## 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.

Status | Finished |
---|---|

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

### Funding

- United States-Israel Binational Science Foundation (2018201)

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