nLab
field with one element

Overview

Various identities coming from algebraic geometry (and particularly algebraic groups) over finite fields ${𝔽}_q$ turn out to make perfect sense as expressions in $q$ when extrapolated to the case $q=1$, and to reflect interesting (combinatorial, representation theoretical…) facts, even though, of course, there is no actual field with a single element.

Motivated by such observations, Jacques Tits envisioned a new kind of geometry adapted to the explanation of these identities. Christophe Soulé then expanded on Tits’ ideas by introducing the notion of field with one element and studying its fine arithmetic invariants. While there is no field with a single element in the standard sense of field, the idea is that there is some other object, denoted ${𝔽}_1$, such that it makes sense to speak of “geometry over ${𝔽}_1$”. Following the French pronunciation one also writes $F_{\mathrm{un}}$ (and is thus led to the inevitable pun).

In the relative point of view the $S$-schemes are schemes with a morphism of schemes over a base scheme $S$; but every $S$-scheme is a scheme over $\mathrm{Spec}(\mathbb{Z})$. In absolute algebraic geometry all “generalized schemes” should live over $\mathrm{Spec}(F_1)$ and $\mathrm{Spec}(F_1)$ should live below $\mathrm{Spec}(\mathbb{Z})$; this is similar to the fact that the quotient stacks like $[*/G]$ live below the single point $*$ (there is a direct image functor from sheaves on a point to sheaves over $[*/G]$. One of the principal and very bold hopes is that the study of $F_{\mathrm{un}}$ should lead to a natural proof of Riemann conjecture; namely some attractive strategies to such a proof already exist (see MathOverflow here).

Contributors

After the very first observations by Tits, pioneers were Soulé and Kapranov and Smirnov. More recently there are extensive works by Alain Connes and Katia Consani, Nikolai Durov, James Borger and Oliver Lorscheid.

References and links

See also Lambda-ring, blue scheme and tropical geometry.

expositions

• Lieven Le Bruyn, Looking for $F_{\mathrm{un}}$, blog

A survey of the various competing theories is

• Javier López Peña, Oliver Lorscheid, Mapping $F_1$-land:An overview of geometries over the field with one element, arXiv/0909.0069

• John Baez This Week’s Finds 259 (html blog)

• Alain Connes, Fun with $F_1$, 5 min. video

• Lieven Le Bruyn, The field with one element, 3 lectures at the Seminar of Algebra and Geometry group at Antwerp, pdfs: 1, 2, 3

• Vladimir Berkovich, Algebraic geometry over $F_1$, video

more details

• Yuri Manin, Cyclotomy and analytic geometry over $F_1$, arxiv/0809.1564

• Alain Connes, Caterina Consani, Matilde Marcolli, Fun with $F_1$, arxiv/0806.2401

• Alain Connes, Caterina Consani, On the notion of geometry over $F_1$, arxiv/0809.2926; Schemes over $F_1$ and zeta functions, arxiv/0903.2024; Characteristic one, entropy and the absolute point, arxiv/0911.3537; From monoids to hyperstructures: in search of an absolute arithmetic, ; On the arithmetic of the BC-system, arxiv/1103.4672

• M. Marcolli, R. Thomgren, Thermodynamical semirings, arXiv: 1108.2874

• Bora Yalkinoglu, On Endomotives, Lambda-rings and Bost-Connes systems, With an appendix by Sergey Neshveyev, arxiv/1105.5022

An approach in terms of Lambda-rings is

• James Borger?, Lambda-rings and the field with one element, arXiv/0906.3146