Explicit Formulas for Hilbert Pairings on Formal Groups

Transcription

1 CHAPTER 8 Explicit Formulas for Hilbert Pairings on Formal Groups The method of the previous chapter possesses a valuable property: it can be relatively easily applied to derive explicit formulas for various generalizations of the Hilbert symbol. This chapter explains how to establish explicit formulas for the generalized Hilbert pairing associated to a formal group of Lubin Tate type or more generally of Honda type. Section 1 briefly recalls the theory of Lubin Tate groups and their applications to local class field theory. In section 2 we discuss for Lubin Tate formal groups a generalization of the exponential and logarithm maps E X and l X of Ch. VI and the arithmetic of the points of formal module. Then we describe explicit formulas for the Hilbert pairing. In section 3 we discuss the arithmetic and explicit formulas in the case of Honda formal groups. The presentation in this chapter is more concise than in the rest of the book. 1. Formal Groups (1.1). Let A be a commutative ring with unity. A formal power series F (X, Y ) over A is said to determine the commutative formal group F over A if F (X, 0) = F (0, X) = X, F (F (X, Y ), Z) = F (X, F (Y, Z)) F (X, Y ) = F (Y, X) (associativity), (commutativity). Natural examples of such formal groups are the additive formal group and the multiplicative formal group F + (X, Y ) = X + Y F (X, Y ) = X + Y + XY = (1 + X)(1 + Y ) 1. Other examples will be exposed below and in Exercises. The definition implies that F (X, Y ) = X + Y + i+j 2 a ijx i Y j, a ij A. A formal power series f(x) XA[[X]] is called a homomorphism from a formal group F to a formal group G if f(f (X, Y )) = G(f(X), f(y )). 267

2 268 VIII. Explicit Formulas on Formal Groups f is called an isomorphism if there exists a series g = f 1 inverse to it with respect to composition, i.e. such that (f g)(x) = (g f)(x) = X. The set End A (F ) of all homomorphisms of F to F has a structure of a ring: Lemma. f(x) F g(x) = F (f(x), g(x)), f(x) g(x) = f(g(x)). There exists a uniquely determined homomorphism Z End A (F ) : n [n] F. Proof. Put [0] F (X) = 0, [1] F (X) = X, [n + 1] F (X) = F ([n] F (X), X) for n 0. Now we will verify that there exists a formal power series [ 1] F (X) XA[[X]] such that F (X, [ 1] F (X)) = 0. Put ϕ 1 (X) = X and assume that F (X, ϕ i (X)) 0 mod deg i + 1 for 1 i m. Let F (X, ϕ m (X)) c m+1 X m+1 we obtain mod deg m + 2, c m+1 A. Then for ϕ m+1 (X) = ϕ m (X) c m+1 X m+1 F (X, ϕ m+1 (X)) = X + ϕ m (X) c m+1 X m+1 + i+j 2 a ij ϕ m (X) j F (X, ϕ m (X)) c m+1 X m+1 0 mod deg m + 2. The limit of ϕ m (X) in A[[X]] is the desired series [ 1] F (X). Finally, we put [n] F (X) = F ([n + 1] F (X), [ 1] F (X)) for n 2. This completes the proof. From now on let A = K be a field of characteristic 0. Proposition. Any formal group F over K is isomorphic to the additive group F +, i.e. there exists a formal power series λ(x) XK[[X]], λ(x) X mod deg 2 such that Proof. F (X, Y ) = λ 1 (λ(x) + λ(y )). Denote the partial derivative F Y (X, Y ) by F 2 (X, Y ). First we show that To do this, we write F 2 (X, F (Y, Z))F 2 (Y, Z) = F 2 (F (X, Y ), 0) = F 2 (X, Y )F 2 (Y, 0). F (X, F (Y, Z)) = Z and put Z = 0. Now let λ(x) = X + i 2 c ix i be such that λ (X) = 1 + n 2 nc n X n 1 = F (F (X, Y ), Z), Z 1 F 2 (X, 0) = X + i 1 a i1x i.

3 Then 1. Formal Groups 269 Y λ(f (X, Y )) = F 2 (X, Y ) F 2 (F (X, Y ), 0) = 1 F 2 (Y, 0) = λ(y ). Y Therefore, (λ(f (X, Y )) λ(y )) = 0 Y and λ(f (X, Y )) = λ(y )+g(x) for some formal power series g(x) K[[X]]. Setting Y = 0, we get λ(x) = λ(f (X, 0)) = g(x). Thus, we conclude that F (X, Y ) = λ 1 (λ(x) + λ(y )). The series λ(x) is called the logarithm of the formal group F. We will denote it by log F (X). The series inverse to it with respect to composition is denoted by exp F (X). Then F (X, Y ) = exp F (log F (X) + log F (Y )). The theory of formal groups is presented in [Fr], [Haz3]. (1.2). From now on we assume that K is a local number field. For such a field the Lubin Tate formal groups play an important role. Let F denote the set of formal power series f(x) O K [[X]] such that f(x) X mod deg 2, f(x) X q mod, where is a prime element in K and q is the cardinality of the residue field K. The following assertion makes it possible to deduce a number of properties of the Lubin Tate formal groups. Lemma. Let f(x), g(x) F and α i O K for 1 i m. Then there exists a formal power series h(x 1,..., X m ) K[[X 1,..., X m ]] uniquely determined by the conditions: h(x 1,..., X m ) α 1 X α m X m mod deg 2, f(h(x 1,..., X m )) = h(g(x 1 ),..., g(x m )). Proof. It is immediately carried out putting h 1 = α 1 X 1 + +α m X m and constructing polynomials h i K[X 1,..., X m ] such that h i h i 1 mod deg i, f(h i (X 1,..., X m )) h i (g(x 1 ),..., g(x m )) mod deg i + 1. Then h = lim h i is the desired series. Proposition. O K such that Let f(x) F. Then there exists a unique formal group F = F f over F f (f(x), f(y )) = f(f f (X, Y )). For each α O K there exists a unique [α] F End OK (F ) such that [α] F (X) αx mod deg 2.

4 270 VIII. Explicit Formulas on Formal Groups The map O K End OK (F ) : α [α] F is a ring homomorphism, and f = [] F. If g(x) F and G = F g is the corresponding formal group, then F f and F g are isomorphic over O K, i.e., there is a series ρ(x) K[[X]], ρ(x) X mod deg 2, such that ρ(f f (X, Y )) = F g (ρ(x), ρ(y )). Proof. All assertions follow from the preceding Lemma. For instance, there exists a unique F f (X, Y ) K[[X, Y ]] such that F f (X, Y ) X + Y mod deg 2, F f (f(x), f(y )) = f(f f (X, Y )). Then F f (X, 0) = F f (0, X) = X. Both series F f (X, F f (Y, Z)) and F f (F f (X, Y ), Z) satisfy the conditions for h: h(x, Y, Z) X + Y + Z mod deg 2, h(f(x), f(y ), f(z)) = f(h(x, Y, Z)). Therefore, by the Lemma F f (X, F f (Y, Z)) = F f (F f (X, Y ), Z). In the same way we get F f (X, Y ) = F f (Y, X). This means that F f is a formal group. The formal group F f is called a Lubin Tate formal group. Note that the multiplicative formal group F is a Lubin Tate group for = p. (1.3). Let F = F f, f F, be a Lubin Tate formal group over O K, K a local number field. Let L be the completion of an algebraic extension over K. On the set M L of elements on which the valuation takes positive values one can define the structure of O K -module F (M L ): α + F β = F (α, β), a α = [a] F (α), a O K, α, β M L. Let κ n denote the group of n -division points: κ n = {α M K sep : [ n ] F (α) = 0}. It can be shown (see Exercise 5) that κ n is a free O K / n O K -module of rank 1, O K / n O K is isomorphic to End OK (κ n ), and U K /U n,k is isomorphic to Aut OK (κ n ). Define the field of n -division points by L n = K(κ n ). Then one can prove (see Exercise 6) that L n /K is a totally ramified abelian extension of degree q n 1 (q 1) and Gal(L n /K) is isomorphic to U K /U n,k. Put K = n 1 L n and let Ψ K be the reciprocity map (see section 4 Ch. IV). The significance of the Lubin Tate groups for class field theory is expressed by the following Theorem. The field L n is the class field of U n,k and the field K is the class field of.

5 1. Formal Groups 271 The group Gal(K ab /K) is isomorphic to the product Gal(K ur /K) Gal(K /K) and Ψ K ( a u)(ξ) = [u 1 ] F (ξ) for ξ n 1 κ n, a Z, u U K. See Exercise 7. Exercises. 1. a) Let A = F p [Z]/(Z 2 ). Show that F (X, Y ) = X + Y + ZXY p determines a noncommutative formal group over A. b) Let A be a commutative ring with unity and let 2 be invertible in A. Show that F α (X, Y ) = X (1 Y 2 )(1 α 2 Y 2 ) + Y (1 X 2 )(1 α 2 X 2 ) 1 + α 2 X 2 Y 2 with α A, determines a formal group over A (this is the addition formula for the Jacobi functions for elliptic curves). c) Let F (X, Y ) Z[X, Y ]. Show that F determines a formal group over Z if and only if F (X, Y ) = X + Y + αxy for some α Z. 2. a) Show that log F (X) = n 1 d a n n X n and exp F (X) = n 1 d b n n! X n for some a n O K, b n O K. b) Let F be a Lubin Tate formal group over O K. Show that log F induces an isomorphism of O K -module F (M m K ) onto O K -module F a (M m K ), where m is an integer, m > v K (p)/(p 1). c) Let F be as in b). Let M be the maximal ideal of the completion of the separable closure of K. Show that the kernel of the homomorphism F (M) K sep induced by log F coincides with κ = κ n. 3. Show that the homomorphism O K End OK (F f ) of Proposition (1.2) is an isomorphism. 4. a) Let F be a formal group over O K and a prime element in K. Assume that log F (X) 1 log F (X q ) O K [[X]]. Put f(x) = exp F ( log F (X)), f i (X) = f(x) i X qi, for i 1. Show that if log F (X) = i 1 c ix i, c 1 = 1, then c i f i (X) 0 mod. i 1 Deduce that f 1 (X) 0 mod. Since f(x) X f F and F is a Lubin Tate formal group. b) Show that the series log Fah (X) = X + Xq determines the Lubin Tate formal group + Xq F ah (X, Y ) = log 1 F ah (log Fah (X) + log Fah (Y )) mod deg 2, this means that

6 272 VIII. Explicit Formulas on Formal Groups over O K. c) Using Proposition (1.2), show that if F = F f is a Lubin Tate formal group over O K and f F, then the series E (X) = exp F (log Fah (X)) belongs to O K [[X]] and determines an isomorphism of F 0 onto F. The series E (X) is a generalization of the Artin Hasse function considered in (9.1) Ch. I. d) Show using Lemma (7.2) Ch. I that if F is as in c), then log F (X) 1 log F (X q ) O K [[X]]. Thus, a formal group F over O K is a Lubin Tate formal group over O K if and only if log F (X) 1 log F (X q ) O K [[X]]. 5. a) Let f, g F. Show that κ n associated to f is isomorphic to κ n associated to g. Taking g = X + X q show that κ n = q n. b) Let ξ κ n \ κ n 1. Using the map O K κ n, a [a] F (ξ) show that κ n is isomorphic to O K / n O K. c) Using the map O K End OK (κ n ), a (ξ [a] F (ξ)) show that O K / n O K is isomorphic to End OK (κ n ) and U K /U n,k is isomorphic to Aut OK (κ n ). 6. Let ξ κ n \ κ n 1. Define the field of n -division points L n = K(ξ). Using Exercise 5 show that L n /K is a totally ramified abelian extension of degree q n 1 (q 1), N Ln /L( ξ) = and Gal(L n /K) is isomorphic to U K /U n,k. 7. a) Define a linear operator φ acting on power series with coefficients in the completion of the ring of integers Ô of the maximal unramified extension of K as φ( a i X i ) = a ϕ K i X i. Let u U K and u = v φ 1 for some v Ô according to Proposition (1.8) in Ch. IV. Let f F and g F u. Using the method of (1.2) show that there is a unique h(x) Ô[[X]] such that h(x) vx mod deg 2 and f h = h φ g. b) Let u U K and let σ Gal(L n /K) be such that σ(ξ) = [u 1 ] F (ξ). Denote by Σ the fixed field of σ = ϕ Ln σ Gal(L ur n /K). Show that Σ is the field of n -division points of F g. c) Let h be as in a). Show that h(ξ) is a prime element of Σ. Deduce using sections 2 and 3 Ch. IV that ϒ Ln /K(σ) N Σ/K ( h(ξ)) = u u mod N Ln /KL n. Thus, Ψ Ln /K( a u)(ξ) = [u 1 ] F (ξ). d) Deduce that N Ln /KL n = U n,k. e) Show that K ab = K ur K where K = n 1 L n. 2. Generalized Hilbert Pairing for Lubin Tate Groups In this section K is a local number field with residue field F q, is a prime element in K, F = F f is a Lubin Tate formal group over O K for f F. Let L/K be a finite extension such that the O K -module κ n of n -division points is contained in L. Let O T be the ring of integers of T = L K ur and let O 0 be the ring of integers of L Q ur p. Put e = e(l Q p ), e 0 = e(k Q p ).

7 2. Generalized Hilbert Pairing for Lubin Tate Groups 273 (2.1). Define the generalized Hilbert pairing (, ) F = (, ) F,n : L F (M L ) κ n by the formula (α, β) F = Ψ F (α)(γ) + F [ 1] F (γ), where γ F (M K sep) is such that [ n ] F (γ) = β. If F = F, = p, then (α, β) F,n coincides with the Hilbert symbol (α, 1 + β) p n. Proposition. The generalized Hilbert pairing has the following properties: (1) (α 1, α 2, β) F = (α 1, β) F,n + F (α 2, β) F, (α, β 1 + F β 2 ) F = (α, β 1 ) F + F (α, β 2 ) F ; (2) (α, β) F = 0 if and only if α N L(γ)/L L(γ), where [ n ] F (γ) = β ; (3) (α, β) F = 0 for all α L if and only if β [ n ] F F (M L ) ; (4) (α, β) F in the field E coincides with (N E/L (α), β) F in the field L for α E, β F (M L ), where E is a finite extension of L ; (5) (σα, σβ) F in the field σl coincides with σ(α, β) F, where (α, β) F is considered to be taken in the field L, σ Gal(K sep /K). Proof. It is carried out similarly to the proof of Proposition (5.1) Ch. IV. Now we shall briefly discuss a generalization of the relevant assertions of Chapters VI, VII to the case of formal groups. (2.2). For α i in the completion of the maximal unramified extension K ur of K put ( αi X i) = ϕ K (α i )X qi, where ϕ K is the continuous extension of the Frobenius automorphism of K, and q is the cardinality of K. Let Ô be the ring of integers in the completion of Kur. Let F (XÔ[[X]]) denote the O K -module of formal power series in XÔ[[X]] with respect to operations f + F g = F (f, g), a f = [a] F (f), a O K. Analogs of the maps E X, l X of section 2 Ch. VI are the following E F = E F,X, l F = l F,X : E F (f(x)) = exp F (( l F (g(x)) = ) ) f(x), f(x) XÔ[[X]] ( 1 ) (logf (g(x)) ), g(x) XÔ[[X]]. Then E F is a O K -isomorphism of XÔ[[X]] onto F (XÔ[[X]]) and l F is the inverse one. This assertion can be proved in the same way as Proposition (2.2) Ch. VI, using

8 274 VIII. Explicit Formulas on Formal Groups the equality E F (θx m ( ) = exp F θx m + (θxm ) q +... ) = exp F (log Fah (θx m )), where θ is an l th root of unity, (l, p) = 1, and log Fah is the logarithm of the Lubin Tate formal group F ah, defined in Exercise 4 section 1. (2.3). Let Π be a prime element in L. Let ξ be a generator of the O K -module κ n. To ξ we relate a series z(x) = c 1 X + c 2 X , c i O T, such that z(π) = ξ. Put s m (X) = [ m ] F (z(x)), s(x) = s n (X). An element α M L is called n -primary if the extension L(γ)/L is unramified where [ n ](γ) = α. As in sections 3 and 4 of Ch. VI one can prove that is a n -primary element and, moreover, ω(a) = E F (a s(x)) X=Π, a O T, (, ω(a)) F = [Tr a](ξ), where Tr = Tr T/K (see [V3]). The O K -module Ω of n -primary elements is generated by an element ω(a 0 ) with Tr a 0 / O K, a 0 O T. An analog of the Shafarevich basis considered in section 5 Ch. VI can be stated as follows: every element α F (M L ) can be expressed as α = i (F )E F (a i X i ) X=Π + F ω(a), a i, a O T, where 1 i < qe/(q 1), i is not divisible by q. The element α belongs to [ n ] F F (M L ) if and only if Tr a n O K, a i n O T. There are also other forms of generalizations of the Shafarevich basis; see [V2 3]. (2.4). To describe formulas for the generalized Hilbert pairing, we introduce the following notions. For a i O 0 put δ ( ai X i) = ϕ(a i )X pi, where ϕ = ϕ Qp is the Frobenius automorphism of Q p. For the series α(x) = θx m ε(x), where θ is a l th root of unity, l is relatively prime to p, ε(x) 1 + XO 0 [[X]], put, similar to Ch. VII, l(α(x)) = l(ε(x)) = ( 1 δ p ) (log(ε(x))) = 1 p log ( α(x) p α(x) δ ), and l m (α(x)) = l m (ε(x)) = L(α(X)) = (1 + δ + δ )l(α(x)), ( 1 ) (log (ε(x))) = 1 ( ) α(x) q q q log α(x).

9 2. Generalized Hilbert Pairing for Lubin Tate Groups 275 Let α L, β F (M L ). Let α = α(x) X=Π, β = β(x) X=Π, where α(x) is as just above, β(x) XO T [[X]]. Put and Φ α(x),β(x) = α(x) α(x) l F (β(x)) l m (α(x)) ( log F (β(x))) Φ (1) α(x),β(x) = 2 ( ( ml F (β(x)) + L(α(X))Xε(X) )) q (Xε(X)) (log F (β(x))), Φ (2) α(x),β(x) (l(α(x))( = )l F (β(x))), ( Φ (3) α(x),β(x) = ( L(α(X))( )l F (β(x)) )) 2 (concerning the form of Φ (3) see Exercise 2 section 2 Ch. VII keeping in mind the restriction on the series α(x), β(x) above). Similarly to (2.1) Ch. VII we can introduce an appropriate pairing, X on power series using the series 1/s(X) instead of V (X). Similarly to (2.2) Ch. VII we can introduce a pairing, on L F (M L ) and then prove that it coincides with the generalized Hilbert pairing. Thus, there are the following explicit formulas for the generalized Hilbert pairing. If p > 2 then If p = 2 and q > 2, then If p = 2, q = 2, e 0 > 1, then (α, β) F = [Tr res X Φ α(x),β(x) /s(x)](ξ n ) (α, β) F = [Tr res X (Φ α(x),β(x) + Φ (1) α(x),β(x) )/s(x)](ξ n) (α, β) F = [Tr res X (Φ α(x),β(x) + Φ (2) α(x),β(x) )r(x)/s(x)](ξ n) If p = 2, q = 2, e 0 = 1, then (α, β) F = [Tr res X (Φ α(x),β(x) + Φ (3) α(x),β(x) )r(x)/s(x)](ξ n) For odd p see [V2 4]. For p = 2 see [VF], [Fe1], and for full proofs [Fe2, Ch.II]. Here for q = 2, e 0 > 1, we put r(x) = 1 + n 1 r 0 (X) and the polynomial r 0 (X) is determined by the congruence 2 r 0 + (1 + ( n 1 1)s) r 0 + sr 0 ( 2 s n 1 s)/ n modev(, deg X 4e ) (modev is as in (3.4) Ch. VI).

10 276 VIII. Explicit Formulas on Formal Groups For q = 2, e 0 = 1, we put r(x) = 1 + n 1 r 0 (X), where the polynomial r 0 (X) is determined by the congruence 2 r 0 + (1 + ( n 1 1)s n 1 ) r 0 + s n 1 r 0 ( s n 1 s)/ n modev(, deg X 2e ). (2.5). Remarks. 1. These formulas can be applied to deduce the theory of symbols on Lubin Tate formal groups; see [V3]. For a review of different types of formulas see [V11]. 2. If in the case p > 2 the series α(x) is chosen in O 0 (X)), then the series 1/s(X) should be replaced with V (X) = 1/s(X) + c/( 2 ) where c is the coefficient of X 2 in [](X) = X + cx In particular, if [](X) = X + X q, then c = 0 and V (X) = 1/s(X). 3. In connection with Remark 4 in (5.3) Ch. VII we note that no syntomic theory related to formal groups, which could provide an interpretation of explicit formulas discussed in this chapter, is available so far. 3. Generalized Hilbert Pairing for Honda Groups We assume in this section that p > 2. Let K be a local field with residue field of cardinality q = p f and L be a finite unramified extension of K. Let be a prime element of K. In this section we put ϕ = ϕ K which differs from the notation in section 2. (3.1). Let be defined in the same way as in (2.2). The set of operators of the form i 0 a i i, where a i O L, form a noncommutative ring O L [[ ]] of series in in which a = a ϕ for a O L. Definition. A formal group F O L [[X, Y ]] with logarithm log F (X) L[[X]] is called a Honda formal group if u log F 0 mod for some operator u = + a 1 + O L [[ ]]. The operator u is called the type of the formal group F. Every 1-dimensional formal group over an unramified extension of Q p is a Honda formal group [Hon]. Types u and v of a formal group F are called equivalent if u = ε v for some ε O L [[ ]], ε(0) = 1. Let F be of type u. Then v = + b 1 + O L [[X]] is a type of F if and only if v is equivalent to u.

11 3. Generalized Hilbert Pairing for Honda Groups 277 Using the Weierstrass preparation theorem for the ring O L [[ ]], one can prove [Hon] that for every formal Honda group F there is a unique canonical type (*) u = a 1 a h h, a 1,..., a h 1 M L, a h O L. This type determines the group F uniquely up to isomorphism. Here h is the height of F. If F and G are Honda formal groups of types u and v respectively, then Hom OL (F, G) = {a O L : au = va}, End OL (F ) = O K. Along with ( ) we can use the following equivalent type where ũ = C 1 u, C = 1 a 1 ũ = a h h a h+1 h+1..., a h 1 h 1, i.e., ũ = ( 1 (u + a h h )) 1 u = ( 1 (u + a h h )) 1 a h h = a h h a h+1 h Now we state O.Demchenko classification theorems that connect Honda formal groups with Lubin Tate groups [De1]. Theorem 1. Let F be a Honda formal group of type ũ = a h h a h+1 h+1..., a i O L, where a h is invertible in O L. Let u = a 1 a h 1 h 1 a h h be the canonical type of F, a 1,..., a h 1 M L. Let λ = log F be the logarithm of F. Put λ 1 = B 1 λ ϕh, where B 1 = 1 + a h+1 a h + a h+2 a h (i.e., ũ = a h B 1 h ). Then (1) λ 1 is the logarithm of the Honda formal group F 1 of type ũ 1 = a 1 h ũa h and of canonical [ ] type u 1 = a 1 h ua h ; (2) f = Hom OL (F, F 1 ) and f(x) X q h mod. a h F,F 1 Examples. 1. ï A formal Lubin Tate group F has type u =, its height is h = 1 and F 1 = F. 2. A relative Lubin Tate group F has type u = a 1, where a 1 = /, h = 1, and F 1 = F ϕ. Theorem 2 (converse to Theorem 1). relations Let f O L [[X]] be a series satisfying f(x) X qh mod, f(x) a h X mod deg 2,

12 278 VIII. Explicit Formulas on Formal Groups where a h is an invertible element of O L. Let u = a 1 a h h, where a 1,..., a h 1 M L. Let C = 1 a 1 a h 1 h 1 and ũ = C 1 u = a h h a h+1 h Then there exists a[ unique ] Honda formal group F of type ũ and of canonical type u such that f = is a homomorphism from F to the formal group F 1 defined and given by Theorem 1. Remarks. 1. If λ and λ 1 are the logarithms of F and F 1 respectively, then [ ] ( ) f = = λ 1 a 1 λ. h F,F 1 a h 2. Theorem 2 can be viewed as a generalization of Proposition (1.2). These theorems allow one to define on the set of Honda formal groups over the ring O L the invertible operator A: F F 1. Define the sequence of Honda formal groups (**) F f F 1 f 1... f n 1 F n, where F m = A m F. Let λ m = log Fm be the logarithm of F m and let u m be the canonical type of F m. Put (***) 1 = /a h, m = ϕh(m 1) 1 = /a ϕh(m 1) m (m) 1 = i = m/ a 1+ϕh + +ϕ h(m 1) h. i=1 h, Then u m (m) 1 = (m) 1 u. Denote f (m) = f m 1 f m 2 f 1 f. From Theorem 1 one can deduce that f m 1 (X) m X mod deg 2, f (m) (X) (m) 1 X mod deg 2. (3.2). Define the generalized Hilbert pairing for a Honda formal group. Let E be a finite extension of L which contains all elements of n -division points κ n = ker [ n ] F. Along with the generalized Hilbert pairing (, ) F = (, ) F,n : E F (M E ) κ n, (α, β) F = Ψ E (α)(γ) F γ, where Ψ E is the reciprocity map, γ is such that [ n ] F (γ) = β, we also need another generalization that uses the homomorphism f (n) : {, } F = {, } F,n : E F (M E ) κ n, {α, β} F = Ψ E (α)(δ) F δ, a h F,F 1

13 3. Generalized Hilbert Pairing for Honda Groups 279 where δ is such that f (n) (δ) = β. Then (α, β) F = {α, [ (n) 1 /n ] F,Fn (β)} F. We get the usual norm property for both (, ) F and {, } F. (3.3). We introduce a generalization for Honda formal modules of the maps E F, l F defined in (2.2). Let T be the maximal unramified extension of K in E. Denote by F (XO T [[X]]) the O K -module whose underlying set is XO T [[X]] and operations are given by f + F g = F (f, g); a f = [a] F (f), a O K. The class of isomorphic Honda formal groups F contains the canonical group F ah of type u = a 1 a h h, a i,..., a h 1 M L, a h O L with Artin Hasse type logarithm log Fah = (u 1 )(X) = X + α 1 X q + α 2 X q2 +..., α i L. Define the map E F and its inverse l F as follows: E F (g) = log 1 F (1 + α 1 +α )(g) ( l F (g) = 1 a 1 a h h) (log F g), where g XO T [[X]]. We also need similar maps for the formal group F n = A n F with logarithm λ n = log Fn defined in the previous section. Let u n = b 1 b h h be the canonical type of F n. Consider the canonical formal group F b of type u n whose logarithm is λ b = (u 1 n )(X) = X + β 1 X q + β 2 X q2 +..., β i L. The groups F n and F b are isomorphic because they have the same type u n. Now we define the functions E Fn (g) = λ 1 n (u 1 n l Fn (g) = (u n 1 )(λ n ψ) = )(g) = λ 1 n (1 + β 1 +β )(g) ( 1 b 1 b h h ) (λ n g). The functions E F and l F yield inverse isomorphisms between XO T [[X]] and F (XO T [[X]]), and the functions E Fn and l Fn yield inverse isomorphisms between XO T [[X]] and F n (XO T [[X]]), see [De2].

14 280 VIII. Explicit Formulas on Formal Groups (3.4). We discuss an analog of the Shafarevich basis for a Honda formal module. Let Π be a prime element of E. First we construct primary elements. An element ω F (M E ) is called n -primary if the extension E(ν)/E is unramified, where [ n ] F (ν) = ω. The O K -module module κ n has h generators [Hon], [De 2]. Fix a set of generators ξ 1,..., ξ h. Let z i (X) O T [[X]] be the series corresponding to an expansion of ξ i into a power series in Π, i.e. z i (Π) = ξ i. Similarly define z i (X). Put s (i) = f (n) z i (X), 1 i h. Fix an element b O T and put b = b + b ϕ + + b ϕh 1. Let Tr be the trace map for the extension K h /K where K h extension of K of degree h; note that b K h. is the unramified Proposition. The element ω i (b) = E Fn ( bλ n s (i) ) X=Π is well-defined. It belongs to F n (M E ), and it is n -primary. Moreover, See [De2], [DV2]. Further, let g 0 (X) = n 1 X + X qh {Π, ω i (b)} F = [Tr b] F (ξ i ). g ρ,a (X) = n 1 X + n 1 ax pρ + X qh, a O T, 1 ρ < fh. Let u n 1 be the type of the formal group F n 1 from the sequence ( ). By Theorem 2 in (3.1) there exist unique Honda formal groups G 0 and G ρ,a of type u n 1 which correspond to g 0 (X) and g ρ,a (X) respectively. Then AG 0 and AG ρ,a are of the same type as F n. Denote by E 0 n: AG 0 F n and E ρ,a n : AG ρ,a F n the corresponding isomorphisms. Theorem. Let R be the set of multiplicative representatives in T. Elements {ω i (b); b O T, 1 i h}, {E 0 n(θπ i ); θ R, 1 i < q h e/(q h 1), (i, p) = 1}, {E ρ,a n (θπ i ); θ R, a O T, 1 ρ < fh, 1 i < q h e/(q h 1), (i, p) = 1} form a set of generators of the O K -module F n (M E ). Furthermore, {Π, E 0 n(θπ i )} F = {Π, E ρ,a n (θπ i )} F = 0, {Π, ω i (b)} F = [Tr b] F (z i ). See [De2], [DV2].

15 3. Generalized Hilbert Pairing for Honda Groups 281 (3.5). Similarly to the case of multiplicative groups discussed Ch. VII and the case of formal Lubin Tate groups in section 2 one can introduce a pairing on formal power series, check its correctness and various properties and then prove that when X is specialized to Π it gives explicit formulas for the generalized Hilbert pairing. For a monomial d i X i T ((X)) put ν(d i X i ) = v T (d i ) + i/q h where v T is the discrete valuation of T. Denote by L the T -algebra of series L = { d i X i : d i T, inf ν(d i X i ) >, i lim ix i ) = + }. i + i Z Since the O K -module κ n has h generators, we are naturally led to work with h h matrices. Denote the ring of integers of the maximal unramified extension of Q p in E by O 0. Theorem. For α E let α(x) be a series in {X i θε(x) : θ R, ε 1 + XO 0 [[X]]}. For β F (M E ) let β(x) be a series in XO T [[X] such that β(π) = β. The generalized Hilbert symbol (, ) F is given by the following explicit formula: (α, β) F = h j=1 (F )[Tr res ΦV j ] F (ξ j ), where Φ(X)V j (X) belongs to L, V j = A j / det A, 1 j h, n λ z 1 (X)... n λ z h (X) A = n (λ z 1 (X))... n (λ z h (X)), n h 1 (λ z 1 (X))... n h 1 (λ z h (X)) A j is the cofactor of the (j, 1)-element of A, Φ = α(x) α(x) l F (β(x)) 1 See [DV2]. Remarks. h i=1 ( ) a i 1 i (log ) ε(x) i q i (λ β(x)). 1. The formula above can be simplified in the case of n = 1, see [BeV1]. 2. The first explicit formula for the generalized Hilbert pairing for formal Honda group and arbitrary n in the case of odd p under some additional assumptions on the field E was obtained by V A. Abrashkin [Ab6] using the link between the Hilbert pairing and the Witt pairing via an auxiliary construction of a crystalline symbol as a generalization of his method in [Ab5] (see Remark 6 in (5.3) Ch. VII).