On the field with one element

by C.Soulé

CNRS and IHES

1. Let G be a Chevalley group scheme over Z, and F_{q} the finite field with q elements. It has been noticed for some time that, when q tends to 1, the cardinality of the group of points of G in F_{q} behaves as follows:

card G(F_{q})\begin{matrix}∼ \\ q → 1 \end{matrix} (q − 1)^{r} × card W,

where r is the rank of G and W is its Weyl group. In 1957, Tits [1] proposed to think of W as the group of points of G in ”the field of characteristic one”:

(1)  G(F_{1}) := W .

He also argued that the finite geometries attached to each group G have a limit when q goes to 1, namely the finite geometry attached to the Coxeter group W.

In 1993, Manin [2] wrote some lectures on zeta functions where he mentions the field with one element. He proposes to develop algebraic geometry over F_{1}, and predicts that varieties over that field have simple zeta functions. For instance

(2)  ζ_{P_{F_{1}}^{N}} (s) = s(s − 1)(s − 2) ⋯ (s − N).

He also notes that the equation (1) for G = SL_{N} leads to the fact that the higher K-theory of F_{1} must be the homotopy groups of spheres. Indeed, by the Barratt-Priddy-Quillen theorem, we get

(3)  K_{m}(F_{1}) = π_{m}BGL(F_{1})^{+} := π_{m}BΣ_{∞}^{+} = π_{m}^{s}.

Later on Smirnov and Kapranov-Smirnov (unpublished preprints) studied the question further. Among other things, they developped linear algebra over F_{1} (a vector space being a pointed finite set), and obtained in this way a description of the Gauss reciprocity law similar to the description given earlier by Arbarello, De Concini and Kac of the Weil reciprocity law on curves by means of determinants of vector spaces.

2. The analogy between number fields and function fields finds a basic limitation with the lack of a ground field. One says that Spec(Z) (with a point at infinity added, as is familiar in Arakelov theory) is like a (complete) curve; but over which field? In particular, one would dream of having an object like

Spec(Z) ×_{Spec(F_{1})} Spec(Z),

since Weil’s proof of the Riemann hypothesis for a curve over a finite field makes use of the product of two copies of this curve. I have nothing to say about that question. The mere fact that the Riemann zeta function has infinitely many zeroes indicates that Spec(Z) cannot be viewed as an object of finite type over F_{1}, hence Spec(Z)×_{Spec(F_{1})}Spec(Z) cannot be an honest algebraic variety over Z.

On the other hand, if X is a variety of finite type over F_{1}, its base change X_{Z} = X ⊗_{F_{1}} Z to Z must be a variety of finite type over Z. So there is hope to describe such objects X using usual algebraic geometry, and we can ask:

Question: Which varieties over Z are obtained by base change from F_{1} to Z?

For instance, when X is smooth over F_{1}, X_{Z} will be smooth over Z, which is already a strong constraint.

3. Our starting point for a tentative definition of varieties over F_{1} is the short definition of schemes which says that a scheme is a covariant functor from rings to sets which is locally representable by a ring. In this language, the base change from a field k to a k-algebra Λ is described by the following easy lemma. Given a variety X over k (resp. a k-algebra A), let X_{Λ} = X ⊗_{k} Λ (resp. A_{Λ} = A ⊗_{k} Λ).

Lemma 1:

i) There is a canonical inclusion X(A) ⊂ X_{Λ}(A_{Λ}).

ii) Given any variety S over Λ and any natural transformation of functors (from k-algebras to sets)

ϕ : (A ↦ X(A)) ⟶ (A ↦ S(A_{Λ}))

there exists a unique algebraic morphism

ϕ̃: X_{Λ} ↦ S

inducing ϕ on each set X(A).

We would like to have a similar situation with k = F_{1} and Λ = Z. But nobody is telling us which are the Λ-algebras coming from k.

4. A definition:

4.1. For any integer n ≥ 1, a finite field F_{q} has a finite extension F_{q^{n}} of degree n, obtained by adjoining roots of unity. As Kapranov and Smirnov suggest, F_{1} should have an extension F_{1^{n}} of degree n, and we decide that

F_{1^{n}} ⊗_{F_{1}} Z = Z[T]/(T^{n} − 1).

In other words, F_{1^{n}} ⊗_{F_{1}} Z is the ring R_{n} of functions on the affine group scheme of n-th roots of unity.

Let R be the full subcategory of Rings with objects the rings R_{n}, n ≥ 1, and their finite tensor products. Let also R′ be the full subcategory of Rings with objects the tensor products of rings in R with the rings Z[1/N], N ≥ 1.

Definition: A variety over F_{1} is a covariant functor X from R to finite sets, equipped with natural inclusions X(R) ⊂ X_{Z}(R), R ∈ ObR, where X_{Z} is a variety over Z, and such that the following property (U) holds: for any variety S over Z and any ”good” transformation of functors (from R to sets)

ϕ : (R ↦ X(R)) ⟶ (R ↦ S(R))

there exists a unique algebraic map

ϕ̃: X_{Z} ↦ S

inducing ϕ on each set X(R), R ∈ ObR.

4.2. Here are three possibilities for what ”good’ could mean in the previous definition:

(G0) Let X(C) be the union of the subsets σ(X(R)) ⊂ X_{Z}(C), where R runs over all rings in R and σ over all ring morphisms from R to C. There is a continuous map ϕ_{C} from the topological closure of X(C) (in X_{Z}(C)) to S(C) such that, for all R and σ as above,

σ ∘ ϕ = ϕ_{C} ∘ σ .

(G1) The same as (G0), but now ϕ_{C} is holomorphic on the holomorphic hull of X(C) in X_{Z}(C).

(G2) The functor X extends to a functor from R′ to sets, and ϕ extends to a transformation of functors on R′. Then (G0) holds and, furthermore, the same statement is true when C (resp. R) gets replaced by any finite extension of a p-adic field (resp. by R′).

5. Examples:

5.0. When X_{Z} = Spec(Z) and X(R) = X_{Z}(R), we say that X = Spec(F_{1}).

5.1. When X_{Z}(R) = R^{∗} and X(R) is the set μ(R) of roots of unity in R, we say that X = G_{m}/F_{1}.

Lemma 2: The property (U) holds when S is affine and ϕ satisfies (G0).

Sketch of proof: We prove the existence of ϕ̃ when S is the affine line (looking at coordinates reduces the proof to that case). For any n ≥ 1 the map ϕ sends the element T ∈ μ(R_{n}) to a polynomial ∑_{i=0}^{n−1} a_{i}(n)T^{i} in R_{n}. On the other hand, the topological closure of X(C) is the circle S^{1}. The continuous map ϕ_{C} from the circle to C can be written as a Laurent series

ϕ_{C}(z) = ∑_{−∞}^{+∞} α_{i}z^{i} .

As n goes to infinity the coefficient

a_{i}(n) = ∑_{ζ^{n}=1} ϕ_{C}(ζ)ζ^{−i}/n

tends to

α_{i} = ∫_{S^{1}} ϕ_{C}(z)z^{−i}dθ .

Since all the a_{i}(n)’s are integers, the sequence a_{i}(n), n ≥ 1, must be stationary, hence ϕ_{C} is a Laurent polynomial ϕ̃ ∈ Z[T, T^{−1}] = Hom(G_{m}, A_{1}). q.e.d.

5.2. When, for all R ∈ R, X_{Z}(R) = R and X(R) is the set μ(R)∪{0}, we say that X = A_{1}/F_{1}. When R ∈ Ob(R′), if Σ = Hom(R, C) has cardinality N, we let X(R) be the set of elements x ∈ R such that

∑_{σ∈Σ} |σ(x)|^{2} ≤ N.

Property (U) holds if S is affine and ϕ satisfies (G1) or (G2).

5.3. More generally, if X_{Z} is a smooth toric variety over Z, we may write it as a disjoint union of products of copies of A_{1} with copies of G_{m}. By imposing that the corresponding coordinates lie in μ(R) ∪ {0}, we get a subset X(R) ⊂ X_{Z}(R), which satisfies (U) when S is affine and ϕ satisfies (G1) or (G2).

5.4. Let E ≃ Z^{N} be a lattice equipped with an hermitian scalar product h on E ⊗ C, and ||.|| the corresponding norm. It is customary in Arakelov theory to view the pair Ē = (E, h) as a bundle on the curve Spec(Z) ∪ {∞}, the global sections of which are the elements of E of norm less than one.

If we define X(R) as the set of elements x ∈ E ⊗ R such that

∑_{σ∈Σ} ||σ(x)||^{2} ≤ N,

one can again show that property (U) holds (for some variety X_{Z}) if S is affine and ϕ satisfies (G1) or (G2).

5.5. When X_{Z} = G is a Chevalley group as in §1, it seems natural to define X(R) as the set of elements g ∈ G(R) which are mapped to the standard maximal compact subgroup of G(C) by all σ ∈ Σ. The group G(F_{1}) will then be an extension of W by a finite abelian elementary 2-group. I have not checked if property (U) holds.

6. Zeta functions:

Let X be a variety over F_{1}. For any n ≥ 1, let q = 2n + 1, and

N(q) := card X(R_{n}).

In all cases considered in §5, N(q) happens to be a polynomial in q with integral coefficients (this is easy to show, but somewhat surprising in case 5.4). We may then define

Z(q, T) = exp(∑_{k≥1} N(q^{k})T^{k}/k).

Replacing T by q^{−s} and letting q go to 1, we get

Z(q, q^{−s}) \begin{matrix}∼ \\ q → 1 \end{matrix} (q − 1)^{χ}ζ_{X}(s),

where χ is the Euler-Poincaré characteristic of X_{Z}(C) and

ζ_{X}(s) = P (s)/Q(s),

where P and Q are polynomials with both integral coefficients and integral roots.

For instance, when X = P_{F_{1}}^{N} (defined as in 5.3, i.e. X(R) consists of points in P^{N}(R) having homogeneous coordinates in μ(R) ∪ {0}) the formula (2) holds.

7. Motives:

7.1. Let T be a split torus over C, M a T -toric variety, and r > 1 an integer. The endomorphism which maps t ∈ T to its r-th power extends to an endomorphism Φ_{r} of M. Totaro noticed in a recent paper [3] that the subspace of the rational cohomology of M where Φ_{r} acts by multiplication by r^{i} provides a canonical splitting of the weight filtration (of degree i).

In terms of mixed motives (still a conjectural notion), M gives rise to (several) extensions of Tate motives (because of the stratification mentioned in §5.3), and their classes e ∈ Ext(Z(i), Z(i+ n)) are killed by the greatest common divisor of r^{i+n} −r^{i}, r > 1, i.e. (essentially) the denominator of B_{n}/n, where B_{n} is the n-th Bernoulli number.

7.2. On the other hand, Beilinson proposed the formula

K_{2n−1}(Z) = Ext_{M/Z}(Z, Z(n))

describing the K-theory of Z as extensions of Tate motives over Spec(Z). Because of (3) one might then speculate that

π_{2n−1}^{s} = Ext_{M/F_{1}}(Z, Z(n)).

If the toric variety M is defined and smooth over Z, the example of §5.3 indicates that the extension classes e above should lie in the image of the morphism π_{2n−1}^{s} → K_{2n−1}(Z). It is a theorem of Quillen and Mitchell that this image is (up to 2-torsion) the cyclic group Im(J)_{2n−1}, the order of which is the denominator of B_{n}/n. This is quite consistent with Totaro’s remark.

It would be interesting to check if this bound on e is sharp, i.e. if all elements in Im(J)_{2n−1} can be interpreted in terms of toric varieties.

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[1] Tits, J.: Sur les analogues algébriques des groupes semi-simples complexes. Colloque d’algèbre supérieure, Bruxelles, 1956, pp. 261–289. Centre Belge de Recherches Mathématiques. Gauthier-Villars, (1957).

[2] Manin, Y.: Lectures on zeta functions and motives (according to Deninger and Kurokawa). Columbia University Number Theory Seminar (New York, 1992). Astérisque No. 228 (1995), 4, 121–163.

[3] Totaro, B. : Chow groups, Chow cohomology, and linear varieties, preprint (1998).

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