field with one element



Various phenomena in the context of algebraic geometry/arithmetic geometry (and particularly in the context of algebraic groups) over finite fields 𝔽 q\mathbb{F}_q turn out to make perfect sense as expressions in qq when extrapolated to the case q=1q=1, and to reflect interesting (combinatorial, representation theoretical…) facts, even though, of course, there is no actual field with a single element (since in a field by definition the elements 1 and 0 are distinct).

Motivated by such observations, Jacques Tits envisioned in (Tits 57) 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\mathbb{F}_1, such that it does make sense to speak of “geometry over 𝔽 1\mathbb{F}_1”. Following the French pronunciation one also writes F unF_{un} (and is thus led to the inevitable pun).

In the relative point of view the SS-schemes are schemes with a morphism of schemes over a base scheme SS; but every SS-scheme is a scheme over Spec(Z). In absolute algebraic geometry all “generalized schemes” should live over Spec(F 1)Spec(F_1) and Spec(F 1)Spec(F_1) should live below Spec()Spec(\mathbb{Z}); this is similar to the fact that the quotient stacks like [*/G][*/G] live below the single point ** (there is a direct image functor from sheaves on a point to sheaves over [*/G][*/G]). One of the principal and very bold hopes is that the study of F unF_{un} should lead to a natural proof of Riemann conjecture (see also MathOverflow here). It was originally suggested by (Manin 95) that the Riemann hypothesis might be solved by finding an 𝔽 1\mathbb{F}_1-analogue of André Weil’s proof for the case of arithmetic curves over the finite fields 𝔽 q\mathbb{F}_q.

A first proposal for what a variety “over 𝔽 1\mathbb{F}_1” ought to be is due to (Soulé 04). After that a plethora of further proposals appeared,including (Connes-Consani 08).

Borger’s absolute geometry

Maybe an emerging consensus is that the preferred approach is Borger's absolute geometry (Borger 09). Here the structure of a Lambda-ring on a ring RR, hence on Spec(R)Spec()Spec(R) \to Spec(\mathbb{Z}), is interpreted as a collection of lifts of all Frobenius morphisms and hence as descent data for descent to Spec(𝔽 1)Spec(\mathbb{F}_1) (which is defined thereby). This definition yields an essential geometric morphism of gros etale toposes

Et(Spec())Et(Spec(𝔽 1)), Et(Spec(\mathbb{Z})) \stackrel{\overset{}{\longrightarrow}}{\stackrel{\overset{}{\longleftarrow}}{\underset{}{\longrightarrow}}} Et(Spec(\mathbb{F}_1)) \,,

where on the right the notation is just suggestive, the topos is a suitable one over Lambda-rings. Here the middle inverse image is the forgetful functor which forgets the Lambda structure, and its right adjoint direct image is given by the arithmetic jet space construction (via the ring of Witt vectors construction).

This proposal seems to subsume many aspects of other existing proposals (see e.g. Le Bruyn 13) and stands out as yielding an “absolute base toposEt(Spec(𝔽 1))Et(Spec(\mathbb{F}_1)) which is rich and genuinely interesting in its own right.

Function field analogy

function field analogy

number fields (“function fields of curves over F1”)function fields of curves over finite fields 𝔽 q\mathbb{F}_q (arithmetic curves)Riemann surfaces/complex curves
affine and projective line
\mathbb{Z} (integers)𝔽 q[t]\mathbb{F}_q[t] (polynomials, function algebra on affine line 𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q})𝒪 \mathcal{O}_{\mathbb{C}} (holomorphic functions on complex plane)
\mathbb{Q} (rational numbers)𝔽 q(t)\mathbb{F}_q(t) (rational functions)meromorphic functions on complex plane
pp (prime number/non-archimedean place)x𝔽 px \in \mathbb{F}_pxx \in \mathbb{C}
\infty (place at infinity)\infty
Spec()Spec(\mathbb{Z}) (Spec(Z))𝔸 𝔽 q 1\mathbb{A}^1_{\mathbb{F}_q} (affine line)complex plane
Spec()place Spec(\mathbb{Z}) \cup place_{\infty} 𝔽 q\mathbb{P}_{\mathbb{F}_q} (projective line)Riemann sphere
genus of the rational numbers = 0genus of the Riemann sphere = 0
formal neighbourhoods
p\mathbb{Z}_p (p-adic integers)𝔽 q[[tx]]\mathbb{F}_q[ [ t -x ] ] (power series around xx)[[tx]]\mathbb{C}[ [t-x] ] (holomorphic functions on formal disk around xx)
Spf( p)×Spec()XSpf(\mathbb{Z}_p)\underset{Spec(\mathbb{Z})}{\times} X (“pp-arithmetic jet space” of XX at pp)formal disks in XX
p\mathbb{Q}_p (p-adic numbers)𝔽 q((tx))\mathbb{F}_q((t-x)) (Laurent series around xx)((tx))\mathbb{C}((t-x)) (holomorphic functions on punctured formal disk around xx)
𝔸 = pplace p\mathbb{A}_{\mathbb{Q}} = \underset{p\; place}{\prod^\prime}\mathbb{Q}_p (ring of adeles)𝔸 𝔽 q((t))\mathbb{A}_{\mathbb{F}_q((t))} ( adeles of function field ) x((tx))\underset{x \in \mathbb{C}}{\prod^\prime} \mathbb{C}((t-x)) (restricted product of holomorphic functions on all punctured formal disks, finitely of which do not extend to the unpunctured disks)
𝕀 =GL 1(𝔸 )\mathbb{I}_{\mathbb{Q}} = GL_1(\mathbb{A}_{\mathbb{Q}}) (group of ideles)𝕀 𝔽 q((t))\mathbb{I}_{\mathbb{F}_q((t))} ( ideles of function field ) xGL 1(((tx)))\underset{x \in \mathbb{C}}{\prod^\prime} GL_1(\mathbb{C}((t-x)))
zeta functions
Riemann zeta functionGoss zeta function
branched covering curves
KK a number field (K\mathbb{Q} \hookrightarrow K a possibly ramified finite dimensional field extension)KK a function field of an algebraic curve Σ\Sigma over 𝔽 p\mathbb{F}_pK ΣK_\Sigma (sheaf of rational functions on complex curve Σ\Sigma)
𝒪 K\mathcal{O}_K (ring of integers)𝒪 Σ\mathcal{O}_{\Sigma} (structure sheaf)
Spec an(𝒪 K)Spec()Spec_{an}(\mathcal{O}_K) \to Spec(\mathbb{Z}) (spectrum with archimedean places)Σ\Sigma (arithmetic curve)ΣP 1\Sigma \to \mathbb{C}P^1 (complex curve being branched cover of Riemann sphere)
genus of a number fieldgenus of an algebraic curvegenus of a surface
formal neighbourhoods
vv prime ideal in ring of integers 𝒪 K\mathcal{O}_KxΣx \in \SigmaxΣx \in \Sigma
K vK_v (formal completion at vv)((t x))\mathbb{C}((t_x)) (function algebra on punctured formal disk around xx)
𝒪 K v\mathcal{O}_{K_v} (ring of integers of formal completion)[[t x]]\mathbb{C}[ [ t_x ] ] (function algebra on formal disk around xx)
𝔸 K\mathbb{A}_K (ring of adeles) xΣ ((t x))\prod^\prime_{x\in \Sigma} \mathbb{C}((t_x)) (restricted product of function rings on all punctured formal disks around all points in Σ\Sigma)
𝒪\mathcal{O} xΣ[[t x]]\prod_{x\in \Sigma} \mathbb{C}[ [t_x] ] (function ring on all formal disks around all points in Σ\Sigma)
𝕀 K=GL 1(𝔸 K)\mathbb{I}_K = GL_1(\mathbb{A}_K) (group of ideles) xΣ GL 1(((t x)))\prod^\prime_{x\in \Sigma} GL_1(\mathbb{C}((t_x)))
Galois and class field theory
Galois groupπ 1(Σ)\pi_1(\Sigma) fundamental group
Galois representationflat connection (“local system”) on Σ\Sigma
automorphy and bundles
GL 1(K)\GL 1(𝔸 K)GL_1(K)\backslash GL_1(\mathbb{A}_K) (idele class group)
GL 1(K)\GL 1(𝔸 K)/GL 1(𝕆)GL_1(K)\backslash GL_1(\mathbb{A}_K)/GL_1(\mathbb{O})Cech cocycles mod coboundaries for holomorphic line bundles with respect to cover by formal disks = moduli stack of line bundles Bun GL 1(Σ)Bun_{GL_1}(\Sigma)
GL n(K)\GL n(𝔸 K)//GL n(𝒪)GL_n(K) \backslash GL_n(\mathbb{A}_K)//GL_n(\mathcal{O}) (constant sheaves on this stack form unramified automorphic representations)Bun GL n()(Σ)Bun_{GL_n(\mathbb{C})}(\Sigma) (moduli stack of bundles on the curve Σ\Sigma)
number field Langlands correspondencefunction field Langlands correspondencegeometric Langlands correspondence
Tamawa-Weil for number fieldsTamagawa-Weil for function fields
zeta functions
Dedekind zeta functionWeil zeta functionzeta function of a Riemann surface


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

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


A survey of the various competing theories is

more details

The approach in terms of Lambda-rings due to

with details in

More discussion relating to this includes

Revised on July 25, 2014 03:38:05 by Urs Schreiber (