Dynamical systems for arithmetic schemes

2018/07/17 by Deninger, Christopher
#Algebraic Geometry (math.AG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.06400

Abstract

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space W_rat (X) to every scheme X. We also define R-valued points W_rat (X) (R) of W_rat (X) for every commutative ring R. For normal schemes X of finite type over spec ℤ, using W_rat (X) (ℂ) we construct infinite dimensional ℝ-dynamical systems whose periodic orbits are related to the closed points of X. Various aspects of these topological dynamical systems are studied. We also explain how certain p-adic points of W_rat (X) for X the spectrum of a p-adic local number ring are related to the points of the Fargues-Fontaine curve. The new intrinsic construction of the dynamical systems generalizes and clarifies the original extrinsic construction in v.1 and v.2. Many further results have been added.

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