# Project Details

### Description

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.

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.

Status | Active |
---|---|

Effective start/end date | 10/1/19 → 9/30/23 |

### Funding

- United States-Israel Binational Science Foundation (2018201)