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On Dynamical Systems in the Étale Topology

Lars Andersen

Abstract

We discuss $\mathcal{D}$-modules and dynamical systems in the étale topology. We introduce the differential scheme associated to a morphism $f: X\to S$ of schemes of the same dimension. We introduce differential inertia group $I_{diff}^i$ which act trivially on the special fibers of these schemes. As an application we discuss the problem of counting points on elliptic curves which we show is connected to vanishing cycles of singularities and to closed orbits of dynamical systems.

On Dynamical Systems in the Étale Topology

Abstract

We discuss -modules and dynamical systems in the étale topology. We introduce the differential scheme associated to a morphism of schemes of the same dimension. We introduce differential inertia group which act trivially on the special fibers of these schemes. As an application we discuss the problem of counting points on elliptic curves which we show is connected to vanishing cycles of singularities and to closed orbits of dynamical systems.
Paper Structure (7 sections, 4 theorems, 26 equations)

This paper contains 7 sections, 4 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Differential Schemes
  3. The étale topology
  4. Differential schemes
  5. The differential inertia groups
  6. The sheaf of differential cycles
  7. Appendix: Collatz Conjecture

Key Result

Lemma 1

The differential inertia groups act trivially on $V_s$.

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Example 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Lemma 1
  • Lemma 2
  • ...and 8 more