Explicit Form of the Hilbert Symbol for Polynomial Formal Groups

Algebra i analiz St. Petersburg Math. J. Tom 26 (2014), 5 Vol. 26 (2015), No. 5, Pages 785–796 http://dx.doi.org/10.1090/spmj/1358 Article electronically published on July 27, 2015

EXPLICIT FORM OF THE HILBERT SYMBOL FOR POLYNOMIAL FORMAL GROUPS

S. VOSTOKOV AND V. VOLKOV

Abstract. Let K be a local ﬁeld, c a unit in K,andFc(X, Y )=X + Y + cXY a polynomial formal group that gives rise to a formal module Fc(M) on the maximal

ideal in the ring of integers of K. Assume that K contains the group μFc,n of the n · · ∗ × → roots of isogeny [p ]c(X). The natural Hilbert symbol ( , )c : K Fc(M) μFc,n is deﬁned over the module F (M). An explicit formula for ( · , · )c is constructed.

§1. Introduction The present paper continues the papers [5, 6], which were devoted to Hilbert pairing over the so-called polynomial formal groups. It is easy to show that the groups of the form Fc = X + Y + cXY are the only formal groups defined by polynomials. We call such groups polynomial formal groups and investigate the Hilbert pairing over them. The case where the element c belongs to the ring of Witte vectors, i.e., the case of polynomial Honda groups, was treated in [5]. There, the main results in the unramified case were formulated together with proof sketches. A similar pairing over the universal formal group Fx = X + Y + xXY (where x is an independent variable) was constructed and studied in [6]. Now we consider an arbitrary polynomial formal group Fc = X + Y + cXY ,wherec is a unit in some local ring (finite extension of Qp) and find an explicit formula for the Hilbert symbol defined over the formal module of the group Fc. Similar formulas were constructed previously for the formal Lubin–Tate groups in [9, 10], for Honda groups in [11, 12, 14], and for the relative formal Lubin–Tate groups in [13]. The case considered in the present paper provides the first example of a formula for the Hilber symbol of groups whose ring of endomorphisms embeds in, but is not isomorphic to the ring where the group is defined. We use the method suggested in [8] to derive the claimed explicit formula.

§2. Notation We denote: • p ≥ 3 is a prime number; • ζ is a certain fixed primitive root of unity of degree pn; • K is a finite extension of the field Qp that contains ζ, with the ring of integers OK ; • c is a unit of the local field K; • T is the inertia subfield of K, with the ring of integers O = OT ; • e is the ramification degree of K/T; • e1 = e/(p − 1) is the relative ramification degree of K/T(ζ1); n−1 • e0 = e1/p ;

2010 Mathematics Subject Classiﬁcation. Primary 11S31; Secondary 14L05. Key words and phrases. Polynomial formal groups, formal groups, Hilbert symbol, local rings. Supported by RFBR (grant no. 14-01-00393).

c 2015 American Mathematical Society 785

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• is the Frobenius automorphism in T/Qp; • tr is the trace in T/Qp; • π is a prime element of the field K; • M is a maximal ideal of OK ; • R is a set of Teichmuller representatives of the residue field Ks in the ring OK ; • Fc = X + Y + cXY is a polynomial formal group; • Fc(M)istheformalZp-module on M, determined by the group Fc; • λc(X) is the logartihm of the formal group Fc; • Kβ is the extension of the field K constructed by adding roots of the equation n [p ]c(X)=β for some β ∈ Fc(M); n • ξ is a generating element of the kernel of [p ]c(X); • d is the operator of differentiation by X in O((X)). It is easily seen that −1 Fc(X, Y )=c (1 + cX)(1 + cY ) − 1 , −1 −1 whence λc(X)=c log(1 + cX)andξ = c (ζ − 1). We denote the completion of the maximal unramified p-extension of T by Tr, and the Frobenius automorphism of the topological Galois group Gal(T/Tr )byϕ: Qp T T (ζ) T (ζ,c) K

ϕ ϕ ϕ ϕ

Tr Tr(ζ) Tr(ζ,c) TKr We fix certain series c, ζ,andξ = c−1(ζ − 1) in O[[X]] such that c(π)=c, ζ(π)=ζ, and ξ(π)=ξ. ∗ Consider the group Hm = O((X)) and the formal Zp-module Hc = XO[[X]]. The structure of a Zp-module is defined on Hc by the formal group Fc = X +Y +cXY , −1 which is an analog of the group Fc, and the series λc = c log(1 + cX) is an analog of its logarithm. We introduce an operator on the Laurent series ring: pi i ∈O f(x)= ai X , where f(x)= aiX ,ai . Next,wedefinetheseries n −1 pn • sc =[p ]c(ξ)=c (1 + c · ξ) − 1 ; n−1 • sn−1,c =[p ]c(ξ); • uc = sc/sn−1,c. pn pn−1 Note that if we further define s =(ζ − 1) and sn−1 =(ζ − 1), where ζ =1+c · ξ, then

uc = sc/sn−1,c = s/sn−1 = u,

where s, sn−1,andu are the same as in [1, §3]. Consider the following functions E and : 1 αp (α)= log , where α ∈H , p α m 2 E(β)=exp 1+ + + ... β, where β ∈ XO[[X]] p p2 (see [1, §1]).

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The deﬁnition given above agrees with that in [1, §1], because for β ∈ XO[[X]] we have 1 (1 + β)p 1 (1 + β)= log =log(1+β) − log(1 + β)= 1 − log(1 + β). p (1 + β) p p

Consider the analogs of these functions for the formal group Fc:

−1 c(β)=c (1 + cβ), where β ∈Hc, −1 Ec(β)=c (E(cβ) − 1), where β ∈ XO[[X]]. The corresponding result on the functions and E (see [1, §1, Proposition 1]) implies the following statement.

Proposition 1. The functions c and Ec are mutually inverse isomorphims between the additive Zp-module XO[[X]] and the formal Zp-module Hc.

In what follows, we use the notation Fc for the formal group Fc and λc for its logarithm λc, since it is always clear whether we are currently dealing with numbers or formal series. One can easily check the following relations: (1) dα = pX−1(Xdα);d(α)=α−1dα − X−1(Xα−1dα).

We also recall the deﬁnition of the Hilbert symbol ( · , · )c of the formal group Fc: σ(α) − (α, β)c = B Fc B, ∗ n ∗ where α ∈ K , β ∈ Fc(M), B is the solution of the equation [p ]c(B)=β,andσ : K → Gal(Kab/K) is the Artin–Frobenius automorphism.

§3. Pairing over the formal Zp-module Hc ∗ For α ∈ K and β ∈ Fc(M), we introduce the pairing · · H ×H →O n , c : m c /p , n α, β → resX Φ(α, β) /sc mod p , where Φ(α, β)= (β)α−1dα − (α)c−1d cλ (β). c p c

Proposition 2. The pairing ·, ·c is Zp-linear, i.e., a α1α2,βc = α1,βc + α2,βc; α ,βc = aα, βc, α, β1 +Fc β2 c = α, β1 c + α, β2 c; α, [a]c(β) c = a α, β c. Proof. Linearity over the ﬁrst argument follows from that of the logarithmic derivative and the function (α). Linearity over the second argument follows from that of the functions c and λc.

Proposition 3. The pairing ·, ·c satisﬁes the following Steinberg relation: pn−1 ∈ O \{ } α, c α c =0, where α X [[X]] 0 . Proof. First we prove the congruence (2) αmp − αm ≡ mpαm mod (mp)2, for α ∈O[[X]]∗.

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By the deﬁnition of we have (α)=log(αp/α)/p, i.e., mp·(α) = log αmp−m, whence αmp − αm = αm(αmp−m − 1) = αm(exp(log αmp−m) − 1) (mp)i = αm(exp(mp · (α)) − 1) = αm mp(α)+ (α)i i! i≥2 (mp)i = αmmp(α)+αm (α)i. i! i≥2 Note that if i ≥ 2andp ≥ 3, then (mp)i/i! ≡ 0mod(mp)2. Therefore, (2) is true. The identities in (1) imply αpm − αm αm ψ = α−1 dα − (α)d =dχ , m pm pm m (3) αpm − αm αm where χ = − (α). m (pm)2 pm

Moreover, the series χm has integral coeﬃcients, in accordance with (2). Any series β ∈O[[x]] satisﬁes the congruence

n n−1 (4) βp m ≡ βp m mod pn.

n n−1 Indeed, βp ≡ β mod p, and easy induction shows that βp ≡ βp mod pn, yielding (4). n Now we calculate the series Φ(α, cp −1α). Directly from the deﬁnition, we deduce the relation

n n n (5) Φ(α, cp −1α)= (cp −1 α)α−1 dα − (α)c−1 d (cλ(cp −1α)). c p

Consider the ﬁrst summand and apply the deﬁnition of c to it: n n (cp −1α)α−1dα = c−1 1 − log(1 + cp α)α−1 dα c p n cp mαm = c−1 (−1)m−1 α−1 dα (6) m pm n+1 n cp mαpm − cp mαm + c−1 (−1)m−1 α−1 dα. pm m≥1 It is easily seen that if p m,then

n n cp mαm cp mαm α−1 dα ≡ d mod pn, m m2 so that the ﬁrst summand in (6) takes the form

n n cp mαm cp mαm c−1 (−1)m−1 α−1 dα ≡ c−1 dη mod pn, where η = (−1)m−1 . m m2 pm pm Here the series η has integral coeﬃcients. Finally, substituting this in (6) and taking (4) into account, we get pm m n n+1 α − α (7) (cp −1α)α−1 dα = c−1 dη + c−1 (−1)m−1cp m α−1 dα. c pm m≥1

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Now we consider the second summand in (5): pnm m n c α (α)c−1 d (cλ(cp −1α)) = (α)c−1 d (−1)m−1 . p pm m≥1 We use congruence (4) to get pn+1m m n c α (α)c−1 d (cλ(cp −1α)) ≡ (α)c−1 d (−1)m−1 p pm m≥1 m n+1 α ≡ (α)c−1 (−1)m−1cp m d mod pn. pm m≥1 Hence, recalling (7), we see that pm m m n n+1 α − α α Φ(α, cp −1α) ≡ c−1 dη + (−1)m−1cp m α−1 dα − (α)d . pm pm m≥1

The deﬁnition of the function χm (see (3)) implies pn−1 −1 m−1 pn+1m n Φ(α, c α) ≡ c dη + (−1) c dχm mod p , m≥1 and the series ϕ and χ have integral coeﬃcients. Thus, m pn−1 pn−1 ≡ pn+1m ≡ n α, c α c =resX Φ(α, c α)/sc resX d η + c χm s 0modp , m≥1 −1 n because sc = c s and d(1/s) ≡ 0modp . This proves the Steinberg relation is question.

§4. Primary elements In the multiplicative case, the pn-primary elements in the field K were constructed ∗ n and investigated√ in [1]. Recall that an element ω ∈ K is said to be p -primary if the extension K( pn ω)/K is unramified. Two types of primary elements were defined in [1]. The first type will be called the Hasse type; we construct it in the following way. Let a ∈Oand A ∈O be elements such that A − A = a. Then the element H(a)= Tr E(pnA(ζ)) is pn-primary, and for the Hilbert symbol ( · , · ): K∗ ×K∗ →ζ in the X=π field K we have (π, H(a)) = ζtr a (see [1, §4, Lemma 8]). The second type of pn-primary elements, as constructed in [1], has the form ω(a)= n n E(a(ζp − 1)) ;itisrelatedtoH(a)byω(a) ≡ H(a)mod(K∗)p (see [1, §4, item 3]). X=π The main difference between these types is that in the construction of the second only elements of initial field K are used. Denote H(a)= pn H(a), ω(a)= pn ω(a). Then H(a)andω(a) lie in the finite unramified extension of the field T (ζ). Now we define primary elements in the formal module Fc(M). An element ωc ∈ Fc(M) n 1 is said to be p -primary if the extension K( n ωc)/K is unramified. Define [p ]c H (a)=c−1(H(a) − 1) = E (pnA (ξ)) ; c c c X=π −1 − ωc(a)=c (ω(a) 1) = Ec(asc(X)) X=π. ∈ 1 It is clear that Hc(a), ωc(a) T (ζ,c), and the elements H (a)= n Hc(a), ω (a)= c [p ]c c 1 n ωc(a) lie in the unramified extension of the field T (ζ,c)=T (ξ,c). [p ]c ϕ ϕ − ∈ Z Theorem 1. Hc(a) = Hc(a)+Fc [aϕ](ξ),whereA A = aϕ p.

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Proof. Since a similar statement is true for the multiplicative case (see [1, §4, proof of Lemma 8]), and cϕ = c,wehave − − H (a)ϕ = c 1(H (a)ϕ − 1) = c 1(H (a)ζaϕ − 1) c −1 a = c E(A(ζ)) · ζ ϕ − 1 X=π −1 −1 a = c 1+cE (Ac (ζ)) · ζ ϕ − 1 c X= π −1 a = c 1+cE (A (ξ)) · ζ ϕ − 1 c c X=π − 1 · aϕ − = c (1 + cHc(a)) ζ 1 = Hc(a)+Fc [aϕ](ξ).

f ϕ Remark 1. In our case ϕ = with f =(T : Qp), whence A − A =tra ⇒ aϕ =tra.

Corollary. (π, Hc(a))c =[tra]c(ξ). ∈ ∈ Proof. Hc(a) T (ζ,c), Hc(a) Tr(ζ,c). By the deﬁnition of the Hilbert symbol, we get ϕ − (π, Hc(a))c = Hc(a) Fc Hc(a) = [tr a]c(ξ). n Proposition 4. ωc(a) ≡ Hc(a)mod[p ]c(Fc(M)). Proof. A similar proposition for the multiplicative case (see [1, §4] or [3, Chapter V, pn item 4]) gives ω(a)=H(a) · ε ,whereε is some unit in OK such that ε ≡ 1modπ. Thus, pn n −1 ω(a)=H(a) · ε =(1+cHc(a)) · 1+c[p ]c(c (ε − 1)) . −1 Now, since ωc(a)=c (ω(a) − 1), we get −1 n −1 ωc(a)=c (1 + cHc(a)) · 1+c[p ]c(c (ε − 1)) − 1 n −1 − = Hc(a)+Fc [p ]c(c (ε 1)).

− e0 ∈O Proposition 5. Let ζ 1=γe0 π + ...,whereγe0 .Then − n n ≡ ≡ 1 p p pe1 pe1+1 n (8) ωc(a) Hc(a) c a γe0 π mod (π , [p ]c(Fc(M))). Proof. It is known that

n pn−1 pn pe pe +1 H(a)=E(p A(ζ)) ≡ 1+a γ π 1 mod π 1 X=π e0

(see [2, p. 121, (14)]). Now the deﬁnition of Hc(a) directly implies the formula − − n−1 n 1 − ≡ 1 p p pe1 pe1+1 Hc(a)=c (H(a) 1) c a γe0 π mod π . The ﬁnal result follows from Proposition 4.

n Proposition 6. The primary element ωc(a)mod[p ]c(Fc(M)) does not depend on the choice of the prime element π and the form of the series expansion for the root ξ of the n kernel [p ]c(x). Proof. The proof is much similar to that in the multiplicative case (see [3, Chapter V, item 4]).

§5. Main lemma First, we prove some preliminary assertions. Lemma 1. For ψ ∈ XO[[X]], the following congruence holds true: (9) log(1 + cuψ) s ≡ log(1 + pcψ) s mod (pn, deg 1). p p

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Proof. For series u and s,wehave

ur pr−1 s ≡ s mod (pn, deg 0), where r ≥ 1 rp r (see [4, proof of Lemma 4]). Multiplying both sides of this congruence by (−1)r−1(cψ)r and summing over r, we arrive at (9). Lemma 2. For ψ ∈ XO[[X]], the following congruence holds true: 1 (10) log(1 + cuψ)/s ≡ log(1 + pcψ)/s − mod deg 1. p n 1 r r−1 Proof. We use the congruence u /s ≡ p /sn−1 mod deg 0, where r ≥ 1 (see [4, proof of (cψ)r Lemma 3]). Multiplying by (−1)r and summing over r, we complete the proof. r Corollary. (11) (log(1 + cuψ)/s) ≡ log(1 + pcψ) s mod (pn, deg 1). p ≡ Proof. It suffices to apply the operator to both sides of (10) and to recall that 1/sn−1 1/s mod pn (see [3, Chapter VI, item 3]). Lemma 3. For ψ ∈ XO[[X]], the following congruence holds true: (12) log(1 + cuψ)d(1/s) ≡ 0mod(pn, deg 0). p Proof. Note that d(1/s) ≡ 0modpn (see [3, Chapter VI, item 3]). Since the series 1 − log(1 + cuψ) has integral coefficients, we have p 1 − log(1 + cuψ)d(1/s) ≡ 0modpn. p Now the claim is equivalent to the following congruence: (13) log(1 + cuψ)d(1/s) ≡ 0mod(pn, deg 0). The definition of the series u (see [1, §3]) shows that u ∈ pO[[X]] + XeO[[X]] ⇒ cuψ = pa + Xeb, where a, b ∈O[[X]]. Plugging this in the series for log(1 + X), we get pe p2e p3e e X X X (14) log(1 + cuψ)=pa− + a X + a + a + a + ..., 1 0 1 p 2 p2 3 p3

where ai ∈O[[X]]. From the deﬁnition of s and ζ and the properties of ζ (see [2, p. 121]) it follows that n n e0 p e0−e p e0 ζ ∈ 1+pO[[X]] + X O[[X]] ⇒ s = pX ε1(X)+X ε2(X), ∗ n where ε1, ε2 ∈O{{X}} . This identity and the congruence ds ≡ 0modp (see [3, Chapter VI, item 3]) imply that n n n 2 n −2p e0 −2p e0−e −2p e0−2e 2 (15) d(1/s)=−ds/s = p b0X + b1X p + b2X p + ... , m n where bi ∈O[[X]]. The expansions (14) and (15) with the inequality p e ≥−2p e0 +me for m ≥ 1 prove the desired congruence (13).

Lemma 4 (main lemma). For α ∈Hm and ψ ∈ XO[[X]], the following congruence holds true: ≡ − n ∈O α, uψ c (1 )d0 mod p , where d0 .

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Proof. By the definition of the pairing ·, ·c,wehave (16) α, uψ =res ( (uψ)α−1 dα)/s − (α)d log(1 + cuψ) s. c X c c p Consider the second summand. Since (α) ∈O[[X]], the obvious relation res d (α) log(1 + cuψ) s =0 X p and congruence (12) imply − res (α)d log(1 + cuψ) s ≡ res d(α) log(1 + cuψ) s. X p X p Directly from the definition of we see that d(α)=α−1 dα − X−1(Xα−1 dα). Substituting the resulting representation of the second summand in (16), we obtain ≡ −1 α, uψ c resX (α dα) c(uψ)/sc + log(1 + cuψ) s p − X−1(Xα−1 dα) log(1 + cuψ) s p −1 ≡ resX (α dα) (1 + cuψ)+ log(1 + cuψ) s p − X−1(Xα−1 dα) log(1 + cuψ) s p ≡ [using the definition of and formulas (11) and (9)] ≡ res (α−1 dα)log(1+cuψ)/s − X−1 Xα−1 dα(log(1 + cuψ))/s X −1 −1 n ≡ resX X (1− ) Xα dα(log(1 + cuψ))/s ≡ (1− )d0 mod p ,

−1 where d0 ∈Ois the constant term of the series Xα dα(log(1 + cuψ))/s.

§6. Hensel generators of the formal module Fc(M)

Lemma 5. The Hilbert symbol ( · , · )c satisﬁes the following Steinberg type relation: pn−1 (α, c α)c =0for any α ∈ M \{0}. Proof. We need to check that α is a multiplicative norm in the extension K(β)/K,where n pn−1 β is a solution of the equation [p ]c(X)=c α.Notethat n −1 pn n 2 pn−1 pn−1 pn [p ]c(X)=c (1 + cX) − 1 = p X + c2X + ...+ cpn−1X + c X ,

where ci ∈ pOK ; hence β solves the equation pn 1−pn pn−1 n 1−pn X + c cpn−1X + ...+ p c X − α =0, and, therefore, α is a norm in the extension K(β)/K.

Proposition 7. The elements pn−1 i εi,θ = θc π , where θ ∈ R,p i, and ωc(a) with a ∈O

generate the formal Zp-module Fc(M). Moreover, the Hilbert symbol ( · , · )c satisﬁes

(17) (π, εi,θ)c =0, (π, ωc(a)) = [tr a]c(ξ).

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Proof. The ﬁrst part of the proposition is a simple generalization of the Hensel theorem (see [7, 2]) to the formal Zp-module Fc(M) , with regard to the congruence (8). Now we check the relation (π, εi,θ) = 0. Indeed, Lemma 5 shows that

i pn−1 i pn−1 i pn−1 i pn−1 i 0=(θπ ,c (θπ ))c =(θ, c θπ )c +Fc [i]c(π, c θπ )c =[i]c(π, c θπ )c, pn−1 i because θ is a p-divisible element of the Teichm¨uller set R. Therefore, (π, c θπ )c =0 whenever p i. The second relation (π, ωc(a))c =[tra]c(ξ) is proved in the corollary to Theorem 1 and Proposition 4.

Proposition 8. Let α ∈Hm, a ∈O.Then ≡ · n α, Ec(asc(X)) c deg α a mod p . Proof. First, we consider the case where α(X)=X. By deﬁnition, we have −1 −1 X, Ec(asc(X)) c =resX c Ec(asc(X)) X dX/sc =resX ascX /sc = a.

Since the pairing ·, ·c is multiplicative in the first argument, it sufices to prove that for ε(X) ∈O[[X]]∗ we have (18) ε(X),Ec(asc(X)) c =0. Using the definition, we get Φ(ε(X),E (as (X)))/s = aε(X)−1 dε(X) − (ε(X)) d log 1+cE (as (X)) /s. c c c p c c Trivially, we have aε(X)−1 dε(X) ∈O[[X]], so that the residue of this element is zero. Consider the second summand: (ε(X)) d log(1 + cE (as (X)))/s = (ε)d log(E(as))/s p c c p m (as) m = (ε) · (1/s)d ≡ (ε) · (1/s) · X−1 (Xads) ≡ 0modpn. pm m≥1 The last congruence is valid because ds ≡ 0modpn and all other factors have integral coefficients. This proves formula (18) and the lemma.

§7. Pairing over the formal module Fc(M)

Now we introduce the pairing {·, ·}c on numbers by using the pairing ·, ·c on series. Deﬁnition 1. Let {·, ·} : K∗ × F (M) →ξ, c c { } α, β c = tr α, β c c(ξ),

where α ∈Hm and β ∈Hc are such that α(π)=α, β(π)=β. In the following lemmas we check the consistency of this deﬁnition, i.e., the indepen- dence of the choice of the series α and β.

Lemma 6. Let series β1,β2 ∈Hc be such that β1(π)=β2(π). Then for any series ∈H α m we have tr α, β1 c c(ξ)= tr α, β2 c c(ξ).

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− Proof. Consider the series β = β1 Fc β2. By assumption, β(π)=0.Hence(see[1, Lemma 6]), β = uψ for some series ψ ∈ XO[[X]]. The main lemma (Lemma 4) implies that ⇒ α, β c =0 tr α, β c c(ξ)=0. Hence, by the linearity of the pairing ·, ·c, we get the claim. Lemma 7 (change of variables). Let g(Y ) ∈O[[Y ]] be such that deg g =1,andlet g(π)=π be the uniformizing parameter of OK . Then for any α ∈Hm and β ∈Hc we have α(X),β(X) c,X,π = α(g(Y )),β(g(Y )) c,Y,π . In the pairing on the right-hand side the series and residue are regarded with respect to the variable Y , and the corresponding series sm and u are constructed with the help of expansions of ξ, ζ,andc in π-series (for the prime element π). Proof. We denote by c the π-series expansions of the element c. First, we consider ther case where α(X)=X. By Lemma 6 and Proposition 7, we may pn−1 i replace the series β by the formal sum of the series θc X and Ec(asc(X)). By the linearity of pairings, it suffices to check that pn−1 i pn−1 i X, θc X = g(Y ),θc g(Y ) , where p i, c,X,π c,Y,π r X, Ec(asc(X)) c,X,π = g(Y ),Ec(asc(Y )) c,Y,π . The first congruence follows immediately from the Steinberg relation (Proposition 3), and the second from Proposition 8. Now we address the case of α(X)=ε(X) ∈O[[X]]∗. It suffices to check only this case, by multipilicativity in the first argument. We need to prove that ε(X),β(X) c,X,π = ε(g(Y )),β(g(Y )) c,Y,π . Denoting f(X)=Xε(X), π = f(π), Z = f(X), and h(Y )=f(g(Y )), we write ε(X),β(X) = f(X),β(X) −X, β(X) , c,X,π c,X,π c,X,π − ε(g(Y )),β(g(Y )) c,Y,π = f(g(Y )),β(g(Y )) c,Y,π g(Y ),β(g(Y )) c,Y,π . It suffices to prove that f(X),β(X) c,X,π = f(g(Y )),β(g(Y )) c,Y,π . Thecaseconsideredaboveshowsthat −1 f(X),β(X) = Z, β(f (Z)) c,X,π c,Z,π −1 = h(Y ),β(f (h(Y ))) c,Y,π = f(g(Y )),β(g(Y )) c,Y,π . The lemma is proved.

Lemma 8. Let series α1,α2 ∈Hm be such that α1(π)=α2(π). Then for any series β ∈Hc we have tr α ,β (ξ)= tr α ,β (ξ). 1 c c 2 c c Proof. By multiplicativity, it suﬃces to prove that α1/α2,β c =0.

Consider the series g(X)=X · (α1(X)/α2(X)); since g(π)=π, Lemmas 6 and 7 show that · X, β(X) c = g(X),β(g(X)) c = g(X),β(X) c = X (α1(X)/α2(X)),β(X) c, which immediately implies the claim.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use EXPLICIT FORM OF THE HILBERT SYMBOL 795

We unite and reshape these results as follows.

Theorem 2. The pairing {·, ·}c is well defined, i.e., it does not depend of the choice of ∗ a uniformizing element π ∈OK and the way of expansion of elements α ∈ K and β ∈ Fc(M) in the π -series α ∈Hm and β ∈Hc. It is multiplicative in the first argument and linear in the second; moreover, it satisfies the Steinberg relation pn−1 {α, c α}c =0, where α ∈ Fc(M) \{0}.

§8. Hilbert pairing ∗ Theorem 3. For any elements α ∈ K and β ∈ Fc(M), the values of the pairings {·, ·}c and ( · , · )c are equal: {α, β}c =(α, β)c. Proof. First, we consider case of prime α = π.Expandβ in Hensel generators. By linearity of the pairings with respect to the second argument, it suﬃces to prove the theorem in the cases where β = εi,θ, β = ωc(a) (see Proposition 7). The Steinberg relations immediately imply (see the proof of Proposition 7) that

{π, εi,θ}c =(π, εi,θ)c =0. Propositions 7 and 8 also show that

{π, ωc(a)}c =(π, ωc(a))c =[tra](ξ).

The case of prime α is proved. Now, an arbitrary unit of the ﬁeld ε ∈OK can be rewriten in the form ε = τ/π,whereτ and π are prime elements. Then { } { } − { } − ε, β c = τ,β c Fc π, β c =(τ,β)c Fc (π, β)c =(ε, β)c. By multplicativity in the ﬁrst argument, we get the desired claim for arbitrary α ∈ K∗.

Thus, we have obtained an explicit formula for the symbol ( · , · )c. References

[1]S.V.Vostokov,An explicit form of the reciprocity law, Izv. Akad. Nauk SSSR Ser. Mat. 42 (1978), no. 6, 1288–1321; English transl., Math. USSR-Izv. 13 (1978), no. 3, 557–588. MR522940 (80f:12014) [2]I.R.Shafarevich,A general reciprocity law, Mat. Sb. 26 (1950), no. 1, 113–146. (Russian) MR0031944 (11:230f) [3] I. V. Fesenko and S. V. Vostokov, Local ﬁelds and their extensions, 2nd ed., Trans. Math. Monogr., vol. 121, Amer. Math. Soc., Providence, RI, 2002. MR1915966 (2003c:11150) [4]S.V.Vostokov,The Hilbert symbol in a discretely valued ﬁeld, Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 94 (1979), 50–69; English transl., J. Sov. Math. 19 (1982), no. 2, 1006–1019. MR571515 (81g:12019) [5] S. V. Vostokov and E. V. Ferens-Sorotskiy, Hilbert pairing for the polynomial formal groups,Vestnik S.-Peterburg. Univ. Mat. 2010, vyp. 1, 23–27; English transl., Vestnik St. Petersburg Univ. Math. 43 (2010), no. 1, 18–22. MR2662405 (2011e:14085) [6] S. V. Vostokov, V. V. Volkov, and G. K. Pak, The Hilbert symbol for polynomial formal groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 400 (2012), 127–131; English transl., J. Math. Sci. (N.Y.) 192 (2013), no. 2, 196–199. MR3029567 [7] K. Hensel, Die multiplicative Dars ellung der algebraischen Zahlen fur den Bereich eines beliebigen Prin teil,J.ReineAngew.Math.136 (1916). [8] S. V. Vostokov, On I. R. Shafarevich’s paper “A general reciprocity law”, Mat. Sb. 400 (2013), no. 6, 3–22; English transl., Sb. Math. 204 (2013), no. 5–6, 781–800. MR3113452 [9] , Normed pairing in formal modules, Izv. Akad. Nauk SSSR Ser. Mat. 43 (1979), no. 4, 765–794; English transl., Math. USSR-Izv. 15 (1980), no. 1, 25–51. MR548504 (81b:12017) [10] , Symbols on formal groups, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 5, 985–1014; English transl., Math. USSR-Izv. 19 (1982), no. 2, 261–284. MR637613 (83c:12016)

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[11] D. G. Benua and S. V. Vostokov, Norm pairing in formal groups and Galois representations, Algebra i Analiz 2 (1990), no. 6, 69–97; English transl., Leningrad Math. J. 2 (1991), no. 6, 1221– 1249. MR1092526 (92j:11141) [12] V. A. Abrashkin, Explicit formulas for the Hilbert symbol of a formal group over Witt vectors,Izv. Ross. Akad. Nauk Ser. Mat. 61 (1997), no. 3, 3–56; English transl., Izv. Math. 61 (1997), no. 3, 463–515. MR1478558 (98j:14060) [13] S. V. Vostokov and O. V. Demchenko, Explicit form on the Hilbert pairing for relative formal Lubin– Tate groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 227 (1995), 41–44; English transl., J. Math. Sci. (N.Y.) 89 (1998), no. 2, 1105–1107. MR1374555 (97b:11142) [14] , An explicit formula for the Hilbert pairing of formal Honda groups, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 272 (2000), 86–128; English transl., J. Math. Sci. (N.Y.) 116 (2003), no. 1, 2926–2952. MR1811794 (2002k:11213)

Mathematics and Mechanics Department, St. Petersburg State University, Universitetski˘ı pr. 28, Petrodvorets, St. Petersburg 198504, Russia E-mail address: [email protected] Mathematics and Mechanics Department, St. Petersburg State University, Universitetski˘ı pr. 28, Petrodvorets, St. Petersburg 198504, Russia E-mail address: [email protected] Received 10/JAN/2014

Translated by V. VOLKOV

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