Cyclotomy and Analytic Geometry over F 1

Geometry over non–existent “field with one element” F1 conceived by Jacques Tits [Ti] half a century ago recently found an incarnation, in several 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 F1. The paper combines a focused survey of various approaches with some new constructions. To Alain Connes, for his sixtieth anniversary 0. Introduction: many faces of cyclotomy 0.1. Roots of unity and field with one element. The basics of algebraic geometry over an elusive “field with one element F1” were laid down recently in [So], [De1], [De2], [TV], fifty years after a seminal remark by J. Tits [Ti]. There are many motivations to look for F1; a hope to imitate Weil’s proof for Riemann’s zeta is one of them, cf. [CCMa3], [Ku], [Ma1]. An important role in the formalization of F1–geometry was played by the suggestion made in [KS] that one should simultaneously consider all the “finite extensions” F1n . This resulted in the approach of [So], where a geometric object, say a scheme, V over F1, acquired flesh after a base extension to Z, and the F1– geometry of V was reflected in (and in fact, formally defined in terms of) the geometry of “cyclotomic” points of an appropriate ordinary scheme VZ. In [De1] and [TV], schemes over F1 are defined in categorical terms independently of cyclotomy, but the latter reappears soon: see the Definition 1.7.1 below and the following discussion. All these ideas are interrelated but lead to somewhat different versions of basic definitions, and develop the initial intuition in different directions, so that their divergence can be fruitfully exploited. With this goal in mind, I have chosen the topics to be discussed in sec. 1, where four approaches to the definition of F1– geometry are sketched and compared. Of course, roots of unity appear naturally in many different geometric contexts, not motivated by geometry over F1: some of these contexts are reviewed below in 1