Cyclotomy and analytic geometry over F_1

TL;DR

This paper explores the role of roots of unity in geometry over the hypothetical field with one element, proposing a new concept of analytic functions over F_1 and combining survey and novel constructions.

Contribution

It introduces a new approach to defining analytic functions over F_1, emphasizing roots of unity and providing updated constructions and references.

Findings

Roots of unity are crucial in F_1 geometry

Proposed a new notion of analytic functions over F_1

Updated constructions and references in the theory

Abstract

Geometry over non--existent "field with one element" $F_1$ conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in at least two related but different guises. In this paper I analyze the crucial role of roots of unity in this geometry and propose a version of the notion of "analytic functions" over $F_1$. The paper combines a focused survey with some new constructions. In new version, several local additions and changes are made, references added.