Perfectoid spaces arising from arithmetic jet spaces

TL;DR

This paper constructs perfectoid spaces from arithmetic jet spaces associated with smooth schemes and explores their applications to arithmetic differential equations, including elliptic curves and modular curves.

Contribution

It introduces a novel method of attaching perfectoid spaces to arithmetic jet spaces and $ ext{delta}$-morphisms, expanding the toolkit for studying arithmetic differential equations.

Findings

Construction of perfectoid spaces from arithmetic jet spaces.

Application to $ ext{delta}$-characters of elliptic curves.

Analysis of the $ ext{delta}$-period map on modular curves.

Abstract

Using arithmetic jet spaces, we attach perfectoid spaces to smooth schemes and to $\delta$-morphisms of smooth schemes. We also study perfectoid spaces attached to arithmetic differential equations defined by some of the remarkable $\delta$-morphisms appearing in the theory such as the $\delta$-characters of elliptic curves and the $\delta$-period map on modular curves.