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Dynamical systems for arithmetic schemes

Christopher Deninger

Abstract

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space $W_{\mathrm{rat}} (X)$ to every scheme $X$. We also define $R$-valued points $W_{\mathrm{rat}} (X) (R)$ of $W_{\mathrm{rat}} (X)$ for every commutative ring $R$. For normal schemes $X$ of finite type over spec $\mathbb{Z}$, using $W_{\mathrm{rat}} (X) (\mathbb{C})$ we construct infinite dimensional $\mathbb{R}$-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 $W_{\mathrm{rat}} (X)$ 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.

Dynamical systems for arithmetic schemes

Abstract

Motivated by work of Kucharczyk and Scholze, we use sheafified rational Witt vectors to attach a new ringed space to every scheme . We also define -valued points of for every commutative ring . For normal schemes of finite type over spec , using we construct infinite dimensional -dynamical systems whose periodic orbits are related to the closed points of . Various aspects of these topological dynamical systems are studied. We also explain how certain -adic points of for the spectrum of a -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.

Paper Structure

This paper contains 15 sections, 78 theorems, 692 equations.

Table of Contents

  1. Rational Witt vectors
  2. Rational Witt spaces
  3. ${\mathbb{C}}$-valued points of $W_{\mathrm{rat}} (X)$
  4. Classes of characters and subsystems
  5. The structure of the ${\mathbb{Q}}^{> 0}_0$-orbits
  6. The ${\mathbb{R}}^{> 0}$-dynamical system $X_0$
  7. Topology
  8. The closure of the periodic orbits in $X_0$
  9. Connectedness of the ${\mathbb{R}}^{> 0}$-dynamical system
  10. The $0$-th leafwise cohomology of the ${\mathbb{R}}^{> 0}$-dynamical system
  11. (Generalized) functions on $\check{\mathfrak{X}} ({\mathbb{C}})$ and $\check{\mathfrak{X}}_0 ({\mathbb{C}})$
  12. Invertible functions on $\mathfrak{X} , \mathfrak{X}_0$ as sections on $X , X_0$
  13. $\mathfrak{o}$-valued points of $W_{\mathrm{rat}} (X)$
  14. $A_{\inf}$-rings as $\mathfrak{o}$-valued functions on subspaces of $\check{W}_{\mathrm{rat}} (\mathfrak{X}) (\mathfrak{o})$
  15. A relation to the Fargues-Fontaine curve

Key Result

Proposition 1.1

a) If $A$ is an integral domain, $\omega$ is injective b) For an integrally closed domain $A$ with algebraically closed quotient field, we have an isomorphism:

Theorems & Definitions (190)

  • Proposition 1.1
  • Remark 1.2
  • proof
  • Proposition 1.3
  • proof
  • Proposition 1.4
  • proof
  • Theorem 1.5: KS
  • proof
  • Corollary 1.6
  • ...and 180 more