\(\mathbb{F}_{1}\) for everyone. (English) Zbl 1430.14005
The article contains a very nice and lucidly written introduction to the various approaches and algebraic and geometrical concepts connected with the so-called “field with one element”, denoted by \(\mathbb{F}_1\) (which, in the author’s words, “in most cases [...] is not a field and has two elements”). It is aimed at a general mathematical audience but assumes a basic background in algebraic geometry.
After a short introduction in Section 1, Section 2 reports on the history, Section 3 on various approaches, and Section 4 on the impact of the topic. Problem areas guiding the reader through the article are the interplay between algebraic and combinatorial geometry (as envisioned by J. Tits in [in: Centre Belge Rech. math., Colloque d’Algèbre supérieure, Bruxelles du 19 au 22 déc. 1956, 261–289 (1957; Zbl 0084.15902)]) and some of the efforts that went towards attempts for proving the Riemann hypothesis (in particular, trying to transfer the proof of the Hasse-Weil theorem from function fields to the case of \(\mathbb{Q}\)). Section 4 also discusses some relevant aspects of and recent developments in tropical geometry.
The article is written in a conversational, yet stringent and clearly presented tone, and thus makes for great recreational reading, as well as serving as a overview of and introduction to both classical and more recent developments in the field.
After a short introduction in Section 1, Section 2 reports on the history, Section 3 on various approaches, and Section 4 on the impact of the topic. Problem areas guiding the reader through the article are the interplay between algebraic and combinatorial geometry (as envisioned by J. Tits in [in: Centre Belge Rech. math., Colloque d’Algèbre supérieure, Bruxelles du 19 au 22 déc. 1956, 261–289 (1957; Zbl 0084.15902)]) and some of the efforts that went towards attempts for proving the Riemann hypothesis (in particular, trying to transfer the proof of the Hasse-Weil theorem from function fields to the case of \(\mathbb{Q}\)). Section 4 also discusses some relevant aspects of and recent developments in tropical geometry.
The article is written in a conversational, yet stringent and clearly presented tone, and thus makes for great recreational reading, as well as serving as a overview of and introduction to both classical and more recent developments in the field.
Reviewer: Fabian Müller (Berlin)
MSC:
| 14A23 | Geometry over the field with one element |
| 11G25 | Varieties over finite and local fields |
| 14A15 | Schemes and morphisms |
| 14G10 | Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) |
| 14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |
| 14T25 | Arithmetic aspects of tropical varieties |
Keywords:
field with one element; Riemann hypothesis; zeta function; monoid schemes; blueprints; blue schemes; tropical geometryCitations:
Zbl 0084.15902References:
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