Higher Aspects of Counting Points on Schemes

Project: Research project

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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Counting
Ring
Finite Rings
Isomorphism Classes
Tangent Space
Groupoid
Galois field
Range of data