Cyclotomy and analytic geometry over F_1

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.

30 pages

References in corpus (7)

• New Approach to Arakelov Geometry
• Les variétés sur le corps à un élément
• On the notion of geometry over $\F_1$
• The canonical strip, I
• Galois theory and integral models of Lambda-rings
• Smoothness in Relative Geometry
• Relative Zariski Open Objects

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• Quantum F_un: the q=1 Limit of Galois Field Quantum Mechanics, Projective Geometry, and the Field with One Element
• Quantum field theory over F1
• Quantum Statistical Mechanics of the Absolute Galois Group
• Moduli Operad over F1
• Endomotives of toric varieties
• Sur les π-exponentielles I : Vecteurs de Witt annulés par Frobenius et Algorithme de (leur rayon de convergence)
• Arithmetic theory of q-difference equations (G_q-functions and q-difference modules of type G, global q-Gevrey series)
• (non)commutative f-un geometry
• Cyclotomy and endomotives
• Counting in times of fake fields
• On the unification of quantum 3-manifold invariants