Higher Aspects of Counting Points on Schemes

Project: Research project

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.
StatusActive
Effective start/end date10/1/199/30/23

Funding

  • United States-Israel Binational Science Foundation (2018201)

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