ON CYCLOTOMIC NUMBERS AND REDUCTION

MAP FOR K-THEORY OF THE INTEGERS

by Wojciech Gajda

Abstract. We apply recently proven compatibility of Beilinson and Soulé elements in K-theory to investigate density of rational primes p, for which the reduction map

K_{2n+1}(ℤ) → K_{2n+1}(𝔽_{p})

is nontrivial. Here n is an even, positive integer and 𝔽_{p} denotes the field of p elements. In the proof we use arithmetic of the cyclotomic numbers

u_{k}^{(n)} = ∏_{(a,l)=1; 1≤a<l^{k}}(1 − ξ_{l^{k}}^{a})^{a^{n}},

where l is an odd prime, k ≥ 1 and ξ_{l^{k}} is a primitive root of unity of order l^{k}. We investigate divisibility of the number u_{k}^{(n)} in the group of certain S-units of the cyclotomic field ℚ(ξ_{l^{k}}). It turns out that the divisibility of u_{k}^{(n)} is bounded from above by the maximal power of l dividing the index of the subgroup of K_{2n+1}(ℤ) generated by a nontorsion element which maps to the Beilinson element. We reformulate the Vandiver conjecture on the class group of the cyclotomic field ℚ(ξ_{l}) in terms of divisibility properties of u_{k}^{(n)}. Using the density computation on the reduction map we describe an infinite family of elements of the Brauer group of the field ℚ(ξ_{l^{k}}).

1. Introduction

It is known that the problem of computing K-groups of the rational integers is closely related to deep questions in number theory, such as the Vandiver conjecture and the Iwasawa cyclicity conjecture (cf. [K], section 3, [BG1], section 5 and [BG2]). On the other hand, investigation of the K-groups of ℤ can provide some valuable information about class groups of cyclotomic fields. Let A and A^{[i]} denote the Sylow l-subgroup of the ideal class group of ℚ(ξ_{l}) and the ith eigensubmodule of A under

2000 Mathematics Subject Classification. Primary 19F27; Secondary 19Fxx. Part of this work was done while the author was visiting the SFB 343 ,,Diskrete Strukturen in

der Mathematik” in Bielefeld and the Max-Planck-Institute für Mathematik in Bonn. He would like to thank both institutions for hospitality and support. In Poznań, the author was partially supported by the KBN grant 2 P03A 023 17.

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the action of the Galois group G(ℚ(ξ_{l})/ℚ), respectively. Kurihara observed in [K], Cor. 3.8 that the vanishing of K_{4}(ℤ) (proven by Rognes, Soulé and Voevodsky), implies that the eigenspace A^{[l−3]} is trivial. Soulé using K-theory and Voronoi’s reduction theory proved in [So3] that for a fixed number n the eigenspace A^{[l−n]} vanishes for primes l bigger than a certain large but explicitly computed number. We refer the reader to [BG1] and to [BG2], where the subgroup of infinitely divisible elements in K_{2m}(ℚ) was investigated in connection with the classical conjectures mentioned above.

Our first objective in this paper is to use K-theory to investigate certain lunits u_{k}^{(n)}, u_{k}^{(n)+} ∈ ℤ[ξ_{l^{k}}] (see Theorem B and Definition 2.1, below) which emerge naturally when one explores the structure of the group K_{2n+1}(ℤ) for n even and positive. The group has a nontorsion element b_{2n+1} which is provided by the compatibility of the Beilinson element in K_{2n+1}(ℤ) ⊗ ℚ with the Soulé elements in K_{2n+1}(ℤ)⊗ℤ_{l} for all odd primes l. The compatibility follows by results of Beilinson and Deligne, [BD], and also Huber and Wildeshaus, [HW] who proved Conjecture 6.2, [BK] of Bloch and Kato. Using standard techniques of K-theory we show that the element b_{2n+1} maps onto the element u_{k}^{(n)} ⊗ ξ_{l^{k}}^{⊗n} of the group of twisted Sunits of the cyclotomic field E = ℚ(ξ_{l^{k}}) (see Lemma 2.9, below). This implies the following result about the numbers u_{k}^{(n)} and u_{k}^{(n)+}.

Theorem A. [Th. 2.14] Assume that the l-torsion part of the group K_{2n+1}(ℤ) is trivial. Let l^{k_{0}} be the largest power of l which divides the index of the subgroup of K_{2n+1}(ℤ) generated by the element b_{2n+1}. Let k be a natural number not smaller than k_{0}.

(1) If u_{k}^{(n)} = η^{l^{k_{1}}} , for some η ∈ ℤ[ξ_{l^{k}}], then k_{1} ≤ k_{0}.

(2) The same property holds true for u_{k}^{(n)+}, if we assume in addition that l does not divide (n+1)C_{n}, where C_{n} denotes the product of denominators of the divided Bernoulli numbers \frac{B_{2}}{2} , \frac{B_{4}}{4} , …, \frac{B_{n}}{n} (written in the lowest terms).

It is well known that divisibility properties of cyclotomic units are related to the Vandiver conjecture on the ideal class group of the field ℚ(ξ_{l}) (see Section 8.3 of Washington’s book [W]). Such a relation exists also in the case of the divisibilty property considered in Theorem A. This is indicated by the following proposition which is probably known to experts.

Theorem B. [Prop. 2.15] Let l be an odd prime and let n be an even interger such that 2 ≤ n ≤ l−3. Let k be a positive integer. The following statements are equivalent.

(1) The number

u_{k}^{(n)} = ∏_{(a,l)=1; 1≤a<l^{k}}(1 − ξ_{l^{k}}^{a})^{a^{n}}

is not l-divisible in the group O_{E,S_{k}}^{×}. Here, S_{k} is any finite set of primes containing primes over l and such that the class group of O_{E,S_{k}} vanishes.

(2) The eigensubmodule A^{[l−1−n]} is trivial.

We deduce Theorem B from the formula for the index of the subgroup of the group H_{cts}^{1}(ℤ[\frac{1}{l}], ℤ_{l}(n+1)) generated by the Soulé cyclotomic element, which was proven by Bloch and Kato in [BK] using the main conjecture in Iwasawa theory. Note that Thaine shoved (see [Th], Prop. 2, p. 305), without using main conjecture, that A^{+} = ⨁_{even} A^{[i]} = 0, (i.e., the Vandiver conjecture holds true for l), if and only if none of u_{1}^{(n)} is an lth power in ℤ[ξ_{l}], for all n even and 2 ≤ n ≤ l−3.

Our second goal in this paper is to apply the arithmetic of the numbers u_{k}^{(n)} to investigate K_{∗}(ℤ). We consider the reduction map in K-theory of ℤ

K_{2n+1}(ℤ) ⟶ K_{2n+1}(𝔽_{p})_{l},

where l is a fixed odd prime and p is another prime such that l|(p^{n+1}−1). It follows by [AB], Theorem 1, p. 68 that the reduction map is nontrivial for infinitely many primes p, if n is an even, positive integer. Using the arithmetic of the Kummer fields E( \sqrt[l^{k}]{u_{k}}) and the theorem of Chebotarev we refine the result of [AB] in the case of the group K_{2n+1}(ℤ).

Theorem C. [Th. 3.6] Let l be an odd prime, n an even, positive integer such that l is relatively prime to n+1. Assume that l does not divide the index of the subgroup of K_{2n+1}(ℤ) generated by the element b_{2n+1}. Let k be a fixed positive integer. Denote by M_{k} the set of rational primes p which satisfy the following two conditions:

(1) the l-torsion part of the group K_{2n+1}(𝔽_{p}) is isomorphic to ℤ/l^{k}

(2) the reduction map K_{2n+1}(ℤ) ⟶ K_{2n+1}(𝔽_{p})_{l} is nontrivial.

Then the set M_{k} has positive density, which equals \frac{l^{k}−1}{l^{2k}}.

Observe that the set M = ∪_{k=1}^{∞}M_{k} consists of primes p for which the reduction map K_{2n+1}(ℤ) → K_{2n+1}(𝔽_{p})_{l} has nontrivial image. Since the sets M_{k} for k ≥ 1 are pairwise disjoint, it is natural to expect that the set M has Dirchlet density and that it is equal to ∑_{k=1}^{∞} \frac{l^{k}−1}{l^{2k}} = \frac{l}{l^{2}−1}. By definition, the density of M should be

lim_{x→∞} \frac{#\{p ≤ x: p ∈ M\}}{#\{p ≤ x\}}.

It does not seem to be known whether the limit exists.

In the last part of the paper we prove the following consequence of Theorem C.

Theorem D. [Prop 3.15] Let n, l and k be as in the assumptions of Theorem C. Then the following statements hold.

(1) The set of primes p for which the l-torsion part of the group K_{2n+1}(𝔽_{p}) is isomorphic to ℤ/l^{k} and such that the class of the central simple algebra A_{ξ_{l^{k}}} (u_{k}^{(n)}, p) is a nontrivial element of the Brauer group Br(E) has positive density which equals \frac{l^{k}−1}{l^{2k}}.

(2) Assume in addition that l does not divide the number (n+1)C_{n}. Then the set of primes for which the l-torsion part of the group K_{2n+1}(𝔽_{p}) is isomorphic to ℤ/l^{k} and such that the class of the algebra A_{ξ_{l^{k}}} (u_{k}^{(n)+}, p) in Br(E^{+}) is nontrivial has positive density which equals \frac{l^{k}−1}{l^{2k}}.

It is clear that the numbers u_{k}^{(n)} considered in this paper are implicitly present in the work of Soulé on cyclotomic elements in K-theory and étale cohomology (cf.

[So2] and proof of Lemma 2.9 given below). For some other applications of u_{k}^{(n)} (where k ≥ 1) to the arithmetic geometry, we refer the reader to papers by Ihara, see for example [I].

Acknowledgements

During the work on this paper I benefited from discussions or from correspondence with Grzegorz Banaszak, Radan Kućera and B.Z. Moroz. I am greatful to all of them. I would like to thank Don Zagier for suggesting Remark 2.17.

Notation

n is a positive, even integer l is a rational prime which is assumed to be odd, if not stated otherwise l^{k} is a fixed power of l which will be specified later ξ_{l^{k}} = e^{\frac{2πi}{l^{k}}} is the fixed primitive root of 1 of order l^{k}. E = ℚ(ξ_{l^{k}}) is the cyclotomic field

ℚ(ξ_{l^{k}})^{+} denotes the maximal real subfield of E A denote the Sylow l-subgroup of the ideal class group of ℚ(ξ_{l}) A^{[i]} for 0 ≤ i ≤ l−2 is the ith component of A cf. [W], p. 101 l^{k}||m means that l^{k} is the largest power of l which divides the natural number m

B_{l} denotes the lth torsion part of an abelian group B B[l^{k}] denotes the subgroup of the abelian group B consisting of elements od order

dividing l^{k} O_{F} denotes the ring of integers of the number field F O_{F,S} is the ring of S-integers of the number field F, defined for a set of primes S

K_{m}(R) denotes the mth algebraic K-theory group of the commutative ring R with

identity, as was defined by Quillen.

H_{cts}^{i}(−, −) denotes the ith continuous étale cohomology group, as was defined by Jannsen in [J].

2. Some computations with cyclotomic elements

Definition 2.1. Let k be a positive integer.

(1) We put

u_{k}^{(n)} = ∏_{1≤a<l^{k}, (a,l)=1} (1 − ξ_{l^{k}}^{a})^{a^{n}}

u_{k}^{(n)+} = ∏_{1≤a< \frac{l^{k}}{2} , (a,l)=1} [−ξ_{l^{k}}^{−a}(\frac{1 − ξ_{l^{k}}^{a}}{1 − ξ_{l^{k}}} )^{2}]^{a^{n}}.

¿From now on, in order to simplify notation we will not indicate that u_{k}^{(n)} and u_{k}^{(n)+} depend on n and denote them simply by u_{k} and u_{k}^{+}, respectively. This should not lead to any confusion, since most of the time the number n will be fixed.

(2) We denote by C_{n} the product of denominators of the numbers \frac{B_{2}}{2} , \frac{B_{4}}{4} , …, \frac{B_{n}}{n} (put in the lowest terms), where B_{m} denotes the mth Bernoulli number.

Our basic reference for the theory of cyclotomic fields is the book of Washington, [W]. For some more information on the numbers u_{k} we refer the reader to [Th], where the case k = 1 is treated and also to [I], where the case of arbitrary k is considered. Note that the number C_{n} defined above depends only on n. In order to check this, let us write B_{2i} = \frac{N_{2i}}{D_{2i}} and \frac{B_{2i}}{2i} = \frac{N′_{2i}}{D′_{2i}} with D_{2i} > 0, D′_{2i} > 0 and (D_{2i}, N_{2i}) = 1, (D′_{2i}, N′_{2i}) = 1, for 1 ≤ i ≤ \frac{n}{2}. It follows by the theorem proven by Glaisher [G] that a prime l divides the number D_{2i} if and only if it divides D′_{2i}. The theorem of Clausen-von Staudt ([W], Theorem 5.10. p. 56) implies that D_{2i} is a square free integer such that its prime factors are primes l for which l−1 divides 2i.

Lemma 2.2.

(1) The number u_{k} is an l-unit and the number u_{k}^{+} is a unit in the ring ℤ[ξ_{l^{k}}].

(2) Assume that l is relatively prime to the number (n+1)C_{n}. Then u_{k}^{+} ∈ ℤ[ξ_{l^{k}}+ξ_{l^{k}}^{−1}], i.e., the unit u_{k}^{+} is real.

Proof. (1) See [W], Chapter 8.

(2) Let σ_{−1} denote the complex conjugation. It is easy to check that

(2.3)  σ_{−1}(u_{k}^{+}) = ξ_{l^{k}}^{2 ∑′ a^{n}}u_{k}^{+},

where the summation is over a such that 1 ≤ a < \frac{l^{k}}{2} and (a, l) = 1. Observe that

2∑′a^{n} ≡ ∑_{1≤a<l^{k}, (a,l)=1} a^{n} mod l^{k},

hence it suffices to show that the last sum is divisible by l^{k}. Using the classical formula due to Jacob Bernoulli we obtain

∑_{1≤a<l^{k}, (a,l)=1} a^{n} = ∑_{1≤a≤l^{k}} a^{n} − ∑_{1≤a≤l^{k−1}} l^{n}a^{n}

= \frac{1}{n+1}[B_{n+1}(l^{k}) − B_{n+1}] − \frac{l^{n}}{n+1}[B_{n+1}(l^{k−1}) − B_{n+1}],

where

B_{m}(x) = ∑_{i=0}^{m} \binom{m}{i}B_{i}x^{m−i}.

By a simple calculation we obtain

∑_{1≤a<l^{k}, (a,l)=1} a^{n} = \frac{1}{n+1} ∑_{i=0}^{n} \binom{n+1}{i}B_{i}(l^{k(n+1−i)} − l^{(k−1)(n+1−i)+n})

= \frac{(l−1)l^{k(n+1)−1}}{n+1} + ∑_{i=1}^{n} \binom{n}{i−1}l^{k(n+1−i)}(1−l^{i−1})\frac{B_{i}}{i} . (2.4)

The claim (2) follows, since by assumption l^{k} divides the number on right hand side of the equality (2.4). □

Lemma 2.5. Let v_{l}(−) denote the l-adic valuation. Then we have

v_{l}(p^{n+1} − 1) = v_{l}(n+1) + v_{l}(p−1).

Proof. We use the l-adic logarithm. Let us put m = n+1. Then we have

log_{l}(p^{m}) = m log_{l}(1+(p−1)) = m l^{v_{l}(p−1)}ϵ_{1}

log_{l}(p^{m}) = log_{l}(1+(p^{m}−1)) = l^{v_{l}(p^{m}−1)}ϵ_{2},

where ϵ_{1} and ϵ_{2} are l-adic units. □

Consider the following sequence of maps.

(2.6)  \begin{matrix}K_{2n+1}(ℤ) \\ α^{0} & ↓ \\ (K_{2n+1}(ℤ) ⊗ ℤ_{l})/l^{k} \\ α^{1} & ↓ \\ H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n + 1))/l^{k} \\ α^{2} & ↓ \\ H_{cts}^{1}(O_{E,S_{k}}; ℤ_{l}(n + 1))/l^{k} \\ α^{3} & ↓ \\ H_{ét}^{1}(O_{E,S_{k}}; ℤ/l^{k}(n + 1)) \\ α^{4} & ↓ & ≅ \\ O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n) \end{matrix}

We use étale cohomology and the continuous étale cohomology H_{cts}^{∗}(−) in the sense of Jannsen, [J]. Note that

\begin{matrix}H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n + 1)) = \underleftarrow{lim} H_{ét}^{1}(ℤ[\frac{1}{l}]; ℤ/l^{m}(n + 1)) \\ H_{cts}^{1}(O_{E,S_{k}}; ℤ_{l}(n + 1)) = \underleftarrow{lim} H_{ét}^{1}(O_{E,S_{k}}; ℤ/l^{k}(n + 1)), \end{matrix}

since the lim^{1}-term vanishes, where the inverse system is over reduction of coefficients maps, [J], p.216, (3.1). Now we will describe maps in the diagram (2.6).

(i) The number k is a positive integer. It will be specified in Theorem 2.14 and further in the beginning of section 3.

(ii) The map labeled α_{0} is induced by the map

K_{2n+1}(ℤ) → K_{2n+1}(ℤ) ⊗ ℤ_{l}

which sends x to x ⊗ 1. (iii) The map α_{1} is induced by the Dwyer-Friedlander map from K-theory to

continous étale cohomology, [DF], Proposition 4.4 and Remark 8.8. It is a surjection by [DF], Theorem 8.7 and Remark 8.8. The map α_{1} is an isomorphism for primes l such that K_{2n+1}(ℤ)_{l} = 0.

(iv) We put E = ℚ(μ_{l^{k}}) and we choose S_{k} to be a finite set of primes of E such that it contains primes over l and that the class group of the ring of S_{k}-integers O_{E,S_{k}} vanishes.

(v) The arrow α_{2} is an imbedding which comes from the Hochschild-Serre spectral sequence of the injection ℤ[\frac{1}{l}] → O_{E,S_{k}}.

(vi) The map α_{3} is an imbedding which comes from the long exact sequence in cohomology induced by the exact sequence of l-adic sheaves

0 ⟶ ℤ_{l}(n+1) ⟶^{×l^{k}} ℤ_{l}(n+1) ⟶ ℤ/l^{k}(n+1) ⟶ 0.

(vii) Finally, the map α_{4} is an isomorphism provided by the Kummer exact

sequence, since by (iv) P ic O_{E,S_{k}} = 0. Consider the cyclotomic element

(2.7)  c_{l} = (c_{l^{m}}) ∈ H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n + 1)) = \underleftarrow{lim} H_{ét}^{1}(ℤ[\frac{1}{l}]; ℤ/l^{m}(n + 1))

constructed in cohomology by Soulé [So2], (4.1)-(4.3), for l odd. It is defined as the norm compatible system of elements

(2.8)  c_{l^{m}} = T r_{ℚ(μ_{l^{m}})/ℚ}((1 − ξ_{l^{m}}) ∪ ξ_{l^{m}}^{⊗n}),

where (1) ξ_{l^{m}} = e^{\frac{2πi}{l^{m}}} , 1 − ξ_{l^{m}} ∈ H^{1}(ℤ[ξ_{l^{m}}, \frac{1}{l}]; ℤ/l^{m}(1))

(2) ξ_{l^{m}}^{⊗n} ∈ H^{0}(ℤ[ξ_{l^{m}}, \frac{1}{l}]; ℤ/l^{m}(n))

(3) ∪ is the cup product and T r_{ℚ(μ_{l^{m}})/ℚ} is the transfer map in étale cohomology

T r_{ℚ(μ_{l^{m}})/ℚ}: H_{ét}^{1}(ℤ[ξ_{l^{m}}, \frac{1}{l}]; ℤ/l^{m}(n + 1)) ⟶ H_{ét}^{1}(ℤ[\frac{1}{l}]; ℤ/l^{m}(n + 1)).

In a similar way, using produtcs, Bott elements and transfer maps in K-theory with finite coefficients, Soulé constructed cyclotomic elements in K_{2n+1}(ℤ) ⊗ ℤ_{l}, for l odd, [So4], p. 384. Note that the cyclotomic elements of Soulé exist also when l = 2 in which case one has to modify the construction at the level m = 1, see [ABG], Def. 2.2 and Def. 1.5. The group K_{2n+1}(ℤ) has rank one by results of Borel, [Bo]. There is a nontrivial element in K_{2n+1}(ℤ) ⊗ ℚ which was constructed by Beilinson [Be], (7.1.5). Corollary 9.8, [HW] implies that there is a nontorsion element b_{2n+1} ∈ K_{2n+1}(ℤ) which maps to the Beilinson element in K_{2n+1}(ℤ) ⊗ ℚ and such that for l odd, the image of b_{2n+1} by

K_{2n+1}(ℤ) ⟶ K_{2n+1}(ℤ) ⊗ ℤ_{l} ⟶ H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n + 1))

is the cyclotomic element c_{l}. The second map above is the Dwyer-Friedlander map. Note that the element b_{2n+1}⊗1 is uniquely determined in the group K_{2n+1}(ℤ)⊗ℤ[\frac{1}{2}]. By some abuse of terminology, in the sequel we call b_{2n+1} ∈ K_{2n+1}(ℤ) the Beilinson element. In order to simplify notation we will use the same symbol for element of the group O_{E,S_{k}}^{×} and its image in the quotient group O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}}; what is meant will be clear from the context.

Lemma 2.9.

(1) The composition of maps in the diagram (2.6) sends the Beilinson element b_{2n+1} onto the element u_{k} ⊗ ξ_{l^{k}}^{⊗n} of the group O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n).

(2) Assume that l does not divide (n+1)C_{n}, where C_{n} is the number from Definition 2.1 (3). Then we have u_{k} ⊗ ξ_{l^{k}}^{⊗n} = u_{k}^{+} ⊗ ξ_{l^{k}}^{⊗n} in the group O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n).

Proof. (1) This is a strightforward calculation. We give details of the proof for the convenience of the reader. By the naturality of the long exact sequence in cohomology, the imbedding ℤ[\frac{1}{l}] → O_{E,S_{k}} induces the following commutative diagram.

(2.10)  \begin{matrix}H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n + 1))/l^{k} & ⟶^{α′_{3}} & H_{ét}^{1}(ℤ[\frac{1}{l}]; ℤ/l^{k}(n + 1)) \\ ^{α_{2}}↓ & & ^{α′_{2}}↓ \\ H_{cts}^{1}(O_{E,S_{k}}; ℤ_{l}(n + 1))/l^{k} & ⟶^{α_{3}} & H_{ét}^{1}(O_{E,S_{k}}; ℤ/l^{k}(n + 1)) \end{matrix}

It follows that the image of the element b_{2n+1} under the composition (2.6) is the same as if we replace the map α_{2} by α′_{3} and the map α_{3} by α′_{2}, respectively. This and the definition of c_{l} gives

α_{3}α_{2}α_{1}α_{0}(b_{2n+1}) = α′_{2}(T r_{ℚ(μ_{l^{k}})/ℚ}((1 − ξ_{l^{k}}) ∪ ξ_{l^{k}}^{⊗n}).

By the basic property of the transfer map, [So1], III.1.5, the last element equals

(2.11)  ∑_{σ_{a}∈G}(1 − ξ_{l^{k}})^{σ_{a}} ∪ ξ_{l^{k}}^{⊗nσ_{a}} ∈ H_{ét}^{1}(O_{E,S_{k}}; ℤ/l^{k}(n + 1)).

Here, we have put G = Gal(E/ℚ). By the bilinearity of the cup product the element (2.11) can be identified with

(2.12)  ∑_{1≤a<l^{k}, (a,l)=1}(1 − ξ_{l^{k}}^{a})^{a^{n}} ∪ ξ_{l^{k}}^{⊗n}.

Applying the isomorphism α_{4} to (2.12) gives

(2.13)  ∏_{1≤a<l^{k}, (a,l)=1}(1 − ξ_{l^{k}}^{a})^{a^{n}} ⊗ ξ_{l^{k}}^{⊗n} ∈ O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n).

To prove (2) observe that in the group O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}}

∏_{1≤a<l^{k}, (a,l)=1}(1 − ξ_{l^{k}}^{a})^{a^{n}} ≡ (1 − ξ_{l^{k}})^{∑ a^{n}} ∏_{1≤a<l^{k}, (a,l)=1}[\frac{1 − ξ_{l^{k}}^{a}}{1 − ξ_{l^{k}}} ]^{a^{n}}

≡ (1 − ξ_{l^{k}})^{∑ a^{n}} ∏_{1≤a< \frac{l^{k}}{2} , (a,l)=1}[\frac{1 − ξ_{l^{k}}^{a}}{1 − ξ_{l^{k}}} ]^{a^{n}}[\frac{1 − ξ_{l^{k}}^{(l^{k}−a)}}{1 − ξ_{l^{k}}} ]^{(l^{k}−a)^{n}}

≡ (1 − ξ_{l^{k}})^{∑ a^{n}} ∏_{1≤a< \frac{l^{k}}{2} , (a,l)=1} [−ξ_{l^{k}}^{−a}(\frac{1 − ξ_{l^{k}}^{a}}{1 − ξ_{l^{k}}} )^{2}]^{a^{n}},

where the sum is over a such that 1 ≤ a < l^{k} and (a, l) = 1. The claim (2) follows, since the assumption on l implies that l^{k} divides ∑ a^{n}, as was checked in the proof of Lemma 2.2 (2). □

As a consequence of the compatibility of the Beilinson and the Soulé elements we obtain the following divisibility property of the cyclotomic numbers u_{k} and u_{k}^{+}. It would be interesting to know whether Theorem 2.14 can be proven by an argument which does not rely on K-theory.

Theorem 2.14. Assume that the l-torsion part of the group K_{2n+1}(ℤ) is trivial. Let l^{k_{0}} be the largest power of l which divides the index of the subgroup of K_{2n+1}(ℤ) generated by the Beilinson element. Let k be a natural number which is not smaller than k_{0}.

(1) If u_{k} = η^{l^{k_{1}}} , for some η ∈ ℤ[ξ_{l^{k}}], then k_{1} ≤ k_{0}.

(2) The same property holds true for the real unit u_{k}^{+}, if we assume in addition that l does not divide (n+1)C_{n}.

Proof. The map α_{0} from (2.6) is a surjection because the group K_{2n+1}(ℤ) is finitely generated. Since b_{2n+1} is nontorsion and l^{k} ≥ l^{k_{0}}, the element α_{0}(b_{2n+1}) is divisible by at most l^{k_{0}} in (K_{2n+1}(ℤ) ⊗ ℤ_{l})/l^{k}. Let x be the generator of K_{2n+1}(ℤ) ⊗ ℤ_{l}. Observe that the last group equals ℤ_{l}, since by assumption K_{2n+1}(ℤ)_{l} = 0. Then b_{2n+1} ⊗ 1 = l^{k′}x, for some k′ and it is clear that l^{k_{0}} ≥ l^{k′}. The map α_{1} is an isomorphism, by [DF], Theorem 8.7 and Remark 8.8, and since K_{2n+1}(ℤ)_{l} = 0.

We prove (1). The order of the images of x by compositions of the imbeddings α_{1}, α_{2}, α_{3} and α_{4} is l^{k}, because all the groups in (2.6), except for the top one have exponent l^{k}. Hence, the image of the Beilinson element by (2.6) is divisible in the group

O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n)

by at most l^{k_{0}}. However, by Lemma 2.9 (1) u_{k} ⊗ ξ_{l^{k}}^{⊗n} is the image of the Beilinson element by the map (2.6). This implies (1). The second claim follows by (1) and Lemma 2.9 (2). □

In some sense the following proposition can be thought of as an analogue for groups of units of the corollary to the theorem of Furtwängler ([W], Cor. 10.6 and Cor. 10.5, p. 187) which shows that the Vandiver conjecture for the prime l is equivalent to vanishing of the l-Sylow subgroup of the class group of the field ℚ(ξ_{l^{k}})^{+}, for all k ≥ 1 .

Proposition 2.15. Let l be an odd prime and let n be an even interger such that 1 ≤ n ≤ l−3. Let k be a positive integer. The following statements are equivalent.

(1) The number

u_{k} = ∏_{(a,l)=1; 1≤a<l^{k}}(1 − ξ_{l^{k}}^{a})^{a^{n}}

is not l-divisible in the group O_{E,S_{k}}^{×}.

(2) The eigensubmodule A^{[l−1−n]} is trivial.

Proof. It follows by [K], Corollary 1.5, p. 226 that for an even n as in the assumptions above we have A^{[l−1−n]} = 0, if and only if, H_{cts}^{2}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)) = 0. Let C denote the subgroup of H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)) generated by the Soulé element c_{l}. Using the main conjecture of Iwasawa theory Bloch and Kato proved that the number of elements of the group H_{cts}^{2}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)) equals the index [H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)) ; C], ([BK], Theorem 6.1 (i), see formulas (6.8) and (6.9), p. 388). We prove that (1) implies (2). If A^{[l−1−n]} ≠ 0, then by results of Kurihara and Bloch and Kato loc. cit. we obtain that c_{l} = lx for some nonzero x ∈ H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)). It was shown in the proof of Lemma 2.9 (1) that the map (2.16)

H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)) ⟶ H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1))/l^{k} ⟶ O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n),

(where the second map is the composition of maps α_{2}, α_{3} and α_{4} from (2.6)) sends the element c_{l} to u_{k} ⊗ξ_{l^{k}}^{⊗n}. This gives u_{k} = d^{l}y^{l^{k}} = (dy^{l^{k−1}})^{l} for some d, y ∈ O_{E,S_{k}}^{×},

where d ⊗ ξ_{l^{k}}^{⊗n} ∈ O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n) is the image of x by the map (2.16). This contradicts (1).

We prove that (2) implies (1). Assume that A^{[l−1−n]} = 0. It follows by the results of Kurihara and Bloch and Kato that c_{l} generates the group H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)). At this point we can repeat the proof of Theorem 2.14 (1) in which we replace: the group K_{2n+1}(ℤ) by the group H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)), the element b_{2n+1} by the element c_{l} and the map α_{1} by the identity. Since c_{l} generates H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)), and since here k > k_{0} = 0, the same argument as in the proof of Theorem 2.14 (1) shows that u_{k} ∉ O_{E,S_{k}}^{×l}. □

Remark 2.17. I am indebted to Don Zagier for the following observation. There is a direct way of showing that for every k > 1 the following three statements are eequivalent

(1) u_{1} is an lth power in the group ℚ(ξ_{l})^{×}

(2) u_{1} is an lth power in the group ℚ(ξ_{l^{k}})^{×}

(3) u_{k} is an lth power in the group ℚ(ξ_{l^{k}})^{×}.

In order to see this, observe that for every n and k we have the following equalities in the group ℚ(ξ_{l^{k}})^{×}/ℚ(ξ_{l^{k}})^{×l}.

u_{k} ≡ ∏_{1≤b<l} [  ∏_{a≡b mod l; 1≤a<l^{k}}(1 − ξ_{l^{k}}^{a})]^{b^{n}}

≡ ∏_{1≤b<l} [ ∏_{1≤j≤l^{k−1}}(1 − ξ_{l^{k}}^{b+lj})]^{b^{n}}

≡ ∏_{1≤b<l} [ ∏_{λ^{l^{k−1}} =1}(1 − λξ_{l^{k}}^{b})]^{b^{n}}

≡ ∏_{1≤b<l} [(1 − (ξ_{l^{k}}^{b})^{l^{k−1}})]^{b^{n}}

≡ ∏_{1≤b<l}(1 − ξ_{l}^{b})^{b^{n}} = u_{1}

This shows that (2) and (3) are equivalent. Also, (1) clearly implies (2). To check that (2) implies (1), let us assume that u_{1} ∉ ℚ(ξ_{l})^{×l}. Consider the Kummer extension L = ℚ(ξ_{l},\sqrt[l]{u_{1}}). By assumption it has degree l over ℚ(ξ_{l}). Also, the Galois group G(L/ℚ) is not abelian (use Proposition 3.2, below for k = 1.) Since G(ℚ(ξ_{l^{k}})/ℚ) is obviously abelian, we see that L ⊄ ℚ(ξ_{l^{k}}), hence \sqrt[l]{u_{1}} ∉ ℚ(ξ_{l^{k}}).

Remark 2.18. According to the Quillen-Lichtenbaum conjecture the Dwyer-Fried-lander map

K_{2n+1}(ℤ) ⊗ ℤ_{l} ⟶ H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1))

should be an isomorphism for l odd. If we assume the conjecture, then by the result of Bloch and Kato loc. cit., the number l^{k_{0}} from Theorem 2.14 equals #H_{cts}^{2}(ℤ[\frac{1}{l}]; ℤ_{l}(n+1)). Hence, using again [K], Corollary 1.5, p. 226 we conclude that the following two statements are equivalent, if the Quillen-Lichtenbaum conjecture is true for ℤ and all odd primes.

(1) The Vandiver conjecture holds true for all odd primes l.

(2) The element b_{2n+1} ⊗ 1 generates the group K_{2n+1}(ℤ) ⊗ ℤ[\frac{1}{2}] for all even, positive n.

Remark 2.19. Recall that G. Banaszak and the author showed that the following statements are equivalent for a fixed odd prime l (see [BG1], Prop. 8 and Prop. 9. pp. 248).

(1) The Vandiver conjecture holds true for l.

(2) The groups of infinitely l-divisible elements D_{n+1}(ℚ) ⊂ K_{2n}(ℚ)_{l} vanish for all n, even and positive.

(3) The étale K-theory group K_{4}^{et}(ℤ[\frac{1}{l}, ξ_{l}+ξ_{l}^{−1}]) is trivial. We refer the reader to [BG2], [K], [K-NQD] and to [So3] for other results relating the conjectures on the class groups of the cyclotomic fields to the arithmetic of the groups K_{∗}(ℤ).

Remark 2.20. Note that Ichimura and Sakaguchi in [IS], Cor., p. 62 proposed another reformulation of the Vandiver conjecture in terms of Kummer character which is defined using the cyclotomic numbers u_{k}^{(n)} (see also [K-NQD], Cor. 4.7 (ii), p. 444).

3. Reduction map and densities of primes

As in the previous sections n is an even, positive integer. ¿From now on we assume that the odd prime l does not divide the number n+1 and that l is relatively prime to the index of the subgroup of K_{2n+1}(ℤ) generated by the Beilinson element b_{2n+1}. The last assumption means that the number k_{0} from Theorem 2.14 equals 0. Hence, in particular K_{2n+1}(ℤ)_{l} = 0. We consider the image of the reduction map

ϕ_{p,l}: K_{2n+1}(ℤ) ⟶ K_{2n+1}(𝔽_{p})_{l}

onto the l-torsion part of the K-theory of the finite field 𝔽_{p}, where p is an odd prime.

Consider the diagram of field extensions

L = E( \sqrt[l^{k}]{u_{k}})

E = ℚ(ξ_{l^{k}})

ℚ

where L denotes the splitting field of the polynomial x^{l^{k}}−u_{k}. By the assumption on l we have l^{k_{0}} = 1. It follows by Theorem 2.14 (1) that u_{k} is not an lth power, hence the polynomial x^{l^{k}}−u_{k} is irreducible over E. We have the exact sequence of Galois groups

(3.1)  0 ⟶ G(L/E) ⟶ G(L/ℚ) ⟶ G(E/ℚ) ⟶ 0.

For 1 ≤ a < l^{k} and (a, l) = 1 we denote by σ_{a} the element of G(E/ℚ) ≅ (ℤ/l^{k})^{×} such that

σ_{a}(ξ_{l^{k}}) = ξ_{l^{k}}^{a}.

Every element γ of G(L/E) ≅ ℤ/l^{k} is determined by its action on \sqrt[l^{k}]{u_{k}}. Let

γ( \sqrt[l^{k}]{u_{k}}) = ξ(γ) \sqrt[l^{k}]{u_{k}},

for an l^{k}th root of unity ξ(γ). It is clear that the order of γ in the group G(L/E) equals the order of the root ξ(γ). We consider the action of G(E/ℚ) on G(L/E) by conjugation. With this action the group G(L/ℚ) becomes a semidirect product G(L/E) ⋊ G(E/ℚ).

Proposition 3.2.

(1) For γ ∈ G(L/E) and σ_{a} ∈ G(E/ℚ) as above, we have σ_{a}γσ_{a}^{−1} = γ^{a^{n+1}}.

(2) Let γσ_{a} be an element of G(L/ℚ), where σ_{a} and γ are as in (1). Then for any integer m we have

(γσ_{a})^{m} = \{ \begin{matrix}γ^{x_{m}}σ_{a^{m}} & if a^{n+1} ≢ 1 mod l^{k} \\ γ^{m}σ_{a^{m}} & if a^{n+1} ≡ 1 mod l^{k}, \end{matrix}

where x_{m} = \frac{a^{m(n+1)}−1}{a^{(n+1)}−1}.

Proof. (1) To simplify notation, we put ξ_{0} = ξ(γ). Lift σ_{a} to σ̃_{a} ∈ G(L/ℚ). Let b be such that ab ≡ 1 mod l^{k}. Then in the group O_{E}^{×}/O_{E}^{×l^{k}} we have

σ_{a}^{−1}(u_{k}) ≡ σ̃_{b}(u_{k}) ≡ ∏_{1≤a<l^{k}, (a,l)=1} (1 − ξ_{l^{k}}^{ab})^{(ab)^{n}a^{n}} ≡ u_{k}^{a^{n}}.

Hence, there exists z ∈ O_{E}[\frac{1}{l}]^{×} such that equality

σ_{a}^{−1}(u_{k}) = u_{k}^{a^{n}}z^{l^{k}}

holds in the group O_{E}[\frac{1}{l}]^{×}. This gives the following equality in O_{L}[\frac{1}{l}]^{×}

σ_{a}^{−1}( \sqrt[l^{k}]{u_{k}}) = ( \sqrt[l^{k}]{u_{k}})^{a^{n}}zξ_{1},

where ξ_{1} ∈ O_{E}^{×} is a root of unity which depends on a. Now we compute

σ̃_{a}γσ_{a}^{−1}( \sqrt[l^{k}]{u_{k}}) = σ̃_{a}γ(( \sqrt[l^{k}]{u_{k}})^{a^{n}}zξ_{1})
= σ̃_{a}(ξ_{0}^{a^{n}}( \sqrt[l^{k}]{u_{k}})^{a^{n}}zξ_{1})
= σ̃_{a}(ξ_{0}^{a^{n}}) σ̃_{a}(( \sqrt[l^{k}]{u_{k}})^{a^{n}}zξ_{1})
= ξ_{0}^{a^{n+1}} \sqrt[l^{k}]{u_{k}}

(2) follows by (1), since by the multiplication rule in the semidirect product G(L/ℚ) = G(L/E) ⋊ G(E/ℚ) we have (γσ_{a})^{m} = γ^{1+a^{n+1}+...+a^{(n+1)(m−1)}}σ_{a^{m}}. □

Propostion 3.3. Let n and l be as specified above. Let l^{k} be the largest power of l which divides p−1. Let S_{k} be the finite set of primes of O_{E} from (2.6) (cf. condition (iv) below (2.6)), which we construct for p and l^{k}. Let w ∉ S_{k} be a prime of E over p and let w̃ be a prime of L = E( \sqrt[l^{k}]{u_{k}}) lying over w. We denote by F r_{w} the Frobenius element which generates the subgroup G(κw̃/κ_{w}) of G(L/E). Then the number of elements of the image of the reduction map

ϕ_{p,l}: K_{2n+1}(ℤ) → K_{2n+1}(𝔽_{p})_{l}

equals the order of the Frobenius element F r_{w}.

Proof. The proof is based on an idea of Rubin [Ru], p. 319, see also [AB], p. 67 or [BG1], section 4.4. Because v_{l}(n+1) = 0, it follows by Lemma 2.5 and by results of Quillen [Q2] that K_{2n+1}(𝔽_{p})_{l} = ℤ/l^{k}. Expanding the diagram (3.1) by adding reduction maps as horizontal arrows we obtain the following commutative diagram.

(3.4)  \begin{matrix}K_{2n+1}(ℤ) & ⟶^{ϕ_{p,l}} & K_{2n+1}(𝔽_{p})_{l} \\ ^{α_{0}}↓ & & ↓^{=} \\ (K_{2n+1}(ℤ) ⊗ ℤ_{l})/l^{k} & ⟶ & (K_{2n+1}(𝔽_{p}) ⊗ ℤ_{l})/l^{k} \\ ^{α_{1}}↓≅ & & ↓≅ \\ H_{cts}^{1}(ℤ[\frac{1}{l}]; ℤ_{l}(n + 1))/l^{k} & ⟶ & H^{1}(𝔽_{p}; ℤ_{l}(n + 1))/l^{k} \\ ^{α_{2}}↓ & & ↓ \\ H_{cts}^{1}(O_{E,S_{k}}; ℤ_{l}(n + 1))/l^{k} & ⟶ & H^{1}(κ_{w}; ℤ_{l}(n + 1))/l^{k} \\ ^{α_{3}}↓ & & ↓≅ \\ H_{ét}^{1}(O_{E,S_{k}}; ℤ/l^{k}(n + 1)) & ⟶ & H^{1}(κ_{w}; ℤ/l^{k}(n + 1)) \\ ^{α_{4}}↓≅ & & ↓≅ \\ O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⊗ ℤ/l^{k}(n) & ⟶^{ψ_{p,l}} & κ_{w}^{×}/κ_{w}^{×l^{k}} ⊗ ℤ/l^{k}(n) \end{matrix}

In the diagram (3.4) w ∉ S_{k} is a prime which lies over p and κ_{w} denotes its residue field. Let π′_{w}: O_{E,S_{k}} ⟶ κ_{w} be the reduction at the ideal w ∉ S_{k}. Let

(3.5)  π_{w}: O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⟶ κ_{w}^{×}/κ_{w}^{×l^{k}}

denote the map induced by π′_{w}. For x ∈ O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} we put ψ_{p,l}(x ⊗ ξ_{l^{k}}^{⊗n}) = π_{w}(x) ⊗ ξ_{l^{k}}^{⊗n}. Remaining horizontal arrows in (3.4) are the maps induced on Ktheory and on cohomology by reduction at p and at w. The right vertical arrows are imbeddings by the choices of l^{k}, S_{k} and since the cohomological dimension of a finite field is one. Let H denote the subgroup of O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} generated by the image of the number u_{k}. By assumptions on l and k the element α_{0}(b_{2n+1}) is a generator of (K_{2n+1}(ℤ) ⊗ ℤ_{l})/l^{k}. It follows by Lemma 2.9 (1) that the image of K_{2n+1}(ℤ) by the map in the left column of (3.4) equals H ⊗ ℤ/l^{k}(n). Since all vertical maps in the diagram (3.4), except α_{0} are injections, the image of the reduction map ϕ_{p,l} is isomorphic to the group π_{w}(H). Note that κw̃ = κ_{w}( \sqrt[l^{k}]{u_{k}}), because for local rings we have O_{L,w̃} = O_{E,w}[ \sqrt[l^{k}]{u_{k}}] what follows by a simple argument using the different (cf. [A], p. 85), since u_{k} is an l-unit and its minimal polynomial over O_{E,w} divides x^{l^{k}}−u_{k}. The Frobenius element F r_{w} generates the group G(κw̃/κ_{w}) ⊂ G(L/E) ≅ ℤ/l^{k}. On the other hand, by the Kummer theory, [L], Chapter VIII.8, Theorem 13, we have

π_{w}(H) ≅ Hom(G(κw̃/κ_{w}); ℤ/l^{k}(1)).

This establishes the claim. □

Theorem 3.6. Let n and l be as specified in the beginning of this section. Let k be a fixed natural number. Denote by M_{k} the set of rational primes p which satisfy the following two conditions:

(1) the l-torsion part of the group K_{2n+1}(𝔽_{p}) is isomorphic to ℤ/l^{k}

(2) the reduction map K_{2n+1}(ℤ) ⟶ K_{2n+1}(𝔽_{p})_{l} is nontrivial.

Then the set M_{k} has positive density, which equals \frac{l^{k}−1}{l^{2k}}.

In the proof of the theorem we will need some notation and a lemma. Consider the following field extensions

[figure: K; β; E′; L; w̃; E; w; ℚ; p]

where E = ℚ(ξ_{l^{k}}), E′ = ℚ(ξ_{l^{k+1}}) and L = E( \sqrt[l^{k}]{u_{k}}). We denote by K the composite of the fields E′ and L. For a prime p we choose a prime w in E lying over p and primes w̃ in L, and β in K such that β|w̃|w as is shown at the picture above.

Lemma 3.7. Let n be an even, positive integer and l be an odd prime. Assume that l^{k} is the largest power of l which divides p−1. Then

(1) E′ ∩ L = E

(2) G(K/E) = G(E′/E) ⊕ G(L/E).

(3) [K : ℚ] = l^{2k}(l−1).

Proof of Lemma 3.7. Observe that the group G(K/ℚ) is a semidirect product of G(K/E) by G(E/ℚ), where the latter group acts on the former by conjugation. It is clear that (1) implies (2) and (3). We sketch the proof of (1) which is a standard exercise in Galois theory. Since [E′ : E] = l, it is enough to show that E′ ⊄ L. If E′ ⊂ L, then the group G(L/ℚ) would contain an element of order ϕ(l^{k+1}), where ϕ(−) is the Euler function. However, an easy check based on the formula from Proposition 3.2 (2) shows that orders of elements in G(L/ℚ) divide either ϕ(l^{k}) or l^{k}. The assumptions on numbers l and n are essential for the last conclusion.

Proof of Theorem 3.6. We use the Chebotarev density theorem, [Se1], Theorem 1, p. 331. Let us fix k′ such that 1 ≤ k′ ≤ k. It follows by Proposition 3.3 that the image of reduction map on the l^{k}-torsion of K_{2n+1}(𝔽_{p}) has order l^{k′}, if and only if the following conditions hold

(1) p splits completely in E, i.e., restriction of the Frobenius element F r_{β} ∈ G(K/ℚ) to the field E is trivial

(2) F r_{β}|_{L} has order l^{k′}

(3) restriction of F r_{β} to E′ is nontrivial. Since G(E′/E) = ℤ/l, the last condi-

tion means that the order of F r_{β}|_{E′} is l. We count the number of elements of G(K/ℚ) which satisfy conditions (1)-(3). By Lemma 3.7 and Proposition 3.2 (1) the group G(K/ℚ) is a semidirect product of G(K/E) = G(E′/E) ⊕ G(L/E) by G(E/ℚ), where the action is given by

σ_{a}(τ, γ)σ_{a}^{−1} = (τ, γ^{a^{n+1}})

for τ ∈ G(E′/E), γ ∈ G(L/E) and σ_{a} ∈ G(E/ℚ). Let γ ∈ G(L/E) have order l^{k′}. The conjugacy class of (τ, γ) in G(K/ℚ) has

#{a^{n+1}: a ∈ (ℤ/l^{k′})^{×}}

elements. From this and Lemma 3.7 it follows easily that in G(K/ℚ) there are ϕ(l^{k′})(l−1) elements g such that

(a) g|_{E} = 1,

(c) g|_{L} has order l^{k′} and

(b) g|_{E′} has order l. Using this and the Chebotarev density theorem for the extension K/ℚ we obtain that the set of rational primes p for which the Frobenius F r_{β} meets conditions (1)-(3) for 1 ≤ k′ ≤ k, has Dirichlet density which equals

\frac{∑_{k′=1}^{k} ϕ(l^{k′})(l−1)}{l^{2k}(l−1)} = \frac{l^{k} − 1}{l^{2k}}. □

Remark 3.8. Assume that we fix k′ such that 1 ≤ k′ ≤ k. The proof of Theorem 3.6 shows that for l, p and n as in assumptions, the density of the primes p for which the image of the reduction map

K_{2n+1}(ℤ) ⟶ K_{2n+1}(𝔽_{p})_{l}

has order exactly l^{k′}, equals \frac{ϕ(l^{k′})(l−1)}{l^{2k}}.

Remark 3.9. The set

M = ∪_{k=1}^{∞}M_{k}

consists of primes p for which the reduction map K_{2n+1}(ℤ) → K_{2n+1}(𝔽_{p})_{l} has nontrivial image. Since the sets M_{k} for k ≥ 1 are pairwise disjoint, it seemes natural to expect that the set M has density

∑_{k=1}^{∞} \frac{l^{k}−1}{l^{2k}} = \frac{l}{l^{2} − 1}.

By definition, the density of M should be

lim_{x→∞} \frac{#\{p ≤ x: p ∈ M\}}{#\{p ≤ x\}}.

It is an interesting question whether the above limit exists.

Theorem 3.6 has the following consequence for K_{2} of the cyclotomic field E = ℚ(ξ_{l^{k}}). Let l and p be two odd primes. Assume that n and l meet conditions formulated in the beginning of this section. If l^{k}||p^{n+1}−1 for k > 0, then l^{k}||p−1 by Lemma 2.5 and under the assumptions we have K_{2n+1}(𝔽_{p})_{l} = ℤ/l^{k} by [Q2]. We consider the Galois symbol map

A_{ξ_{l^{k}}}: K_{2}(E) ⟶ Br(E)

which to a symbol {a, b} associates the class of the central simple algebra A_{ξ_{l^{k}}} (a, b) ∈ Br(E) cf. [M], Chapter 15. It follows by the main result of [MS] that the map A_{ξ_{l^{k}}} induces an isomorphism

(3.10)  K_{2}(E)/l^{k} ⟶^{≅} Br(E)[l^{k}].

Let S_{k} be the same set of primes as in condition (iv) below the diagram (2.6). Choose a prime w ∉ S_{k} in E which lies over p. Consider the symbol {u_{k}, p}, where u_{k} is the l-unit from Definition 2.1. Let v_{w}(−) denote the valuation at the prime w. For l and p as above v_{w}(u_{k}) = 0 and v_{w}(p) = 1. Hence, the value of the tame symbol map ∂′_{w} at {u_{k}, p} is

(3.11)  ∂′_{w}\{u_{k}, p\} = (−1)^{v_{w}(u_{k})v_{w}(p)}π′_{w}(\frac{u_{k}^{v_{w}(p)}}{p^{v_{w}(u_{k})}} ) = π′_{w}(u_{k}),

where π′_{w}: O_{E,S_{k}}^{×} → κ_{w}^{×} denotes the reduction map at the ideal w. Clearly,

∂_{w}{u_{k}, p} = π_{w}(u_{k}),

where ∂_{w}: K_{2}(E)/l^{k} ⟶ κ_{w}^{×}/κ_{w}^{×l^{k}} and π_{w}: O_{E,S_{k}}^{×}/O_{E,S_{k}}^{×l^{k}} ⟶ κ_{w}^{×}/κ_{w}^{×l^{k}} are the maps induced by ∂′_{w} and π′_{w}, respectively.

Lemma 3.12. The following conditions are equivalent.

(1) A_{ξ_{l^{k}}} (u_{k}, p) ≠ 0 in the group Br(E).

(2) There exists a prime w of E such that w|p and π_{w}(u_{k}) ≠ 0 in κ_{w}^{×}/κ_{w}^{×l^{k}}.

Proof. By the calculation of the tame symbol above, it remains to show that (1) implies (2). Consider the commutative diagram.

(3.13)  \begin{matrix}K_{2}(E)/l^{k} & ⟶ & ⨁_{v} K_{2}(E_{v})/l^{k} \\ ≅↓ & & ≅↓ \\ Br(E)[l^{k}] & ⟶ & ⨁_{v} Br(E_{v})[l^{k}] \end{matrix}

The direct sums in (3.13) are over finite places of E. The vertical arrows are isomorphisms of [MS]. The upper horizontal arrow is given by the power residue symbols (,)_{v} cf. [M], Theorem 16.1. The lower horizontal map in (3.13) is induced by the localization map. It is an injection because, by the class field theory we have the short exact sequence

(3.14) 0 ⟶ Br(E)[l^{k}] ⟶ ⨁_{v} Br(E_{v})[l^{k}] ⟶^{Σ} ℤ/l^{k} ⟶ 0.

The direct sum in (3.14) is over finite places, since the field E is totally imaginary. Assume that A_{ξ_{l^{k}}} (u_{k}, p) ≠ 0 and denote by A_{v} its image in the Brauer group Br(E_{v}) by the lower horizontal map of (3.13). Observe that the symbol (u_{k}, p)_{v} may be non trivial only at v|p or v|l. This follows easily by the formula (3.11) and [Se2], Corollary 8, p. 211. Since there is exactly one prime over l in E, the exactness of (3.14) implies that there is at least one prime w|p such that (u_{k}, p)_{w} is nontrivial. By commutativity of (3.13) we obtain A_{w} ≠ 0. Using again (3.11) and [Se2] loc. cit. we conclude that ∂_{w}{u_{k}, p} = π_{w}(u_{k}) ≠ 0. □

Theorem 3.6 gives the density of primes p for which π_{w}(u_{k}) is nontrivial. Thus Lemma 3.11 and Theorem 3.6 imply the following result. Claim (2) of Proposition 3.15 below follows by an easy transfer argument, if we replace u_{k} by u_{k}^{+} in the calculation and apply Lemma 2.9 (2).

Proposition 3.15.

(1) The set of primes p for which the l-torsion part of the group K_{2n+1}(𝔽_{p}) is isomorphic to ℤ/l^{k} and such that the class of the algebra A_{ξ_{l^{k}}} (u_{k}, p) is a nontrivial element of the Brauer group Br(E) has positive density which equals \frac{l^{k}−1}{l^{2k}}.

(2) Assume in addition that l does not divide the number (n+1)C_{n}. Then the set of primes for which the l-torsion part of the group K_{2n+1}(𝔽_{p}) is isomorphic to ℤ/l^{k} and such that the class of the algebra A_{ξ_{l^{k}}} (u_{k}^{+}, p) in Br(E^{+}) is nontrivial has positive density which equals \frac{l^{k}−1}{l^{2k}}.

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Department of Mathematics, Adam Mieckiewicz University, Matejki 48/49, 60769 Poznań, POLAND

   E-mail address: GAJDA@math.amu.edu.pl