Absolute Derivations and Zeta Functions

Dedicated to Professor Kazuya Kato’s 50th birthday

Nobushige Kurokawa, Hiroyuki Ochiai, and Masato Wakayama

Received: September 24, 2002 Revised: December 17, 2002

Abstract. Just as the function ring case we expect the existence of the coefficient field for the integer ring. Using the notion of one element field in place of such a coefficient field, we calculate absolute derivations of arithmetic rings. Notable examples are the matrix rings over the integer ring, where we obtain some absolute rigidity. Knitting up prime numbers via absolute derivations we speculate the arithmetic landscape. Our result is only a trial to a proper foundation of arithmetic.

2000 Mathematics Subject Classification: 11R27, 11R42, 14G10 Keywords and Phrases: absolute derivations, absolute mathematics, absolute schemes

Kronecker and many excellent arithmeticians attempted to study the arithmetic geometry by looking at the intimate analogy between Z and F_{p}[T]. Although these two objects are similar in some respects, there exists a quite clear difference: the non-existence (or “invisibility”) of the constant (coefficient) field of Z. Zeta functions suggest to compare

ζ̂_{Z}(s) = \frac{det(R − (s − \frac{1}{2}))}{s(s − 1)}

and

ζ̂_{F_{p}[T]}(s) = \frac{1}{(1 − p^{−s})(1 − p^{−(s−1)})},

where ζ̂ denotes the “completed zeta function”; in the latter case we know good cohomologies with dim H^{0}(F_{p}[T]) = 1, dim H^{1}(F_{p}[T]) = 0, dim H^{2}(F_{p}[T]) = 1 and H^{i}(F_{p}[T]) = 0 for i > 2. Up to now, we have not come across a cohomology theory such as dim H^{0}(Z) = 1, dim H^{1}(Z) = ∞ with a skew-hermitian operator R : H^{1}(Z) → H^{1}(Z), dim H^{2}(Z) = 1 and H^{i}(Z) = 0 for i > 2. Yet we can try to figure out the nature of the “constant field” F_{1} of Z (Manin [9], Deninger [1], Kurokawa [6]). As a first little step we calculate F_{1}- derivations (in other words, “absolute derivations”) of Z and allied objects here.

The authors thank Professor Kazuya Kato for his patient listening to our primitive tales in old days. [One of our friends indicates the appearance of KAZUYA by looking at the leading alphabets of sentences in the introduction: the readers are welcome to find such an accidental coincidence.]

1 Absolute derivations

We define an absolute derivation of a ring R as a map D : R → R satisfying the condition (Leibniz rule)

D(ab) = D(a)b + aD(b) for all a, b ∈ R.

If an absolute derivation D satisfies moreover the additivity property

D(a + b) = D(a) + D(b),

it is called a derivation of R. Here the word “absolute” indicates objects over “the one element field F_{1}”. Elements of the absolute mathematics are briefly described in § 2 below. We denote by Der_{F_{1}}(R) the set of all absolute derivations of R, and by Der_{Z}(R) the set of all derivations of R.

1.1

We first determine the absolute derivations of the most simple but the fundamental case R = Z. For each prime p, define a map \frac{∂}{∂p} : Z → Z by

\frac{∂}{∂p}(x) = \frac{x}{p} · ord_{p}(x).

Here ord_{p}(x) denotes the p-order of x ∈ Z, that is, the integer ℓ such that x is divisible by p^{ℓ} but is not divisible by p^{ℓ+1}. Namely we have

\frac{∂}{∂p}(x) = \begin{cases}0 & if p ∤ x \\ ℓp^{ℓ−1} · m & if x = p^{ℓ} · m (ℓ ≥ 1, p ∤ m) \end{cases}

and put \frac{∂}{∂p}(0) = 0. It is easy to see that \frac{∂}{∂p} satisfies the Leibniz rule;

\frac{∂}{∂p}(xy) = \frac{∂}{∂p}(x)y + x\frac{∂}{∂p}(y),

whence \frac{∂}{∂p} ∈ Der_{F_{1}}(Z). In fact, since ord_{p}(xy) = ord_{p}(x) + ord_{p}(y), we see that

\frac{∂}{∂p}(xy) = \frac{xy}{p} · ord_{p}(xy)
= \frac{xy}{p} (ord_{p}(x) + ord_{p}(y))
= (\frac{x}{p} · ord_{p}(x)) y + x (\frac{y}{p} · ord_{p}(y))
= \frac{∂}{∂p}(x)y + x\frac{∂}{∂p}(y).

The following theorem shows these \frac{∂}{∂p}’s (p: prime numbers) span the set of the absolute derivations of Z.

Theorem 1 We have the following direct product decomposition:

Der_{F_{1}}(Z) = ; ⨁_{p:prime}Z\frac{∂}{∂p} := \{ ∑_{p} c_{p} \frac{∂}{∂p} ; c_{p} ∈ Z\} ⊂ End_{F_{1}}(Z),

where End_{F_{1}}(Z) = Map(Z, Z).

Proof: Note first that the infinite sum ∑_{p} c_{p} \frac{∂}{∂p}(x) ∈ ˆ⨁pZ\frac{∂}{∂p} is well-defined since for each x ∈ Z, \frac{∂}{∂p}(x) = 0 except the finite number of p. It is also easy to see that such an expression is unique. The fact that the sum of absolute derivations is also an absolute derivation, shows clearly that

Der_{F_{1}}(Z) ⊃ ˆ ⨁_{p}Z\frac{∂}{∂p}.

It is therefore enough to prove that any D ∈ Der_{F_{1}}(Z) can be written as

D = ∑_{p} D(p)\frac{∂}{∂p}.

In order to see this we show that the absolute derivation D is completely determined by its values D(p) on prime numbers p = 2, 3, 5, . . .. By successive use of the Leibniz rule it is obvious to see that

D(p^{ℓ}) = ℓp^{ℓ−1}D(p).

Remark also that D(0) = D(1) = D(−1) = 0. Actually, the Leibniz rule shows

D(0) = D(0 · 0) = D(0) · 0 + 0 · D(0) = 0,
D(1) = D(1 · 1) = D(1) · 1 + 1 · D(1) = 2D(1),

whence it follows that D(1) = 0. Further,

0 = D(1) = D((−1) · (−1)) = D(−1)(−1) + (−1)D(−1) = −2D(−1).

Since any non-zero x ∈ Z can be written as x = ±p_{1}^{i_{1}}p_{2}^{i_{2}} ⋯ p_{ℓ}^{i_{ℓ}} by primes p_{j}, using the Leibniz rule again the assertion follows, that is, D is completely determined once the values D(p) are given. To confirm the relation D = ∑_{p} D(p)\frac{∂}{∂p} holds it suffices to show D(q) = ( ∑_{p} D(p)\frac{∂}{∂p})(q) for each prime q. Since

\frac{∂}{∂p}(q) = \begin{cases}1 & (p = q) \\ 0 & (p ≠ q), \end{cases}

it is actually immediate to have

( ∑_{p} D(p)\frac{∂}{∂p})(q) = ∑_{p} D(p)\frac{∂}{∂p}(q) = D(q).

This completes the proof.

Remark 1 It is easy to see that Der_{Z}(Z) = 0. In fact, since D(0) = D(1) = D(−1) = 0 as above, the additivity asserts D(m) = 0 for m ∈ Z.

Remark 2 We have for primes p, q

[\frac{∂}{∂p}, q] = δ_{pq},

where q in the left-hand side is regarded as a multiplication operator, since

[\frac{∂}{∂p}, q](x) = \frac{∂}{∂p}(qx) − q \frac{∂}{∂p}(x)
= (\frac{∂}{∂p}(q)x + q \frac{∂}{∂p}(x)) − q \frac{∂}{∂p}(x) = \frac{∂}{∂p}(q)x = δ_{pq} · x.

1.2

We have the similar statement for Z[i] and F_{p}[T]. Let Z[i] = \{m + ni; m, n ∈ Z\} (i = \sqrt{−1}) be the ring of Gaussian integers. Let {π} be a complete set of representatives of the prime elements in Z[i]. Define the map \frac{∂}{∂π} ∈ End_{F_{1}}(Z[i]) by

\frac{∂}{∂π} (x) = \frac{x}{π} · ord_{π}(x).

Then similar to the theorem above we obtain the following:

Theorem 2 We have

Der_{F_{1}}(Z[i]) = ˆ ⨁_{π}Z[i]\frac{∂}{∂π} ⊂ End_{F_{1}}(Z[i]).

Moreover, for prime elements π, π′ we have

[\frac{∂}{∂π} , π′] = δ_{ππ′}.

Now we consider the case R = F_{p}[T], the polynomial ring over the finite field F_{p}. Any f ∈ F_{p}[T] can be factorized uniquely as

f = c · h_{1}^{e_{1}} · h_{2}^{e_{2}} ⋯ h_{ℓ}^{e_{ℓ}},

where c ∈ F_{p}, e_{i} ∈ Z_{≥0} and h_{1}, h_{2}, …, h_{ℓ} are monic irreducible polynomials, that is, the prime elements in F_{p}[T]. We define the order ord_{h}(f) in an obvious way; ord_{h_{i}}(f) = e_{i}. Quite similarly as Theorem 1 we have the following theorem.

Theorem 3 For a monic irreducible polynomial h ∈ F_{p}[T], define a map \frac{∂}{∂h} : F_{p}[T] → F_{p}[T] by

\frac{∂}{∂h}f(T) = \frac{f(T)}{h(T)} · ord_{h}(f).

Then we have

Der_{F_{1}}(F_{p}[T]) = ⨁_{h:monic irred.}F_{p}[T]\frac{∂}{∂h}.

Remark 3 We notice that \frac{∂}{∂h} is different from the usual derivation. For example, \frac{∂}{∂T} (T^{2}) = 2T and \frac{∂}{∂T} (T^{2} + 1) = 0 here.

1.3

For some general unique factorization domains, we have the following result:

Theorem 4 Let R be a commutative unique factorization domain whose unit group R^{×} is a finitely generated abelian group. Fix a set of representative P_{0}(R) of irreducible elements of R \ (R^{×} ∪ {0}) modulo R^{×}, and a set of generators P_{1}(R) of R^{×} modulo R_{tor}^{×}, where R_{tor}^{×} is the subgroup of torsion elements. Put P (R) = P_{0}(R) ∪ P_{1}(R). Each element a ∈ R \ {0} can be uniquely written as

a = u ∏_{π∈P (R)} π^{m(π)},

where m(π) ∈ Z_{≥0} if π ∈ P_{0}(R), m(π) ∈ Z if π ∈ P_{1}(R), and u ∈ R_{tor}^{×}. We define

\frac{∂}{∂π} (a) = m(π)\frac{a}{π}

and

\frac{∂}{∂π} (0) = 0.

Then

Der_{F_{1}}(R) = ; ⨁_{π∈P (R)}R\frac{∂}{∂π} .

Proof. Each a ∈ R \ {0} is uniquely written as

a = u′ ∏_{π∈P_{0}(R)} π^{m(π)}

with u′ ∈ R^{×}. Since R^{×}/R_{tor}^{×} is a free abelian group, we can write

u′ = u ∏_{π∈P_{1}(R)} π^{m(π)}

with u ∈ R_{tor}^{×} uniquely, where P_{1}(R) is a finite set. It is clear that \frac{∂}{∂π} ∈ Der_{F_{1}}(R) by definition. Now, take any X ∈ Der_{F_{1}}(R). We show

X = ∑_{π∈P (R)} X(π)\frac{∂}{∂π} .

Take an a ∈ R \ {0}. Express it as

a = u ∏_{π∈P (R)} π^{m(π)}

with u ∈ R_{tor}^{×}. Then, using X(u) = 0 we have

X(a) = ∑_{π∈P (R)} m(π)\frac{a}{π} X(π) = ∑_{π∈P (R)} X(π)\frac{∂}{∂π} (a).

Hence,

X = ∑_{π∈P (R)} X(π)\frac{∂}{∂π} .

Example 1 (1) If R = Z[\sqrt{2}], then P_{1}(R) = \{1 + \sqrt{2}\}.

(2) If R = Z[T_{1}, …, T_{n}], then P_{1}(R) = ∅.

1.4

We note on a special subset of Der_{F_{1}}(R) for R = Z and Z[i].

Theorem 5 Let p be a prime. Then 𝔤_{p} = Z\frac{∂}{∂p} is closed under the Lie bracket defined by the commutator [·, ·] of End_{F_{1}}(Z). Similarly, for a prime element π in Z[i], the subset 𝔤_{π} = Z[i]\frac{∂}{∂π} of Der_{F_{1}}(Z[i]) is closed under the commutator of End_{F_{1}}(Z[i]).

Proof: The first assertion is easily confirmed by using the relations

[p^{ℓ} \frac{∂}{∂p}, p^{m} \frac{∂}{∂p}] = (m − ℓ)p^{ℓ+m−1} \frac{∂}{∂p}.

The assertion for Z[i] can be proved similarly.

Remark 4 Define H = p∂, E = −p^{2}∂, F = ∂ with ∂ = \frac{∂}{∂p}. Then the formula above implies the following commutation relations

[H, E] = E, [H, F] = −F, [E, F] = 2H.

These commutation relations coincide with those of standard generators {H, E, F } of the simple Lie algebra 𝔰𝔩_{2}. Hence, the operator C := 2H^{2} + EF + F E can be considered as an “absolute” Casimir operator (cf. [7]). This element C commutes with each H, E, F, and moreover it is not hard to see that C vanishes under the map 𝔤_{p} → End_{F_{1}}(Z). It would be interesting to study the

“absolute Virasoro algebra” extending 𝔤_{p} with its physical implications.

1.5

Let R be a (non-commutative) ring. For an element a ∈ R, we define an inner derivation D_{a} by D_{a}(b) = ab − ba. It is easy to see that D_{a} ∈ Der_{Z}(R). We denote the set of all inner derivations by InnDer_{Z}(R). We have

InnDer_{Z}(R) ⊂ Der_{Z}(R) ⊂ Der_{F_{1}}(R).

The main result of this subsection is the following:

Theorem 6 Der_{F_{1}}(M_{2}(Z)) = InnDer_{Z}(M_{2}(Z)).

Let D be an absolute derivation of M_{2}(Z).

Lemma 1 D(0) = D(I_{2}) = D(−I_{2}) = 0.

Proof: Actually, D(0) = D(00) = 0, D(I_{2}) = D(I_{2}I_{2}) = 2D(I_{2}), 0 = D((−I_{2})(−I_{2})) = −2I_{2}D(−I_{2}).

Let N = ( \begin{matrix}0 & 1 \\ 0 & 0 \end{matrix} ), H = ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ), N′ = ( \begin{matrix}0 & 0 \\ 1 & 0 \end{matrix} ). These elements

satisfy the following relations HN = N, NH = −N, H^{2} = I_{2}, HN′ = −N′, N′H = N′, NN′ = E_{11}, N′N = E_{22}.

Lemma 2 (i) There exist a, b ∈ Z such that D(N) = aH + bN.

(ii) There exist a′, b′ ∈ Z such that D(N′) = a′H + b′N′.

Proof: 0 = D(0) = D(N^{2}) = D(N)N + ND(N). Then (i) follows easily. The argument for N′ is the same.

Lemma 3 D(H) = −2a′N − 2aN′.

Proof: D(N) = D(HN) = HD(N) + D(H)N. Then D(H)N = a(H − I_{2}). Also,

−D(N) = D(−N) = D(NH) = D(N)H + ND(H).

Then ND(H) = −a(H + I_{2}). By these two conditions, the diagonal entries of D(H) are both zero, and the (2, 1)-entry of D(H) is −2a. By D(HN′) = D(−N′), we have D(H)N′ = −a′(I +H). Then the (1, 2)-entry of D(H) is −2a′.

Lemma 4 There exists q ∈ Z such that for all c ∈ Z

\begin{aligned}D(H + cN) & = & D(H) + c(aH + qN), \\ D(I_{2} + cN) & = & c(aH + qN). \end{aligned}

Proof: Let A_{c} := H + cN for c ∈ Z. Since

D(N) = D(A_{c}N) = A_{c}D(N) + D(A_{c})N

and (i) of Lemma 2, we have D(A_{c})N = acN + a(H − I_{2}). Also,

−D(N) = D(−N) = D(NA_{c}) = D(N)A_{c} + ND(A_{c}).

Then ND(A_{c}) = −a(H +I_{2})−acN. By these two conditions, we have D(A_{c}) = q_{c}N + acH − 2aN′. By setting c = 0, we have D(H) = q_{0}N − 2aN′. Then q_{0} = −2a′. Notice

D(I_{2} + (c′ − c)N) = D(A_{c}A_{c′})
= A_{c}D(A_{c′}) + D(A_{c})A_{c′}
= (c′ − c)aH + (q_{c′} − q_{c})N.

Then q_{c′+c} − q_{c} = q_{c′} − q_{0}. This means that Z ∋ c ↦ q_{c} − q_{0} ∈ Z is an additive map. We set q = q_{1} − q_{0}, then q_{c} = cq + q_{0}. This proves the lemma.

Lemma 5 We have

\begin{aligned}D(H + cN′) & = & D(H) + c(a′H − qN′), \\ D(I_{2} + cN′) & = & c(a′H − qN′). \end{aligned}

Proof: By a similar argument, there exists q′ ∈ Z, independent of c, such that

\begin{aligned}D(H + cN′) & = & D(H) + c(a′H + q′N′), \\ D(I_{2} + cN′) & = & c(a′H + q′N′) \end{aligned}

for all c ∈ Z. In fact, let A′_{c} := H + cN′ for c ∈ Z. Since

D(−N′) = D(A′_{c}N′) = A′_{c}D(N′) + D(A′_{c})N′

and (ii) of Lemma 2, we have D(A′_{c})N′ = −a′cN′ − a′(H + I_{2}). Also,

D(N′) = D(N′) = D(N′A′_{c}) = D(N′)A′_{c} + N′D(A′_{c}).

Then N′D(A′_{c}) = a′(H−I_{2})+a′cN′. By these two conditions, we have D(A′_{c}) = q′_{c}N′ + a′cH − 2a′N. By setting c = 0, we have D(H) = q′_{0}N′ − 2a′N. Then q′_{0} = −2a. The remaining is similar to the previous lemma. Now we put B = (I_{2} + N)(I_{2} − N′). Then, det(B) = 1 and tr(B) = 1 show that B^{3} = −I_{2}. Hence

0 = D(B^{3}) = D(B)B^{2} + BD(B)B + B^{2}D(B).

By multiplying −B from the left, we have

BD(B)B^{−1} + B^{−1}D(B)B + D(B) = 0.

Taking the trace, 3tr(D(B)) = 0. On the other hand, calculate as

D(B) = D(I_{2} + N)(I_{2} − N′) + (I_{2} + N)D(I_{2} − N′)
= (aH + qN)(I − N′) − (I + N)(a′H + q′N′),

then tr(D(B)) = −q′ − q. Thus q′ = −q.

Proof of Theorem 6: We use the notation above, especially, a, a′, q ∈ Z. Let Y = a′N − aN′ + qE_{11} ∈ M_{2}(Z). We note that

[Y, H] = −2a′N − 2aN′,
[Y, N] = aH + qN,
[Y, N′] = a′H − qN′.

We consider the corresponding inner derivation D_{Y} ∈ InnDer_{Z}(M_{2}(Z)). Then D(H) = D_{Y} (H), D(I_{2} + cN) = D_{Y} (I_{2} + cN) and D(I_{2} + cN′) = D_{Y} (I_{2} + cN′) for all c ∈ Z. Recall the fact that the group GL(2, Z) is generated by {H, I_{2} + cN, I_{2} +cN′ | c ∈ Z}. Then we know that D(A) = D_{Y} (A) for all A ∈ GL(2, Z). We put Z := D−D_{Y} ∈ Der_{F_{1}}(M_{2}(Z)). Then we have proved that Z(A) = 0 for all A ∈ GL(2, Z), Z(N) = bN for some b ∈ Z. We set K = N +N′ ∈ GL(2, Z). Then NKN = N implies that

Z(N)KN + NKZ(N) = Z(N).

This shows bN = 0 and so b = 0. Now we take a non-zero integer λ ∈ Z. Consider a matrix A ∈ GL(2, Z) with (2, 1)-entry of A is divisible by λ. We define A′ ∈ GL(2, Z) by

(E_{11} + λE_{22})A = A′(E_{11} + λE_{22}).

Then

Z(E_{11} + λE_{22})A = A′Z(E_{11} + λE_{22}),
(E_{11} + λE_{22})^{−1}Z(E_{11} + λE_{22})A = A(E_{11} + λE_{22})^{−1}Z(E_{11} + λE_{22})

for all A as above. Such an A is so many, this equality holds for all A ∈ M_{2}(Z). This means that (E_{11} + λE_{22})^{−1}Z(E_{11} + λE_{22}) is a scalar matrix since it commutes with all the right multiplication. Therefore, there exists τ(λ) ∈ Z such that

Z(E_{11} + λE_{22}) = τ(λ)(E_{11} + λE_{22}).

On the other hand, we have (E_{11} + λE_{22})N = N. Since Z(N) = 0, we have Z(E_{11} + λE_{22})N = 0. This shows that τ(λ) = 0. We have proved Z(E_{11} + λE_{22}) = 0 for all non-zero λ ∈ Z. By considering K(E_{11} + λE_{22})K, we know that Z(λE_{11} + E_{22}) = 0. This proves that Z(A) = 0 for all A ∈ M_{2}(Z) with det(A) ≠ 0. For any matrix C in M_{2}(Z) of rank one, there exist A, A′ ∈ M_{2}(Z) with det A ≠ 0, det A′ ≠ 0 such that C = ANA′. This shows that Z(C) = 0. Finally we have Z(0) = 0.

This result would suggest a kind of rigidity or semi-simplicity of M_{2}(Z) as an absolute algebra, but it is not quite sure, at this moment, that such a notion can be formulated rigorously. We also remark that some argument can be extended to other non-commutative algebras. For example, the following result is proved in [10]: Let R ∋ 1 be a ring contained in the algebraic closure Q̄. Then for each n ≥ 2,

Der_{F_{1}}(M_{n}(R)) = InnDer_{Z}(M_{n}(R)) = {D_{a} | a ∈ M_{n}(R)}.

1.6 Absolute Hochschild cohomology

Theorem 6 can be stated as

H_{F_{1}}^{1}(M_{2}(Z), M_{2}(Z)) = 0,

where the left-hand side indicates the absolute Hochschild cohomology in the following sense. Let R be a ring and M be an R-bimodule. Let C_{F_{1}}^{0} = M, C_{F_{1}}^{1} = Map(R, M), C_{F_{1}}^{2} = Map(R × R, M), C_{F_{1}}^{n} = Map(R^{n}, M). We define a derivation δ^{n} : C_{F_{1}}^{n} → C_{F_{1}}^{n+1} by

(δ^{0}m)(a) = am − ma,
(δ^{1}f)(a_{1}, a_{2}) = a_{1}f(a_{2}) − f(a_{1}a_{2}) + f(a_{1})a_{2},
(δ^{n}f)(a_{1}, …, a_{n+1}) = a_{1}f(a_{2}, …, a_{n+1}) + ∑_{i=1}^{n}(−1)^{i}f(a_{1}, …, a_{i}a_{i+1}, …, a_{n+1})
+(−1)^{n+1}f(a_{1}, …, a_{n})a_{n+1}.

Then (δ^{∗}, C_{F_{1}}^{∗}) is a chain complex of abelian groups. We call the cohomology groups H_{F_{1}}^{n}(R, M) = Kerδ^{n}/Imδ^{n−1} of this chain complex by the absolute Hochschild cohomology (cf. [8]). In particular,

H_{F_{1}}^{0}(R, R) = Kerδ^{0} = {b ∈ R | ba = ab for all a ∈ R},

the center of R, and

H_{F_{1}}^{1}(R, R) = Der_{F_{1}}(R)/InnDer_{Z}(R).

For example, Theorem 1 says

H_{F_{1}}^{1}(Z, Z) = ; ⨁_{p:prime}Z\frac{∂}{∂p}.

2 Absolute mathematics

We explain the background material of absolute mathematics, i.e., the mathematics over F_{1}. As noted in Manin [9] the first appearance of F_{1} seems to be in GL_{n}(F_{1}) = S_{n} where S_{n} is the symmetric group of order n. This might be a folklore, but a much precise reference was supplied by Soulé [11] citing a paper [12] by Tits. There it seems that Tits conjectured G(F_{1}) = W (G) for each algebraic group G, where W (G) is the Weyl group; in the case G = GL_{n} we get GL_{n}(F_{1}) = W (GL_{n}) = S_{n} again. We consider that GL_{n}(F_{1}) = S_{n} suggests to identify the category Mod(F_{1}) of F_{1}-modules with the category Set of sets. Let denote the free F_{1}-module over a set X by F_{1}^{(X)}. Then the more precise expectation is as follows: for objects X and Y of Set, the corresponding objects of Mod(F_{1}) are F_{1}^{(X)} and F_{1}^{(Y)} respectively with the corresponding morphisms

Hom_{Set}(X, Y) ≅ Hom_{Mod(F_{1})}(F_{1}^{(X)}, F_{1}^{(Y)}).

Hence, especially for X = {1, 2, …, n} it would hold that

M_{n}(F_{1}) = End_{Mod(F_{1})}(F_{1}^{n}) = End_{Set}({1, 2, …, n})

and GL_{n}(F_{1}) = Aut_{Mod(F_{1})}(F_{1}^{n}) = Aut_{Set}({1, 2, …, n}) = S_{n}, which give #M_{n}(F_{1}) = n^{n} and #GL_{n}(F_{1}) = n!. For example

M_{2}(F_{1}) = \{( \begin{matrix}1 & 1 \\ 0 & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , ( \begin{matrix}0 & 0 \\ 1 & 1 \end{matrix} )\}

and

GL_{2}(F_{1}) = \{( \begin{matrix}1 & 0 \\ 0 & 1 \end{matrix} ) , ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} )\} = S_{2}.

(Here we may omit 0’s respecting F_{1} = {1}.) Furthermore we identify the category Alg(F_{1}) of F_{1}-algebras with the category Monoid of monoids according to the following picture:

( \begin{matrix}& Alg(Z) \\ ↙ & & ↘ \\ Mod(Z) & & Alg(F_{1}) \\ ↘ & & ↙ \\ & Mod(F_{1}) \end{matrix} ) = ( \begin{matrix}& & Ring \\ & ↙ & & ↘ \\ Ab & & & Monoid \\ & ↘ & & ↙ \\ & & Set \end{matrix} )

under the forgetful functors

(R, ×, +)
↙  ↘
(R, +)  (R, ×)
↘  ↙
R.

Then, the absolute derivations Der_{F_{1}}(R) of a ring R studied in §1 would be understood by looking at the multiplicative monoid structure of the ring R. (Actually we do not forget completely the additive structure.) Now we state a problem to which the absolute mathematics may be applicable. Let X be a (projective smooth) scheme of finite type over Spec(Z). The Hasse zeta function is defined as

ζ_{X}(s) = ∏_{x∈X_{0}}(1 − N(x)^{−s})^{−1}

where x runs over the set X_{0} of closed points (0-dimensional points) of X and N(x) is the cardinality of the residue field at x. It is expected that there exists the so-called gamma factor Γ_{X}(s) and that the completed zeta function ζ̂_{X}(s) = ζ_{X}(s)Γ_{X}(s) is meromorphic in s ∈ C with the functional equation s ↔ dim(X) − s. For our purpose it is convenient to formulate the following conjecture (cf. Kurokawa [4], Deninger [2]): There would exist cohomologies H^{m}(X) for m = 0, 1, …, 2 dim(X) with skew-hermitian operators R^{m} : H^{m}(X) → H^{m}(X) satisfying

ζ̂_{X}(s) = ∏_{m=0}^{2 dim(X)} det (R^{m} − (s − \frac{m}{2}))^{(−1)^{m+1}} .

We refer to Deninger [1], [2] for various investigations. For example, in the case X = Spec(Z), we expect

ζ̂(s) = ζ(s)π^{−s/2}Γ(\frac{s}{2}) = \frac{det(R^{1} − (s − \frac{1}{2}))}{s(s − 1)},

where we consider that H^{0}(Spec(Z)) and H^{2}(Spec(Z)) are one dimensional with the trivial R^{0} and R^{2}. Of course H^{1}(Spec(Z)) should be infinite dimensional.

Problem Can such a cohomology H^{m}(X) and an operator R^{m} : H^{m}(X) → H^{m}(X) be constructed via the absolute mathematics ? Is H^{m}(X) interpreted as a cohomology of the associated absolute scheme X^{abs} → Spec(F_{1}) ?

Some trials will be made in §3 and §4 below.

3 Zeta functions for absolute derivations

3.1

For a map X : Z → C, we define the zeta function attached to X by the Dirichlet series:

ζ(s, X) := ∑_{n=1}^{∞} \frac{X(n)}{n^{s}}.

Lemma 6

ζ(s, \frac{∂}{∂p}) = \frac{ζ(s − 1)}{p^{s} − p}.

Proof: We start with

\frac{∂}{∂p}(n) = \frac{n · ord_{p}(n)}{p}.

Then

ζ(s, \frac{∂}{∂p}) = \frac{1}{p} ∑_{n=1}^{∞} ord_{p}(n)n^{−(s−1)}.

We express n = p^{k}m with (p, m) = 1, and k ≥ 0.

ζ(s, \frac{∂}{∂p}) = \frac{1}{p} ∑_{m:(p,m)=1} m^{−(s−1)} ∑_{k=0}^{∞} kp^{−k(s−1)}

= \frac{1}{p} × ζ(s − 1)(1 − p^{−(s−1)}) × \frac{p^{−(s−1)}}{(1 − p^{−(s−1)})^{2}}
= ζ(s − 1) × \frac{1}{p^{s} − p}.

Documenta Mathematica · Extra Volume Kato (2003) 565–584

Theorem 7 For an X ∈ Der_{F_{1}}(Z) of finite type, i.e., X(p) = 0 for all but finitely many primes p,

ζ(s, X) := ∑_{n=1}^{∞} \frac{X(n)}{n^{s}} = ζ(s − 1) ∑_{p:primes} \frac{X(p)}{p^{s} − p}.

This can be extended to a meromorphic function on the whole complex plane. The special value at s = 0 is given by

ζ(0, X) = \frac{1}{12} ∑_{p} \frac{X(p)}{p − 1}.

Proof: It follows directly from Lemma 6 and X = ∑_{p} X(p)\frac{∂}{∂p}.

3.2 Quantum noncommutativity

We introduce the noncommutativity of primes as “ζ(0, [\frac{∂}{∂p}, \frac{∂}{∂q}])”. First we give a rather explicit formula of the zeta of the commutator of absolute derivations.

Lemma 7

ζ(s, [\frac{∂}{∂p}, \frac{∂}{∂q}]) = \frac{1}{pq} ζ(s − 1) ((1 − q^{−(s−1)}) ∑_{k=1}^{∞} p^{k} \frac{q^{−p^{k}(s−1)}}{(1 − q^{−p^{k}(s−1)})^{2}}

−(1 − p^{−(s−1)}) ∑_{k=1}^{∞} q^{k} \frac{p^{−q^{k}(s−1)}}{(1 − p^{−q^{k}(s−1)})^{2}} ) .

Proof: We start with

\frac{∂}{∂p}(\frac{∂}{∂q} (n)) = \frac{n}{pq} (ord_{q}(n)ord_{p}(n) + ord_{q}(n)ord_{p}(ord_{q}(n))) .

Then

[\frac{∂}{∂p}, \frac{∂}{∂q}](n) = \frac{n}{pq} (ord_{q}(n)ord_{p}(ord_{q}(n)) − ord_{p}(n)ord_{q}(ord_{p}(n))) .

We set n = q^{k}m with (q, m) = 1, and k ≥ 0. Then

ζ(s, [\frac{∂}{∂p}, \frac{∂}{∂q}]) = \frac{1}{pq} (∑_{n=1}^{∞} ord_{q}(n)ord_{p}(ord_{q}(n))n^{−(s−1)} − ⋯)

= \frac{1}{pq} ( ∑_{m:(q,m)=1} m^{−(s−1)} ∑_{k=1}^{∞} k ord_{p}(k)q^{−k(s−1)} − ⋯)

= \frac{1}{pq} ((1 − q^{−(s−1)})ζ(s − 1) ∑_{k=1}^{∞} k ord_{p}(k)q^{−k(s−1)} − ⋯)

Absolute Derivations and Zeta Functions 579

We set k = p^{l}m′ with (p, m′) = 1 and l ≥ 0. Then

∑_{k=1}^{∞} k ord_{p}(k)q^{−k(s−1)} = ∑_{l=1}^{∞} ∑_{m′:(p,m′)=1} p^{l}m′lq^{−p^{l}m′(s−1))}
= ∑_{l=1}^{∞} lp^{l} ( ∑_{m′=1}^{∞} m′q^{−p^{l}m′(s−1)} − ∑_{m′=1}^{∞}(pm′)q^{−p^{l+1}m′(s−1)})
= ∑_{l=1}^{∞} lp^{l} (\frac{q^{−p^{l}(s−1)}}{(1 − q^{−p^{l}(s−1)})^{2}} − p\frac{q^{−p^{l+1}(s−1)}}{(1 − q^{−p^{l+1}(s−1)})^{2}} )
= ∑_{l=1}^{∞} lp^{l} \frac{q^{−p^{l}(s−1)}}{(1 − q^{−p^{l}(s−1)})^{2}} − ∑_{l=0}^{∞} lp^{l+1} \frac{q^{−p^{l+1}(s−1)}}{(1 − q^{−p^{l+1}(s−1)})^{2}}
= ∑_{l=1}^{∞} p^{l} \frac{q^{−p^{l}(s−1)}}{(1 − q^{−p^{l}(s−1)})^{2}} .

This proves the lemma.

Remark 5 Notice a partial functional equation under s ↔ 2 − s. Remark 6 Let ϕ_{p}(z) = ∑_{n=1}^{∞} ord_{p}(n)z^{n}. Then the Mellin transform of ϕ_{p} is pζ(s + 1, \frac{∂}{∂p}), and the series in the lemma above is obtained as

∑_{l=1}^{∞} p^{l} \frac{q^{−p^{l}(s−1)}}{(1 − q^{−p^{l}(s−1)})^{2}} = ∑_{l=1}^{∞} p^{l} \frac{z^{p^{l}}}{(1 − z^{p^{l}})^{2}} |_{z=q^{−(s−1)}}
= z \frac{∂}{∂z} ∑_{l=1}^{∞} \frac{z^{p^{l}}}{1 − z^{p^{l}}} |_{z=q^{−(s−1)}}
= zϕ′_{p}(z)|_{z=q^{−(s−1)}} .

Now we give a rigorous definition of the quantum noncommutativity motivated by the lemma above. Definition The quantum noncommutativity, abbreviated as QNC, of p and q is defined by

QNC(p, q) = \frac{1}{12pq} ((q − 1) ∑_{k=1}^{∞} p^{k} \frac{q^{−p^{k}}}{(1 − q^{−p^{k}})^{2}} − (p − 1) ∑_{k=1}^{∞} q^{k} \frac{p^{−q^{k}}}{(1 − p^{−q^{k}})^{2}} ) .

Numerically,

Documenta Mathematica · Extra Volume Kato (2003) 565–584

QNC(2, 3) = 0.00220482..,
QNC(2, 5) = 0.00172077..,
QNC(2, 7) = 0.00124803..,
QNC(3, 5) = 0.00031155...

Note that QNC(p, p) = 0 and it seems that QNC(p, q) > 0 for p < q. We remark that QNC(p, q) has a neat expression using the Jackson integral (q-integral). We recall the following standard notions in q-analysis [3]. For an appropriate function f(x), we define the Jackson integral

∫_{1}^{∞} f(x)d_{q}x := ∑_{k=1}^{∞}(q^{k} − q^{k−1})f(q^{k})

with a base q. For a real number x, we define the corresponding q-number

[x]_{q} := \frac{q^{x/2} − q^{−x/2}}{q^{1/2} − q^{−1/2}} .

Then the quantum noncommutativity is expressed as

QNC(p, q) = \frac{1}{12pq} ((q − 1) ∑_{k=1}^{∞} p^{k} \frac{q^{−p^{k}}}{(1 − q^{−p^{k}})^{2}} − (p − 1) ∑_{k=1}^{∞} q^{k} \frac{p^{−q^{k}}}{(1 − p^{−q^{k}})^{2}} )

= \frac{1}{12pq} (\frac{(q − 1)}{(q^{1/2} − q^{−1/2})^{2}} ∑_{k=1}^{∞} p^{k} \frac{1}{[p^{k}]_{q}^{2}} − \frac{(p − 1)}{(p^{1/2} − p^{−1/2})^{2}} ∑_{k=1}^{∞} q^{k} \frac{1}{[q^{k}]_{p}^{2}} )

= \frac{1}{12(p − 1)(q − 1)} (∫_{1}^{∞} \frac{d_{p}x}{[x]_{q}^{2}} − ∫_{1}^{∞} \frac{d_{q}x}{[x]_{p}^{2}} ) .

Question (A) Let R = (r_{pq})_{p,q:primes} with r_{pq} =“ζ(0, [\frac{∂}{∂p}, \frac{∂}{∂q}])”= QNC(p, q). Then does it hold that

ζ̂_{Z}(s) = \frac{det(1 − R(s − \frac{1}{2}))}{s(s − 1)}?

Here we may recall that

H_{F_{1}}^{1}(Z, Z)_{C} = ˆ⊕_{p}C\frac{∂}{∂p}.

(B) For a sheaf (automorphic or Galois) ρ of Z, let R_{ρ} = (r_{pq}(ρ)) with

r_{pq}(ρ) = (\frac{ρ(p) + ρ(q)^{∗}}{2} ) r_{pq},

where ρ(q)^{∗} is the adjoint of ρ(p). Then does it hold that

L̂(s, ρ) = \frac{det(1 − R_{ρ}(s − \frac{1}{2}))}{s^{m(ρ)}(s − 1)^{m(ρ)}}

where m(ρ) is the multiplicity of the trivial representation in ρ ?

Documenta Mathematica · Extra Volume Kato (2003) 565–584

Remark 7 Let P be a set of “generalized primes” with the zeta function

ζ_{P} (s) = ∏_{p}(1 − N(p)^{−s})^{−1},

whose conjectual functional equation is s ↦ d(P) − s. Let R : V → V be a linear operator on a complex vector space V. Assume that there is a basis {v_{p} | p ∈ P } of V indexed by P. Let {e_{μ} | μ ∈ Spect(R)} be an R-eigen basis of V with Re_{μ} = μe_{μ}. Thus

⊕_{μ}Ce_{μ} = V = ⊕_{p}Cv_{p}.

Take a suitable test function W such that W (R) has a trace. Then we have (under a suitable condition) a so-called “trace formula”

∑_{μ} W (μ) = ∑_{p} M(p)

where M(p) = M(p, p) is given by

W (R)v_{p} = ∑_{q} M(q, p)v_{q}.

When

W (μ) = log(μ − (s − \frac{d(P)}{2}))

and

M(p) = log((1 − N(p)^{−s})^{−1})

we would obtain the determinant expression

ζ_{P} (s) = ∏_{p}(1 − N(p)^{−s})^{−1}

= ∏ ∐_{μ}(μ − (s − \frac{d(P)}{2}))

= Det(R − (s − \frac{d(P)}{2})).

By this way we would have the analytic continuation of the zeta functions and the L-functions.

4 Towards absolute schemes

We try to set up the first step to F_{1}-schemes. Recall that the usual scheme is coming from the affine scheme Spec_{Z}(A) = Spec(A) for a (commutative) ring A, where Spec_{Z}(A) is the set of the prime ideals of A with the Zariski (Stone-Jacobson-Gelfand) topology. Since we consider an F_{1}-algebra as a monoid, we define Spec(M) for a monoid M. A typical example is the case M = (A, ×) for a ring A. Generalizing a bit, let X be an algebraic system having an associative multiplication with the identity element 1. We assume that X has the zero element 0 (satisfying 0 · x = x · 0 = 0 for all x ∈ X). An equivalence relation α on X is called a congruence if α is compatible with the algebraic operations on X. For example, if x ≡ x′ and y ≡ y′ mod α, then xy ≡ x′y′ mod α. In other words, such an α is associated to the residue (quotient) algebraic system X/α. We denote by Cong(X) the set of all congruences on X.

Example 2 A congruence on a (not necessary commutative) ring corresponds to a two-sided ideal.

Example 3 A congruence on a group corresponds to a normal subgroup.

We say that a congruence α on X is a prime congruence if X/α is integral in the sense that X/α has no (non-zero) zero divisors: if x ≢ 0, y ≢ 0 mod α, then xy ≢ 0 mod α. We denote by Spec(X) the set of the prime congruences on X with the following topology: the closed subsets are

V (β) = {α ≤ β | α is a prime congruence}

for β ∈ Cong(X). Here α ≤ β means that x ≡ 0 mod β implies x ≡ 0 mod α. As in the usual case, it is checked that such V (β)’s satisfy the needed conditions for closed sets by

V (id) = X, V (triv) = ∅,

V (β_{1}) ∪ ⋯ ∪ V (β_{n}) = V (β_{1} ∧ ⋯ ∧ β_{n}),

⋂_{λ∈Λ} V (β_{λ}) = V (∑_{λ} β_{λ}),

where X/triv = {1}, X/id = X, and ∑_{λ} β_{λ} denotes the congruence generated by β_{λ}’s. For a ring A, we define

(Spec_{Z}(A))^{abs} = Spec_{F_{1}}(A) = Spec((A, ×)).

[Notice that Spec((A, ×)) is \underline{not} the set of usual “ideals” of (A, ×).] Then we have the natural map

Spec_{Z}(A) ⟶ Spec_{F_{1}}(A)

since each congruence on the ring A induces a congruence on the multiplicative monoid (A, ×). The “local-global” map

A ⟶ ∏_{α∈Spec_{F_{1}} (A)}(A/α)

refines the usual local global map

A ⟶ ∏_{γ∈Spec_{Z}(A)}(A/γ).

For example, Spec_{F_{1}}(Z) and ∏_{α∈Spec_{F_{1}} (Z)}(Z/α) are both big and unusual.

Remark 8 The absolute fundamental group π_{1}(Spec_{F_{1}}(Z)) would be interesting from the view point of the Langlands conjecture since for each unramified automorphic representation π of GL_{n}(A) (A being the adele ring of Q) we may have an n-dimensional representation (local system)

ρ : π_{1}(Spec_{F_{1}}(Z)) → GL_{n}(C)

satisfying

L(s, π) = L(s, ρ).

Now we notice on the F_{1}-tensor product. We define the F_{1}-tensor product of rings A and B as

A ⊗_{F_{1}} B = (A, ×) ∗ (B, ×)

the free product of monoids under the identification

1_{A} = 1_{B} = 1 and 0_{A} = 0_{B} = 0.

Theorem 8

Spec_{F_{1}}(A ⊗_{F_{1}} B) = Spec_{F_{1}}(A) × Spec_{F_{1}}(B).

Proof: We show

Spec(M ∗ N) ≅ Spec(M) × Spec(N)

by sending α ∗ β to (α, β), where M and N are multiplicative monoids having 1 and 0. Here α ∗ β is the natural congruence on M ∗ N coming from α and β. Since it is easy to see that α ∗ β is a prime congruence on M ∗ N if α and β are prime congruences on M and N respectively by

(M ∗ N)/(α ∗ β) ≅ (M/α) ∗ (N/β),

it is sufficient to show that the map is surjective. Let γ be a prime congruence on M ∗ N. Then we obtain a congruence α on M and a congruence β on N by restricting γ to M and N respectively, and it holds that γ = α ∗ β. Since

(M ∗ N)/γ ≅ (M/α) ∗ (N/β),

we know that α (resp. β) is a prime congruence on M (resp. N).

Thus we have

Spec_{F_{1}}(Z ⊗_{F_{1}} Z) = Spec_{F_{1}}(Z) × Spec_{F_{1}}(Z)

where Z ⊗_{F_{1}} Z = Z ∗ Z, as expected in [4]–[7] and [9].

References

[1] C. Deninger: Local L-factors of motives and regularized determinants. Invent. Math. 107 (1992), no. 1, 135–150.

[2] C. Deninger: Some analogies between number theory and dynamical systems on foliated spaces, Proceedings of the ICM, Vol. I (Berlin, 1998) Doc. Math. 1998, Extra Vol. I, 163–186.

[3] V. Kac and P. Cheung: Quantum Calculus, Springer-Verlag, Berlin, 2002.

[4] N. Kurokawa: On some Euler products. I, Proc. Japan Acad. Ser. A Math. Sci. 60(1984), no. 9, 335–338.

[5] N. Kurokawa: Multiple zeta functions: an example, Adv. Stud. Pure Math. 21(1992), 219–226.

[6] N. Kurokawa: On F_{1}. Proceedings of the Symposium on Algeraic Geometry at Kinosaki in 1994 Oct., pp38–46.

[7] N. Kurokawa and M. Wakayama: Absolute Casimir Elements (in Japanese), Iwanami, Tokyo, 2002.

[8] J. L. Loday: Cyclic Homology, second ed., Springer-Verlag, Berlin, 1998.

[9] Yu. I. Manin: Lectures on zeta functions and motives (according to Deninger and Kurokawa). Astérisque 228 (1995) 121–163.

[10] H. Ochiai: Monoidal absolute derivations. A talk at the Conference “Zetas and Trace Formulas” in Okinawa 2002.

[11] C. Soulé: On the field with one element, Max-Planck-Institut für Mathematik, Arbeitstagung 1999, 5pages; IHES prépublications M/55, 1999.

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

Nobushige Kurokawa Department of Mathematics Tokyo Institute of Technology Oh-okayama Meguro-ku, Tokyo 152-8551 Japan kurokawa@math.titech.ac.jp Hiroyuki Ochiai Department of Mathematics Tokyo Institute of Technology Current address: Department of Mathematics, Nagoya University Chikusa, Nagoya 464-8602, Japan ochiai@math.nagoya-u.ac.jp

Masato Wakayama Faculty of Mathematics Kyushu University Hakozaki Higashi-ku, Fukuoka 812-8581 Japan wakayama@math.kyushu.u-ac.jp