ALGEBRAIC K-THEORY AND TWISTED RECIPROCITY LAWS

BRUNO KAHN

Abstract. We extend the classical reciprocity laws of algebraic number theory to higher odd K-groups of algebraic integers. Moreover, we show that a suitable renormalisation of the corresponding “twisted Legendre symbols” is independent “in the average” from the chosen prime when the number field considered is Galois over Q. This seems new even in the classical case.

To Hyman Bass for his 70th birthday

Introduction

In this paper we generalise the classical reciprocity laws of algebraic number theory from K_{1} (the units) to higher odd K-groups of rings of algebraic integers. Classically these laws concern “symbols” with values in roots of unity, while our symbols take their values in twisted roots of unity: for this reason we like to think of the laws governing them as twisted reciprocity laws.

Recall that, if K is a number field, 𝔭 is a finite prime of K and a ∈ K^{∗} is such that v_{𝔭}(a) = 0, the corresponding generalised Legendre symbol is defined by the formula

(\frac{a}{𝔭}) = f^{\frac{N(𝔭)−1}{m}} (mod 𝔭)

where m is the number of roots of unity of K. In fact this is an m-th root of unity in the residue field κ(𝔭), but it lifts canonically to an m-th root of unity of K and is thought of as such. The generalised Legendre symbol is then extended into a generalised Jacobi symbol by

(\frac{a}{𝔞}) = ∏_{𝔭}(\frac{a}{𝔭})^{v_{𝔭}(𝔞)}

provided that the fractional ideals (a) and 𝔞 are coprime. A weak form of the reciprocity law (actually equivalent to the strong form) is that, for any a, there is a module 𝔪, i.e. a linear combination of primes of K with nonnegative integer coefficients, such that

Date: 22-7-2003.

(*)  (\frac{a}{(f)}) = 1 if f ≡ 1 (mod^{∗} 𝔪).

The most conceptual way to prove this formula is to interpret (\frac{a}{𝔭})

as the Hilbert symbol (a, π)_{𝔭}, where π is a uniformising parameter of the completion K_{𝔭}. Hilbert symbols are now defined at all 𝔭 (including infinite places), not just those 𝔭 prime to a, and (*) follows from the Hilbert reciprocity law

∏_{𝔭}(a, b)_{𝔭} = 1

for any a, b ∈ K^{∗}. This gives a more precise form of (*). An excellent and synthetic exposition of all this is given in the exercises of Cassels-Fröhlich [4].

We carry this over in the following way: we fix a rational prime number l. For i > 1, we associate to each finite prime 𝔭 ∤ l a twisted Legendre symbol (see Definition 2.7 a))

(\frac{−}{𝔭}) : K_{2i−1}(K) → H^{0}(K, Q_{l}/Z_{l}(i))

and prove a reciprocity law for them (Theorem 2.6). Actually, for l = 2 we are obliged to assume that K is totally imaginary for technical reasons (see Remarks 2.11 and 4.4). The proof is very parallel to the one in the classical context: we introduce twisted Hilbert symbols and prove a reciprocity formula for them (Theorem 4.3), from which Theorem 2.6 follows. In order to define the wild Hilbert symbols, i.e. those at places above l, we have to use Dwyer-Friedlander’s étale K-theory: this makes our approach somewhat less elementary than one might hope at the outset. The twisted Hilbert reciprocity law then follows easily from (a small part of) the Poitou-Tate exact sequence.

The most intriguing result however comes when we modify the twisted Legendre symbols by changing their normalisation, i.e. multiplying

them by certain integers prime to l: we get variants (\frac{−}{−})^{(l)} (see Def-

inition 2.7 b)). Then, in Theorem 2.9, we show that (\frac{−}{𝔭})^{(l)} is inde-

pendent of 𝔭 “in the average”, at least when K is Galois over Q. This seems new even in the classical case i = 1. In the proof, the l-adic integer exp(l) (exp(4) for l = 2) appears mysteriously… This contrasts starkly with the result of Arlettaz-Banaszak [1] which states that the map K_{2i−1}(K) → ∏_{𝔭} K_{2i−1}(κ(𝔭)) has finite kernel. We don’t know the exact significance of Theorem 2.9; Soulé has suggested that it may have to do with the “field(s) with one element”. Trying to prove it was motivated by [11] (see E.3 at the end of that paper, and Scholium 2.12 in the present one).

This paper is structured as follows. In Section 1 we give an easier, finite characteristic motivational analogue of our characteristic 0 results. In that case the results (Theorems 1.2 and 1.8) are much better, more general and easier to prove: they are closely related to Suslin’s rigidity theorem [25]. In Section 2 we define the twisted Legendre and Jacobi symbols and their variants and state our main results, Theorems 2.6 and 2.9. The next sections up to Section 7 are devoted to the proofs. Finally, in Section 8, we give a computation of the twisted Hilbert symbol in terms of the Artin symbol, which connects our definition with a more classical approach of the Hilbert symbol.

We hope in the future to mix the results of Sections 1 and 2 to get twisted reciprocity laws for (at least some) smooth projective varieties over a number field.

Theorems 2.6 and 2.9 (the latter in the cyclotomic case) were found in the early nineties and explained in an Oberwolfach K-theory conference; Sections 1 and 8 are new.

I thank T. Nguyen Quang Do and M. Kolster for both recalling me that the generalised Moore reciprocity law of Theorem 4.3 is originally due to G. Banaszak [2], a fact I had embarrassingly forgotten in the preliminary version of this paper! I also wish to thank the referee for helpful comments.

1. The situation in characteristic p

We start by recalling Suslin’s rigidity theorem [24]: let k be an algebraically closed field and X a smooth, projective, connected curve over k. Let n be an integer prime to char k. Then, for any two closed points x, y ∈ X, the specialisation maps

K_{∗}(X, Z/n) \begin{matrix}i_{x}^{∗} \\ ⇉ \\ i_{y}^{∗} \end{matrix} K_{∗}(k, Z/n)

are equal.

Keep the same situation but relax the assumptions that dim X = 1 and k is algebraically closed. For x ∈ X_{(0)}, we define the specialisation map at x

s_{x} : K_{∗}(X, Z/n) → K_{∗}(k, Z/n)

as the composition

K_{∗}(X, Z/n) ⟶^{i_{x}^{∗}} K_{∗}(k(x), Z/n) ⟶^{N_{k(x)/k}} K_{∗}(k, Z/n).

These specialisation maps extend by linearity to a pairing

Z_{0}(X) × K_{∗}(X, Z/n) → K_{∗}(k, Z/n).

By homotopy invariance of algebraic K-theory, this pairing factors as another pairing

(1.1)  CH_{0}(X) × K_{∗}(X, Z/n) → K_{∗}(k, Z/n).

Recall that the proof of Suslin’s theorem is just that, when k is algebraically closed (and dim X = 1), (1.1) factors through the degree map Pic(X) → Z because Pic^{0}(X) is divisible as the group of k-points of an abelian variety. The same argument, using [3, Lemma 1.4], gives:

1.1. Proposition. If k is algebraically closed, the pairing (1.1) factors through the degree map deg : CH_{0}(X) → Z. □

Of course we have an analogous pairing without coefficients:

(1.2)  CH_{0}(X) × K_{∗}(X) → K_{∗}(k).

1.2. Theorem. Suppose that k is algebraic over F_{p} and that X satisfies the Beilinson-Parshin conjecture: K_{i}(X) is torsion for i > 0 (e.g. X is a curve, or an abelian variety on which the Tate conjecture holds, cf. [14]). Then (1.2) factors through the degree map.

Proof. Let k̄ be an algebraic closure of k. Suppose first ∗ > 0. By Quillen [19], K_{∗}(k) → K_{∗}(k̄) is injective; hence we reduce to the case where k = k̄. The argument is then the same as Suslin’s.

For ∗ = 0, the theorem still holds, simply because Ker(deg) is torsion (Kato-Saito, [15]) and K_{0}(k) = Z. □

1.3. Remark. Assume that X is geometrically connected, i.e. k is its field of constants. Recall that X has a 0-cycle of degree 1 [23, Lemma 1 p. 325] Any such 0-cycle z defines a retraction s : K_{∗}(X) → K_{∗}(k) of the map K_{∗}(k) → K_{∗}(X). A reformulation of Theorem 1.2 is that s does not depend on z and that, for any x ∈ X_{(0)}, one has

s_{x} = deg(x)s.

From now on we assume that X is geometrically connected, as in Remark 1.3. We are going to prove a similar but more bizarre theorem. First:

1.4. Lemma. Let F_{q} be a finite field and n ≥ 1, i ≥ 1 two integers. Then the composition

K_{2i−1}(F_{q^{n}}) ⟶^{N} K_{2i−1}(F_{q}) ⟶^{ι} K_{2i−1}(F_{q^{n}})

equals multiplication by \frac{q^{ni} − 1}{q^{i} − 1} = \frac{|K_{2i−1}(F_{q^{n}})|}{|K_{2i−1}(F_{q})|}.

Proof. By Quillen [19], K_{2i−1}(F_{q}) → K_{2i−1}(F_{q^{n}}) is injective and Frobenius acts on K_{2i−1}(F_{q^{n}}) by multiplication by q^{i}. □

1.5. Lemma. Let l be a prime number and u ∈ C_{l} be such that

v(\frac{u − 1}{2}) > 0, where v is the valuation of C_{l}. Then, for any integer n > 0, we have

\frac{u^{n} − 1}{n(u − 1)} ≡ 1 (mod \frac{u − 1}{2}O_{l})

where O_{l} is the valuation ring of C_{l}. In particular, v(\frac{u^{n} − 1}{u − 1}) = v(n).

Proof. Let a = u − 1 and b = u^{s} − 1. The identity

\frac{(1 + a)^{st} − 1}{sta} = \frac{(1 + b)^{t} − 1}{tb} · \frac{(1 + a)^{s} − 1}{sa}

and an induction on the number of prime factors of n reduces us to two cases:

• (n, l) = 1;

• n = l. In the first case, we have

\frac{(1 + a)^{n} − 1}{a} = n + \binom{n}{2}a + ⋯ ≡ n (mod \frac{a}{2}O_{l})

hence the claim.

In the second case, we have

\frac{(1 + a)^{l} − 1}{la} = 1 + (l − 1)\frac{a}{2} + ⋯ ≡ 1 (mod \frac{a}{2}O_{l})

again. □

1.6. Corollary. With notation as in Lemma 1.4, assume that

v_{l}(\frac{q^{i} − 1}{2}) > 0. Then, we have for any x ∈ K_{2i−1}(F_{q^{n}})\{l\}:

2(ιN(x) − nx) = 0.

This follows from Lemmas 1.4 and 1.5. □

1.7. Definition. Let l be a prime number and x ∈ X_{(0)}.

a) We write

deg_{l}(x) = | deg(x)|_{l}^{−1}

where | · |_{l} denotes the normalised l-adic absolute value (it is just the l-part of deg(x)).

b) We define

s_{x}^{(l)} : K_{2i−1}(X) → K_{2i−1}(k){l}

by

s_{x}^{(l)}(a) = ι^{−1} |\frac{N(x)^{i} − 1}{q^{i} − 1}|_{l}^{−1} i_{x}^{∗}a.

Note that, by Lemma 1.5 and Corollary 1.6, we have

s_{x}^{(l)}(a) = ι^{−1} deg_{l}(x)i_{x}^{∗}a
π_{l}(2(ιs_{x}(a) − deg(x)i_{x}^{∗}(a))) = 0

provided that v_{l}(\frac{q^{i} − 1}{2}) > 0 (where π_{l} is the projection on the lprimary part).

1.8. Theorem. a) If l ≠ 2, then for any x ∈ X:

s_{x}^{(l)} = deg_{l}(x)s

where s is as in Remark 1.3.

b) If l = 2, then then for any x ∈ X:

s_{x}^{(l)} = deg_{l}(x)s + s′

where s is as in Remark 1.3 and 2s′ = 0. In particular s_{x}^{(l)} does not depend on x for deg_{l}(x) = 1.

Proof. If v_{l}(\frac{q^{i} − 1}{2}) > 0, this follows from Remark 1.3 and the remarks

preceding the statement of the theorem. If v_{l}(\frac{q^{i} − 1}{2}) = 0, when l > 2 we have K_{2i−1}(k){l} = 0 and the statement of a) if empty. When l = 2, this group is reduced to Z/2 and the statement of b) is still empty (take s′ = s_{x}^{(l)} − deg_{l}(x)s). So Theorem 1.8 is true in all cases. □

1.9. Remark. Suppose X is a curve. According to the (still unproven!) Quillen-Lichtenbaum conjecture, the natural map

K_{2i−1}(k) ⊗ K_{0}(X) → K_{2i−1}(X)

should induce an isomorphism

K_{2i−1}(k){l} ⊕ K_{2i−1}(k){l} ⟶^{∼} K_{2i−1}(X){l}

for any l ≠ char k, via the canonical isomorphism K_{0}(X) = Z⊕Pic(X) and the degree map deg : Pic(X) → Z. At least, it is immediate to check that for a point x ∈ X, the composition of s_{x} (resp. s_{x}^{(l)}) with this homomorphism is (deg(x), 0) (resp. (deg_{l}(x), 0)). Therefore, Theorems 1.2 and 1.8 are not particularly interesting in this case once the conjecture is proven. Nevertheless we have given full proofs as a motivation for the sequel of this article, the number field case. We shall get an analogue to (1.2) (Theorem 2.6) and a partial analogue to Theorem 1.8 (Theorem 2.9).

2. The number field case

From now on, we fix a prime number l. If F is a field of characteristic ≠ l and i > 0, we write

(2.1)  m_{i}(F) = |H^{0}(F, Q_{l}/Z_{l}(i))|.

Then m_{i}(F) is finite in the following cases:

• F is finitely generated.

• F is a p-adic field.

• F = R and i is odd; then m_{i}(F) = \begin{cases}1 & if l ≠ 2 \\ 2 & if l = 2. \end{cases}

Suppose first that F is finite with q elements. It follows from [19] that one has isomorphisms

(2.2)  K_{2i−1}(F){l} ≃ H^{0}(F, Q_{l}/Z_{l}(i)).

For the sequel, we need to fix such isomorphisms explicitly. Quillen’s theorem implies that the map

K_{2i−1}(F) → H^{0}(F, K_{2i−1}(F̄))

is an isomorphism, where F̄ is an algebraic closure of F. Therefore it suffices to produce an explicit isomorphism K_{2i−1}(F̄){l} →^{∼} Q_{l}/Z_{l}(i).

If (n, q) = 1, we have an isomorphism

K_{2i}(F̄, Z/n) ⟶^{∼} _{n}K_{2i−1}(F̄)

since K_{2i}(F̄) = 0. For i = 1, this gives an isomorphism

K_{2}(F̄, Z/n) ⟶^{∼} μ_{n} = Z/n(1).

On the other hand, the product in K-theory with coefficients gives an isomorphism (take n large enough to avoid nonexistent problems with the product…)

K_{2}(F̄, Z/n)^{⊗i} ⟶^{∼} K_{2i}(F̄, Z/n).

Putting this together, we get isomorphisms for all n prime to q

nK_{2i−1}(F̄) ⟶^{∼} Z/n(i)

hence the desired isomorphisms in the limit, since the groups K_{2i−1}(k̄) are torsion prime to q. These isomorphisms obviously commute with extension and restriction of scalars.

Let now F be a finite extension of Q_{p} (p ≠ l), O_{F} its valuation ring and k its residue field. There are canonical maps

(2.3)  K_{∗}(O_{F}) → K_{∗}(k)

hence a canonical map

(2.4)  i_{F}^{∗} : K_{2i−1}(O_{F}) → H^{0}(k, Q_{l}/Z_{l}(i))

defined as the composition of (2.3), the projection of K_{2i−1}(k) onto its l-primary component and the explicit isomorphism (2.2) described above.

Let finally K be a number field and O_{K} its ring of integers. Throughout the paper we write

R_{K} := O_{K}[1/l].

Let 𝔭 a prime of R_{K}. Composing i_{K_{𝔭}} with the canonical map K_{2i−1}(R_{K}) → K_{2i−1}(O_{K_{𝔭}}), we get a canonical map

(2.5)  i_{𝔭}^{∗} : K_{2i−1}(R_{K}) → H^{0}(κ(𝔭), Q_{l}/Z_{l}(i)).

On the other hand, there is a canonical injection (2.6)

ι : H^{0}(K, Q_{l}/Z_{l}(i))(←^{∼})^{−1}H^{0}(R_{K}, Q_{l}/Z_{l}(i)) ⟶ H^{0}(κ(𝔭), Q_{l}/Z_{l}(i))

that we use to identify H^{0}(K, Q_{l}/Z_{l}(i)) to a subgroup of H^{0}(κ(𝔭), Q_{l}/Z_{l}(i)).

2.1. Lemma. Let m_{i} = m_{i}(K) and m_{i}(𝔭) = m_{i}(κ(𝔭)). Then, if

v_{l}(m_{i}/2) > 0, the quotient \frac{m_{i}(𝔭)}{m_{i}} does not depend on the choice of

i. This happens as soon as l − 1 | i (l odd) or 2 | i (l = 2).

Proof. Let G_{K} be the absolute Galois group of K, G_{l} its maximal lramified quotient, G_{𝔭} the absolute Galois group of κ(𝔭) and κ : G_{K} → Z_{l}^{∗} the cyclotomic character. Since the maximal l-cyclotomic extension of K is l-ramified, κ factors through G_{l}. Let Γ_{K} = κ(G_{K}) = κ(G_{l}) ⊆ Z_{l}^{∗} and Γ_{𝔭} = κ(G_{𝔭}). Then Γ_{𝔭} is a subgroup of finite index of Γ_{K}.

If l > 2, we may write

Γ_{K} = μ_{d} × (1 + l^{n}Z_{l})
Γ_{𝔭} = μ_{d_{𝔭}} × (1 + l^{n_{𝔭}}Z_{l})

with d, d_{𝔭} prime to l. If l = 2, we may write

Γ_{K}^{2} = 1 + 2^{n+1}Z_{l}
Γ_{𝔭}^{2} = 1 + 2^{n_{𝔭}+1}Z_{l}

with 2 ≤ n ≤ n_{𝔭}.

For l > 2, a simple computation shows that m_{i} ≠ 1 ⟺ d | i (resp. m_{i}(𝔭) ≠ 1 ⟺ d_{𝔭} | i), and in these cases

m_{i} = l^{n+v_{l}(i)}
m_{i}(𝔭) = l^{n_{𝔭}+v_{l}(i)}

hence

\frac{m_{i}(𝔭)}{m_{i}} = l^{n_{𝔭}−n}.

The reasoning for l = 2 is similar. □

2.2. Definition. For 𝔭 prime to l, we denote the integer of Lemma 2.1 by deg_{l}(𝔭). For a fractional ideal 𝔞 prime to l, we define

deg_{l}(𝔞) = ∑_{𝔭∤l}v_{𝔭}(𝔞) deg_{l}(𝔭).

2.3. Remarks. 1) Keep the notation in the proof of Lemma 2.1. Then q = N(𝔭) generates Γ_{𝔭}. Hence for l odd we may compute d_{𝔭}, n_{𝔭} as follows:

d_{𝔭} = order of q in (Z/l)^{∗}
n_{𝔭} = v_{l}(q^{d_{𝔭}} − 1) = v_{l}(q^{l−1} − 1)

while for l = 2

n_{𝔭} = v_{2}(q^{2} − 1) − 1.

2) It is probably not true that for a given 𝔭, the set of primes l such that deg_{l}(𝔭) ≠ 1 is finite, but at least this set seems “small”. For example, if K = Q and p ≠ l is a rational prime, then (assuming l odd), with notation as in the proof of Lemma 2.1, we have d = l − 1, n = 1 (Gauß). Taking the previous remark into account, this shows that deg_{l}(p) ≠ 1 if and only if

p^{l−1} ≡ 1 (mod l^{2}).

This condition appears in proofs of the first case of Fermat’s Last Theorem by Wieferich et al. in the early 20th century; it has been checked numerically that it occurs very rarely. A heuristic argument shows that the number of such l ≤ x should be asymptotic to log log x.

On the other hand, for a given l, the set of 𝔭 such that deg_{l}(𝔭) ≠ 1 has a density (∈]0, 1[ except when l = 2 and K is exceptional) by Čebotarev’s density theorem, cf. [8].

2.4. Definition. a) Let 𝔭 ∤ l. Let m_{i} = m_{i}(K) and N(𝔭) the norm of 𝔭 (the cardinality of the residue field). For a ∈ K_{2i−1}(R_{K}), we set

s_{𝔭}(a) = (\frac{a}{𝔭}) = ι^{−1} (\frac{N(𝔭)^{i} − 1}{m_{i}}i_{𝔭}^{∗}(a)) ∈ H^{0}(K, Q_{l}/Z_{l}(i))

where i_{𝔭}^{∗} (resp. ι) is the map of (2.4) (resp. (2.6)).

b) Let 𝔞 be a fractional ideal prime to l. We set

(\frac{a}{𝔞}) = ∑_{𝔭}v_{𝔭}(𝔞) (\frac{a}{𝔭}) .

These are the twisted Legendre symbol and the twisted Jacobi symbol.

2.5. Example. For i = 1 we recover the classical generalised Legendre and Jacobi symbols (with an additive notation).

2.6. Theorem. Suppose that either l is odd or K is totally imaginary.Then there is a modulus 𝔪_{i} with support on prime divisors of l

such that, for all a ∈ K_{2i−1}(R_{K}), (\frac{a}{(f)}) = 0 for any f ∈ K^{∗} such that

f ≡ 1 (mod^{∗} 𝔪_{i}).

2.7. Definition. a) For a prime 𝔭 ∤ l and a ∈ K_{2i−1}(R_{K}), we set

s_{𝔭}^{(l)}(a) = (\frac{a}{𝔭})^{(l)} = ι^{−1} \frac{m_{i}(𝔭)}{m_{i}}i_{𝔭}^{∗}(a) ∈ H^{0}(K, Q_{l}/Z_{l}(i))

so that we have

s_{𝔭}(a) = \frac{N(𝔭)^{i} − 1}{m_{i}(𝔭)}s_{𝔭}^{(l)}(a).

b) Let 𝔞 be a fractional ideal prime to l. We set

(\frac{a}{𝔞})^{(l)} = ∑_{𝔭}v_{𝔭}(𝔞) (\frac{a}{𝔭})^{(l)} .

Note that, by Lemma 2.1, we have

s_{𝔭}^{(l)}(a) = ι^{−1} deg_{l}(𝔭)i_{𝔭}^{∗}(a)

as soon as v_{l}(\frac{m_{i}}{2} ) > 0.

Note also that working with s_{𝔭}(a) or s_{𝔭}^{(l)}(a) is equivalent in the sense

that \frac{N(𝔭)^{i} − 1}{m_{i}(𝔭)} is prime to l. The same is not true for (\frac{a}{𝔞}) and (\frac{a}{𝔞})^{(l)}

when 𝔞 is not prime. 2.8. Definition. Let l be a prime number. An extension L/K is lcyclotomic if L ⊆ K(μ_{l^{∞}}). 2.9. Theorem. Let K be a Galois extension of Q and K_{0} the maximal l-cyclotomic subextension of K. Then

a) If l is odd, there exists a homomorphism s : K_{2i−1}(R_{K}) → H^{0}(K, Q_{l}/Z_{l}(i)) such that, for any rational number r prime to l, one has identically

(\frac{a}{(rR_{K})})^{(l)} = \frac{deg_{l}(rR_{K})}{[K : K_{0}]}s(a).

b) If l = 2 and K_{0} is totally imaginary, there exist two homomorphisms s_{1}, s_{2} : K_{2i−1}(R_{K}) → H^{0}(K, Q_{2}/Z_{2}(i)), with 2s_{1} = 0, such that, for any rational number r with v_{2}(r) = 0, one has identically

(\frac{a}{rR_{K}} )^{(2)} = \frac{r − 1}{2}s_{1}(a) − \frac{deg_{2}(rR_{K})}{2[K : K_{0}]}s_{2}(a).

(In the course of the proof it will be shown that \frac{deg_{l}(rR_{K})}{2[K : K_{0}]} is an

l-integer for all l, which gives a meaning to the formulas.) 2.10. Corollary. In Theorem 2.9, assume that K = K_{0} and that r is a prime number p. Let n = v_{l}([K : Q]) + 1.

a) If l is odd, n ≥ 2 and p is inert in Theorem 2.9, then s_{p}^{(l)}(a) does not depend on p.

b) If l = 2, n ≥ 3 and p ≡ ±5 (mod 8) in Theorem 2.9, then s_{p}^{(l)}(a)

only depends on the classical Legendre symbol (\frac{−1}{p}).

Indeed, in a), p inert means that pR_{K} is prime and the hypothesis n ≥ 2 implies that deg_{l}(pR_{K}) = 1. Similarly, in b), n ≥ 3 implies deg_{2}(pR_{K}) = 2. □

Theorem 2.6 will be proven in Section 5 as a corollary of a reciprocity formula (Theorem 5.5). Theorem 2.9 will be proven in Section 7; moreover, explicit values of s, s_{1} and s_{2} will be given in the proof.

2.11. Remark. Theorems 2.6, 2.9 and 5.5 are consequences of Theorem 4.3 which states that a certain sequence is a complex. The only reason why the formally real case is excluded from the first three theorems for l = 2 is that we cannot prove Theorem 4.3 in this case. (See Remark 4.4 in this respect.) Granting it, let us indicate what the statements of Theorem 2.6 and 2.9 become in the real case:

• Theorem 2.6 remains the same for i ≢ 1 (mod 4). For i ≡ 1 (mod 4), the condition on f must be strengthened by requiring that f is totally positive.

• In the formula of Theorem 2.9 b), two changes occur: the factor

\frac{deg_{2}(rR_{K})}{2[K : K_{0}]} must be replaced by \frac{deg_{2}(rR_{K})}{[K : K_{0}]} (this is explained

by formulas (7.1)) and when i ≡ 1 (mod 4) a third summand of the form sgn(r)s_{3}(a) appears, where sgn(r) is the sign of r (viewed in Z/2) and 2s_{3} = 0.

2.12. Scholium. It follows from the work of Harris-Segal [8] and Dwyer-Friedlander-Mitchell [6] that, for l odd, there are canonical homomorphisms

β^{i} : H^{0}(K, Q_{l}/Z_{l}(i)) → K_{2i−1}(R_{K}).

This is still true if l = 2 and \sqrt{−1} ∈ K. (See [11] for a homotopical strengthening of this result and [7] for an extension to the case l = 2, K non-exceptional.) These homomorphisms are number-theoretic analogues of the homomorphisms K_{∗}(k) → K_{∗}(X) where X is a smooth curve over a field k, as in Section 1. By its very construction, β^{i} is a retraction of s_{𝔭}^{(l)} when 𝔭 is a prime of K such that κ(𝔭) has the “same cyclotomy” as K, i.e. deg_{l}(𝔭) = 1. In general, one checks easily that

s_{𝔭}^{(l)} ∘ β^{i} = deg_{l}(𝔭).

Theorem 2.9 says that s_{𝔭} is independent of 𝔭 “in the average”, at least for Galois extensions K of Q (strictly speaking, this interpretation is correct only when K is cyclotomic). In this sense it is a weak analogue of Theorem 1.8.

3. Comparing pairings

For a topological Z_{l}^{∗}-module A and i ∈ Z, we denote by A(i) the topological Z_{l}^{∗}-module with new action ρ^{i} given by ρ^{i}(u)a = u^{i}ρ(u)a, where ρ is the original action.

Let i ≠ 0, and let Γ be an open subgroup of Z_{l}^{∗}. The groups H^{0}(Γ, Q_{l}/Z_{l}(i)) and H^{0}(Γ, Q_{l}/Z_{l}(−i)) are then finite and cyclic of the same order. Let m be an integer dividing this order. We have a perfect duality of finite groups

(3.1) α_{m} : H^{0}(Γ, Z/m(−i)) × H^{0}(Γ, Z/m(i)) → H^{0}(Γ, Z/m) = Z/m.

These pairings are related as follows:

3.1. Lemma. Let m | m′ dividing the order of H^{0}(Γ, Q_{l}/Z_{l}(i)). Then, under the pairings α_{m} and α_{m′}, the transpose of the inclusion map H^{0}(Γ, Z/m(i)) ↪ H^{0}(Γ, Z/m′(i)) is multiplication by m′/m. □

3.2. Proposition. Let χ_{i} : Γ → Z/m be the character given by the formula

χ_{i}(γ) = \frac{γ^{i} − 1}{m} (mod m)

where m divides |H^{0}(Γ, Q_{l}/Z_{l}(i))|. Then the Bockstein homomorphism β : H^{0}(Γ, Z/m(i)) → H^{1}(Γ, Z/m(i)) associated to the exact sequence of Γ-modules

0 → Z/m(i) → Z/m^{2}(i) → Z/m(i) → 0

equals cup-product by χ_{i}.

Proof. Let x ∈ H^{0}(Γ, Z/m(i)). A 1-cocycle c representing βx may be described as follows: lift x to x̃ ∈ Z/m^{2}(i). For γ ∈ Γ, we then have (γ − 1)x̃ = mc(γ) for a well-defined c(γ) ∈ Z/m(i) = mZ/m^{2}(i). Proposition 3.2 follows. □

Let now F be a finite extension of Q_{p}, with p ≠ l. We denote by k its residue field, and let q := |k|. For m | m_{i}(F) (see (2.1)), we have a composite isomorphism θ_{m}:

(3.2) H^{0}(F, Z/m(i)) ⟶^{∼} H^{0}(F, Z/m(−i))^{∗} ⟶^{∼} H^{2}(F, Z/m(i + 1))

where the first isomorphism stems from (3.1) and the second one comes from local Tate duality.

3.3. Remark (compare [18, Remark 15.9]). The isomorphism θ_{m} may also be described as cup-product by e_{m}, where e_{m} ∈ _{m}Br(F) = H^{2}(F, Z/m(1)) is the generator such that inv(e_{m}) = 1 ∈ Z/m, where inv : _{m}Br(F) → Z/m is the isomorphism stemming from class field theory, with the normalisation that Frobenius elements correspond to uniformising parameters. This is obvious.

Finally, we have a residue isomorphism

∂ : H^{2}(F, Z/m(i + 1)) ⟶^{∼} H^{1}(k, Z/m(i))

a specialisation isomorphism

(3.3)  s : H^{0}(F, Z/m(i)) ⟶^{∼} H^{0}(k, Z/m(i))

and a Bockstein isomorphism

β : H^{0}(k, Z/m(i)) ⟶^{∼} H^{1}(k, Z/m(i))

already considered in Proposition 3.2.

3.4. Proposition. For l ≠ p, the following diagram commutes:

H^{0}(F, Z/m(i)) ⟶^{θ_{m}} H^{2}(F, Z/m(i + 1))
s↓  ^{\frac{q^{i}−1}{m_{i}(F)} ∂}↓
H^{0}(k, Z/m(i)) ⟶^{β} H^{1}(k, Z/m(i)).

Proof. We have a commutative diagram of pairings

H^{0}(F, Z/m(−i)) × H^{2}(F, Z/m(i + 1))→ _{m}Br(F) ⟶^{inv} Z/m
s×∂↓  _{∂′}↓  ||
H^{0}(k, Z/m(−i)) × H^{1}(k, Z/m(i)) →H^{1}(k, Z/m) ⟶^{inv_{0}} Z/m

where inv_{0} carries a character χ to χ(F rob), where F rob is the arithmetic Frobenius. The result follows from this and Proposition 3.2, since the cyclotomic character of F rob is q. □

4. Generalised Moore reciprocity

We shall use Dwyer-Friedlander’s étale K-theory [5]. Recall that, for any Z[1/l]-scheme X of finite étale l-cohomological dimension, there are groups K_{∗}^{ét}(X), maps K_{∗}(X) → K_{∗}^{ét}(X) and a spectral sequence [5, Prop. 5.1]

E_{2}^{p,q} = H_{ét}^{p}(X, Z_{l}(−q/2)) ⇒ K_{−p−q}^{ét}(X)

where the E_{2}-terms are continuous étale cohomology groups^{1}.

If k is a finite field, then the Dwyer-Friedlander spectral sequence degenerates into isomorphisms

K_{2i−1}^{ét}(k) ⟶^{∼} H^{1}(k, Z_{l}(i)) (i ∈ Z).

4.1. Lemma. For i > 0, let ρ_{1} : K_{2i−1}(k) → H^{0}(k, Q_{l}/Z_{l}(i)) be the homomorphism from (2.2) and ρ_{2} the composite homomorphism K_{2i−1}(k) → K_{2i−1}^{ét}(k) → H^{1}(k, Z_{l}(i)). Then the diagram

1To avoid fringe effects we work with the non-connective version of Dwyer-Friedlander’s étale K-theory, although this makes in fact no difference in the cases we’ll consider.

H^{0}(k, Q_{l}/Z_{l}(i))
ρ_{1} ↗
K_{2i−1}(k)  ^{β}↓
ρ_{2} ↘
H^{1}(k, Z_{l}(i))

commutes, where β is the Bockstein homomorphism from the exact sequence

0 → Z_{l}(i) → Q_{l}(i) → Q_{l}/Z_{l}(i) → 0.

Proof. The Dwyer-Friedlander spectral sequence may be interpreted as the descent spectral sequence associated to the weak equivalence

K^{ét}(k) ⟶^{∼} ℍ_{ét}^{·}(k, K^{ét})

where we use Thomason’s hypercohomological notation. We may smash this weak equivalence with another spectrum and get a new spectral sequence. Smashing it in particular with the homotopy fibre sequence

(4.1)  𝕊 → M(Z[1/l]) → M(Q_{l}/Z_{l}) → Σ𝕊

where 𝕊 is the sphere spectrum and M(Z[1/l]), M(Q_{l}/Z_{l}) are Moore spectra, we get a commutative diagram of homotopy fibre sequences

K^{ét}(k)  ⟶^{∼}  ℍ_{ét}^{·}(k, K^{ét})
↓  ↓
K^{ét}(k) ∧ M(Z[1/l]) ⟶^{∼} ℍ_{ét}^{·}(k, K^{ét} ∧ M(Z[1/l]))
↓  ↓
K^{ét}(k) ∧ M(Q_{l}/Z_{l}) ⟶^{∼} ℍ_{ét}^{·}(k, K^{ét} ∧ M(Q_{l}/Z_{l}))
↓  ↓
ΣK^{ét}(k)  ⟶^{∼}  ℍ_{ét}^{·}(k, ΣK^{ét})

and the bottom square shows that the diagram of edge homomorphisms

K_{2i}^{ét}(k, Q_{l}/Z_{l}) ⟶ H^{0}(k, Q_{l}/Z_{l}(i))
δ^{ét}↓  ^{β}↓
K_{2i−1}^{ét}(k) ⟶ H^{1}(k, Z_{l}(i))

commutes, where δ^{ét} is the Bockstein coming from (4.1).^{2} In view of the commutative diagram

K_{2i}(k, Q_{l}/Z_{l}) ⟶ K_{2i}^{ét}(k, Q_{l}/Z_{l})
δ↓^{≀}  _{δ^{ét}}↓
K_{2i−1}(k){l} ⟶ K_{2i−1}^{ét}(k)

we are reduced to showing that the composition of δ^{−1} with the map K_{2i}(k, Q_{l}/Z_{l}) → K_{2i}^{ét}(k, Q_{l}/Z_{l}) → H^{0}(k, Q_{l}/Z_{l}(i)) coincides with the map defined at the beginning of Section 2. For this, we may further replace k by its algebraic closure k̄: then it is clear by construction (see [5, Proofs of Prop. 5.1 and Prop. 5.2] for the identification of the E_{2}-terms in the Dwyer-Friedlander spectral sequences). □

For i > 0 and if X is essentially of finite type over Z[1/l] or a p-adic field, we have H_{ét}^{0}(X, Z_{l}(i)) = 0 and this yields canonical homomorphisms

(4.2)  ch_{i+1} : K_{2i}(X) → K_{2i}^{ét}(X) → H_{ét}^{2}(X, Z_{l}(i + 1))

which are natural in X.

For X = Spec F with F a finite extension of Q_{p}, we get in this way a composite map

K_{2i}(F) → H^{2}(F, Z_{l}(i + 1)) → H^{2}(F, Z/m(i + 1))

when m | m_{i}(F).

Composing with the inverse of the isomorphism θ_{m} of (3.2), we get a homomorphism

h_{m} : K_{2i}(F) → H_{ét}^{0}(F, Z/m(i)).

By Tate duality and Lemma 3.1, these homomorphisms are related as follows:

4.2. Lemma. Let m | m′ | m_{i}(F). Then h_{m} = \frac{m′}{m} h_{m′}. □

We shall write h^{(i)} for the composition of h_{m_{i}(F)} with the isomor-

phism H^{0}(F, Z/m_{i}(F)(i)) →^{∼} H^{0}(F, Q_{l}/Z_{l}(i)).

Suppose X = Spec R. If l is odd, we define h^{(i)} to be the zero map. If l = 2, we cannot define a map h^{(i)} as above by the same method. However, by Suslin [25, Th. 4.9], K_{2i}(R) is uniquely divisible for i ≢ 1 (mod 4) (i > 0) and is the direct sum of a group of order 2 and a uniquely divisible group for i ≡ 1 (mod 4). We define h^{(i)} to be 0 for i ≢ 1 (mod 4) and to be the unique nontrivial homomorphism to Z/2 = H^{0}(R, Q_{2}/Z_{2}(i)) for i ≡ 1 (mod 4).

2Note that, for any Noetherian site X of finite cohomological dimension, any sheaf of spectra F on X and any spectrum T, the natural map ℍ^{·}(X, F) ∧ T → ℍ^{·}(X, F ∧ T) is a weak equivalence.

Coming back to our number field K, we get for each noncomplex place v a homomorphism

h^{(i)}(v) : K_{2i}(K) → H^{0}(K_{v}, Q_{l}/Z_{l}(i)).

4.3. Theorem (Moore [17, Th. 7.4] for i = 1, Banaszak [2, Th. 1] for l odd). Let m_{i} = m_{i}(K) and, for any noncomplex place v of K, m_{i}(v) = m_{i}(K_{v}). Then, at least in the following cases:

• l is odd

• l = 2 and K is totally imaginary

the sequence

K_{2i}(K){l} ⟶^{(h^{(i)}(v))} ⨁_{v noncomplex} H^{0}(K_{v}, Q_{l}/Z_{l}(i))

⟶^{( \frac{m_{i}(v)}{m_{i}})}  H^{0}(K, Q_{l}/Z_{l}(i)) → 0

is exact, where \frac{m_{i}(v)}{m_{i}} is interpreted as 0 if m_{i}(v) is infinite (this

happens exactly when v is real and i is even).

(The exactness will not be used in the proof of Theorems 2.6 and 2.9: only the fact that this sequence is a complex.)

For l odd, this theorem is due to Banaszak (loc. cit.), but he does

not make explicit the constants \frac{m_{i}(v)}{m_{i}} which are crucial for the sequel.

We give an alternate, slightly simpler prof. Proof. We shall get this exact sequence as a direct limit of exact sequences

K_{2i}(O_{S})\{l\} ⟶^{(h^{(i)}(v))} ⨁_{v∈S} H^{0}(K_{v}, Q_{l}/Z_{l}(i)) ⟶^{( \frac{m_{i}(v)}{m_{i}})} H^{0}(K, Q_{l}/Z_{l}(i)) → 0

where S runs through the finite sets of primes of K containing all infinite primes and all primes dividing l.

For such an S, as long as cd_{l}(O_{S}) < ∞ we have the maps (4.2) at our disposal: this happens exactly when either l is odd or l = 2 and K is totally imaginary. Hence a commutative diagram in these cases

K_{2i}(O_{S})  ⟶  ⨁_{v∈S} K_{2i}(K_{v})
↓  ↓
H^{2}(O_{S}, Z_{l}(i + 1)) ⟶ ⨁_{v∈S} H^{2}(K_{v}, Z_{l}(i + 1)).

In this diagram the left vertical map is surjective by Dwyer-Fried-lander [5, Th. 8.7] (assuming \sqrt{−1} ∈ K if l = 2) and by Rognes-Weibel [22] for l = 2 in general. Hence it suffices to prove that the sequence

H^{2}(O_{S}, Z_{l}(i + 1)) → ⨁_{v∈S} H^{2}(K_{v}, Z_{l}(i + 1))

≃ ⨁_{v∈S} H^{0}(K_{v}, Q_{l}/Z_{l}(i)) → H^{0}(K, Q_{l}/Z_{l}(i)) → 0

is exact to get that the K-theory sequence above is exact. This follows by a passage to the inverse limit from a part of the Poitou-Tate exact sequences [16, p. 70, Th. 4.10]

H^{2}(O_{S}, Z/l^{ν}(i + 1)) → ⨁_{v∈S} H^{2}(K_{v}, Z/l^{ν}(i + 1))

→ H^{0}(K, Z/l^{ν}(−i))^{∗} → 0

since all groups involved in this sequences are finite, provided we iden-

tify the last map with the collection of \frac{m_{i}(v)}{m_{i}}. But this map is the dual

of the natural map

H^{0}(K, Z/l^{ν}(−i)) → ∏_{v∈S} H^{0}(K_{v}, Z/l^{ν}(−i))

hence the claim by Lemma 3.1. □

4.4. Remark. The only use we make of the surjectivity results of Dwyer-Friedlander and Rognes-Weibel is to prove that the complex of Theorem 4.3 is exact. For l = 2 and K formally real, it is likely that the sequence of Theorem 4.3 is still a complex; however we cannot use the maps (4.2) to prove it in this case. On the other hand it is unlikely that this sequence is still exact. The correct point of view would be to replace algebraic K-groups by motivic cohomology groups (defined for example by means of Bloch’s cycle complexes, generalised to the case of Dedekind domains by Levine). Then the analogue of (4.2) is easy to construct in full generality. The fact that the resulting complex is exact is a consequence of the Milnor conjecture (Voevodsky). The defect of exactness for algebraic K-theory would correspond to the difference between the latter and motivic cohomology in the real case (see [12]).

To make this approach compatible with the current one, however, would involve comparing the natural map from motivic cohomology to étale cohomology with the edge homomorphism (4.2) of Dwyer-Friedlander. This appears quite nontrivial and would go far beyond the relatively elementary approach of this paper.

5. Proof of Theorem 2.6

5.1. Definition. Let F be a local field, a ∈ K_{2i−1}(F) and f ∈ F^{∗}. We set

(a, f) = h^{(i)}(a · {f}) ∈ H^{0}(F, Q_{l}/Z_{l}(i)).

This is the generalised Hilbert symbol. If 𝔭 is a prime of K and a ∈ K_{2i−1}(K), f ∈ K^{∗}, we set

(a, f)_{𝔭} = (ϕ_{∗}a, ϕ_{∗}f)

where ϕ is the inclusion K → K_{𝔭}.

5.2. Proposition. Let p be a prime number ≠ l and F be a finite extension of Q_{p}, with normalised valuation v, valuation ring O_{F} and residue field k. Let m be the order of H^{0}(k, Q_{l}/Z_{l}(i)) and q the cardinality of k. Then, for any f ∈ F^{∗}, the diagram

K_{2i−1}(O_{F}) ⟶^{·{f}}  K_{2i}(O_{F})
v(f)i_{F}^{∗} ↓  _{h^{(i)}}↓
H^{0}(k, Q_{l}/Z_{l}(i)) ⟵^{\frac{m}{q^{i}−1} s} H^{0}(F, Q_{l}/Z_{l}(i))

commutes, where s is the map of (3.3) and i_{F}^{∗} is the map of (2.5).

Proof. This follows from the functoriality and multiplicative properties of the Dwyer-Friedlander spectral sequence, and Proposition 3.4. □

5.3. Corollary. Let K be a number field and 𝔭 a prime of K not dividing l. Let i > 0. Then, for any (a, f) ∈ K_{2i−1}(R_{K}) × K^{∗}, one has

\frac{m_{i}(𝔭)}{m_{i}}(a, f)_{𝔭} = v_{𝔭}(f) (\frac{a}{𝔭}) .

5.4. Lemma. Let F be a finite extension of Q_{l}. Then, for any a ∈ K_{2i−1}(O_{F}), the map

F^{∗} → H^{0}(F, Q_{l}/Z_{l}(i))
f ↦ (a, f)

is continuous.

Proof. This is obvious, since the target group is a finite l-group. (Use the fact that the topology of U_{F}^{(1)} coincides with the topology given by l^{n}-th powers.) □

5.5. Theorem (Reciprocity). Suppose that either l is odd or K is totally imaginary. Then, for any a ∈ K_{2i−1}(R_{K}) and any f ∈ K^{∗} with divisor prime to l, we have

(\frac{a}{(f)}) = − ∑_{𝔩|l} \frac{m_{i}(𝔩)}{m_{i}}(a, f)_{𝔩}.

(This is a higher analogue of formula (**) of [4, ex. 2.10 p. 353].) Proof. Apply Theorem 4.3 to a · {f} and use Corollary 5.3. □

Proof of Theorem 2.6. This follows immediately from Theorem 5.5 and Lemma 5.4. □

5.6. Remark. The proof of Theorem 2.6 gives an easy way to estimate the moduli 𝔪_{i}: for 𝔩 | l, it suffices to choose v_{𝔩}(𝔪_{i}) large enough so that 1 + 𝔪_{𝔩}^{v_{𝔩}(𝔪_{i})} ⊆ K_{𝔩}^{∗m_{i}}, where 𝔪_{𝔩} is the maximal ideal of the valuation ring of K_{𝔩}. This shows that 𝔪_{i} depends on i, at least a priori.

6. Somme horrible computations

6.1. Lemma. a) Let u ∈ U^{(1)}, where U^{(1)} is the group of principal units

of C_{l}. Suppose that v(u − 1) ≥ inf(\frac{1}{l − 2}, \frac{3}{2}), where v is the valuation of C_{l}. Then we have

u ≡ exp(u − 1) (mod \frac{(u − 1)^{2}}{2} O_{l})

where O_{l} is the valuation ring of C_{l}.

b) Let u be a unit of C_{l}. Assume that v(u^{d} − 1) ≥ inf(\frac{1}{l − 2}, \frac{3}{2}) for a suitable integer d prime to l. Then, for any integer n > 0 multiple of d, we have

u ≡ exp(\frac{u^{n} − 1}{n}) (mod^{∗} (U^{(v_{1})})^{l^{a}})

where v_{1} = v (\frac{(u^{d} − 1)^{2}}{2} ) and l^{a} is the largest power of l dividing n.

c) Suppose l = 2. Let u be a unit of Z_{2} and let ν = v_{2}(u^{2} − 1) − 1. Then, for any integer n > 0, we have

u ≡ (−1)^{\frac{u−1}{2}} exp(\frac{u^{2n} − 1}{2n}) (mod^{∗} U^{2^{a+2ν−1}})

where 2^{a} is the largest power of 2 dividing n.

(We shall use this lemma only for u ∈ Q_{l}.) Proof. a) Write

exp(u − 1) = u + \frac{(u − 1)^{2}}{2} + ⋯ + \frac{(u − 1)^{k}}{k!} + …

First note that, if v(u − 1) ≥ inf(\frac{1}{l − 2}, \frac{3}{2}), then v(u − 1) > \frac{1}{l − 1} and the exponential series converges. It is now sufficient to show that,

under the former assumption, v (\frac{(u − 1)^{k}}{k!} ) ≥ v (\frac{(u − 1)^{2}}{2} ) for all

k > 2. The left hand side is

v (\frac{(u − 1)^{k}}{k!} ) = kv(u − 1) − \frac{k − S_{l}(k)}{l − 1}

where S_{l}(k) is the sum of the digits of k written in base l. We now have

v (\frac{(u − 1)^{k}}{k!} ) − v (\frac{(u − 1)^{2}}{2} ) = (k − 2)(v(u − 1) − \frac{1}{l − 1}) + \frac{S_{l}(k) − 2}{l − 1}.

If k is not a power of l, S_{l}(k) ≥ 2 and this is ≥ 0 by assumption. If k is a power of l with l odd, the right hand side equals

(k − 2)(v(u − 1) − \frac{1}{l − 1}) − \frac{1}{l − 1} ≥ (l − 2)(v(u − 1) − \frac{1}{l − 1}) − \frac{1}{l − 1}

= (l − 2)v(u − 1) − 1

and this is still ≥ 0. Finally, if l = 2 and k is a power of 2, then

(k − 2)(v(u − 1) − 1) − 1 ≥ 2(v(u − 1) − 1) − 1

and this is ≥ 0 again.

b) We may write u = ζu_{0}, with ζ ∈ μ_{d} and u_{0} ∈ U^{(1)}. Since ζ ≡ 1 (mod^{∗} U^{l^{a}}), we are reduced to the case u ∈ U^{(1)}.

Applying a) to u^{n} instead of u, we get

u^{n} ≡ exp(u^{n} − 1) (mod \frac{(u^{n} − 1)^{2}}{2} O_{l})

or equivalently

u^{n} ≡ exp(u^{n} − 1) (mod^{∗} U^{(v)})

where

v = v (\frac{(u^{n} − 1)^{2}}{2} ) = 2v(n) + v_{1}

by Lemma 1.5, with v_{1} = v (\frac{(u − 1)^{2}}{2} ).

Recall the following well-known fact: if s > \frac{1}{l − 1}, raising to the l-th

power induces an isomorphism U^{(s)} →^{∼} U^{(s+1)}. Writing n = l^{a}n_{0} with n_{0} prime to l, we find first

u^{n_{0}} ≡ exp(\frac{u^{n} − 1}{l^{a}}) (mod^{∗} U^{(v_{1}+a)})

and, since U^{(1)} is uniquely n_{0}-divisible

u ≡ exp(\frac{u^{n} − 1}{n}) (mod^{∗} U^{(v_{1}+a)})

which may be rewritten

u ≡ exp(\frac{u^{n} − 1}{n}) (mod^{∗} (U^{(v_{1})})^{n}).

c) Note that u^{2} ≡ 1 (mod 8). Applying b) to u^{2}, we get

u^{2} ≡ exp(\frac{u^{2n} − 1}{n}) (mod^{∗} (U_{Z_{2}}^{(2ν+1)})^{n})

where U_{Z_{2}}^{(r)} := U^{(r)} ∩ Z_{2}. Since u ≡ (−1)^{\frac{u−1}{2}} (mod U_{Z_{2}}^{(3)}) and U_{Z_{2}}^{(2ν+1)} = (U_{Z_{2}})^{2^{2ν−1}}, we get the desired congruence. □

6.2. Lemma. Let E/F be a finite extension of finite extensions of Q_{p} and i > 0. Then

a) The diagram

K_{2i}(E) ⟶^{h^{(i)}(E)} H^{0}(E, Q_{l}/Z_{l}(i))
N_{L/K} ↓  \frac{m_{i}(E)}{m_{i}(F)} ↓
K_{2i}(F) ⟶^{h^{(i)}(F)} H^{0}(F, Q_{l}/Z_{l}(i))

commutes.

b) If p ≠ l, the diagram

K_{2i−1}(O_{E}) ⟶^{i_{E}^{∗}} H^{0}(ℓ, Q_{l}/Z_{l}(i))
N_{L/K} ↓  ^{e \frac{|ℓ|^{i}−1}{|k|^{i}−1}} ↓
K_{2i−1}(O_{F}) ⟶^{i_{F}^{∗}} H^{0}(k, Q_{l}/Z_{l}(i))

commutes, where k (resp. ℓ) is the residue field of F (resp. of E) and e is the ramification index of E/F.

Proof. a) By the covariance of the Dwyer-Friedlander spectral sequence for proper morphisms, the diagram

K_{2i}(E) ⟶ H^{2}(E, Z_{l}(i + 1))
N_{E/F} ↓  ^{Cor}^{E/F} ↓
K_{2i}(F) ⟶ H^{2}(F, Z_{l}(i + 1))

commutes. On the other hand, Tate duality exchanges corestriction and restriction. The claim then follows from Lemma 3.1.

b) This reduces to proving the commutation of the two diagrams

\begin{matrix}K_{2i−1}(O_{E}) & ⟶ & K_{2i−1}(ℓ) \\ ^{N_{L/K}} ↓ & & ^{eN_{ℓ/k}}↓ \\ K_{2i−1}(O_{F}) & ⟶ & K_{2i−1}(k) \end{matrix} \begin{matrix}K_{2i−1}(ℓ) & ⟶ & H^{0}(ℓ, Q_{l}/Z_{l}(i)) \\ ^{N_{ℓ/k}}↓ & & \frac{|l|^{i}−1}{|k|^{i}−1} ↓ \\ K_{2i−1}(k) & ⟶ & H^{0}(k, Q_{l}/Z_{l}(i)). \end{matrix}

The first one is obvious in view of the functoriality of algebraic Ktheory, and the second one follows from Lemma 1.4. □

6.3. Lemma. a) With notation as in Lemma 6.2, let a ∈ K_{2i−1}(E) and f ∈ F^{∗}. Then

(N_{E/F} (a), f) = \frac{m_{i}(E)}{m_{i}(F)}(a, f).

b) Let L/K be an extension of number fields, a ∈ K_{2i−1}(R_{L}) and 𝔞 a fractional ideal of R_{K}. Then

(\frac{N_{L/K}(a)}{𝔞}) = \frac{m_{i}(L)}{m_{i}(K)} (\frac{a}{𝔞R_{L}} ) .

Proof. a) immediately follows from Lemma 6.2 a) and the projection formula in algebraic K-theory. We deduce b) from Lemma 6.2 b) as follows: first we reduce to the case where 𝔞 = 𝔭 is prime. Now, developing both sides, we have to prove the identity

\frac{N(𝔭)^{i} − 1}{m_{i}(K)}i_{𝔭}^{∗}N_{L/K}(a) = \frac{m_{i}(L)}{m_{i}(K)} ∑_{𝔓|𝔭}e_{𝔓} \frac{N(𝔓)^{i} − 1}{m_{i}(L)}i_{𝔓}^{∗}a.

But Lemma 6.2 b) gives the identity

i_{𝔭}^{∗}N_{L/K}(a) = ∑_{𝔓|𝔭}e_{𝔓} \frac{N(𝔓)^{i} − 1}{N(𝔭)^{i} − 1}i_{𝔓}^{∗}a

which immediately implies the former. □

6.4. Lemma. Let L/K be an extension of number fields and 𝔭 a prime ideal of R_{K} and i be such that v_{l}(m_{i}/2) > 0. Then

a) One has

deg_{l}(𝔭R_{L}) = \frac{m_{i}(K)}{m_{i}(L)} ∑_{𝔓|𝔭}e_{𝔓}|f_{𝔓}|_{l}^{−1} deg_{l}(𝔭)

where e_{𝔓} and f_{𝔓} denote respectively the ramification index and the residue degree at 𝔓. When L/K is Galois, this formula simplifies in

deg_{l}(𝔭R_{L}) = (f_{𝔭}^{0})^{−1}[L_{0} : K_{0}]^{0} \frac{[L : K]}{[L_{0} : K_{0}]} deg_{l}(𝔭)

where f_{𝔭} := f_{𝔓} for any prime 𝔓 | 𝔭 and, for an integer a ≠ 0, a^{0} denotes the prime-to-l part of a and K_{0}, L_{0} denote respectively the maximal l-cyclotomic subfield of K and L. Moreover

\frac{m_{i}(𝔓)}{f_{𝔭}i} = \begin{cases}(i^{0})^{−1}m_{l−1}(L)[L : K]^{−1} deg_{l}(𝔭R_{L}) & if l > 2 \\ (i^{0})^{−1}m_{2}(L)[L : K]^{−1} deg_{l}(𝔭R_{L})/2 & if l = 2. \end{cases}

b) Let a ∈ K_{2i−1}(R_{L}). If L/K is Galois, one has

2 ((\frac{N_{L/K}(a)}{𝔭})^{(l)} − f_{𝔭}^{0} \frac{m_{i}(L)}{m_{i}(K)} (\frac{a}{𝔭R_{L}} )^{(l)}) = 0.

Proof. a) We have

deg_{l}(𝔭R_{L}) = ∑_{𝔓|𝔭}e_{𝔓} deg_{l}(𝔓) = ∑_{𝔓|𝔭}e_{𝔓} \frac{m_{i}(𝔓)}{m_{i}(L)}

= \frac{m_{i}(K)}{m_{i}(L)} ∑_{𝔓|𝔭}e_{𝔓} \frac{m_{i}(𝔓)}{m_{i}(𝔭)} \frac{m_{i}(𝔭)}{m_{i}(K)} = \frac{m_{i}(K)}{m_{i}(L)} ∑_{𝔓|𝔭}e_{𝔓}|f_{𝔓}|_{l}^{−1} deg_{l}(𝔭).

Suppose now that L/K is Galois. Then all f_{𝔓} are equal (to f_{𝔭}, by definition), and we may write

\frac{m_{i}(K)}{m_{i}(L)} ∑_{𝔓|𝔭}e_{𝔓}|f_{𝔓}|_{l}^{−1} deg_{l}(𝔭) = \frac{m_{i}(K)}{m_{i}(L)}(f_{𝔭}^{0})^{−1} ∑_{𝔓|𝔭}e_{𝔓}f_{𝔓} deg_{l}(𝔭)

= \frac{m_{i}(K)}{m_{i}(L)}(f_{𝔭}^{0})^{−1}[L : K] deg_{l}(𝔭)

= \frac{m_{i}(K_{0})}{m_{i}(L_{0})}[L_{0} : K_{0}](f_{𝔭}^{0})^{−1} \frac{[L : K]}{[L_{0} : K_{0}]} deg_{l}(𝔭)

= [L_{0} : K_{0}]^{0}(f_{𝔭}^{0})^{−1} \frac{[L : K]}{[L_{0} : K_{0}]} deg_{l}(𝔭)

(the equality \frac{m_{i}(K_{0})}{m_{i}(L_{0})}[L_{0} : K_{0}] = [L_{0} : K_{0}]^{0} is easy to check).

For the second formula,

\frac{m_{i}(𝔓)}{f_{𝔭}i} = \frac{m_{i}(L)}{f_{𝔭}i} deg_{l}(𝔓) = \frac{m_{i}(L)}{e_{𝔓}f_{𝔭}i} deg_{l}(𝔭R_{L}) = \frac{m_{i}(L)}{[L : K]i} deg_{l}(𝔭R_{L})

= \begin{cases}(i^{0})^{−1}m_{l−1}(L)[L : K]^{−1} deg_{l}(𝔭R_{L}) & if l > 2 \\ (i^{0})^{−1}m_{2}(L)[L : K]^{−1} deg_{l}(𝔭R_{L})/2 & if l = 2. \end{cases}

b) From Lemma 6.3 b) we have

(\frac{N_{L/K}(a)}{𝔭})^{(l)} = \frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1} (\frac{N_{L/K}(a)}{𝔭}) = \frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1} \frac{m_{i}(L)}{m_{i}(K)} (\frac{a}{𝔭R_{L}} )

= \frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1} \frac{m_{i}(L)}{m_{i}(K)} ∑_{𝔓|𝔭}e_{𝔓} (\frac{a}{𝔓})

= \frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1} \frac{m_{i}(L)}{m_{i}(K)} ∑_{𝔓|𝔭}e_{𝔓} \frac{N(𝔓)^{i} − 1}{m_{i}(𝔓)} (\frac{a}{𝔓})^{(l)} .

Since L/K is Galois, we have N(𝔓) = N(𝔭)^{f_{𝔓}} for any 𝔓 | 𝔭 and, by Lemma 1.5

\frac{N(𝔓)^{i} − 1}{N(𝔭)^{i} − 1} ≡ f_{𝔓} (mod \frac{N(𝔓)^{i} − 1}{2}Z_{l}).

Moreover, we have

f_{𝔓} = f_{𝔭} = f_{𝔭}^{0} \frac{m_{i}(𝔓)}{m_{i}(𝔭)}

for any 𝔓 | 𝔭. Hence

2 (\frac{N_{L/K}(a)}{𝔭})^{(l)} = 2f_{𝔭} \frac{m_{i}(𝔭)}{m_{i}(𝔓)} \frac{m_{i}(L)}{m_{i}(K)} ∑_{𝔓|𝔭}e_{𝔓} (\frac{a}{𝔓})^{(l)}

= 2f_{𝔭}^{0} \frac{m_{i}(L)}{m_{i}(K)} ∑_{𝔓|𝔭}e_{𝔓} (\frac{a}{𝔓})^{(l)} = 2f_{𝔭}^{0} \frac{m_{i}(L)}{m_{i}(K)} (\frac{a}{𝔭R_{K}} )^{(l)}

as requested. □

7. Proof of Theorem 2.9

Suppose that v_{l}(m_{i}/2) = 0. If l is odd, then H^{0}(K, Q_{l}/Z_{l}(i)) = 0 and the statement of Theorem 2.9 a) is empty. Similarly, if l = 2 then H^{0}(K, Q_{l}/Z_{l}(i)) = Z/2 and the statement of Theorem 2.9 b) is also empty, since its left hand side is additive in a and multiplicative in r (we may take s_{2} = 0). Hence, in this section we assume v_{l}(m_{i}/2) > 0. This will allow us to use Lemma 6.4.

By additivity of both sides, we may assume that r is a prime p ≠ l. Note first that by Lemma 6.4 we have

\frac{deg_{l}(pR_{K})}{2[K : K_{0}]} = (f_{p}^{0})^{−1} \frac{1}{2}[K_{0} : Q] deg_{l}(pZ)

where f_{p} is the residue degree at p in the Galois extension K/Q. This is an l-integer for l > 2. This is still the case for l = 2 except perhaps if K_{0} = Q. But K_{0} = Q is real, hence excluded from the statement of Theorem 2.9.

Suppose first that l is odd. We are going to prove Theorem 2.9 a) with

s(a) = −(i^{0})^{−1}(N_{K/K_{0}}(a), exp(\frac{l}{δ}))_{𝔩}

where [K_{0} : Q] = δl^{n−1}, 𝔩 is the unique prime of K_{0} dividing l and i^{0} is the largest divisor of i that is prime to l. By Lemma 6.4, we immediately reduce to the case where K = K_{0} (notice that m_{i}(K) = m_{i}(K_{0}) for all i).

Then l is the only ramified prime in K/Q and it is totally ramified, so that

m_{i} = m_{i}(𝔩) = l^{n+v_{l}(i)} if δ | i

where 𝔩 is the unique prime of K above l.

By Lemma 6.1 b), we have

p ≡ exp(\frac{p^{fi} − 1}{fi}) (mod (U^{(v_{1})})^{|fi|_{l}^{−1}})

where v_{1} = 2v_{l}(p^{l−1} − 1) = 2v_{l}(m_{l−1}(pZ)). Since U^{(v_{1})} = (U^{(1)})^{l^{v_{1}−1}}, this gives

p ≡ exp(\frac{p^{fi} − 1}{fi}) (mod (U^{|fim_{l−1}(pZ)^{2}/l|_{l}^{−1}}).

On the other hand,

v_{l}(fim_{l−1}(pZ)^{2}/l) − v_{l}(m_{i}) = v_{l}(fm_{l−1}(pZ)^{2}/l) − v_{l}(m_{l−1})

= v_{l}(\frac{m_{l−1}(𝔭)}{m_{l−1}(pZ)} \frac{m_{l−1}(pZ)^{2}}{l}) − v_{l}(m_{l−1}) = v_{l}(\frac{m_{l−1}(𝔭)}{m_{l−1}} \frac{m_{l−1}(pZ)}{l})

= v_{l}(deg_{l}(𝔭)) + v_{l}(\frac{m_{l−1}(pZ)}{l})) ≥ 0.

Hence, using Theorem 5.5, we get

(\frac{a}{pR_{K}} )^{(l)} = \frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1} (\frac{a}{pR_{K}} ) = −\frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1}(a, p)_{𝔩}

= −\frac{m_{i}(𝔭)}{p^{fi} − 1}(a, p)_{𝔩} = −\frac{m_{i}(𝔭)}{p^{fi} − 1}(a, exp(\frac{p^{fi} − 1}{fi}))_{𝔩}

= −(a, exp(\frac{m_{i}(𝔭)}{fi}))_{𝔩}.

By Lemma 6.4 a), we have

\frac{m_{i}(𝔭)}{fi} = (i^{0})^{−1}m_{l−1}(K)[K : Q]^{−1} deg_{l}(pR_{K}) = (i^{0})^{−1} deg_{l}(pR_{K})\frac{l}{δ}

hence

(\frac{a}{pR_{K}} )^{(l)} = −(i^{0})^{−1} deg_{l}(pR_{K})(a, exp(\frac{l}{δ}))_{𝔩}

as requested.

Suppose now that l = 2. We shall prove Theorem 2.9 b) with

s_{1}(a) = (N_{K/K_{0}}(a), −1)_{𝔩}
s_{2}(a) = −(i^{0})^{−1}(N_{K/K_{0}}(a), exp(4))_{𝔩}.

As above, we reduce to K = K_{0} and r = p ≠ 2 (a prime number). Then K is classified by an open subgroup Γ_{K} of Z_{2}^{∗}. Given an integer n ≥ 1, there are exactly three such subgroups of index 2^{n}, namely:

• Γ_{K} = Γ^{n} = 1 + 2^{n+1}Z_{2}, corresponding to K = Q(μ_{2^{n+1}}) (cyclo-

tomic case); (Γ_{K})^{2} = Γ^{n+1}.

• Γ_{K} = Γ_{+}^{n} = μ_{2} ×(1+2^{n+2}Z_{2}), corresponding to K = Q(ζ +ζ^{−1}) where ζ is a generator of μ_{2^{n+2}}(Q̄) (totally real case); (Γ_{K})^{2} =

Γ^{n+2}.

• Γ_{K} = Γ_{−}^{n} = −1 + 2^{n+1}Z_{2}, corresponding to K = Q(ζ − ζ^{−1})

(non-exceptional case); (Γ_{K})^{2} = Γ^{n+1}. We include K = Q in the totally real series: Γ_{Q} = Z_{2}^{∗} and (Γ_{Q})^{2} =

Γ^{2}. Using these computations, we find that

(7.1)  m_{2}(K) = (Z_{2}^{∗} : (Γ_{K})^{2}) = \begin{cases}2^{n+3} & if K is totally real \\ 2^{n+2} & otherwise. \end{cases}

Recall that the totally real case is excluded from the statement of Theorem 2.9. Henceforth we assume K totally imaginary.

Let p^{∗} = (−1)^{\frac{p−1}{2}} p. By Lemma 6.1 c), we have

p ≡ (−1)^{\frac{p−1}{2}} exp(\frac{(p^{∗})^{fi} − 1}{fi}) (mod U^{|fi/2|_{2}^{−1}2^{2v_{1}−3}})

where v_{1} = v_{2}(p^{2} − 1) = v_{2}(m_{2}(pZ)). On the other hand,

v_{2}(fim_{2}(pZ)^{2}/16) − v_{2}(m_{i}) = v_{2}(fm_{2}(pZ)^{2}/16) − v_{2}(m_{2}) + 1

= v_{2}(\frac{m_{2}(𝔭)}{m_{2}(pZ)} \frac{m_{2}(pZ)^{2}}{8} ) − v_{2}(m_{2}) = v_{2}(\frac{m_{2}(𝔭)}{m_{2}} \frac{m_{2}(pZ)}{8})

= v_{2}(deg_{2}(𝔭)) + v_{2}(\frac{m_{2}(pZ)}{8})) ≥ 0.

Suppose first that fi is even or that p ≡ 1 (mod 4). Then (p^{∗})^{fi} = p^{fi}. Using Theorem 5.5, we get

(\frac{a}{pR_{K}} )^{(2)} = \frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1} (\frac{a}{pR_{K}} ) = −\frac{m_{i}(𝔭)}{N(𝔭)^{i} − 1}(a, p)_{𝔩}

= −\frac{m_{i}(𝔭)}{p^{fi} − 1}(a, p)_{𝔩} = −\frac{m_{i}(𝔭)}{p^{fi} − 1}(a, (−1)^{\frac{p−1}{2}} exp(\frac{p^{fi} − 1}{fi}))_{𝔩}

= −(a, (−1)^{\frac{p−1}{2}} exp(\frac{m_{i}(𝔭)}{fi}))_{𝔩}.

By Lemma 6.4 a) and (7.1)

\frac{m_{i}(𝔭)}{fi} = (i^{0})^{−1}m_{2}(K)[K : Q]^{−1} deg_{2}(pR_{K})/2 = 2(i^{0})^{−1} deg_{2}(pR_{K}).

Since deg_{2}(pR_{K}) = (f_{p}^{0})^{−1}[K : Q] deg_{2}(pZ), this integer is always divisible by 4. Hence

(7.2) (\frac{a}{pR_{K}} )^{(2)} = \frac{p − 1}{2}(a, −1)_{𝔩} − \frac{deg_{2}(pR_{K}))}{2}(i^{0})^{−1}(a, exp(4))_{𝔩}.

Suppose now that p ≡ −1 (mod 4) and fi is odd, hence f = 1. The condition f = 1 means that p ∈ Γ_{K}. The first condition rules out the cyclotomic case, hence (since the real case is excluded) K is nonexceptional. Then the condition i odd implies that m_{i}(K) = m_{i}(K_{𝔩}) = 2, hence the symbol (a, f)_{𝔩} has order 2 and we have

(\frac{a}{pR_{K}} )^{(2)} = (a, p)_{𝔩} = \frac{p − 1}{2}(a, −1)_{𝔩} + (a, p^{∗})_{𝔩}.

Let [K : Q] = 2^{n}. We must have p ≡ −1+2^{n+1} (mod 2^{n+2}). Then p^{∗} is a square in Z_{2}^{∗} unless n = 1, i.e. K = Q(\sqrt{−2}): then p^{∗} ≡ 5 ≡ exp(4) (mod 8).

On the other hand, note that deg_{2}(pZ) = |\frac{p^{2} − 1}{8}|_{2}^{−1} is > 1 exactly

when p^{∗} is a square. This shows by inspection that \frac{deg_{2}(pR_{K})}{2} is even exactly when p^{∗} is a square.

This discussion shows that formula (7.2) remains valid when p ≡ −1 (mod 4) and fi is odd. □

7.1. Remarks. a) Suppose K/Q l-cyclotomic with [K : Q] = δl^{n−1} as above. Let ν = v_{l}(p^{l−1} − 1) for l > 2 and ν = v_{2}(p^{2} − 1) − 1 for l = 2. By Lemma 6.4 a), we have

deg_{l}(pR_{K}) = (f_{p}^{0})^{−1}[K : Q]^{0} deg_{l}(pZ) = \begin{cases}l^{ν−1} \frac{δ}{gcd(d, δ)} & if l > 2 \\ 2^{ν−1} & if l = 2 \end{cases}

so that deg_{l}(pR_{K}) does not depend on n (however it depends on δ when l is odd).

b) For l = 2, \sqrt{exp(4)} ∈ Q_{2}(\sqrt{5}) − Q_{2}: so there is no way in general

to write the summand \frac{deg_{2}(rR_{K}))}{2[K : K_{0}]}s_{2}(a) in Theorem 2.9 b) in the form

\frac{deg_{2}(rR_{K}))}{[K : K_{0}]}s′_{2}(a) for some s′_{2} : K_{2i−1}(R_{K}) → H^{0}(K, Q_{2}/Z_{2}(i)). I am

grateful to Pierre Colmez for pointing this out.

8. Class field theory

Let F be again a finite extension of Q_{p}.

8.1. Theorem. For i ≥ 1, the Dwyer-Friedlander spectral sequence yields isomorphisms

\begin{aligned}K_{2i−1}(F, Z/l^{ν}) & ⟶^{∼} & H^{1}(F, Z/l^{ν}(i)) \\ K_{2i}(F, Z/l^{ν}) & ⟶^{∼} & H^{0}(F, Z/l^{ν}(i)) ⊕ H^{2}(F, Z/l^{ν}(i + 1)). \end{aligned}

Proof. For l ≠ p, this follows from [10, Th. 4.1]. For l = p > 2, this follows from [9]. For l = p = 2, this follows from e.g. [13, Th. 2] (using inter alia the Milnor conjecture) and the degeneracy of the Dwyer-Friedlander spectral sequence.

8.2. Corollary. For i ≥ 1, we have a long exact sequence

0 → H^{0}(F, Z/l^{ν}(i)) → K_{2i−1}(F) →^{l^{ν}} K_{2i−1}(F) → H^{1}(F, Z/l^{ν}(i))

→ K_{2i−2}(F) →^{l^{ν}} K_{2i−2}(F) → H^{2}(F, Z/l^{ν}(i)) → 0.

Proof. This was proven in [10] under the assumption that Theorem 8.1 above holds (cf. loc. cit., Remark 6.2). We give here an alternate proof.

First, the maps are defined in an obvious way from Theorem 8.1 by using the exact sequences

0 → K_{2i}(F)/l^{ν} → K_{2i}(F, Z/l^{ν}) → _{l^{ν}}K_{2i−1}(F) → 0
0 → K_{2i−1}(F)/l^{ν} → K_{2i−1}(F, Z/l^{ν}) → _{l^{ν}}K_{2i−2}(F) → 0
0 → K_{2i−2}(F)/l^{ν} → K_{2i−2}(F, Z/l^{ν}) → _{l^{ν}}K_{2i−3}(F) → 0.

Exactness is clear, except at the first, second, penultimate and ultimate terms. The composition

K_{2i}(F)/l^{ν} → K_{2i}(F, Z/l^{ν}) → H^{0}(F, Z/l^{ν}(i))

is 0, as one sees by functoriality by passing to the algebraic closure F̄ (by Suslin [25], K_{2i}(F̄)/l^{ν} = 0). This shows that the map K_{2i}(F)/l^{ν} → H^{2}(F, Z/l^{ν}(i + 1)) is injective, which in turn shows that the map H^{0}(F, Z/l^{ν}(i)) → _{l^{ν}}K_{2i−1}(F) is injective.

It remains to see exactness at the second and ultimate terms. By a diagram chase, it is enough to prove it at the ultimate term. We shall use an l-adic argument as follows. Let K̂(F) be the l-completion of the K-theory spectrum K(F). Then Theorem 8.1 implies that the natural map of spectra K̂(F) → K^{ét}(F) induces an isomorphism on π_{i} for all i ≥ 1. In particular, for i > 1,

K̂_{2i−2}(F) ⟶^{∼} K_{2i−2}^{ét}(F) ⟶^{∼} H^{2}(F, Z_{l}(i))

where the second isomorphism comes as usual from the fact that H^{0}(F, Z_{l}(i − 1)) = 0.

In the Milnor exact sequence

0 → \underleftarrow{lim}^{1} K_{2i−1}(F, Z/l^{ν}) → K̂_{2i−2}(F) → \underleftarrow{lim} K_{2i−2}(F, Z/l^{ν}) → 0 the left term vanishes because all terms in the \underleftarrow{lim}^{1} are finite by Theorem 8.1. The right term inserts into an exact sequence

(8.1) 0 → \underleftarrow{lim} K_{2i−2}(F)/l^{ν} → \underleftarrow{lim} K_{2i−2}(F, Z/l^{ν}) → \underleftarrow{lim}_{l^{ν}}K_{2i−3}(F).

But K_{2i−2}(F, Q_{l}/Z_{l}) ≃ H^{0}(F, Q_{l}/Z_{l}(i − 1)) ⊕ H^{0}(F, Q_{l}/Z_{l}(i − 1)) by Theorem 8.1 and Tate duality. Hence this group is finite and K_{2i−3}(F){l} is finite as well. It follows that the right term of (8.1) vanishes. Summing up, we get an isomorphism

\underleftarrow{lim} K_{2i−2}(F)/l^{ν} ⟶^{∼} H^{2}(F, Z_{l}(i))

which is obviously compatible with the last map in the sequence of Corollary 8.2. It remains to observe that the map H^{2}(F, Z_{l}(i)) → H^{2}(F, Z/l^{ν}(i)) is surjective for all ν. This completes the proof of Corollary 8.2. □

Let a ∈ K_{2i−1}(F). Assume that l^{ν} divides |H^{0}(F, Q_{l}/Z_{l}(i))|. Then we have an isomorphism

H^{1}(F, Z/l^{ν}(i)) = H^{1}(F, Z/l^{ν}) ⊗ H^{0}(F, Z/l^{ν}(i))

hence the homomorphism of Corollary 8.2 carries a to an element of the form χ_{a} ⊗ x, where χ_{a} ∈ H^{1}(F, Z/l^{ν}) and x is a generator of H^{0}(F, Z/l^{ν}(i)). The cyclic extension E = F (χ_{a}) does not depend on the choice of χ_{a} and x. Corollary 8.2 implies that a_{E} is of the form l^{ν}b for some b ∈ K_{2i−1}(E). Conversely, if L is an extension of F such that a_{L} is divisible by l^{ν}, then χ_{a} vanishes on L, hence L ⊇ E. This

justifies the notation E = F (\frac{1}{l^{ν}} a).

Let G = Gal(E/F): this is a cyclic group of order dividing l^{ν}. Let g ∈ G and b be as above. Then gb − b ∈ _{l^{ν}}K_{2i−1}(E) ≃ H^{0}(E, Z/l^{ν}(i)), still by Corollary 8.2. Hence we get a 1-cocycle of G with values in H^{0}(E, Z/l^{ν}(i)), or an element of H^{1}(G, H^{0}(E, Z/l^{ν}(i))). Pushing this element into H^{1}(F, Z/l^{ν}(i)), we get back the image of a under the map of Corollary 8.2: this is obvious.

Note that, in fact, G acts trivially on H^{0}(E, Z/l^{ν}(i)) because this group coincides with H^{0}(F, Z/l^{ν}(i)). Hence we have defined above a homomorphism from G to H^{0}(F, Z/l^{ν}(i)), or a homomorphism

ψ_{a} : Gal(F̄/F)^{ab} → H^{0}(F, Z/l^{ν}(i)) = Z/l^{ν}(i).

We now choose l^{ν} = |H^{0}(F, Q_{l}/Z_{l}(i))|. Let f ∈ F^{∗}. Denote by Art(f) its Artin symbol in Gal(F̄/F)^{ab}.

8.3. Theorem. We have

(a, f) = ψ_{a}(Art(f)).

Proof. If χ ∈ H^{1}(F, Z/l^{ν}), then we have by [20, ch. XIV, p. 212, prop. 3]

χ(Art(f)) = inv(f · δ(χ))

where f is viewed as an element of H^{0}(F, 𝔾_{m}) and δ is the Bockstein from the exact sequence 0 → Z → Z → Z/l^{ν} → 0. On the other hand, an easy cocycle computation (which strangely is not in [20], but see loc. cit., ch. XIV, p. 214, proof of prop. 5) shows that

f · δ(χ) = χ · (f)

where (f) is the Kummer class of f in H^{1}(F, Z/l^{ν}(1)).

Twisting this computation by Z/l^{ν}(i), we get

ψ_{a}(Art(f)) = θ_{l^{ν}}^{−1}(ψ_{a} · (f))

where θ_{l^{ν}} is the isomorphism of (3.2) (use Remark 3.3).

On the other hand,

(a, f) = h^{(i)}(a · {f}) = θ_{l^{ν}}^{−1}(ch_{i+1}(a · {f})

where ch_{i+1} is the Dwyer-Friedlander map of (4.2). By the multiplicativity of the Dwyer-Friedlander spectral sequence, we have

ch_{i+1}(a · {f}) = ch_{i}(a) · ch_{1}({f}) = ch_{i}(a) · (f)

where ch_{i} : K_{2i−1}(F) → H^{1}(F, Z/l^{ν}(i)) is the edge homomorphism stemming from this spectral sequence. It remains to see that

ψ_{a} = ch_{i}(a)

which follows tautologically from the definition of the map K_{2i−1}(F) → H^{1}(F, Z/l^{ν}(i)) in Corollary 8.2. □

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  E-mail address: kahn@math.jussieu.fr