Modul: MAT770 Oberseminar: Algebraische Geometrie

## Dynamical systems for arithmetic schemes

Prof. Dr. Christopher Deninger talk

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 07.10.19 **Time:** 13.15 - 14.45 **Room:** Y27H25

Given an arithmetic scheme \(X\), we construct a natural infinite dimensional dynamical system whose periodic orbits come in compact packets \(P = P(x)\) which are in bijection with the closed points \(x\) of \(X\). Here each periodic orbit in \(P\) has length \(\log N(x)\), where \(N(x)\) is the number of elements in the residue field of \(x\). Thus the zeta functions of analytic number theory and arithmetic geometry can be viewed as Ruelle type zeta functions of dynamical systems. We will describe the construction and what is known about these dynamical systems. The "generic fibres" of our dynamical systems are related to an earlier construction by Robert Kucharczyk and Peter Scholze of topological spaces whose fundamental groups realise Galois groups.