Cyclotomy and analytic geometry over F₁

2008/09/09 by Yuri I. Manin, Manin, Yuri I. · 3 citations
Mathematics · #14H10 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.0809.1564

openalex publication_date 2008/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Geometry over non--existent "field with one element" F₁ 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 F₁. The paper combines a focused survey with some new constructions. In new version, several local additions and changes are made, references added.

Citations

• Galois theory and integral models of Lambda-rings
• Relative Zariski Open Objects

Cited by

• Cyclotomy and endomotives
• Lambda-rings and the field with one element
• Arithmetic theory of q-difference equations (G_q-functions and\n q-difference modules of type G, global q-Gevrey series)

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