the collection of isomorphism classes. We will be interested in studying the dependence of other numerical topological invariants of X(R) and the size of R.