TWO-PRIMARY ALGEBRAIC K-THEORY

OF TWO-REGULAR NUMBER FIELDS

John Rognes and Paul Arne Østvær

Abstract. We explicitly calculate all the 2-primary higher algebraic K-groups of the rings of integers of all 2-regular quadratic number fields, cyclotomic number fields, or maximal real subfields of such. Here 2-regular means that (2) does not split in the number field, and its narrow Picard group is of odd order.

Introduction

As a consequence of Voevodsky’s proof [V] of the Milnor Conjecture, the 2primary algebraic K-groups of rings of integers in arbitrary number fields were computed in [RW], using the Bloch–Lichtenbaum spectral sequence of [BL]. Similar results were obtained by Kahn in [K] for totally imaginary number fields. The results of these computations were expressed in terms of the étale cohomology groups of the number ring. Here we make these calculations completely explicit in the case of a large number of quadratic and cyclotomic number fields, and their maximal real subfields.

A number field F is called 2-regular [GJ] if the 2-Sylow subgroup R_{2}(F) of the kernel in K_{2}(F) of the regular symbols attached to the non-complex places of F, is trivial. This is equivalent to the combined hypothesis that the rational prime ideal (2) does not split into distinct prime ideals in O_{F}, and that the narrow Picard group Pic_{+}(R_{F}) of the ring R_{F} = O_{F} [\frac{1}{2}] of 2-integers in F has odd order. (See §2 for definitions and further discussion.)

The 2-regular quadratic number fields were classified by Browkin and Schinzel in [BS]. All their 2-primary algebraic K-groups are determined in Theorems 3.2 and 4.2. More generally, a classification of 2-regular Abelian 2-extensions of ℚ was given by Gras in [G, p. 331].

We also consider the cyclotomic number fields ℚ(ζ_{q}), and their maximal real subfields ℚ(ζ_{q} + ζ̄_{q}). These can only be 2-regular when q = p^{m} is a prime power. We note in Theorems 3.3 and 4.3 that these are all 2-regular when p = 2, and in Examples 3.5, 3.6 and 4.5 that these are all 2-regular when q ≠ 29 is an odd prime power with φ(q) ≤ 66, or a Sophie Germain prime, such that 2 is a primitive root mod q. In the 2-regular cases all the 2-primary algebraic K-groups of the cyclotomic fields and their maximal real subfields are calculated in Theorems 3.3, 3.4, 4.3 and 4.4.

Typeset by AMS-TEX

1991 Mathematics Subject Classification. Primary: 19D50, Secondary: 11R11, 11R18, 11R37, 11R42, 19F27.

Key words and phrases. Algebraic K-group, quadratic number field, cyclotomic number field, 2-regular, Sophie Germain prime, zeta-function.

We remark that in the 2-regular cases K_{4}(O_{F}) = K_{4}(R_{F}) = 0 modulo odd torsion, so the map K_{4}^{M}(F) → K_{4}(F) from Milnor K-theory to algebraic K-theory is zero in degree 4, in these cases.

By Wiles’ proof [Wi] of the Main Conjecture in Iwasawa theory, and Kolster’s appendix in [RW], these calculations of 2-primary algebraic K-groups give 2-primary calculations of the values of the Dedekind zeta functions ζ_{F} (s) at negative odd integers, in the cases when F is totally real Abelian. We present these calculations for the 2-regular real quadratic or maximal real subfields of cyclotomic number fields in Theorems 5.2, 5.3 and 5.4.

1. Notations and Review

Consider a number field F over ℚ, with r_{1} real and 2r_{2} complex embeddings. Units of F with positive image under each real embedding are called totally positive. We say that F is totally imaginary when r_{1} = 0, F is real when r_{1} > 0, and F is totally real when r_{2} = 0. Let O_{F} be the ring of integers in F (a number ring), and let R_{F} = O_{F} [\frac{1}{2}] be the ring of 2-integers.

Recall that the ideal class group Cl(F) of F is the group of fractional ideals of F modulo the principal fractional ideals, the narrow ideal class group Cl_{+}(F) is the group of fractional ideals modulo the totally positive principal fractional ideals, the Picard group Pic(R_{F}) is the quotient group of Cl(F) by the classes of prime ideals above (2), while the narrow Picard group Pic_{+}(R_{F}) is the quotient group of Cl_{+}(F) by the (narrow) classes of prime ideals above (2). Also recall that the 2-rank rk_{2} A of an Abelian group A is the dimension of its exponent 2 subgroup _{2}A as a vector space over ℤ/2.

Let h_{F} be the class number of F, i.e., the order of Cl(F). Let s be the number of primes of O_{F} over the rational prime (2), let t be the 2-rank of the Picard group Pic(R_{F}), and let u be the 2-rank of the narrow Picard group Pic_{+}(R_{F}).

Let ζ_{n} = exp(2π\sqrt{−1}/n) be a primitive nth root of unity. For each integer i, w_{i}^{(2)}(F) is defined as the maximal power 2^{n} of 2 such that Gal(F (ζ_{2^{n}})/F) has exponent dividing i. If no maximal power exists, such as for i = 0, we write w_{i}^{(2)}(F) = 2^{∞}. We say that F is exceptional if the Galois group Gal(F (ζ_{2^{n}})/F) is not cyclic for some n. Any real number field is exceptional.

We write v_{2}(i) for the 2-adic valuation of an integer i, i.e., the greatest number n such that 2^{n} divides i. Let ≅_{(2)} denote isomorphism after 2-localization, which for finite Abelian groups amounts to isomorphism of 2-Sylow subgroups. Let H_{ét}^{n}(R_{F}; M) be the nth étale cohomology group of R_{F} with coefficients in the étale sheaf M.

The following results were obtained in [RW] as applications of Voevodsky’s proof of the Milnor conjecture [V], and the Bloch–Lichtenbaum spectral sequence [BL]. See also [K] and [W2].

Theorem 1.1 [RW, 0.4(a)]. Let E be a totally imaginary number field. Then for all n ≥ 2

K_{n}(O_{E}) ≅_{(2)} K_{n}(R_{E}) ≅_{(2)} \{ \begin{matrix}H_{ét}^{2}(R_{E}; ℤ_{2}(i + 1)) & for n = 2i, \\ ℤ^{r_{2}} ⊕ ℤ/w_{i}^{(2)}(E) & for n = 2i − 1. \end{matrix}

Theorem 1.2 [RW, 0.6]. Let F be a real number field. Then for all n ≥ 2

K_{n}(O_{F}) ≅_{(2)} K_{n}(R_{F}) ≅_{(2)} \{ \begin{matrix}H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 1)) & for n = 8k, \\ ℤ^{r_{1}+r_{2}} ⊕ ℤ/2 & for n = 8k + 1, \\ H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) & for n = 8k + 2, \\ ℤ^{r_{2}} ⊕ (ℤ/2)^{r_{1}−1} ⊕ ℤ/2w_{4k+2}^{(2)}(F) & for n = 8k + 3, \\ (?) & for n = 8k + 4, \\ ℤ^{r_{1}+r_{2}} & for n = 8k + 5, \\ H̃_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) & for n = 8k + 6, \\ ℤ^{r_{2}} ⊕ ℤ/w_{4k+4}^{(2)}(F) & for n = 8k + 7. \end{matrix}

Here H̃_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) is the kernel of the natural surjective map

H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 2)) ⟶ ⨁^{r_{1}} H_{ét}^{2}(ℝ; ℤ_{2}(4k + 2)) ≅ (ℤ/2)^{r_{1}}

induced by the r_{1} real embeddings of F.

The missing group K_{8k+4}(R_{F}) is isomorphic, modulo odd finite groups, to the maximal finite quotient of K_{8k+5}(R_{F}; ℚ_{2}/ℤ_{2}). The latter K-group fits into the short exact sequence

0 → (ℤ/2)^{r_{1}−1} ⟶ K_{8k+5}(R_{F}; ℚ_{2}/ℤ_{2}) ⟶ (ℚ_{2}/ℤ_{2})^{r_{1}+r_{2}} ⊕ H_{ét}^{2}(R_{F}; ℤ_{2}(4k + 3)) → 0.

Theorem 1.3 [RW, 0.7]. Let F be a real number field. For n ≥ 2

rk_{2} K_{n}(R) = \{ \begin{matrix}s + t − 1 & for n = 8k, \\ 1 & for n = 8k + 1, \\ r_{1} + s + t − 1 & for n = 8k + 2, \\ r_{1} & for n = 8k + 3, \\ s + u − 1 & for n = 8k + 4, \\ 0 & for n = 8k + 5, \\ s + u − 1 & for n = 8k + 6, \\ 1 & for n = 8k + 7. \end{matrix}

Recall from e.g. [RW, 1.8, 4.4, 4.6, 6.11 and 6.12]:

Proposition 1.4. Let F be a number field. Then

H_{ét}^{n}(R_{F}; ℚ_{2}/ℤ_{2}(i)) ≅ \{ \begin{matrix}ℤ/w_{i}^{(2)}(F) & for n = 0, \\ (ℚ_{2}/ℤ_{2})^{r} ⊕ H_{ét}^{2}(R_{F}; ℤ_{2}(i)) & for n = 1, i ≠ 0, 1, \\ (ℤ/2)^{r_{1}} & for n ≥ 2, i − n odd, \\ 0 & for n ≥ 2, i − n even. \end{matrix}

Here r = r_{2} for i even, and r = r_{1} + r_{2} for i odd. H_{ét}^{2}(R_{F}; ℤ_{2}(i)) is a finite group for i ≠ 0, 1, with 2-rank

rk_{2} H_{ét}^{2}(R_{F}; ℤ_{2}(i)) = \{ \begin{matrix}r_{1} + s + t − 1 & for i ≠ 0 even, \\ s + t − 1 & for i ≠ 1 odd. \end{matrix}

Also recall from [W1, 6.3]:

Proposition 1.5. Let F be a number field. Let a be maximal such that F (\sqrt{−1}) contains a primitive 2^{a}th root of unity, let i be an integer, and let b = v_{2}(i) be its 2-adic valuation.

(a) If \sqrt{−1} ∈ F then w_{i}^{(2)}(F) = 2^{a+b}.

(b) If \sqrt{−1} ∉ F and i is odd then w_{i}^{(2)}(F) = 2.

(c) If \sqrt{−1} ∉ F, F is exceptional and i is even then w_{i}^{(2)}(F) = 2^{a+b}.

(d) If \sqrt{−1} ∉ F, F is non-exceptional and i is even then w_{i}^{(2)}(F) = 2^{a+b−1}.

2. Two-regular number fields

Let F be a number field. The common kernel of the regular symbols on K_{2}(F) associated to the finite places of F is called the tame kernel, and is isomorphic to K_{2}(O_{F}). Ignoring odd torsion, K_{2}(O_{F}) ≅_{(2)} K_{2}(R_{F}) by the localization sequence. By 1.1 and 1.2, we have a natural isomorphism K_{2}(R_{F}) ≅_{(2)} H_{ét}^{2}(R_{F}; ℤ_{2}(2)). See also [T] and [S, Lemme 10].

In [G], Gras introduced the modified tame kernel R_{2}(F), as the subgroup of the tame kernel where also the regular symbols associated to the r_{1} real embeddings of F vanish. On 2-Sylow subgroups, these symbols are identified with the natural surjection

α^{2} : H_{ét}^{2}(R_{F}; ℤ_{2}(2)) → ⨁^{r_{1}} H_{ét}^{2}(ℝ; ℤ_{2}(2)) ≅ (ℤ/2)^{r_{1}}

and so R_{2}(F) ≅ H̃_{ét}^{2}(R_{F}; ℤ_{2}(2)) in the notation of 1.2.

Definition 2.1. A number field F is 2-regular if the 2-Sylow subgroup of the modified tame kernel R_{2}(F) is trivial.

As noted in 1.3, the 2-rank of K_{2}(R_{F}) is r_{1} + s + t − 1. Fixing r_{1}, the minimal possible 2-rank is thus r_{1}. Totally real fields satisfying 2.2(b) were studied by Berger in [B]. Some of the following characterizations were pointed out to us by an anonymous referee, and improve on our initial results.

Proposition 2.2. For number fields F the following are equivalent:

(a) F is 2-regular.

(b) K_{2}(O_{F}){2} ≅ K_{2}(R_{F}){2} ≅ (ℤ/2)^{r_{1}} is elementary Abelian of minimal 2rank.

(c) The ideal (2) does not split in F and the narrow Picard group Pic_{+}(R_{F}) has odd order, i.e., s = 1 and t = u = 0.

(d)

H_{ét}^{2}(R_{F}; ℤ_{2}(i)) ≅ \{ \begin{matrix}(ℤ/2)^{r_{1}} & for i ≠ 0 even, \\ 0 & for i ≠ 1 odd. \end{matrix}

Proof. Suppose i ≠ 0 is even. Then H_{ét}^{n}(ℝ; ℤ_{2}(i)) is 0 for n odd and ℤ/2 for n > 0 even. We compare the Bockstein sequences in étale cohomology for R_{F} and ℝ under the r_{1} real embeddings:

(2.3)[figure: 0 → H_{ét}^{1}(R_{F}; ℤ_{2}(i))/2; H_{ét}^{1}(R_{F}; ℤ/2(i)); α^{1}; _{2}H_{ét}^{2}(R_{F}; ℤ_{2}(i)) → 0; ⨁^{r_{1}} H_{ét}^{1}(ℝ; ℤ/2(i)); ≅; ⨁^{r_{1}} _{2}H_{ét}^{2}(ℝ; ℤ_{2}(i))]

Similarly:

(2.4)  \xymatrix{H_{ét}^{2}(R_{F}; ℤ_{2}(i))/2 \ar[r]^{≅} \ar[d] & H_{ét}^{2}(R_{F}; ℤ/2(i)) \ar[d] \\ ⨁^{r_{1}} H_{ét}^{2}(ℝ; ℤ_{2}(i))/2 \ar[r]^{≅} & ⨁^{r_{1}} H_{ét}^{2}(ℝ; ℤ/2(i))}

Note that cohomology with coefficients in ℤ/2(i) is independent of i. By the Kummer sequence H_{ét}^{1}(R_{F}; ℤ/2(1)) ≅ R_{F}^{×}/2 ⊕ _{2} Pic(R_{F}) and H_{ét}^{2}(R_{F}; ℤ/2(1)) ≅ Pic(R_{F})/2 ⊕_{2} Br(R_{F}), where Br(R_{F}) is the Brauer group. By the Brauer–Hasse– Noether theorem the natural map

2 Br(R_{F}) → ⨁^{r_{1}} _{2} Br(ℝ) ≅ (ℤ/2)^{r_{1}}

is surjective, so the right-hand vertical map in (2.4) is surjective, which explains why the left-hand vertical map is always surjective.

Now suppose that F is 2-regular. Then R_{2}(F){2} ≅ H̃_{ét}^{2}(R_{F}; ℤ_{2}(2)) = 0, so K_{2}(R_{F}){2} ≅ H_{ét}^{2}(R_{F}; ℤ_{2}(2)) ≅ (ℤ/2)^{r_{1}}. This proves (a) ⇒ (b). The converse (b) ⇒ (a) is equally obvious.

Let R_{F +}^{×} ⊂ R_{F}^{×} denote the groups of totally positive units in R_{F} and units in R_{F}, respectively. Then there is an exact sequence

(2.5)  0 → R_{F +}^{×} → R_{F}^{×} ⟶^{σ} (ℤ/2)^{r_{1}} → Pic_{+}(R_{F}) → Pic(R_{F}) → 0

obtained by applying the snake lemma to the exact sequences defining Pic_{+}(R_{F}) and Pic(R_{F}). Here σ is the signature map.

Now suppose that K_{2}(R_{F}){2} ≅ H_{ét}^{2}(R_{F}; ℤ_{2}(2)) ≅ (ℤ/2)^{r_{1}}. Evaluating 2-ranks we find r_{1} + s + t − 1 = r_{1}, so s = 1 and t = 0. Hence (2) does not split, and Pic(R_{F}) has odd order. The right-hand vertical map in (2.3) is an isomorphism for i = 2, so the middle vertical map

α^{1} : H_{ét}^{1}(R_{F}; ℤ/2(i)) → ⨁^{r_{1}} H_{ét}^{1}(ℝ; ℤ/2(i))

is a surjection for all i. Since _{2} Pic(R_{F}) = 0 we can identify this map with the signature map σ, extended over the natural surjection R_{F}^{×} → R_{F}^{×}/2. Hence σ is surjective and Pic_{+}(R_{F}) ≅ Pic(R_{F}) has odd order. Thus u = 0 and (b) ⇒ (c).

Suppose s = 1 and t = u = 0. Then H_{ét}^{2}(R_{F}; ℤ_{2}(i)) has 2-rank r_{1} for i ≠ 0 even and 0 for i ≠ 1 odd, by 1.4. The image of the signature map σ can be identified with that of α^{1} since t = 0, and is surjective by (2.5). Thus the righthand vertical map in (2.3) is surjective for all even i ≠ 0. Hence the surjection H_{ét}^{2}(R_{F}; ℤ_{2}(i)) → (ℤ/2)^{r_{1}} is in fact an isomorphism. This proves that (c) ⇒ (d).

Finally (d) ⇒ (b) by specialization, which completes the proof. □

We say that an extension E/F of number fields is finitely ramified if a finite (non-Archimedean) prime of F ramifies in the extension.

Proposition 2.6. Let E/F be an extension of number fields that does not contain any nontrivial Abelian finitely ramified subextension. For example, E can be a quadratic finitely ramified extension. Then the narrow norm maps Cl_{+}(E) → Cl_{+}(F) and Pic_{+}(R_{E}) → Pic_{+}(R_{F}) are surjective.

Proof. By class field theory, Cl(F) is the Galois group of the maximal Abelian totally unramified extension H(F) of F, called the Hilbert class field. Likewise Cl_{+}(F) is the Galois group of the maximal Abelian finitely unramified extension H_{+}(F) of F, called the narrow class field of F.

The composite E · H_{+}(F) is Abelian and finitely unramified over E, because H_{+}(F) is Abelian and finitely unramified over F. Hence E · H_{+}(F) is contained in H_{+}(E). By assumption E ∩ H_{+}(F) = F, so we obtain a surjective group homomorphism:

Cl_{+}(E) = Gal(H_{+}(E)/E) → Gal(E · H_{+}(F)/E) ≅ Gal(H_{+}(F)/F) = Cl_{+}(F)

As in [Wa, p. 340] this equals the map induced by the norm on fractional ideals. The surjection Cl_{+}(E) → Pic_{+}(R_{E}) annihilates the classes of ideals of O_{E} lying over (2), and likewise for F. The norm preserves ideals over (2), and so we obtain an induced surjection Pic_{+}(R_{E}) → Pic_{+}(R_{F}), as claimed. □

We now consider cyclotomic fields. Let E = ℚ(ζ_{q}) be the qth cyclotomic field. Its maximal real subfield is F = ℚ(ζ_{q} + ζ̄_{q}).

Proposition 2.7. (a) The qth cyclotomic field E = ℚ(ζ_{q}) is 2-regular if and only if q = 2^{m}, or if q = p^{m} where p an odd prime, 2 is a primitive root mod q and Pic(R_{E}) has odd order.

(b) The maximal real subfield F = ℚ(ζ_{q} + ζ̄_{q}) of the qth cyclotomic field is 2regular if and only if q = 2^{m}, or if q = p^{m} where p is an odd prime, 2 is a primitive root mod q and Pic_{+}(R_{F}) has odd order.

Proof. In both cases it follows easily from the cyclotomic reciprocity law [Wa, p. 14] that the prime ideal (2) does not split in E, resp. in F, if and only if q = 2^{m}, or if q = p^{m} with p an odd prime and 2 a primitive root mod q. Hence in these cases the number field is 2-regular if and only if its narrow Picard group has odd order. Of course Pic(R_{E}) = Pic_{+}(R_{E}) for E totally imaginary.

When q = 2^{m} the class number of E = ℚ(ζ_{2^{m}}) is odd [Wa, 10.5]. In this case the quadratic extension E/F is finitely ramified above (2), so Pic(R_{E}) and Pic_{+}(R_{F}) have odd order by 2.6. □

3. Totally imaginary number fields

Theorem 3.1. Let E be a 2-regular totally imaginary number field. Then for n ≥ 2

K_{n}(O_{E}) ≅_{(2)} K_{n}(R_{E}) ≅_{(2)} \{ \begin{matrix}0 & for n = 2i, \\ ℤ^{r_{2}} ⊕ ℤ/w_{i}^{(2)}(E) & for n = 2i − 1. \end{matrix}

Proof. This is clear from 1.1 and 2.2. □

Theorem 3.2. Let E = ℚ(\sqrt{m}) be a quadratic number field with m < 0 square free. Then E is 2-regular if and only if m = −1, m = −2, m = −p or m = −2p with p ≡ ±3 mod 8 prime. In these cases, for n ≥ 2

K_{n}(ℤ[\sqrt{−1}, \frac{1}{2}]) ≅_{(2)} \{ \begin{matrix}0 & for n = 2i, \\ ℤ ⊕ ℤ/2^{2+v_{2}(i)} & for n = 2i − 1, \end{matrix}

while

K_{n}(ℤ[\sqrt{m}, \frac{1}{2}]) ≅_{(2)} \{ \begin{matrix}0 & for n = 2i, \\ ℤ ⊕ ℤ/2^{2+v_{2}(i)} & for n = 2i − 1, i even, \\ ℤ ⊕ ℤ/2 & for n = 2i − 1, i odd, \end{matrix}

for m ≠ 1.

Proof. The classification of 2-regular imaginary quadratic number fields comes from [BS]. The numbers w_{i}^{(2)}(E) are obtained from 1.5, by noting that the fields ℚ(\sqrt{m}) are non-exceptional for m = −1, −2, and exceptional otherwise. □

Theorem 3.3. Let E = ℚ(ζ_{2^{m}}) with m ≥ 2. Then E is 2-regular. Hence for n ≥ 2

K_{n}(ℤ[ζ_{2^{m}}]) ≅_{(2)} \{ \begin{matrix}0 & for n = 2i, \\ ℤ^{2^{m−2}} ⊕ ℤ/2^{m+v_{2}(i)} & for n = 2i − 1. \end{matrix}

Proof. We use 2.7 and 3.1. The calculation of w_{i}^{(2)}(E) = 2^{m+v_{2}(i)} is immediate from 1.5, since \sqrt{−1} ∈ E. □

The Euler φ-function φ(n) equals the number of units in the ring ℤ/n. This is the degree of ℚ(ζ_{n}) over ℚ.

Theorem 3.4. Let E = ℚ(ζ_{q}) with q an odd prime power. Suppose 2 is a primitive root modulo q and Pic(R_{E}) has odd order. Then for n ≥ 2

K_{n}(ℤ[ζ_{q}]) ≅_{(2)} \{ \begin{matrix}0 & for n = 2i, \\ ℤ^{φ(q)/2} ⊕ ℤ/2^{2+v_{2}(i)} & for n = 2i − 1, i even, \\ ℤ^{φ(q)/2} ⊕ ℤ/2 & for n = 2i − 1, i odd. \end{matrix}

Proof. The discriminant of E is a power of q, so (2) is inert in these cases. We apply 3.1. The numbers w_{i}^{(2)}(E) are obtained from 1.5 by noting that E is exceptional. □

Example 3.5. Let E = ℚ(ζ_{q}) with q an odd prime power. By the table in [Wa, pp. 352–353], the class number h_{E} is odd for all prime powers q with Euler φ-function φ(q) ≤ 66, except q = 29. Hence the hypothesis of 3.4 holds for all odd prime powers q ≠ 29 with φ(q) ≤ 66, such that 2 is a primitive root modulo q.

A Sophie Germain prime p is a prime such that (p − 1)/2 is also prime.

Example 3.6. Let E = ℚ(ζ_{p}) with p a Sophie Germain prime. By a theorem of Estes [E], h_{E} is odd for Sophie Germain primes p such that (2) is inert in the maximal real subfield of E. This is certainly the case if (2) is inert in E. Thus the hypothesis of 3.4 holds for all Sophie Germain primes p with 2 a primitive root modulo p. These are precisely the Sophie Germain primes p ≡ 3 mod 8, and p = 5. The first few examples of such primes are p = 5, 11, 59, 83, 107 and 179.

4. Real number fields

Theorem 4.1. Let F be a 2-regular real number field. Then for n ≥ 2

K_{n}(O_{F}) ≅_{(2)} K_{n}(R_{F}) ≅_{(2)} \{ \begin{matrix}0 & for n = 8k, \\ ℤ^{r_{1}+r_{2}} ⊕ ℤ/2 & for n = 8k + 1, \\ (ℤ/2)^{r_{1}} & for n = 8k + 2, \\ ℤ^{r_{2}} ⊕ (ℤ/2)^{r_{1}−1} ⊕ ℤ/2w_{4k+2}^{(2)}(F) & for n = 8k + 3, \\ 0 & for n = 8k + 4, \\ ℤ^{r_{1}+r_{2}} & for n = 8k + 5, \\ 0 & for n = 8k + 6, \\ ℤ^{r_{2}} ⊕ ℤ/w_{4k+4}^{(2)}(F) & for n = 8k + 7. \end{matrix}

The torsion subgroups in degrees n = 8k + 1 and n = 8k + 2 are detected by the natural maps

K_{n}(R_{F}) ⟶ ⨁^{r_{1}} π_{n}(ℤ × BO) ≅ (ℤ/2)^{r_{1}}

induced by the r_{1} real embeddings.

Proof. The formulas for the algebraic K-groups follow from 1.2, 1.3 and 2.2. Note that there is no extension issue in degree 8k + 4 because the 2-rank there is s + u − 1 = 0. The Adams μ-element μ_{8k+1} in the stable (8k + 1)-stem maps to a class in K_{8k+1}(R_{F}) that is detected by real topological K-theory [A]. The claims in degree 8k + 1 and 8k + 2 follow from this, and 2.2. □

Theorem 4.2. Let F = ℚ(\sqrt{m}) be a quadratic number field with m > 0 square free. Then F is 2-regular if and only if m = 2, m = p or p = 2p with p ≡ ±3 mod 8 prime. In these cases, for n ≥ 2

K_{n}(ℤ[\sqrt{m}, \frac{1}{2}]) ≅_{(2)} \{ \begin{matrix}0 & for n = 8k, \\ ℤ^{2} ⊕ ℤ/2 & for n = 8k + 1, \\ (ℤ/2)^{2} & for n = 8k + 2, \\ ℤ/2 ⊕ ℤ/2w_{4k+2}^{(2)}(F) & for n = 8k + 3, \\ 0 & for n = 8k + 4, \\ ℤ^{2} & for n = 8k + 5, \\ 0 & for n = 8k + 6, \\ ℤ/w_{4k+4}^{(2)}(F) & for n = 8k + 7. \end{matrix}

When i is even, w_{i}^{(2)}(F) = 2^{3+v_{2}(i)} for m = 2, while w_{i}^{(2)}(F) = 2^{2+v_{2}(i)} for m ≠ 2.

Proof. The classification of 2-regular real quadratic number fields also comes from [BS]. To determine the numbers w_{i}^{(2)}(F), note that F contains ζ_{8} only for m = 2, and that F is real, thus exceptional. □

Theorem 4.3. Let F = ℚ(ζ_{2^{m}} + ζ̄_{2^{m}}) with m ≥ 2. Then F is 2-regular. Hence for n ≥ 2

K_{n}(ℤ[ζ_{2^{m}} + ζ̄_{2^{m}}]) ≅_{(2)} \{ \begin{matrix}0 & for n = 8k, \\ ℤ^{2^{m−2}} ⊕ ℤ/2 & for n = 8k + 1, \\ (ℤ/2)^{2^{m−2}} & for n = 8k + 2, \\ (ℤ/2)^{2^{m−2}−1} ⊕ ℤ/2^{m+2} & for n = 8k + 3, \\ 0 & for n = 8k + 4, \\ ℤ^{2^{m−2}} & for n = 8k + 5, \\ 0 & for n = 8k + 6, \\ ℤ/2^{m+2+v_{2}(k+1)} & for n = 8k + 7. \end{matrix}

Proof. We use 2.7 and 4.1. The numbers w_{i}^{(2)}(F^{+}) are obtained from 1.5. □

Theorem 4.4. Let E = ℚ(ζ_{q}) and F = ℚ(ζ_{q} +ζ̄_{q}) with q an odd prime power such that 2 is a primitive root mod q. Suppose Pic(R_{E}) has odd order, or more generally that Pic_{+}(R_{F}) has odd order. Then F is 2-regular, and for n ≥ 2

K_{n}(ℤ[ζ_{q} + ζ̄_{q}]) ≅_{(2)} \{ \begin{matrix}0 & for n = 8k, \\ ℤ^{φ(q)/2} ⊕ ℤ/2 & for n = 8k + 1, \\ (ℤ/2)^{φ(q)/2} & for n = 8k + 2, \\ (ℤ/2)^{φ(q)/2−1} ⊕ ℤ/16 & for n = 8k + 3, \\ 0 & for n = 8k + 4, \\ ℤ^{φ(q)/2} & for n = 8k + 5, \\ 0 & for n = 8k + 6, \\ ℤ/2^{4+v_{2}(k+1)} & for n = 8k + 7. \end{matrix}

In degree n = 8k +3 the unit homomorphism ℤ → ℤ[ζ_{q} +ζ̄_{q}] takes K_{n}(ℤ) ≅_{(2)} ℤ/16 isomorphically to the ℤ/16-summand in K_{n}(ℤ[ζ_{q} + ζ̄_{q}]).

Proof. Write q = p^{e} for an odd prime p. Then (p) is totally ramified in E, hence E/F is finitely ramified at the prime above (p). Thus the assumption that Pic(R_{E}) has odd order and 2.6 imply that Pic_{+}(R_{F}) has odd order. We can therefore quote 4.1. To apply 1.5, note that ℚ(ζ_{q} + ζ̄_{q}, \sqrt{−1}) is contained in ℚ(ζ_{q}, \sqrt{−1}) = ℚ(ζ_{4q}), hence contains ζ_{4} but not ζ_{8}.

The map ℤ → ℤ[ζ_{q} + ζ̄_{q}] induces isomorphisms H_{ét}^{0}(ℤ[\frac{1}{2}]; ℚ_{2}/ℤ_{2}(4k + 2)) → H_{ét}^{0}(R_{F}; ℚ_{2}/ℤ_{2}(4k + 2)), and therefore takes K_{8k+3}(ℤ[\frac{1}{2}]) ≅ ℤ/16 split injectively to a ℤ/16-summand of K_{8k+3}(R_{F}), by naturality of 1.2. The final claim follows. □

Examples 4.5. As noted in 3.5 the hypothesis of this theorem holds for all odd prime powers q ≠ 29 with φ(q) ≤ 66 and 2 a primitive root mod q.

Likewise by Estes’ theorem [E] the hypothesis holds for all Sophie Germain primes p with 2 a primitive root mod p.

5. Values of zeta functions

Let #A denote the order of a finite group A, and let ζ_{F} (s) be the zeta-function of the number field F. Recall the following theorem of Kolster, Rognes and Weibel, combining the results of [RW] with Wiles’ proof [Wi] of the Main Conjecture in Iwasawa theory.

Theorem 5.1 [RW, 0.1]. Let F be a totally real Abelian number field with r_{1} real embeddings. Then for all even i ≥ 2

2^{r_{1}} · \frac{#K_{2i−2}(R_{F})}{#K_{2i−1}(R_{F})} = \frac{#H_{ét}^{2}(R_{F}; ℤ_{2}(i))}{#H_{ét}^{1}(R_{F}; ℤ_{2}(i))} = ζ_{F} (1 − i)

up to odd multiples.

Let v_{2}(q) = v_{2}(a) − v_{2}(b) denote the 2-adic valuation of the rational number q = a/b.

Theorem 5.2. Let F = ℚ(\sqrt{m}). Then for all even i ≥ 2

v_{2}(ζ_{F} (1 − i)) = −1 − v_{2}(i)

when m = 2, while

v_{2}(ζ_{F} (1 − i)) = −v_{2}(i)

when m = p or m = 2p with p ≡ ±3 mod 8 prime.

Theorem 5.3. Let F = ℚ(ζ_{2^{m}} + ζ̄_{2^{m}}) with m ≥ 2. Then for all even i ≥ 2

v_{2}(ζ_{F} (1 − i)) = 2^{m−2} − (m + v_{2}(i)).

Theorem 5.4. Let F = ℚ(ζ_{q} + ζ̄_{q}) with q an odd prime power and 2 a primitive root mod q. Suppose Pic_{+}(R_{F}) has odd order. Then for all even i ≥ 2

v_{2}(ζ_{F} (1 − i)) = φ(q)/2 − (2 + v_{2}(i)).

This accounts for all 2-regular real quadratic or maximal real subfields of cyclotomic number fields. These results may be reformulated in terms of L-functions by the identity [Wa, 4.3]

ζ_{F} (s) = ∏_{χ∈X} L(s, χ)

where X is the group of Dirichlet characters associated to the Abelian field F, and s is complex. In turn the L-function values at negative integers can be expressed in terms of generalized Bernoulli numbers using the formula [Wa, 4.2]

L(1 − n, χ) = −\frac{B_{n,χ}}{n}

for integers n ≥ 1. We omit the easy transcriptions.

References

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[V] V. Voevodsky, The Milnor Conjecture, UIUC K-theory preprint server, no. 170 (1996).

[Wa] L. C. Washington, Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, vol. 83, Springer Verlag, 1982.

[W1] C. Weibel, Étale Chern classes at the prime 2, Algebraic K-theory and Algebraic Topology (P. Goerss and J. F. Jardine, eds.), NATO ASI Series C, vol. 407, Kluwer, 1993, pp. 249–286.

[W2] , The 2-torsion in the K-theory of the integers, C. R. Acad. Sci. Paris 324 (1997), 615–620.

[Wi] A. Wiles, The Iwasawa Conjecture for totally real fields, Annals of Math. 131 (1990), 493–540.

Department of Mathematics, University of Oslo, 0316 Oslo, Norway

   E-mail address: rognes@math.uio.no

Department of Mathematics, University of Oslo, 0316 Oslo, Norway

   E-mail address: paularne@math.uio.no