Sur l'analogie entre le syst\`eme dynamique de Deninger et le topos Weil-\'etale

TL;DR

This paper explores the analogy between Deninger's conjectural dynamical system and the Weil-étale topos, establishing that key dynamical properties are well-defined within the Weil-étale framework and extending this analogy to arithmetic schemes.

Contribution

It demonstrates that the Weil-étale topos satisfies properties of Deninger's dynamical system and constructs morphisms linking these concepts for arithmetic schemes.

Findings

Weil-étale topos captures dynamical properties like flow and fixed points.

The analogy extends to arithmetic schemes over different places.

Constructed morphisms relate Deninger's system to Weil-étale topos.

Abstract

We express some basic properties of Deninger's conjectural dynamical system in terms of morphisms of topoi. Then we show that the current definition of the Weil-\'etale topos satisfies these properties. In particular, the flow, the closed orbits, the fixed points of the flow and the foliation in characteristic $p$ are well defined on the Weil-\'etale topos. This analogy extends to arithmetic schemes. Over a prime number $p$ and over the archimedean place of $\mathbb{Q}$, we define a morphism from a topos associated to Deninger's dynamical system to the Weil-\'etale topos. This morphism is compatible with the structure mentioned above.