The Kneser relation and the Hilbert pairing in multidimensional local field

Sergei Vostokov^{∗}

February 2002

Introduction

In the paper [Kn], M. Kneser obtained a relation (see [Kn], (12)), which allowed him to prove the Shafarevich reciprocity law (see [Sh]) directly from the definition of the Hilbert symbol. In the present paper, we generalize the Kneser relation to multidimensional local fields and obtain an explicit formula for the generalized Hilbert pairing dierctly from the definition of the pairing.

It is well known (see, e.g., [V1]) that the explicit formula for the Hilbert pairing is given by the residue of the product of two series, Φ(X) and V (X), with integer coeffiecents. The first series is calculated by the arguments of the pairing. The second series is directly connected with primary elements. In fact, this series is the inverse of a series, which is substituted in the Artin– Hasse function to obtain a primary element. The Kneser relation, which we generalize to multidimensional local fields in section 2, gives, actually, the first series Φ(X) (see 2.6, (15) below). In section 3, we calculate the series V (X). We find two congruences for V (X) (see 3.3, (22) and 3.4, (24)). The first congruence follows from the explicit form of the primary elements. The second congruence follows from the fact that the Hilbert pairing is independent of the power series expansions of elements with respect to local parameters.

Solving the system of these two congruences, we obtain an explicit expression for V (X) and an explicit formula for the Hilbert pairing. The author is grateful to Tanja Beljaeva and Alex.Zinoviev for several useful remarks.

∗Author thanks the Max-Planck-Institut in Bonn for its valuable support. The paper was also supported by RFBR (grant no. 00-01-00140).

1 Notation and preparatory propositions

1.1

Let K be an n-dimensional local field, i.e. a finite sequence of fields K^{(0)}, K^{(1)}, …, K^{(n)} = K, where K^{(0)} = 𝔽_{q} with q = p^{f} is a finite field, and, for 1 ⩽ i ⩽ n, the field K^{(i)} is a complete discrete valuation field whose residue field is K^{(i−1)}.

The study of multidimensional local fields was initiated by A. Parshin und K. Kato in the seventies of the last century. Now these fields represent a well-developed subject of research (cf. [VZF]).

For an n-dimensional local field K, we have the following structural data:

1) A discrete valuation v : K^{∗} ⟶ ℤ of rank 1

2) A discrete valuation v̄ = (v_{1}, …, v_{n−1}, v_{n}) : K^{∗} ⟶ ℤ^{n} of rank n

3) The chain of discrete valuations v̄_{i} = (v_{n−i+1}, v_{n−i+2}, …, v_{n}) : K^{∗} ⟶ ℤ^{i} 4) The chain 𝔪_{1} ⊂ 𝔪_{2} ⊂ … ⊂ 𝔪_{n} =: 𝔪 of the valuation ideals which belong to the valuations v̄_{i} .

We assume that all n-dimensional local fields considered in this paper are of characteristic zero.

Let 𝔬_{K} denote the ring of integers for the valuation v̄, and let t_{1}, …, t_{n−1}, t_{n} = Π be a system of local parameters for v̄ = (v_{1}, …, v_{n−1}, v_{n} = v). We denote by ℜ the Teichmüller set of representatives of 𝔽_{q} in K. Every α in K has a unique expansion of the form

α = ∑_{r̄∈I} θr̄t_{1}^{r_{1}} … t_{n−1}^{r_{n−1}}Π^{r_{n}} with θr̄ ∈ ℜ,

where r̄ = (r_{1}, …, r_{n}) runs through a certain admissible subset I = I(α) of ℤ^{n} . We call I ⊂ ℤ^{n} admissible, if for all i_{l+1}, …, i_{n} in ℤ, l ⩽ n, there exists an i ∈ ℤ with the following property: If r̄ = (r_{1}, …, r_{l}, i_{l+1}, …, i_{n}) ∈ I, then r_{l} ⩾ i (cf. [VZF]). We consider also expansions of the form

α = ∑_{r̄∈I} ar̄t_{1}^{r_{1}} … t_{n−1}^{r_{n−1}}Π^{r_{n}}

with coefficients ar̄ in

𝔬 := W(K^{(0)}),

the ring of Witt vectors over K^{(0)}; such expansions, however, are no longer unique. If α is contained in the maximal ideal 𝔪_{K} of 𝔬_{K}, then

α = ∑_{\begin{matrix}r̄>¯0 \\ r̄∈I \end{matrix}} ar̄t_{1}^{r_{1}} … t_{n−1}^{r_{n−1}}Π^{r_{n}}.

1.2 The module of Cartier-curves of the multiplicative group of a multidimensional local field

Let K be an n-dimensional local field with last residue field K^{(0)}.We denote by T = Quot(𝔬) the quotient field of the ring of Witt vectors over K^{(0)}, and, as before t_{1}, …, t_{n−1}, Π denotes a system of local parameters of K. Then every α ∈ K is representable in the form

α = ∑_{r̄∈I} ar̄t_{1}^{r_{1}} … t_{n−1}^{r_{n−1}}Π^{r_{n}} with ar̄ ∈ 𝔬,

where r̄ = (r_{1}, …, r_{n}) runs through a certain admissible subset of ℤ^{n} (see 1).

For algebraically independent variables X_{1}, X_{2}, …, X_{n}, we consider the (n + 1)-dimensional local field

M = M(K) := T {{X_{1}}} … {{X_{n−1}}}((X_{n}))

with local parameters X_{1}, …, X_{n−1}, p, X_{n}. Introduce the following notation

𝔬[[X]] = { ∑_{r̄∈I, r̄⩾¯0} ar̄Xr̄ | ar̄ ∈ 𝔬},

where I is an admissible subset in ℤ^{n}. Denote

X𝔬[[X]] = { ∑_{r̄∈I, r̄>¯0} ar̄Xr̄ ∈ 𝔬[[X]] | ar̄ ∈ 𝔬}.

Set

𝔬_{(p)} := 𝔬{{X_{1}}} … {{X_{n−1}}}((X_{n})),
U_{m} := 1 + X𝔬[[X]],
ℋ_{m} := ⟨X_{1}⟩ × … × ⟨X_{n}⟩ × 𝔬^{×} × U_{m} ⊂ 𝔬_{(p)}^{×}.

We call ℋ_{m} the module of Cartier-curves for the multiplicative group of K. We have the following (noncanonical) surjective homomorphism:

η : 𝔬_{(p)} ⟶ K,
α(X) ⟼ α(t_{1}, …, t_{n−1}, Π).

By restriction, it induces a group homomorphism

η : ℋ_{m} ⟶ K^{×},

and the latter homomorphism is also surjective (see 1).

For each α ∈ K^{×}, we choose a preimage \underline{α} of α under η; it satisfies the conditions

\underline{α} ∈ ℋ_{m} and \underline{α}(t_{1}, …, t_{n−1}, Π) = α.  (1)

We set A := 𝔬{{X_{1}}} … {{X_{n−1}}}.

Let Eis(x) = p + a_{1}x + … + a_{e}x^{e} + … ∈ A[[x]] be an Eisenstein series for Π, i.e. such a series that a_{1}, …, a_{e−1} are divisible by p, a_{e} is an unit of the ring A, and

Eis(Π) = 0, i.e., η(Eis(X_{n})) = 0.

Lemma 1. S Let Eis(x) be an Eisenstein series for Π, and let ρ(x) be a series of A[[x]] such that ρ(Π) = 0 (i.e., η(ρ(X_{n})) = 0). Then ρ(x) is divisible by Eis(x) in A[[x]].

Proof. By the Weierstrass Preparation Theorem, there exists an ε(x) ∈ A[[x]] with invertible constant term such that

g(x) := Eis(x) · ε(x)

is an Eisenstein polynomial of degree e over A. Without loss of generality, we may assume that the coefficients of ρ(x) are relatively primer. Now, applying the Weierstrass Preparation Theorem to ρ(x), we obtain a series ε_{1}(x) ∈ A[[x]] with invertible constant term such that ρ_{∗} := ρε_{1} is a polynomial over A that admets Π as a root. It follows that ρ_{∗} = g · φ, where φ ∈ A[x]. Thus, we obtain ρ = Eis(x) · (φ · ε · ε_{1}^{−1}), and the lemma is proved.

It is easy to see that there exists an Eisenstein series for Π. By Lemma 1, such a series is uniquely determined up to units of A[[x]].

2 Kneser’s relation

2.1 The definition of the Hilbert symbol and primary elements

Let K be an n-dimensional local field containing all p^{N}th roots of 1, and let Ω_{N} be the group of p^{N}-primary elements of the field K (an element ω ∈ K is called primary if the extension K( \sqrt[p^{N}]{ω})/K is unramified). We define the character χ on the group Ω_{N} as follows:

χ(ω) := \sqrt[p^{N}]{ω}^{φ−1} ∈ μ_{N},

where φ is the canonical Frobenius automorphism of K( \sqrt[p^{N}]{ω})/K, and μ_{N} is the group of all p^{N}th roots of 1. Let

Ψ_{K} : K_{n}^{top}(K) ⟶ Gal(K^{ab}/K)

be the reciprocity map à la Parshin-Kato from the topological Milnor group in the Galois group of the maximal abelian extension of the field K (cf. [Ka], [P1, 2]).

Definition 1. We define the multidimensional Hilbert symbol as follows:

(·, ·) = (·, ·)_{N,K} : K_{n}^{top}(K) × K^{×} ⟶ μ_{N}  (2)
(α, β)_{N,K} = Ψ_{K}(α)(β̃)/β̃

where β̃ satisfies (β̃)^{p^{N}} = β.

Definition 2. By the Hilbert symbol, we mean a homomorphism

(, …,)_{N,K} : K_{n+1}^{top}(K) ⟶ μ_{N},

normalized by the relation

(t_{1}, …, t_{n}, ω)_{N,K} = χ(ω)

for every set t_{1}, …, t_{n} of local parameters and every p^{N}-primary element ω in the field K.

Proposition 1. The two definitions of the Hilbert symbol are equivalent.

Proof. Let (,)^{(1)} be the Hilbert symbol from the first definition, and (,)^{(2)} be the symbol from the second one.

First, we must verify the relation

(t_{1}, …, t_{n}, β)^{(1)} = (t_{1}, …, t_{n}, β)^{(2)}  (3)

for every system of local parameters and all β ∈ K^{×}.

Using arithmetic properties of K, we can represent β as follows:

β = t_{1}^{a_{1}} … t_{n}^{a_{n}}θ ∏_{\begin{matrix}r̄>¯0 \\ p∤r̄ \end{matrix}}(1 − θr̄t_{1}^{r_{1}} … t_{n}^{r_{n}}) ω_{β},

where θ ∈ ℜ^{×}, θr̄ ∈ ℜ, r̄ = (r_{1}, …, r_{n}) is a set of multi-indices, and ω_{β} is an p^{N}-primary element associated with β. Using the Steinberg relations, we obtain ([V2])

(t_{1}, …, t_{n}, β)^{(1)} = (t_{1}, …, t_{n}, ω_{β})^{(1)} = χ(ω_{β})

and

(t_{1}, …, t_{n}, β)^{(2)} = (t_{1}, …, t_{n}, ω_{β})^{(2)} = χ(ω_{β}),

which gives relation (3).

The general case can be derived from (3) because K_{n+1}^{top} is topologically generated by the symbols of the form {t_{1}, …, t_{n}, β}, where β is a principal unit.

In what follows, we mainly use the second definition.

2.2

Using the relation (x, 1 − x) = 1 for x ≠ 0, 1, and the fact that the Hilbert symbol is trivial if one of the components is a p^{N}th power, Kneser (in [Kn]) obtained the following relation in a one-dimensional local field K:

(1 − x, 1 − y) = ∏_{(i,k)=1} (−x^{ι}y^{κ}, 1 − x^{i}y^{k}), x ≡ y ≡ 0 mod 𝔐,

where ι and κ are determined by the equation

| \begin{matrix}i & ι \\ k & κ \end{matrix} | = 1

(here 𝔐 is the maximal ideal in the ring of integers of K). From this relation, Kneser obtained the following formula:

(E(x), E(y)) = ∏_{s=0}^{∞} (−y^{−1}, E(xy^{p^{s}})) ∏_{r=1}^{∞} (−x, E(x^{p^{r}}y)),

where

E(x) = exp (x + \frac{x^{p}}{p} + \frac{x^{p^{2}}}{p^{2}} + …)

is the Artin-Hasse function.

Using the Kneser method in the case of an n-dimensional local field K, we obtain the following relation for principal units E(x) and E(y), x ≡ y ≡ 0 mod 𝔐:

(α_{1}, …, α_{n−1}, E(x), E(y)) =

∏_{s=0}^{∞}(α_{1}, …, α_{n−1}, y^{−1}, E(xy^{p^{s}})) ∏_{r=1}^{∞}(α_{1}, …, α_{n−1}, x, E(x^{p^{r}}y)), (4)

where α_{1}, …, α_{n−1} ∈ K^{×}

We suppose now that p is odd, then we can drop the minus signs at y^{−1} and x.

2.3

Let x = θt_{1}^{i_{1}} … t_{n}^{i_{n}} and y = θ′t_{1}^{j_{1}} … t_{n}^{j_{n}}, where θ, θ′ ∈ ℜ, be monomials in K; then, using the multiplicativity of the Hilbert symbol, we can represent (4) in the form (α_{1}, …, α_{n−1}, E(x), E(y)) =

∏_{k=1}^{n}(α_{1}, …, α_{n−1}, t_{k}, ∏_{s⩾0} E^{jk}(θθ′^{p^{s}}t_{1}^{i_{1}+p^{s}j_{1}} … t_{n}^{i_{n}+p^{s}j_{n}})^{−1} ×

∏_{r⩾1} E^{ik}(θ^{p^{r}}θ′t_{1}^{p^{r}i_{1}+j_{1}} … t_{n}^{p^{r}i_{n}+j_{n}})) (5)

Further, we introduce the generalized Artin–Hasse function, which was first considered in [V1] and [V2]. Let △ be the Frobenius operator over the ring 𝔬_{(p)} that acts on the coefficients from 𝔬 as usual Frobenius operator and raises the parameters X_{1}, … X_{n} to pth power, i.e.,

△f = f^{△} = ∑_{r̄} Frob(ar̄)X_{1}^{pr_{1}} ⋯ X_{n}^{pr_{n}}

We define the functions

l(α) = \frac{1}{p} log \frac{α^{p}}{α^{△}} , α ∈ ℋ_{m}

and

E(β) = exp (1 + \frac{△}{p} + \frac{△^{2}}{p^{2}} + …) (β), β ∈ X𝔬[[X]].

Remark 1. Each element α ∈ ℋ_{m} can be represented in the form

α = ar̄_{01}X_{1}^{r_{01}} … X_{n}^{r_{0n}}ε, ar̄_{0} ∈ 𝔬^{×},

where ε(X) ∈ U_{m}(see [VZF]). We have

l(α) = l(ar̄_{0}) + (1 − \frac{△}{p}) log ε.

The function l is well defined for α ∈ 𝔬^{×}. This can be proved as follows: α ∈ 𝔬^{×}, then \frac{a^{p}}{a^{△}} ≡ 1 mod p𝔬, i.e. l(a) ∈ 𝔬.

Remark 2. In general the functions E and E are different and they coincides only on the monomials θXr̄, where θ ∈ ℜ; i.e. E(θt_{1}^{r_{1}} ⋯ t_{n}^{r_{n}}) = η(E(θXr̄)). The function E depends on the choice of a system of local parameters of the field K, but the function E does not .

Proposition 2. 1. The function l induces the homomorphism

l : ℋ_{m} → X𝔬[[X]],

and the function E induces the homomorphism

E : X𝔬[[X]] → U_{m}.

2. This homomorphisms are mutually inverse isomorphisms between U_{m} and X𝔬[[X]], i.e.

E(l(ε)) = ε, l(E(α)) = α

for every ε ∈ U_{m} and α ∈ X𝔬[[X]].

3. If α ∈ ℋ_{m} and α = aXr̄ε(X), where a = θa_{1} ∈ 𝔬^{×}, θ ∈ ℜ, a_{1} ≡ 1 mod p, and ε(X) ∈ U_{m}, then

E(l(α)) = a_{1}ε(X)

(for the proof, see [V1, 2]). Remark 3. The function E is a homomorphism from the formal module F (X𝔬[[X]]), where F is a Lubin–Tate formal group with an Artin–Hasse logarithm x + x^{p}/p + x^{p^{2}}/p^{2} + …, to the group U_{m}, i.e.,

E(α +_{F} β) = E(α)E(β),

where α +_{F} β = F (α, β)

2.4

We replace the function E in (5) by the function E and obtain

(α_{1}, …, α_{n−1}, E(x), E(y)) = (α_{1}, …, α_{n−1}, η ∘ E(\underline{x}), η ∘ E(\underline{y})) =

∏_{k=1}^{n} (α_{1}, …, α_{n−1}, t_{k}, α_{n+1}), (6)

with

α_{n+1} = η ∘ E (X_{k} \{\underline{x}∂_{k} (1 + \frac{△}{p} + \frac{△^{2}}{p^{2}} + …) (\underline{y})−

\underline{y}∂_{k} (\frac{△}{p} + \frac{△^{2}}{p^{2}} + …) (\underline{x})\}) ,

where ∂_{k} = \frac{∂}{∂X_{k}}, and \underline{x} and \underline{y} are the preimages of x and y respectively in X𝔬[[X]], i.e. \underline{x} = θX_{1}^{i_{1}} … X_{n}^{i_{n}}, \underline{y} = θ′X_{1}^{j_{1}} … X_{n}^{j_{n}}.

Using the linearity of the function E, one can extend the coefficients θ and θ′ from ℜ up to the coefficients from the ring 𝔬.

According to Proposition 2, one can write for any principal unit ε of the field K the following equality: ε = η ∘ E(l(\underline{ε})), where \underline{ε}(X) is the preimage of ε in U_{m} (note that in general case ε = aε_{1}, where a ∈ 𝔬, a ≡ 1 mod p and ε_{1} ≡ 1 mod 𝔐). From the equality (6) and the additivity of E we derive for principal units ε and ε′ of K:

(α_{1}, …, α_{n−1}, ε, ε′) = (α_{1}, …, α_{n−1}, η ∘ E(l(\underline{ε})), η ∘ E(l(\underline{ε}′))) =

∏_{k=1}^{n}(α_{1}, …, α_{n−1}, t_{k}, η ∘ E(X_{k}Φ_{k}(\underline{ε}, \underline{ε}′))), (7)

where Φ_{k} = l(\underline{ε})∂_{k} log \underline{ε}′ − l(\underline{ε}′)∂_{k} \frac{△}{p} log \underline{ε}.

Now, let α_{n} and α_{n+1} be an arbitrary elements of K^{∗}.Then they can be written as α_{n} = θt_{1}^{a_{1}} … t_{n}^{a_{n}}ε, α_{n+1} = θ′t_{1}^{b_{1}} … t_{n}^{b_{n}}ε′, where θ, θ′ ∈ ℜ and ε and ε′ are the principal units of the field K. From the multiplicativity of the Hilbert symbol, we obtain the following formula:

(α_{1}, …, α_{n}, α_{n+1}) =

∏_{k=1}^{n}(α_{1}, …, α_{n−1}, t_{k}, η ∘ E(X_{k}Φ_{k}(\underline{ε}, \underline{ε}′)\underline{ε}^{−b_{k}} \underline{ε}′^{a_{k}})) ×

∏_{1⩽k_{1}≠k_{2}⩽n} (α_{1}, …, α_{n−1}, t_{k_{1}}, t_{k_{2}})^{δ_{k_{1},k_{2}}} ,

where

δ_{k_{1},k_{2}} = | \begin{matrix}a_{k_{1}} & a_{k_{2}} \\ b_{k_{1}} & b_{k_{2}} \end{matrix} | .

The second product disappears since the Hilbert symbol and the determinants δ_{k_{1},k_{2}} are skew-symmetric. Now we calculate the argument of the function E in the first product by using the following easily verifiable relations:

∂_{k} log \underline{ε} + a_{k}X_{k}^{−1} = \underline{α}_{n}^{−1}∂_{k}\underline{α}_{n},

∂_{k} \frac{△}{p} log \underline{ε}′ + b_{k}X_{k}^{−1} = \frac{1}{p}\underline{α}_{n+1}^{−△}∂_{k}\underline{α}_{n+1}^{△}.

Thus, we have

E(X_{k}Φ_{k}(\underline{ε}, \underline{ε}′)\underline{ε}^{−b_{k}} \underline{ε′}^{a_{k}}) =

E(X_{k}(Φ_{k}(\underline{ε}, \underline{ε}′) + a_{k}l(\underline{ε}′)X_{k}^{−1} − b_{k}l(\underline{ε})X_{k}^{−1})) =
E(X_{k}Φ_{k}(\underline{α}_{n}, \underline{α}_{n+1})),

where

Φ_{k}(\underline{α}_{n}, \underline{α}_{n+1}) = l(\underline{α}_{n+1})\underline{α}_{n}^{−1}∂_{k}\underline{α}_{n} − l(\underline{α}_{n})\frac{1}{p}\underline{α}_{n+1}^{−△}∂_{k}\underline{α}_{n+1}^{△}.

From this and (7), we obtain

(α_{1}, …, α_{n}, α_{n+1}) = ∏_{k=1}^{n} (α_{1}, …, α_{n−1}, t_{k}, η ∘ E(X_{k}Φ_{k}(\underline{α}_{n}, \underline{α}_{n+1}))). (8)

Remark 4. A similar relation is valid for each pair α_{i}, α_{j}.

Remark 5. Since the Hilbert symbol is uniquely determined, the valuations on the right-hand side of (8) are the same despite the fact that the preimages of the elements α_{n}, α_{n+1} in ℋ_{m}(see 1.2, (1)) are different.

2.5

We proceed to the next element α_{n−1} to make clear what might be the induction assumption.

Using Remark 4, we obtain

(α_{1}, …, α_{n}, α_{n+1}) = ∏_{1⩽k_{1}<k_{2}⩽n} (α_{1}, …, α_{n−2}, t_{k_{1}}, t_{k_{2}}, α′_{n+1})

with

α′_{n+1} = η ∘ E (X_{k_{1}}Φ_{k_{1}}(\underline{α}_{n−1}, E(X_{k_{2}}Φ_{k_{2}}(\underline{α}_{n}, \underline{α}_{n+1}))) − X_{k_{2}}Φ_{k_{2}}(\underline{α}_{n−1}, E(X_{k_{1}}Φ_{k_{1}}(\underline{α}_{n}, \underline{α}_{n+1})))) .

Now we calculate the argument of the function E (denoting this sum by S_{k_{1},k_{2}}):

S_{k_{1},k_{2}} = X_{k_{1}}X_{k_{2}}S_{1} − l(\underline{α}_{n−1})(X_{k_{1}}F − X_{k_{2}}G),  (9)

where

S_{1} = Φ_{k_{2}}(\underline{α}_{n}, \underline{α}_{n+1})\underline{α}_{n−1}^{−1}∂_{k_{1}} \underline{α}_{n−1} − Φ_{k_{1}}(\underline{α}_{n}, \underline{α}_{n+1})\underline{α}_{n−1}^{−1}∂_{k_{2}} \underline{α}_{n−1},

F = ∂_{k_{1}} (\frac{△}{p} + \frac{△^{2}}{p^{2}} + …) (X_{k_{2}}Φ_{k_{2}}(\underline{α}_{n}, \underline{α}_{n+1})),

G = ∂_{k_{2}} (\frac{△}{p} + \frac{△^{2}}{p^{2}} + …) (X_{k_{1}}Φ_{k_{1}}(\underline{α}_{n}, \underline{α}_{n+1})).

Using the form of the function Φ, we obtain the following relation for S_{1}:

S_{1} = l(\underline{α}_{n+1})D − l(\underline{α}_{n})D′,  (10)

where

D = | \begin{matrix}\underline{α}_{n−1}^{−1}∂_{k_{1}} \underline{α}_{n−1} & \underline{α}_{n−1}^{−1}∂_{k_{2}} \underline{α}_{n−1} \\ \underline{α}_{n}^{−1}∂_{k_{1}} \underline{α}_{n} & \underline{α}_{n}^{−1}∂_{k_{2}} \underline{α}_{n} \end{matrix} | ,

D′ = | \begin{matrix}\underline{α}_{n−1}^{−1}∂_{k_{1}} \underline{α}_{n−1} & \underline{α}_{n−1}^{−1}∂_{k_{2}} \underline{α}_{n−1} \\ \frac{1}{p} \underline{α}_{n+1}^{−△}∂_{k_{1}} \underline{α}_{n+1}^{△} & \frac{1}{p} \underline{α}_{n+1}^{−△}∂_{k_{2}} \underline{α}_{n+1}^{△} \end{matrix} | .

Then

F = ∑_{r⩾1}∂_{k_{1}} \frac{△^{r}}{p^{r}} \{l(\underline{α}_{n+1})(X_{k_{2}} \underline{α}_{n}^{−1}∂_{k_{2}} \underline{α}_{n}) − l(\underline{α}_{n}) (t_{k_{2}} \frac{1}{p}\underline{α}_{n+1}^{−△}∂_{k_{2}} \underline{α}_{n+1}^{△})\}.

Now, using the relation

(X_{i}\underline{α}^{−1}∂_{i}\underline{α})^{△^{r}} = X_{i} \frac{1}{p^{r}} \underline{α}^{−△^{r}}∂_{i}\underline{α}^{△^{r}}  (11)

and differentiate F with respect to X_{k_{1}} we obtain

F = ∑_{r⩾1}X_{k_{2}} (∂_{k_{1}} \frac{△^{r}}{p^{r}} l(\underline{α}_{n+1})\frac{1}{p^{r}} \underline{α}_{n}^{−△^{r}}∂_{k_{2}} \underline{α}_{n}^{△^{r}} −

∂_{k_{1}} \frac{△^{r}}{p^{r}} l(\underline{α}_{n})\frac{1}{p^{r+1}} \underline{α}_{n+1}^{−△^{r+1}}∂_{k_{2}} \underline{α}_{n+1}^{△^{r+1}}) =

∑_{r⩾1}X_{k_{2}} (\frac{△^{r}}{p^{r}} l(\underline{α}_{n+1})∂_{k−1} (\frac{1}{p^{r}} \underline{α}_{n}^{−△^{r}}∂_{k_{2}} \underline{α}_{n}^{△^{r}}) −

\frac{△^{r}}{p^{r}} l(\underline{α}_{n})∂_{k_{1}} (\frac{1}{p^{r+1}} \underline{α}_{n+1}^{−△^{r+1}}∂_{k_{2}} \underline{α}_{n+1}^{△^{r+1}})) . (12)

One may proceed the calculation for the series series G. Considering X_{k_{1}}F − X_{k_{2}}G and taking into account that

∂_{i} (\frac{1}{p^{r}} \underline{α}^{−△^{r}}∂_{j}\underline{α}^{△^{r}}) = ∂_{j} (\frac{1}{p^{r}} \underline{α}^{−△^{r}}∂_{i}\underline{α}^{△^{r}}) , r ⩾ 0,

we see that the second term in (12) disappears and so

X_{k_{1}}F − X_{k_{2}}G = X_{k_{1}}X_{k_{2}} ∑_{r⩾1}(D_{r} + Dr′),

where

D_{r} = | \begin{matrix}\frac{1}{p^{r}} \underline{α}_{n}^{−△^{r}}∂_{k_{1}} \underline{α}_{n}^{△^{r}} & \frac{1}{p^{r}} \underline{α}_{n}^{−△^{r}}∂_{k_{2}} \underline{α}_{n}^{△^{r}} \\ ∂_{k_{1}} \frac{△^{r}}{p^{r}} l(\underline{α}_{n+1}) & ∂_{k_{2}} \frac{△^{r}}{p^{r}} l(\underline{α}_{n+1}) \end{matrix} | ,

D_{r}′ = | \begin{matrix}∂_{k_{1}} \frac{△^{r}}{p^{r}} l(\underline{α}_{n}) & ∂_{k_{2}} \frac{△^{r}}{p^{r}} l(\underline{α}_{n}) \\ \frac{1}{p^{r+1}} \underline{α}_{n+1}^{−△^{r+1}}∂_{k_{1}} \underline{α}_{n+1}^{△^{r+1}} & \frac{1}{p^{r+1}} \underline{α}_{n+1}^{−△^{r+1}}∂_{k_{2}} \underline{α}_{n+1}^{△^{r+1}} \end{matrix} | .

It is easy to see that

∂_{i} \frac{△^{r}}{p^{r}} l(\underline{α}) = \frac{1}{p^{r}} \underline{α}^{−△^{r}}∂_{i}\underline{α}^{△^{r}} − \frac{1}{p^{r+1}} \underline{α}^{−△^{r+1}}∂_{i}\underline{α}^{△^{r+1}}

Using the latter relation for the second row of the determinant D_{r} and for the first row of D′_{r}, we expand each of the determinants with respect to the corresponding row; then, summing D_{r} + D′_{r} over r, we obtain

X_{k_{1}}F − X_{k_{2}}G = X_{k_{1}}X_{k_{2}}D′′,  (13)

where

D′′ = | \begin{matrix}\frac{1}{p} \underline{α}_{n}^{−△}∂_{k_{1}} \underline{α}_{n}^{△} & \frac{1}{p} \underline{α}_{n}^{−△}∂_{k_{2}} \underline{α}_{n}^{△} \\ \frac{1}{p} \underline{α}_{n+1}^{−△}∂_{k_{1}} \underline{α}_{n+1}^{△} & \frac{1}{p} \underline{α}_{n+1}^{−△}∂_{k_{2}} \underline{α}_{n+1}^{△} \end{matrix} | .

Thus, keeping in the mind the notation of (9), we have

(α_{1}, …, α_{n}, α_{n+1}) = ∏_{1⩽k_{1}<k_{2}⩽n}(α_{1}, …, α_{n−2}, t_{k_{1}}, t_{k_{2}}, α′_{n+1})

with

α′_{n+1} = η ∘ E(X_{k_{1}} … X_{k_{n−m}}Φ_{k_{1},k_{2}}(\underline{α}_{n−1}, \underline{α}_{n}, \underline{α}_{n+1})),

where

Φ_{k_{1},k_{2}} = l(\underline{α}_{n+1})D − l(\underline{α}_{n})D′ + l(\underline{α}_{n−1})D′′

and D, D′, and D′′ are determined in (10) and (13), respectively.

2.6

Now it is clear how to state the induction assumption. We have

(α_{1}, …, α_{m}, α_{m+1}, …, α_{n+1}) =

∏_{1⩽k_{1}<···<k_{n−m}⩽n}(α_{1}, …, α_{m}, t_{k_{1}}, …, t_{k_{n−m}}, α′_{n+1})

with

α′_{n+1} = η ∘ E(X_{k_{1}} … X_{k_{n−m}}Φ^{(m)}(\underline{α}_{m+1}, …, \underline{α}_{n+1})),

where

Φ^{(m)} = l(\underline{α}_{n+1})D_{n+1}^{(n−m)} − l(\underline{α}_{n})D_{n}^{(n−m)} + … + (−1)^{n−m+1}l(\underline{α}_{m+1})D_{m+1}^{(n−m)}, (14)

D_{n+1}^{(n−m)} = | \begin{matrix}\underline{α}_{m+1}^{−1}∂_{k_{1}} \underline{α}_{m+1} & … & \underline{α}_{m+1}^{−1}∂_{k_{n−m}} \underline{α}_{m+1} \\ … & … & … \\ \underline{α}_{n}^{−1}∂_{k_{1}} \underline{α}_{n} & … & \underline{α}_{n}^{−1}∂_{k_{n−m}} \underline{α}_{n} \end{matrix} | ,

D_{r}^{(n−m)} = | \begin{matrix}\underline{α}_{m+1}^{−1}∂_{k_{1}} \underline{α}_{m+1} & … & … & … & … & \underline{α}_{m+1}^{−1}∂_{k_{n−m}} \underline{α}_{m+1} \\ … & … & … & … & … & … \\ \underline{α}_{r−1}^{−1}∂_{k_{1}} \underline{α}_{r−1} & … & … & … & … & \underline{α}_{r−1}^{−1}∂_{k_{n−m}} \underline{α}_{r−1} \\ \frac{1}{p} \underline{α}_{r+1}^{−△}∂_{k1}\underline{α}_{r+1}^{△} & … & … & … & … & \frac{1}{p} \underline{α}_{r+1}^{−△}∂_{k_{n−m}} \underline{α}_{r+1}^{△} \\ … & … & … & … & … & … \\ \frac{1}{p} \underline{α}_{n+1}^{−△}∂_{k1}\underline{α}_{n+1}^{△} & … & … & … & … & \frac{1}{p} \underline{α}_{n+1}^{−△}∂_{k_{n−m}} \underline{α}_{n+1}^{△} \end{matrix} | ,

m + 1 ⩽ r ⩽ n.

We leave to the reader the induction step itself, which can be performed in the same way as in 2.5 but with more cumbersome calculation.

Thus, using the generalized Kneser relation, we obtain the following equation:

(α_{1}, …, α_{n+1}) = (t_{1}, …, t_{n}, η ∘ E(XΦ(α_{1}, …, α_{n}))),  (15)

where

Φ = Φ^{(n)},

see (14).

3 Explicit Formulas

3.1 The series V (X)

Now we proceed to the most difficult part of the paper, i.e. to the calculation of the symbol (t_{1}, …, t_{n}, ε) for the principal unit ε.

As before, let Xm̄ denote the monomial X_{1}^{m_{1}} … X_{n}^{m_{n}} for m̄ = (m_{1}, …, m_{n}).

Lemma 2. For every multiindex m̄ > ¯0 there exists an element vm̄ of 𝔬 such that, for all β ∈ 𝔬, the following equality holds:

(t_{1}, …, t_{n}, η ∘ E(βXm̄)) = ζ^{tr(βvm̄)},

where tr is the trace operator from 𝔬 to ℤ_{p}. The element vm̄ is uniquely determined mod p^{N}. Here we assume that if p ∤ m̄ then vm̄ ≡ 0 mod p^{N}.

Proof. 1) The existence of vm̄. We consider the dual bases α_{1}, …, α_{f} and α1^{′}, …, αf^{′} of the ring 𝔬 over ℤ_{p} with respect to the trace operator. Then

tr α_{i}αj^{′} = δ_{ij}. Let

(t_{1}, …, t_{n}, η ∘ E(α_{i}Xm̄)) = ζ^{x_{i}}, x_{i} ∈ ℤ_{p},

and let

vm̄ = ∑_{i=1}^{f} x_{i}α_{i}′.

If β = y_{1}α_{1} + ⋯ + y_{f}α_{f} is an arbitrary element of 𝔬, then

(t_{1}, …, t_{n}, η ∘ E(βXm̄)) = ζ^{x_{1}y_{1}+···+x_{f} y_{f}} .

On the other hand, from the duality of the bases, we obtain

tr βvm̄ = ∑ x_{i}y_{j} tr α_{i}αj^{′} = ∑_{i=1}^{f} x_{i}y_{i}

2) Uniqueness mod p^{N}. Let ω be another element of 𝔬 satisfying the same conditions. Then, for every β ∈ 𝔬, we have tr β(vm̄ − ω) ≡ 0 mod p^{N}; since the trace is nondegenerate, we obtain vm̄ ≡ ω mod p^{N}.

3) Let p ∤ m̄, and let θ ∈ ℜ. We have

η ∘ E(θXm̄) = E(θT m̄) = ∏_{(i,p)=1} (1 − (θT m̄)^{i})^{−μ(i)/i},

where T m̄ = t_{1}^{m1} … t_{n}^{m_{n}} for the multiindex m̄ = (m_{1}, …, m_{n}).

It follows from the condition p ∤ m̄ that p ∤ m_{k} for some k such that 1 ⩽ k ⩽ n. We have

(t_{1}, …, t_{n}, 1 − (θt_{1}^{m_{1}} … t_{n}^{m_{n}})^{i}) =

(t_{1}, …, (θt_{1}^{m_{1}} … t_{n}^{m_{n}})^{i}, …, t_{n}, 1 − (θt_{1}^{m_{1}} … t_{n}^{m_{n}})^{i})^{\frac{1}{m_{k}i}} = 1.

It follows from the last relation that, for for every β ∈ 𝔬, the following relation holds:

(t_{1}, …, t_{n}, η ∘ E(βXm̄)) = 1, p ∤ m̄.

Then, from the uniqueness of vm̄ mod p^{N}, we obtain vm̄ ≡ 0 mod p^{N}.

We use the elements vm̄ to construct the series

V (X) = ∑_{m̄>¯0} vm̄X^{−m̄} ∈ M(K).

The property of the series V (X) is given in the following statement, which immediately follows from Lemma 2 and the multiplicativity of the function E.

Proposition 3. Let φ(X) ∈ 𝔬[[X]]. Then

(t_{1}, …, t_{n}, η ∘ E(Xφ(X))) = ζ^{tr res φ(X)V (X)},

where res denotes the coefficient at X_{1}^{−1} … X_{n}^{−1}.

From this Proposition and the relation (15), we obtain the following result:

Proposition 4. For each element α = {α_{1}, …, α_{n}} of the K-group K_{n}^{top}(K) and each β ∈ K^{∗}, the Hilbert symbol satisfies the relation :

(α, β)_{N} = ζ^{tr res Φ(α,β)V}

In particular,

(t_{1}, …, t_{n}, ε)_{N} = ζ^{tr res X^{−1}l(\underline{ε})V} ,  (16)

where \underline{ε}(X) ∈ 1 + X𝔬[[X]].

3.2 The explicit form of primary elements

We need to know the explicit form of the primary elements of the field K to calculate the series V (X). Let

s_{m}(X) := \underline{ζ}^{p^{m}} − 1,  (17)

and

s(X) := s_{N}(X).

Proposition 5. For every series φ ∈ 𝔬[[X]], the element

ω(φ) = η ∘ E(φs)  (18)

is the p^{N}-primary and every p^{N}-primary element can be represented in the form (18) with respect to mod (k^{×})^{p^{N}}. Furthermore,

(t_{1}, …, t_{n}, ω(φ)) = ζ^{tr φ(0)}  (19)

(for the proof, see [V1, 2]).

3.3 The first congruence for the series V (X)

From relation (16) in Proposition 4 and relation (18) in Proposition 5, we obtain

(t_{1}, …, t_{n}, ω(φ)) = ζ^{tr res X^{−1}φ s V} ,  (20)

since l(\underline{ω(φ)}) = l(E(φs)) = φs.

Now we use the fact that φ can be chosen arbitrarily. First, let φ(X) = a ∈ 𝔬. Then from (19) and (20), we obtain tr a ≡ tr res aX^{−1} s V mod p^{N}, and so

res X^{−1}s(X)V (X) ≡ 1 mod p^{N}  (21)

Now, we put φ(X) = βXm̄, m̄ > ¯0, β ∈ 𝔬. From (19) and (20), we obtain

tr βam̄ ≡ 0 mod p^{N}, β ∈ 𝔬,

where am̄ is the coefficient of the series s(X)V (X) at X^{−m̄}. Then am̄ ≡ 0 mod p^{N}, and so the congruence s(X)V (X) ≡ 0 mod (p^{N}, deg 0) is valid.

Thus, using the congruence (21), we obtain the first congruence for the series V:

V (X)s(X) ≡ 1 mod (p^{N}, deg 1)

which can be written as

V (X) = 1/s + g/s mod p^{N}  (22)

for some g ∈ X𝔬[[X]].

Remark 6. From the congruence s V ≡ 1 mod (p^{N}, deg 1), we obtain the following relation: sV = 1 + g + p^{N}f, where g ∈ X𝔬[[X]], f ∈ M(K); the series g(X) has integer coefficients since the series s and V have integer coefficients. Then V = (1 + g)/s + p^{N}f/s, hence

V ≡ (1 + g)/s mod p^{N}.

3.4 The second congruence to the series V

To calculate the series g(X) (and so, the series V), we use the following conclusion. For the element ε of K, its preimage \underline{ε}(X) in ℋ_{m} is not uniquely determined, and it may happen that \underline{ε}(X) ≠ 1 but η ∘ \underline{ε} = 1; in this case, for the valuation of the Hilbert symbol, we have, obviously,

(t_{1}, …, t_{n}, η ∘ \underline{ε}(X)) = 1.

On the other hand, by Proposition 2 and relation (16), we obtain

(t_{1}, …, t_{n}, η ∘ \underline{ε}(X)) = ζ^{tr res X^{−1}l(\underline{ε})V}

Thus,

tr res X^{−1}l(ε(X))V (X) ≡ 0 mod p^{N}  (23)

for all ε(X) ∈ ℋ_{m} suchthat η ∘ ε(X) = 1.

It is convenient to represent ε(X) in the following equivalent form:

ε(X) = E(φ(X))

for some φ ∈ X𝔬[[X]].

We can always do this, because, for φ ∈ X𝔬[[X]], we have E(φ) = 1 + H(φ), where H(φ) is the composition of the series exp X − 1 and x + x^{p}/p + . . .. Thus, if f(X) = ε(X) − 1, then we can take H^{−1}(f) as φ, where H^{−1} is the series inverse to H with respect to the composition.

It is obvious that

η ∘ ε(X) = 1 ⇔ η ∘ φ(X) = 0.

However, by Lemma 1, for Eis(X) = u(X) := s/s_{N−1} we have

η ∘ φ(X) = 0 ⇔ φ(X) = u(X)ψ(X),

where ψ is a series such that ψ(X) ∈ X𝔬[[X]]. Then from congruence (23), we obtain the second congruence for V:

tr res X^{−1}l(E(uψ))V ≡ 0 mod p^{N}  (24)

for all series ψ ∈ X𝔬[[X]].

We use the following easily verifiable relation:

tr res X^{−1}h(X)^{△} = tr res X^{−1}h(X),  (25)

which is true for all h(X) in M(K).

From relation (25) and the definition of the function l, we obtain

tr res X^{−1} ∑_{k⩾0}(\frac{(uψ)^{p^{k}}}{p^{k}} − \frac{(uψ)^{p^{k}△}}{p^{k+1}} ) V =

tr res X^{−1} ∑_{k⩾0}(ψ^{pk△} \frac{u^{p^{k}△}}{p^{k+1}} (1 − p△)V ) ≡ 0 mod p^{N}. (26)

Lemma 3. For all m ⩾ 1 and p ≠ 2, the following congruence holds:

\frac{u^{m△}}{pm} (1 − p△)V ≡ \frac{p^{m−1}}{m} \frac{1}{s}(1 − △)g mod (p^{N}, deg 0).

Proof. By the definition of the series u(X), we have

u(X) = Q(s_{N−1}),

where Q(X) = ((1 + x)^{p} − 1)/x. It is obvious that if p ≠ 2, then

Q(−p^{N}h)^{m} ≡ p^{m} mod mp^{N+1},  (27)

where h = (s − s_{N−1}^{△})/p^{N}. Further, for f ∈ 𝔬[[X]], we have

\frac{u^{m△}}{pm} f/s = \frac{Q(s_{N−1}^{△})^{m}}{pm} f/s = \frac{Q(s − p^{N}h)^{m}}{pm} f/s ≡

\frac{Q(−p^{N}h)^{m}}{p^{m}} f/s mod (p^{N}, deg 0).

Using (27), we obtain

\frac{u^{m△}}{pm} f/s ≡ \frac{p^{m−1}}{m} \frac{f}{s} mod (p^{N}, deg 0).  (28)

From the definition of u(X), we obtain

\frac{u^{m△}}{m} f^{△}/s^{△} = \frac{u^{(m−1)△}}{m} f^{△}/s^{△}N − 1 = \frac{Q(s_{N−1}^{△})^{m−1}}{m} f^{△}/s_{N−1}^{△} =

\frac{(p + \binom{p}{2}s_{N−1}^{△})^{m−1}}{m} f^{△}/s_{N−1}^{△} ≡ \frac{p^{m−1}}{m} f^{△}/s_{N−1}^{△} mod deg 0.

Since p^{m−1}/m ∈ ℤ_{p}, m ⩾ 1, the latter congruence and the congruence

1/s ≡ 1/s_{N−1}^{△} mod p^{N}

( see section 3, [V1]) imply

\frac{u^{m△}}{m} f^{△}/s^{△} ≡ \frac{p^{m−1}}{m} f^{△}/s mod (p^{N}, deg 0).  (29)

Substituting f by 1 + gand subtracting congruence (29) from (28), we obtain the congruence of the lemma.

As follows from the lemma, relation (26) takes the following form:

tr res X^{−1} ∑_{k⩾0}(β^{p^{k}△}ψ^{p^{k}△} \frac{p^{p^{k}−1}}{p^{k}} 1/s(1 − △)(g)) ≡ 0 mod p^{N}. (30)

We note that the series ψ, 1/s, g have integer coefficients and p^{p^{k}−p−1} is divisible by p^{N} when k ≥ N, p ≠ 2; therefore, congruence (30) takes the form

tr res X^{−1} ∑_{k=0}^{N−1} ψ^{p^{k}△}p^{p^{k}−k−1}1/s(1 − △)(g) ≡ 0 mod p^{N}.

Using relation (25), we may write the latter congruence in the form

tr res X^{−1} ∑_{k=0}^{N−1} ψ^{p^{k}△^{N−k+1}}p^{p^{k}−k−1}((1 − △)(g)1/s)^{△^{N−k}} ≡ 0 mod p^{N}.

(31)

Lemma 4. Let γ_{0}, …, γ_{M} be elements of the ring 𝔬, and let the following congruence be true for all β ∈ 𝔬:

tr(β^{△^{M}} γ_{0} + β^{p△^{M−1}}γ_{1} + ⋯ + β^{p^{M}} γ_{M}) ≡ 0 mod p^{N}.

Then γ_{0} + γ_{1} + ⋯ + γ_{M} ≡ 0 mod p^{N}.

Proof. We prove the statement if we let β be the generator of the basis of the ring 𝔬 over ℤ_{p} formed by the set of the Teichmüller representatives, which is the lift of the basis of K^{(0)} ≅ 𝔽_{q} over 𝔽_{p}.

Replacing the series ψ(X) in (31) by βψ(X), β ∈ 𝔬 and using Lemma 4, we obtain the following congruence:

res X^{−1} ∑_{k=0}^{N−1} ψ^{p^{k}△^{N−k+1}}p^{p^{k}−k−1}((1 − △)(g)(1/s))^{△^{N−k}} ≡ 0 mod p^{N}.

Using the independence of the choice of ψ, we substitute the powers Xm̄, m̄ > ¯0 for ψ. Then, as in 2.3, we obtain from the latter congruence:

∑_{k=0}^{N−1} p^{p^{k}−k−1}((1 − △)g(1/s))^{△^{N−k}} ≡ 0 mod (p^{N}, deg 0).

This means that

∑_{k=0}^{N−1} p^{p^{k}−k−1}((1 − △)g(1/s))^{△^{N−k}} = h + p^{N}f,

where f ∈ M(K) and h ∈ 𝔬[[X]], i.e. all series on the left-hand side have the coefficients in the ring 𝔬.

We multiply both sides of the latter congruence by s^{△^{N}} and obtain

∑_{k=0}^{N−1} p^{p^{k}−k−1}s^{△^{N} −△^{N−k}}(1 − △)g^{△^{N−k}} ≡ h s^{△^{N}} mod p^{N},  (32)

where h ∈ 𝔬[[X]].

From the congruence s_{m} ≡ s_{m−1}^{△} mod p^{m} (see section 3, [V1] and §1, [V2]), it follows that

s^{△^{N}} ≡ s_{2N} mod p^{N}, s^{△^{N−k}} ≡ s_{2N−k} mod p^{N}.

By the definition of the series s_{m} (see 3.2, (17)), we have s^{△^{N} −△^{N−k}} ≡ s_{2N}/s_{2N−k} =: f_{k} mod p^{N}, where f_{k} ∈ 𝔬[[X]]. Then, (32) transforms into the following congruence:

A(△)(1 − △)g ≡ h s^{△^{N}} mod p^{N},  (33)

where

A(△) = △^{N} + ∑_{k=1}^{N} p^{p^{k}−k−1}fk△^{N−k}

is an operator of the ring 𝔬[[X]][△]. We have

A(△)^{−1} = △^{−N}(1 + ∑_{k=1}^{N} p^{p^{k}−k−1}f_{k}△^{−k})^{−1} ≡

△^{−N} + ∑_{k=1}^{N} a_{k}△^{−N−k} mod p^{N}, a_{k} ∈ p^{k}𝔬[[X]].

We note that the following congruence is valid in the ring 𝔬[[X^{1/p^{m}}]]:

s^{△^{−m}} ≡ s mod p^{N}𝔬[[X^{1/p^{m}}]].  (34)

Indeed, the congruence

s ≡ s_{N+m} ≡ s^{△^{m}} mod p^{N}

means that

s = s^{△^{m}} + p^{N}f, f ∈ 𝔬[[X]];

then

s^{△^{−m}} = s + p^{N}f^{△^{−m}}, f^{△^{−m}} ∈ 𝔬[[X^{1/p^{m}}]].

We apply the operator A(△)^{−1} to congruence (33) and use (34). Thus we obtain

(1 − △)g ≡ A(△)^{−1}(hs^{△^{N}}) ≡ h_{1}s mod p^{N},

where h_{1} ∈ 𝔬[[X^{1/p^{2N}}]]. It remains to apply the operator (1 − △)^{−1} = 1 + △ + △^{2} + … to the both sides and obtain

g ≡ (1 + △ + △^{2} + …)(h_{1}s) ≡ h_{2}s mod p^{N}, h_{2} ∈ 𝔬[[X^{1/p^{2N}}]].

We have

g/s ≡ h_{2} ≡ 0 mod (p^{N}, deg 0),

and therefore

V ≡ 1/s mod (p^{N}, deg 0).

From this congruence and Proposition 4, we obtain the following explicit formula for the Hilbert symbol:

(α, β)_{N} = ζ^{tr res Φ(α,β)/s}

for every

α = {α_{1}, …, α_{n}} ∈ K_{n}^{top}(K), β ∈ K^{×},

where the series Φ defined in 2.6, (15).

References

[Ka] K. Kato, A generalisation of local class field theory by using Kgroups I, J. Fac. Sci. Univ. Tokyo, Sect. IA, 26 (1979), 303-376; II, J. Fac. Sci. Univ. Tokyo, Sect. IA, 27 (1980), 603-683

[Kn] M. Kneser, Zum explitizen Reziprozitätsgesetz von I. R. Shafarevich, Math. Nachr., 6 (1951/1952), 89-96

[P1] A. N. Parshin, Class field theory and algebraic K-theory, Uspekhi Mat. Nauk 30 (1975), 253-254

[P2] A. N. Parshin, Local class field theory, Trudy Mat. Inst. Steklov 165 (1984), 143-170; English transl. in Proc. Steklov Inst. Math. 3 (1985)

[Sh] I. R. Shafarevich, A general reciprocity law, Mat. Sbornik 26 (68) (1950), 113-146; English transl. in Amer. Math. Soc. Transl. Ser. 2.4 (1956), 73-106

[V1] S. V. Vostokov, Explicit form for the reciprocity law, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), 1287-1320; English transl. in Math. USSR Izv. 13 (1979), 557-588

[V2] S. V. Vostokov, An explicit construction of class field theory for multidimensional local fields, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), 283-308; English transl. in Math. USSR Izv. 26 (1986)

[VZF] S. V. Vostokov, I. B. Zhukov, and I. B. Fesenko, On the theory of multidimensional local fields. Methods and constructions, Algebra i Analis 2 (1990), No. 4, 91-118; English transl. in Leningrad Math. J. 2 (1991), No. 4, 775-800