p -Johnson homomorphisms and pro-p groups
aa r X i v : . [ m a t h . N T ] J a n p -Johnson homomorphisms and pro- p groups Masanori Morishita and Yuji Terashima
Dedicated to Professor Kazuya Kato
Abstract.
We propose an approach to study non-Abelian Iwasawa theory,using the idea of Johnson homomorphisms in low dimensional topology. We intro-duce arithmetic analogues of Johnson homomorphisms/maps, called the p -Johnsonhomomorphisms/maps, associated to the Zassenhaus filtration of a pro- p Galoisgroup over a Z p -extension of a number field. We give their cohomological inter-pretation in terms of Massey products in Galois cohomology.
1. Introduction
Let p be an odd prime number, and let µ p n denote the group of p n -throots of unity for a positive integer n and we set µ p ∞ := ∪ n ≥ µ p n . We let k ∞ := Q ( µ p ∞ ) and ˜ k the maximal pro- p extension of k ∞ which is unramifiedoutside p . We let Γ p := Gal( k ∞ / Q ) and F p := Gal(˜ k/k ∞ ), the Galois groupsof the extensions k ∞ / Q and ˜ k/k ∞ , respectively. Classical Iwasawa theorythen deals with the action of Γ p on the Abelianization H ( F p , Z p ) of F p ([Iw1]). A basic problem of non-Abelian Iwasawa theory, with which weare concerned in this paper, is to study the conjugate action of Γ p on F p itself. In terms of schemes, one has the tower of ´etale pro-finite covers (1 .
1) ˜ X p := Spec( O ˜ k [1 /p ]) → X ∞ p := Spec( O k ∞ [1 /p ]) → X p := Spec( Z [1 /p ]) , where O k ∞ and O ˜ k denote the rings of integers of k ∞ and ˜ k , respectively,and the Galois groups(1 .
2) Γ p = Gal( X ∞ p /X p ) , F p = Gal( ˜ X p /X ∞ p ) = π pro − p ( X ∞ p ) , p groups, p -Johnson homomorphisms, Zassenhausfiltration, non-Abelian Iwasawa theory, Massey products. π pro − p stands for the maximal pro- p quotient of the ´etale fundamentalgroup. So the problem is to study the monodromy action of Γ p on the arith-metic pro- p fundamental group F p .Now let us recall the analogy between a prime and a knot(1 .
3) prime knotSpec( F p ) = K ( ˆ Z , ֒ → Spec( Z ) S = K ( Z , ֒ → S Here K ( ∗ ,
1) stands for the Eilenberg-MacLane space and Spec( Z ) := Spec( Z ) ∪{∞} , ∞ being the infinite prime of Q which may be seen as an analogueof the end of R ([De]). This analogy (1.3) opens a research area, called arithmetic topology , which studies systematically further analogies betweennumber theory and 3-dimentional topology ([Ms2]). In particular, there areknown intimate analogies between Iwasawa theory and Alexander-Fox theory([Ma], [Ms2; Chap. 9 ∼ X K → X ∞K → X K := S \ K , for a knot K in S , where X ∞K and ˜ X K denote the infinite cyclic cover andthe universal cover of the knot complement X K , respectively, and the Galoisgroups Γ K := Gal( X ∞K /X K ) , F K := Gal( ˜ X K /X ∞K ) = π ( X ∞K ) , and we have the conjugate action of Γ K on F K .To push our idea further, suppose that K is a fibered knot so that X K isa mapping torus of the monodromy φ : S → S , S being the Seifert surface of K . Then F K = π ( S ) and the conjugate action of Γ K on F K is nothing butthe monodromy action induced by φ on F K (1 . φ ∗ : Γ K −→ Aut( F K ) . Note here that the monodromy φ may be regarded as a mapping class of thesurface S . Thus the action (1.4) can be studied by means of the Johnson2omomorphisms/maps, associated to the lower central series of F K , definedon a certain filtration of the mapping class group for the surface S ([J], [Ki],[Mt]) or, more generally, on the automorphism group Aut( F K ) ([Kw], [Sa]).In this paper, we regard the action of Γ p on F p as an arithmetic analogue ofthe monodromy action (1.4) and propose an approach to study non-AbelianIwasawa theory by introducing arithmetic analogues of the Johnson homo-morphisms/maps, called the p -Johnson homomorphisms/maps, associated tothe Zassenhaus filtration of F p , defined on a certain filtration of the auto-morphism group Aut( F p ). For this, we lay a foundation of a general theoryof p -Johnson homomorphisms/maps in the context of pro- p group.We note that our viewpoint and approach differs from what is called“non-commutative Iwasawa theory” (cf. [CFKSV], [Kt; 3]). The works byM. Ozaki ([O]) and R. Sharifi ([Sh]) are related to ours (see Remark 3.2.7),however, our approach is different from theirs and closer to geometric topol-ogy.Here is the content of this paper. In Section 2, we give a general the-ory of p -Johnson homomorphisms in the context of pro- p groups. We usethe Zassenhaus filtration of a finitely generated pro- p group G in order tointroduce the p -Johnson homomorphisms, defined on a certain filtration ofthe automorphism group of G . In Section 3, we give a framework to studynon-Abelian Iwasawa theory by means of the p -Johnson homomorphisms. InSection 4, we give a theory of Johnson maps for a free pro- p group F byextending the p -Johnson homomorphisms in Section 2 to maps, called the p -Johnson maps, defined on the automorphism group Aut( F ) itself. In Section5, we give a cohomological interpretation of the p -Johnson homomorphismsin terms of Massey products in Galois cohomology. Notation.
For subgroup
A, B of a group G , [ A, B ] stands for the subgroupof G generated by [ a, b ] := aba − b − for all a ∈ A, b ∈ B .
2. Zassenhaus filtration and p -Johnson homomorphisms for apro- p group. In this section, we give a general theory of p -Johnson homomorphisms forpro- p groups. We associate to the Zassenhaus filtration of a finitely generated3ro- p group G a certain filtration on the automorphism group Aut( G ) of G ,and introduce the p -Johnson homomorphisms defined on each term of thefiltration of Aut( G ).Throughout this section, let p be a fixed prime number and G a finitelygenerated pro- p group. For general properties of pro- p groups, we consult[Ko] and [DDMS]. Let F p [[ G ]]be the complete group algebra of G over F p = Z /p Z with the augmentationideal I G := Ker( ǫ F p [[ G ]] ), where ǫ F p [[ G ]] : F p [[ G ]] → F p is the augmentationhomomorphism ([Ko; 7.1]). For each positive integer n , we define the normalsubgroup G n of G by(2 . . G n := { g ∈ G | g − ∈ I nG } . The descending series { G n } n ≥ is called the Zassenhaus filtration of G ([Ko;7.4]). The family { G n } n ≥ forms a full system of neighborhoods of the iden-tity 1 in G and satisfies the following properties(2 . .
2) ( G i ) p ⊂ G pi ( i ≥ . (2 . .
3) [ G i , G j ] ⊂ G i + j ( i, j ≥ . We recall the fact that the abstract commutator subgroup of a finitely gen-erated pro- p group is closed ([DDMS;1.19]).The Zassenhaus filtration is in fact the fastest descending series of G having the properties (2.1.2) and (2.1.3). Namely, it is shown by Jennings’theorem and an inverse limit argument that we have the following inductivedescription of G n :(2 . . G n = ( G [ n/p ] ) p Y i + j = n [ G i , G j ] ( n ≥ , where [ n/p ] stands for the least integer m such that mp ≥ n . ([DDMS; 12.9]).We note by (2.1.3) that elements of G i /G i + j and G j /G i + j commute, inparticular, G n /G n +1 is central in G/G n +1 . The 2nd term G is the Frattinisubgroup G p [ G, G ] of G and we denote by H the Frattini quotient(2 . . H := G/G = G/G p [ G, G ] = H ( G, F p ) . For g ∈ G , we write [ g ] for the image of g in H : [ g ] := g mod G . We notethat each G n is a finitely generated pro- p group ([DDMS; 1.7, 1.14]).4or each n ≥
1, we let gr n ( G ) := G n /G n +1 , which is a finite dimensional F p -vector space. The graded F p -vector space(2 . .
6) gr( G ) := M n ≥ gr n ( G )has a natural structure of a graded Lie algebra over F p by (2.1.3). Here, for a = g mod G i +1 , b = h mod G j +1 ( g ∈ G i , h ∈ G j ), the Lie bracket is definedby [ a, b ] gr( G ) := [ g, h ] mod G i + j +1 . Further, by (2.1.2) again, gr( G ) has the operation [ p ] defined by, for a = g mod G n +1 ∈ gr n ( G ), [ p ]( a ) := g p mod G pn +1 , which makes gr( G ) a restricted Lie algebra over F p ([DDMS; 12.1]).The restricted universal enveloping algebra (abbreviated to universal en-velope ) U (gr( G )) of gr( G ) is given as follows. For each m ≥
0, we letgr m ( F p [[ G ]]) := I mG /I m +1 G . and consider the graded associative algebra over F p :gr( F p [[ G ]]) := M m ≥ gr m ( F p [[ G ]]) . For each m ≥
1, we have an injective F p -linear map θ m : gr m ( G ) −→ gr m ( F p [[ G ]])defined by θ m ( g mod G m +1 ) := g − I m +1 G for g ∈ G m . Putting all θ m together over m ≥
1, we have an injective graded Lie algebrahomomorphism over F p gr( θ ) := M m ≥ θ m : gr( G ) −→ gr( F p [[ G ]]) . F p [[ G ]]) , gr( θ )) is the universal envelope of gr( G ) ([DDMS; 12.8]):(2 . . U gr( G ) = gr( F p [[ G ]]) . p -Johnson homomorphisms. LetAut( G ) denote the group of continuous automorphisms of a finitely generatedpro- p group G . We note that any abstract group homomorphism betweenfinitely generated pro- p groups is always continuous and so Aut( G ) is sameas the group of automorphisms of G (as an abstract group) ([DDMS; 1.21]).We also note that every term G n of the Zassenhaus filtration of G is a char-acteristic subgroup of G , namely, invariant under the action of Aut( G ).Since any automorphism φ of G induces an automorphism [ φ ] m of G/G m +1 for each integer m ≥
0, we have the group homomorphism(2 . .
1) [ ] m : Aut( G ) −→ Aut(
G/G m +1 ) . We then define the normal subgroup A G ( m ) of Aut( G ) by(2 . .
2) A G ( m ) := Ker([ ] m )= { φ ∈ Aut( G ) | φ ( g ) g − ∈ G m +1 } ( m ≥ . We call the resulting descending series { A G ( m ) } m ≥ the Andreadakis-Johnsonfiltration of Aut( G ) associated to the Zassenhaus filtration of G (cf [A], [Sa]).In particular, we set simply [ φ ] := [ φ ] for φ ∈ Aut( G ) and the 1st termA G (1) is called the induced automorphism group of G and denoted by IA( G ):(2 . .
3) IA( G ) := Ker([ ] : Aut( G ) −→ GL( H )) , where GL( H ) denotes the group of F p -linear automorphisms of H = G/G .The family { A G ( m ) } m ≥ forms a full system of neighborhood of the iden-tity id G in Aut( G ) and it can be shown that Aut( G ) is a pro-finite group andIA( G ) is a pro- p group ([DDMS; 5.3, 5.5]). So Aut( G ) is virtually a pro- p group.The next Lemma will play a basic role to introduce the p -Johnson homo-morphisms. Lemma 2.2.4.
For φ ∈ A G ( m ) ( m ≥ and g ∈ G n ( n ≥ , we have φ ( g ) g − ∈ G m + n . roof. We fix m and prove the assertion by induction on n . For n = 1, theassertion φ ( g ) g − ∈ G m +1 is true by definition (2.2.2) of A G ( m ). Assumethat(2 . . . φ ( g ) g − ∈ G m + i if g ∈ G i and 1 ≤ i ≤ n. By (2.1.4), we have G n +1 = ( G [( n +1) /p ] ) p Y i + j = n +1 [ G i , G j ] . Since G n +1 / ( Q i + j = n +1 [ G i , G j ]) is Abelian, we have G n +1 = { a p | a ∈ G [( n +1) /p ] } Y i + j = n +1 [ G i , G j ]and so any element g of G n +1 can be written in the form g = a p [ b , c ] e · · · [ b q , c q ] e q , where a ∈ G [( n +1) /p ] and for each s (1 ≤ s ≤ q ) there are i, j ( i + j = n + 1)such that b s ∈ G i , c s ∈ G j . Since we have φ ( g ) g − = φ ( a ) p φ ([ b , c ]) e · · · φ ([ b q , c q ]) e q [ b q , c q ] − e q · · · [ b , c ] − e a − p , it suffices to show that (cid:26) (2 . . . φ ([ b, c ])[ b, c ] − ∈ G m + n +1 if b ∈ G i , c ∈ G j and i + j = n + 1 , (2 . . . φ ( a ) p a − p ∈ G m + n +1 if a ∈ G [( n +1) /p ] . (2.2.4.2). For simplicity, we shall use the notation: [ ψ, x ] := ψ ( x ) x − and[ x, ψ ] := xψ ( x ) − for x ∈ G and ψ ∈ Aut( G ). By the “three subgrouplemma” and the induction hypothesis (2.2.4.1), we have φ ([ b, c ])[ b, c ] − = [ φ, [ b, c ]] ∈ [ φ, [ G i , G j ]] ⊂ [[ φ, G i ] , G j ][[ G j , φ ] , G i ] ⊂ [ G m + i , G j ][ G m + j , G i ]= G m + i + j = G m + n +1 . t := [( n + 1) /p ] so that pt ≥ n + 1. By (2.1.1) and the inductionhypothesis (2.2.4.1), we have φ ( a ) − a = ( φ ( a ) a − − a ∈ I t + mG . Therefore we have φ ( a ) p a − p − φ ( a ) p − a p ) a − p = ( φ ( a ) − a ) p a − p ∈ I p ( t + m ) ⊂ I m + n +1 . Hence φ ( a ) p a − p ∈ G m + n +1 by (2.1.1). (cid:3) Lemma 2.2.4 yields the following properties of the Andreadakis-Johnson fil-tration { A G ( m ) } m ≥ . Proposition 2.2.5.
We have (1) [A G ( i ) , A G ( j )] ⊂ A G ( i + j ) for i, j ≥ . (2) A G ( m ) p ⊂ A G ( m + 1) if m ≥ . Proof. (1) We use the same notation as in the proof of (2.2.4.2). By Lemma2.2.4, we have [[A G ( j ) , G ] , A G ( i )] ⊂ [ G j +1 , A G ( i )] ⊂ G i + j +1 , [[ G, A G ( i )] , A G ( j )] ⊂ [ G i +1 , A G ( j )] ⊂ G i + j +1 . By the three subgroup lemma, we have[[A G ( i ) , A G ( j )] , G ] ⊂ [A G ( j ) , G ] , A G ( i )][[ G, A G ( i )] , A G ( j )] ⊂ G i + j +1 . By definition (2.2.2), we obtain[A G ( i ) , A G ( j )] ⊂ A G ( i + j ) . (2) Let g ∈ G and φ ∈ A G ( m ). We shall show that for any integer d ≥ . . . φ d ( g ) g − ≡ ( φ ( g ) g − ) d mod G m +1 , from which the assertion follows. In fact, let d = p in (2.2.5.1). Then( φ ( g ) g − ) p ∈ G p ( m +1) by (2.1.2), and G m +1 ⊂ G m +2 because m ≥
1. So φ p ( g ) g − ∈ G m +2 and hence φ p ∈ A G ( m + 1).8e prove (2.2.5.1) by induction on d . For d = 1, it is obviously true.Suppose φ d ( g ) g − ≡ ( φ ( g ) g − ) d mod G m +1 . Note that φ d ( g ) g − ∈ G m +1 , since ( φ ( g ) g − ) d ∈ G m +1 . Then we have φ d +1 ( g ) g − ( φ ( g ) g − ) − ( d +1) = φ d +1 ( g ) φ ( g ) − φ ( g ) g − ( φ ( g ) g − ) − ( d +1) = φ ( φ d ( g ) g − )( φ ( g ) g − ) − d ≡ φ ( φ d ( g ) g − )( φ d ( g ) g − ) − mod G m +1 . Since φ ( φ d ( g ) g − )( φ d ( g ) g − ) − ∈ G m +1 by Lemma 2.2.4, φ d +1 ( g ) g − ≡ ( φ ( g ) g − ) d +1 mod G m +1 and hence the induction holds. (cid:3) Now we are going to introduce the p -Johnson homomorphisms. Let φ ∈ A G ( m ) ( m ≥ g ∈ G , we have φ ( g ) g − ∈ G m +1 . Then we see that φ ( g ) g − mod G m +2 ∈ gr m +1 ( G ) depends only on the class [ g ] ∈ H . In fact,for g ′ = gg with g ∈ G , we have φ ( g ′ ) g ′− = φ ( g ) φ ( g ) g − g − ≡ φ ( g ) g − mod G m +2 , since φ ( g ) g − ∈ G m +2 by Lemma 2.2.4. Thus we have a map τ m ( φ ) : H −→ gr m +1 ( G )defined by(2 . . τ m ( φ )( h ) := φ ( g ) g − mod G m +2 ( h = [ g ]) . Lemma 2.2.7.
For φ ∈ A G ( m ) ( m ≥ , the map τ m ( φ ) is F p -linear.Proof. Let h = [ g ] , h ′ = [ g ′ ] and c ∈ F p . Using the property that G m +1 /G m +2 is central in G/G m +2 , we have τ m ( φ )( h + h ′ ) = τ m ( φ )([ gg ′ ])= φ ( gg ′ )( gg ′ ) − mod G m +2 = φ ( g ) φ ( g ′ ) g ′− g − mod G m +2 = ( φ ( g ) g − )( φ ( g ′ ) g ′− ) mod G m +2 = τ m ( φ )( h ) + τ m ( φ )( h ′ ) , and τ m ( φ )( ch ) = τ m ( φ )([ g c ])= φ ( g c ) g − c mod G m +2 = ( φ ( g ) g − ) c mod G m +2 = cτ m ( φ )( h ) . (cid:3) F p ( H, gr m +1 ( G )) denote the group of F p -linear maps H → gr m +1 ( G ).By Lemma 2.2.7, we have the map τ m : A G ( m ) −→ Hom F p ( H, gr m +1 ( G )) . For m = 0, we easily see by (2.2.6) that τ ( φ ) = [ φ ] − id H for φ ∈ Aut( G ). Theorem 2.2.8.
For m ≥ , the map τ m is a group homomorphism and itskernel is A G ( m + 1) .Proof. Let φ , φ ∈ A G ( m ). For any g ∈ G , we have τ m ( φ φ )([ g ]) = φ ( φ ( g )) g − mod G m +2 = φ ( φ ( g ) g − ) · φ ( g ) g − mod G m +2 . Since φ ( g ) g − ∈ G m +1 , φ ( φ ( g ) g − ) ≡ φ ( g ) g − mod G m +1 by Lemma2.2.4. Since G m +1 ⊂ G m +2 by m ≥
1, we have τ m ( φ φ )([ g ]) = φ ( g ) g − · φ ( g ) g − mod G m +2 = ( τ m ( φ ) + τ m ( φ ))([ g ])for any g ∈ G . Hence the former assertion is proved. The latter assertion onKer( τ m ) is obvious by definition (2.2.6). (cid:3) The homomorphism τ m : A G ( m ) → Hom F p ( H, gr m +1 G )) ( m ≥
1) or theinduced injective homomorphism τ m : gr m (A G ) := A G ( m ) / A G ( m + 1) ֒ → Hom F p ( H, gr m +1 ( G )) ( m ≥ m -th p -Johnson homomorphism .We give some properties of the p -Johnson homomorphisms. Firstly, wenote that the group Aut( G ) acts on both A G ( m ) and Hom F p ( H, gr m +1 ( G ))by the following rules, respectively: (cid:26) ψ.φ := ψ ◦ φ ◦ ψ − ( ψ ∈ Aut( G ) , φ ∈ A G ( m )) , ( ψ.η )( h ) := ψ ( η ([ ψ ] − ( h ))) ( ψ ∈ Aut( G ) , η ∈ Hom F p ( H, gr m +1 ( G )) , h ∈ H ) . Then we have the following 10 roposition 2.2.9.
The p -Johnson homomorphism τ m ( resp. τ m ) is Aut( G ) -equivariant ( resp. Aut( G ) / IA( G ) -equivariant ) .Proof. Let ψ ∈ Aut( G ) and φ ∈ A G ( m ). Then we have, for any g ∈ G , τ m ( ψ.φ )([ g ]) = τ m ( ψ ◦ φ ◦ ψ − )([ g ])= ( ψ ◦ φ ◦ ψ − )( g ) g − mod G m +2 . On the other hand, we have, for any g ∈ G ,( ψ.τ m ( φ ))([ g ]) = ψ ( τ m ( φ )([ ψ ] − ([ g ])))= ψ ( τ m ( φ ))([ ψ − ( g )]))= ψ ( φ ( ψ − ( g ))( ψ − ( g )) − ) mod G m +2 = ( ψ ◦ φ ◦ ψ − )( g ) g − mod G m +2 . Hence τ m is Aut( G )-equivariant. As for τ m , it suffices to note that IA( G )acts trivially on gr m (A G ) = A G ( m ) / A G ( m + 1) by Proposition 2.2.5 (1) andon Hom F p ( H, gr m +1 ( G )) by (2.2.3) and Lemma 2.2.4. (cid:3) Next we compute the p -Johnson homomorphism on inner automorphisms.Let Inn : G → Aut( G ) be the homomorphism defined byInn( x )( g ) := xgx − ( x, g ∈ G ) . The image Inn( G ) is a normal subgroup of Aut( G ) and called the group of inner automorphisms of G . Proposition 2.2.10.
Let m ≥ and x ∈ G m . Then we have Inn( x ) ∈ A G ( m )and τ m (Inn( x ))([ g ]) = [ x, g ] mod G m +2 ( g ∈ G ) . Proof.
For x ∈ G m and g ∈ G , we haveInn( x )( g ) g − = [ x, g ] ∈ G m +1 , from which the assertions follow. (cid:3) p -Johnson homomorphisms on commutators of au-tomorphisms. Lemma 2.2.11.
For ψ ∈ A G ( i ) , φ ∈ A G ( j ) ( k, m ≥ and g ∈ G , wehave, in gr i + j +1 ( G ) ,τ i + j ([ ψ, φ ])([ g ])= ψ ( φ ( g ) g − )( φ ( g ) g − ) − − φ ( ψ ( g ) g − )( ψ ( g ) g − ) − mod G i + j +2 . Proof.
By a straightforward computation, we obtain[ ψ, φ ]( g ) g − = [ ψ, φ ](( φ ( g ) g − ) − ) · ( ψφψ − )(( ψ ( g ) g − ) − ) · ψ ( φ ( g ) g − ) · ψ ( g ) g − . Since [ ψ, φ ] ∈ A G ( i + j ) by Proposition 2.2.5 (1) and φ ( g ) g − ∈ G j +1 byLemma 2.2.4, we have[ ψ, φ ](( φ ( g ) g − ) − ) ≡ ( φ ( g ) g − ) − mod G i +2 j +1 . Similarly, we have( ψφψ − )(( ψ ( g ) g − ) − ) ≡ φ (( ψ ( g ) g − ) − ) mod G i + j +1 . By these three equations together, we have[ ψ, φ ]( g ) g − ≡ ( φ ( g ) g − ) − · φ (( ψ ( g )( g − ) − ) · ψ ( φ ( g ) g − ) · ψ ( g ) g − mod G i + j +2 . Since ψ ( g ) g − ∈ G i +1 , φ ( g ) g − ∈ G j +1 and [ G i +1 , G j +1 ] ⊂ G i + j +2 , we have[ ψ, φ ]( g ) g − ≡ ( φ ( g ) g − ) − · ψ ( φ ( g ) g − ) · φ (( ψ ( g ) g − ) − ) · ψ ( g ) g − mod G i + j +2 . Since we easily see that (cid:26) ( φ ( g ) g − ) − ψ ( φ ( g ) g − ) ≡ ψ ( φ ( g ) g − )( φ ( g ) g − ) − mod G i + j +2 ,φ (( ψ ( g ) g − ) − ) · ψ ( g ) g − ≡ ( φ ( ψ ( g ) g − ) · ( ψ ( g ) g − ) − ) − mod G i + j +2 , we obtain the assertion. (cid:3)
12y Proposition 2.2.5, we can form the graded Lie algebra over F p associ-ated to the Andreadakis-Johnson filtration:gr(A G ) := M m ≥ gr m (A G ) , gr m (A G ) := A G ( m ) / A G ( m + 1) , where the Lie bracket is given by the commutator on the group Aut( G ).Then by Lemma 2.2.11, the direct sum of Johnson homomorphisms τ m overall m ≥ G ) to the derivationalgebra of gr( G ) as follows. Recall that an F p -linear endomorphism of gr( G )is called a derivation on gr( G ) if it satisfies δ ([ x, y ]) = [ δ ( x ) , y ] + [ x, δ ( y )] ( x, y ∈ gr( G )) . Let Der(gr( G )) denote the associative F p -algebra of all derivations on gr( G )which has a Lie algebra structure over F p with the Lie bracket defined by[ δ, δ ′ ] := δ ◦ δ ′ − δ ′ ◦ δ for δ, δ ′ ∈ Der(gr( G )). For m ≥
0, we define the subspaceDer m (gr( G )) of Der(gr( G )), the degree m part, byDer m (gr( G )) := { δ ∈ Der(gr( G )) | δ (gr n ( G )) ⊂ gr m + n ( G ) for n ≥ } so that we have Der(gr( G )) = M m ≥ Der m (gr( G )) . Since a derivation on gr( G ) is determined by its restriction on H = gr ( G ),we have a natural inclusionDer m (gr( G )) ⊂ Hom F p ( H, gr m +1 ( G )); δ δ | H for each m ≥ + (gr( G )) ⊂ M m ≥ Hom F p ( H, gr m +1 ( G )) , where Der + (gr( G )) is the Lie subalgebra of Der(gr( G )) consisting of positivedegree parts. Proposition 2.2.12.
The direct sum of τ m over m ≥ defines the Liealgebra homomorphism gr( τ ) := M m ≥ τ m : gr(A G ) −→ Der + (gr( G )) . roof. (cf. [Da; Proposition 3.18]) By Lemma 2.2.11, it suffices to show thatfor φ ∈ A G ( m ), the map g φ ( g ) g − is indeed a derivation on gr( G ). Let φ ∈ A G ( m ) ( m ≥
1) and g ∈ G i , h ∈ G j . By using the commutator formulas[ ab, c ] = a [ b, c ] a − · [ a, c ] , [ a, bc ] = [ a, b ] · b [ a, c ] b − ( a, b, c ∈ G ) , we obtain φ ([ g, h ])[ g, h ] − = [ φ ( g ) , φ ( h )][ g, h ] − = [ gg − φ ( g ) , φ ( h ) h − h ][ g, h ] − = g ([ g − φ ( g ) , φ ( h ) h − ] · ( φ ( h ) h − )[ g − φ ( g ) , h ]( φ ( h ) h − ) − ) g − · [ g, φ ( h ) h − ]( φ ( h ) h − )[ g, h ]( φ ( h ) h − ) − [ g, h ] − = g ([ g − φ ( g ) , φ ( h ) h − ] · ( φ ( h ) h − )[ g − φ ( g ) , h ]( φ ( h ) h − ) − ) g − · [ g, φ ( h ) h − ][ φ ( h ) h − , [ g, h ]] . Since g − φ ( g ) ∈ G i + m , φ ( h ) h − ∈ G j + m by Lemma 2.2.4, we have[ g − φ ( g ) , φ ( h ) h − ] ∈ G i + j +2 m . Similarly, we have [ φ ( h ) h − , [ g, h ]] ∈ G i +2 j + m . By these three claims together, we have φ ([ g, h ])[ g, h ] − ≡ gφ ( h ) h − [ g − φ ( g ) , h ]( gφ ( h ) h − ) − [ g, φ ( h ) h − ] mod G i + j + m +1 . Noting x [ g − φ ( g ) , h ] x − ≡ [ g − φ ( g ) , h ] mod G i + j + m +1 for x ∈ G , our claimis proved. (cid:3)
3. Non-Abelian Iwasawa theory
In this section, we propose an approach to study non-Abelian Iwasawatheory by means of the Johnson homomorphisms. In the course, we introducesome invariants from a dynamical viewpoint.Throughout this section, a fixed prime number p is assumed to be odd. Let k be a number field of finite de-gree over Q and let k ∞ be a Z p -extension of k , namely, k ∞ /k is a Galois14xtension whose Galois group is isomorphic to the additive group of p -adicintegers Z p . We call k ∞ the cyclotomic Z p -extension of k if k ∞ is the unique Z p -extension of k contained in k ( µ p ∞ ). Let S p denote the set of primes of k lying over p and S a finite set of primes of k containing S p . Note thatthe extension k ∞ /k is unramified outside S p . Let ˜ k S be the maximal pro- p extension of k which is unramified outside S , and let M be a subextensionof ˜ k S /k such that M/k is a Galois extension. We set(3 . .
1) Γ := Gal( k ∞ /k ) , G := Gal( M/k ) and G := Gal( M/k ∞ )so that we have the exact sequence(3 . .
2) 1 −→ G −→ G −→ Γ −→ . We assume that G is a finitely generated pro- p group, in other words, the µ -invariant is zero.We fix a topological generator γ of Γ and its lift ˜ γ ∈ G . We then definethe automorphism φ ˜ γ of G by Inn(˜ γ )(3 . . φ ˜ γ ( g ) = ˜ γg ˜ γ − ( g ∈ G ) . We note that if we choose a different lift ˜ γ ′ of γ , φ ˜ γ ′ differs from φ ˜ γ by aninner automorphism of G :(3 . . φ ˜ γ ′ = Inn( x ) ◦ φ ˜ γ ( x := ˜ γ ′ ˜ γ − ∈ G ) . Let H be the Frattini quotient of G , H = G/G p [ G, G ], as in (2.1.5). The F p -linear automorphism [ φ ˜ γ ] of H induced by φ ˜ γ is independent of the choiceof a lift ˜ γ and so is denoted by [ φ γ ]. Similarly, we let H ∞ be the Abelian-ization of G , H ∞ = G/ [ G, G ], and [ φ γ ] ∞ the Z p -module automorphism of H ∞ induced by φ ˜ γ , which is independent of the choice of a lift ˜ γ of γ . Thereason that we use the Zassenhaus filtrarion instead of the lower central se-ries throughout this paper is that any p -power of φ ˜ γ acts non-trivially on G/ [ G, G ] in general.By the Magnus correspondence γ X , we identify the complete groupalgebra F p [[Γ]] (resp. Z p [[Γ]]) with the power series algebra F p [[ X ]] (resp. Z p [[ X ]]). We set simply Λ := Z p [[ X ]] (Iwasawa algebra) and Λ := F p [[ X ]].Classical Iwasawa theory studies the Λ-module structure of H ∞ , in other15ords, the p -power iterated action of [ φ γ ] ∞ on H ∞ . A fundamental theoremof Iwasawa ([Iw1]), under our assumption on G , tells us that there is a Λ-module homomorphism, called a pseudo-isomorphism ,(3 . . H ∞ −→ s M i =1 Λ / ( f i ( X ))with finite kernel and cokernel, where f i ( X ) is a power of an irreducible dis-tinguished polynomial. Recall that a nonconstant polynomial f ( X ) ∈ Z p [ X ]is called distinguished if f ( X ) has the form X d + a X d − + · · · + a d with all a i ≡ p . The Iwasawa polynomial ( p -adic zeta function) associated to H ∞ is defined by Q si =1 f i ( X ). The set of degrees of f i , { deg( f ) , . . . , deg( f s ) } ,is also an invariant of the Λ-module H ∞ . The Iwasawa λ -invariant λ ( H ∞ ) isdefined by their sum P si =1 deg( f i ).In some cases, the pseudo-isomorphism in (3.1.5) turns out to be an iso-morphism. Then we can describe the p -power iterated action of [ φ γ ] on H in terms of deg( f i )’s. Since H is finite, there is an integer d ≥ φ γ ] p d = [ φ γ pd ] = id H , namely, [ φ γ ] p d ∈ IA( G ). We call such smallest integer d the p -period of [ φ γ ] on H . Proposition 3.1.6.
Suppose that we have a Λ -module isomorphism H ∞ ≃ s M i =1 Λ / ( f i ( X )) , where f i is a distinguished polynomial of degree deg( f i ) . Let d ( H ∞ ) denotethe maximum of deg( f ) , . . . , deg( f s ) . Then we have [ φ γ ] p d = [ φ γ pd ] = id H , namely , [ φ γ ] p d ∈ IA( G ) if and only if p d ≥ d ( H ∞ ) . Hence the p -period of [ φ γ ] is given by the smallest integer ≥ log p d ( H ∞ ) . Proof.
By the assumption, we have a Λ-module isomorphism H ≃ s M i =1 Λ / ( X deg( f i ) ) . φ γ ] p d − id H on H corresponds the multiplication by(1 + X ) p d − X p d , [ φ γ ] p d = id H if and only if X p d ∈ ( X deg( f i ) ) for all i . From this the assertion follows. (cid:3) Example 3.1.7. ∗ Let k := Q ( µ p ) , k ∞ := Q ( µ p ∞ ) and M the maximalunramified pro- p extension of k ∞ . The assumption of Proposition 3.1.5 isthen satisfied if the Vandiver conjecture is true, namely, p does not dividethe class number of the maximal real subfield of k ([Wa; Theorem 10.16]).The Vandiver conjecture is known to be true for p < H ∞ = Λ / ( f ) for p = 37 and H ∞ = Λ / ( f ) ⊕ Λ / ( f )for p = 157, where f, f and f are all distinguished polynomials of degreeone ([IS]). So, the p -period of [ φ γ ] is zero, namely, [ φ γ ] acts trivially on H .Mizusawa made a program to compute the Iwasawa polynomial when k isan imaginary quadratic field Q ( √− D ), k ∞ is the cyclotomic Z p -extensionand M is the maximal unramified pro- p extension of k ∞ . For example, when p = 3 and D = 186 , , , H ∞ = Λ / ( f ) with deg( f ) = 2 and so the3-period of [ φ γ ] is one, and when p = 3 and D = 214 , H ∞ = Λ / ( f ) withdeg( f ) = 4 and so the 3-period of [ φ γ ] is two. Abasic problem in non-Abelian Iwasawa theory is to understand the p -poweriterated action of φ ˜ γ on G , while classical Iwasawa theory deals with that of[ φ γ ] on H ∞ as shown in 3.1. Let { G n } n ≥ be the Zassenhaus filtration of G so that H = G/G , and let [ φ ˜ γ ] m be the automorphism of G/G m +1 inducedby φ ˜ γ as defined in (2.2.1). We aim to study the p -power iterated action of[ φ ˜ γ ] m on G/G m +1 for all m ≥ p -Johnson homomorphismsintroduced in 2.2.First, let us see how a different choice of a lift of γ affects the action of apower of [ φ ˜ γ ] m on G/G m +1 Lemma 3.2.1.
Let ˜ γ , ˜ γ ′ be lifts of γ in G and set x = ˜ γ ′ ˜ γ − ∈ G as in (3 . . . Suppose x ∈ G m . Then, for each integer e ≥ , we have φ e ˜ γ ′ ∈ A G ( m ) ⇐⇒ φ e ˜ γ ∈ A G ( m ) . ∗ We thank Y. Mizusawa for informing us of this example. roof. By (3.1.4), we have φ e ˜ γ ′ ( g ) = yφ e ˜ γ ( g ) y − , y := xφ ˜ γ ( x ) · · · φ e − γ ( x ) ∈ G m , for any g ∈ G . Since elements of G/G m +1 and G m /G m +1 commute, theassertion is shown as follows φ e ˜ γ ′ ∈ A G ( m ) ⇔ φ e ˜ γ ′ ( g ) g − ∈ G m +1 for any g ∈ G ⇔ yφ e ˜ γ ( g ) y − g − ∈ G m +1 for any g ∈ G ⇔ φ e ˜ γ ( g ) g − ∈ G m +1 for any g ∈ G ⇔ φ ˜ γ ∈ A G ( m ) (cid:3) Let gr( G ) = L n ≥ gr n ( G ), gr n ( G ) = G n /G n +1 , be the graded Lie algebraover F p associated to the Zassenhaus filtration of G as in (2.1.6), and let { A G ( m ) } m ≥ be the Andreadakis-Johnson filtration of Aut( G ). For m ≥ τ m : A G ( m ) −→ Hom F p ( H, gr m +1 ( G ))be the p -Johnson homomorphism. The next Corollary follows immediatelyfrom Lemma 3.2.1. Corollary 3.2.2.
Let ˜ γ , ˜ γ ′ be lifts of γ in G and set x = ˜ γ ′ ˜ γ − ∈ G .Suppose x ∈ G m +1 and φ e ˜ γ ∈ A G ( m ) ( e ≥ . Then we have τ m ( φ e ˜ γ ′ ) = τ m ( φ e ˜ γ ) . Proof.
By Lemma 3.2.1, φ e ˜ γ ′ ∈ A G ( m ). Since φ e ˜ γ ′ = Inn( y ) ◦ φ e ˜ γ with y = xφ ˜ γ ( x ) · · · φ e − γ ( x ) ∈ G m +1 , the assertion follows from Theorem 2.2.8and Proposition 2.2.10. (cid:3) We fix a lift ˜ γ ∈ G of γ . Generalizing the p -period of [ φ γ ] on H = G/G m ,we define the p -period d ( m ) of φ ˜ γ acting on G/G m +1 for each m ≥ d ≥ . . φ p d ˜ γ ∈ A G ( m ) . Thus we have non-decreasing sequence { d ( m ) } m ≥ of integers. Lemma 3.2.4.
For each integer m ≥ , we have d ( m + 1) = d ( m ) or d ( m ) + 1 . roof. By definition of d ( m ), we have d ( m + 1) ≥ d ( m ). Suppose φ p d ˜ γ ∈ A G ( m ). Then by Proposition 2.2.5 (2), we have φ p d +1 ˜ γ ∈ A G ( m + 1). Hence d ( m + 1) ≤ d ( m ) + 1. (cid:3) Now we introduce another sequence of integers { m ( d ) } d ≥ as follows. Foreach integer d ≥
0, we define the integer m ( d ) ≥ . . φ p d ˜ γ ∈ A G ( m ( d )) , φ p d ˜ γ / ∈ A G ( m ( d ) + 1) . It is a strictly increasing sequence. In fact, we have
Lemma 3.2.6.
For each integer d ≥ , we have m ( d + 1) ≥ m ( d ) + 1 . Proof.
Since φ p d ˜ γ ∈ A G ( m ( d )) for each d ≥
0, by Proposition 2.2.5 (2), wehave φ p d +1 ˜ γ ∈ A G ( m ( d ) + 1). Hence, by definition (3.2.5), we have m ( d + 1) ≥ m ( d ) + 1. (cid:3) Then the sequence { τ m ( d ) ( φ p d ˜ γ ) } d ≥ in Hom F p ( H, gr m ( d )+1 ( G )) describes theaction of φ p d ˜ γ on G/G m ( d )+1 for all d ≥
0. In Section 5, we give a coho-mological interpretation of τ m ( d ) ( φ p d ˜ γ ) in terms of Massey products in Galoiscohomology. Remark 3.2.7.
Let M the maximal unramified pro- p extension of k ∞ .Ozaki ([O]) studied the Γ-action on the graded pieces associated to the lowercentral series of G = Gal( M/k ∞ ) and obtained arithmetic results. We alsorefer to Sharifi’s paper [Sh] for a related work. Our approach is different fromtheirs. p -Johnson maps for a free pro- p group In this section, following Kawazumi ([Kw]), we extend the p -Johnsonhomomorphisms in Section 2 to maps defined on the whole group of auto-morphisms when G is a free pro- p group.Throughout this section, let F denote a free pro- p group on x , . . . , x r . Afixed prime number p is arbitrary in 4.1 and assumed to be odd in 4.2.19 .1. p -Johnson maps. We keep the same notations as in 2.1, only re-placing G by F . Let F p [[ F ]] be the complete group algebra of F over F p withaugmentation ideal I F . Let { F n } n ≥ be the Zassenhaus filtration defined by F n = F ∩ (1 + I nF ) and let H := F/F = F/F p [ F, F ] be the Frattini quotientof F . We write [ f ] for the image of f ∈ F in H : [ f ] := f mod F . We denote[ x j ] by X j (1 ≤ j ≤ r ) simply so that H is a vector space over F p with basis X , . . . , X r H = F p X ⊕ · · · ⊕ F p X r . As in 2.1, let gr( F ) be the graded restricted Lie algebra over F p associatedto the Zassenhaus filtration { F n } n ≥ of F gr( F ) := M n ≥ gr n ( F ) , gr n ( F ) := F n /F n +1 . It is the free Lie algebra over F p on X , . . . , X r . Its restricted universal en-veloping algebra U gr( F ) is given by the graded associative algebra gr( F p [[ F ]])(cf. (2.1.7)) U gr( G ) = gr( F p [[ F ]]) := M m ≥ gr m ( F p [[ F ]]) , gr m ( F p [[ F ]]) := I mF /I m +1 F together with the injective restricted Lie algebra homomorphismgr( θ ) = M m ≥ θ m : gr( F ) −→ gr( F p [[ F ]]) , where θ m : gr m ( F ) → gr m ( F p [[ F ]]) is given by θ m ( f mod F m +1 ) := f − I m +1 F . By the correspondence x j − I F ∈ gr ( F p [[ F ]]) X j ∈ H , the universalenvelope gr( F p [[ F ]]) is identified with the tensor algebra on H over F p or thenon-commutative polynomial algebra F p h X , . . . , X r i of variables X , . . . , X r over F p U gr( G ) = gr( F p [[ F ]]) = M m ≥ H ⊗ m = F p h X , . . . , X r i . m ( F p [[ F ]]) corresponds to H ⊗ m , the vector spaceover F p with basis X i · · · X i m (1 ≤ i , . . . , i m ≤ r ), monomials of degree m ,and so θ m may be regarded as the injective F p -linear map(4 . . θ m : gr m ( F ) ֒ → H ⊗ m In order to extend the Johnson homomorphisms in 2.2 to the maps definedon the whole automorphism group Aut( F ), we work with the completion b U ofthe universal envelope U gr( F ) = gr( F p [[ F ]]) with respect to I F -adic topology.So b U is the complete tensor algebra on H over F p which is identified withthe F p -algebra F p hh X , . . . , X r ii of non-commutative formal power series ofvariables X , . . . , X r over F p (In [Kw] Kawazumi wrote b T for b U ) b U := Y m ≥ H ⊗ m = F p hh X , . . . , X r ii . Then the composite of θ m in (4.1.1) with the natural inclusion H ⊗ m ֒ → b U isnothing but the restriction to F m of the Magnus embedding (4 . . θ : F ֒ → b U × defined by θ ( x j ) := 1 + X j (1 ≤ j ≤ r ).For n ≥
1, we let b U n := Y m ≥ n H ⊗ m be the two-sided ideal of b U corresponding to formal power series of degree ≥ n . An F p -algebra automorphism ϕ of b U is then called filtration-preserving if ϕ ( b U n ) = b U n for all n ≥ fil ( b U ) the group of filtration-preserving F p -algebra automorphisms of b U . The following useful Lemma,which we call Kawazumi’s lemma , gives a criterion for a F p -algebra endo-morphism of b U to be a filtration-preserving automorphism. Lemma 4.1.3. ( Kawazumi’s lemma ). A F p -algebra endomorphism ϕ of b U is a filtration-preserving automorphism of b U , ϕ ∈ Aut fil ( b U ) , if and only ifthe following conditions are satisfied :(1) ϕ ( b U n ) ⊂ b U n for all n ≥ . the induced F p -linear map [ ϕ ] on b U / b U = H defined by [ ϕ ]( h ) := ϕ ( h ) mod b U ( h ∈ H ) is an isomorphism.Proof. Suppose ϕ ∈ Aut fil ( b U ). Since ϕ is filtration-preserving, the condition(1) holds. To show the condition (2), consider the following commutativediagram for vector spaces over F p with exact rows:0 −→ b U −→ b U −→ H −→ ↓ ϕ | b U ↓ ϕ | b U ↓ [ ϕ ]0 −→ b U −→ b U −→ H −→ . Since ϕ ( b U n ) = b U n for all n ≥
0, we have Coker( ϕ | b U i ) = 0 for i = 1 ,
2, inparticular. Since ϕ is an automorphism, we have Ker( ϕ ) = 0, in particular,Ker( ϕ | b U i ) = 0 for i = 1 ,
2. By snake lemma applied to the above diagram,we obtain Ker([ ϕ ]) = 0 and Coker([ ϕ ]) = 0, hence the condition (2).Suppose that an F p -algebra endomorphism ϕ of b U satisfies the conditions(1) and (2). Let z = ( z m ) be any element of b U with z m ∈ H ⊗ m for m ≥ ϕ is an automorphism, we have only to prove that there existsuniquely y = ( y m ) ∈ b U such that(4 . . . z = ϕ ( y ) . Note by the condition (1) and (2) that ϕ induces an F p -linear automorphismof b U m / b U m +1 = H ⊗ m , which is nothing but [ ϕ ] ⊗ m . Then, writing ϕ ( y i ) j forthe component of ϕ ( y i ) in H ⊗ j for i < j , the equation (4.1.3.1) is equivalentto the following system of equations:(4 . . . z = ϕ ( y ) = y ,z = [ ϕ ]( y ) ,z = [ ϕ ] ⊗ ( y ) + ϕ ( y ) , · · · z m = [ ϕ ] ⊗ m ( y m ) + ϕ ( y ) m + · · · + ϕ ( y m − ) m , · · · Since [ ϕ ] ⊗ m is an automorphism, we can find the unique solution y = ( y m )of (4.1.3.2) from the lower degree. Therefore ϕ is an F p -algebra automor-phism. Furthermore, we can see easily that if z = · · · = z n − = 0, then y = · · · = y n − = 0 for n ≥
1. This means that ϕ − ( b U n ) ⊂ b U n and so ϕ is22ltration-preserving. (cid:3) By Lemma 4.1.3, each ϕ ∈ Aut fil ( b U ) induces an F p -linear automorphism[ ϕ ] of H = b U / b U and so we have a group homomorphism[ ] : Aut fil ( b U ) −→ GL( H ) . We define the induced automorphism group of b U byIA( b U ) := Ker([ ]) . We note that there is a natural splitting s : GL( H ) → Aut fil ( b U ) of [ ] , whichis defined by s ( P )(( z m )) := ( P ⊗ m ( z m )) for P ∈ GL( H ) . In the following, we also regard [ P ] ∈ GL( H ) as an element of Aut fil ( b U )through the splitting s and write simply [ P ] for s ([ P ]). Thus we have thefollowing Lemma 4.1.4.
We have a semi-direct decomposition
Aut fil ( b U ) = IA( b U ) ⋊ GL( H ) given by ϕ = ( ϕ ◦ [ ϕ ] − , [ ϕ ]).Let ϕ ∈ IA( b U ). Since ϕ acts on b U / b U = H trivially, we have ϕ ( h ) − h ∈ b U for any h ∈ H, and so we have a map E : IA( b U ) −→ Hom F p ( H, b U ); ϕ ϕ | H − id H , where Hom F p ( H, b U ) denotes the group of F p -linear maps H → b U . The fol-lowing Proposition will play a key role in our discussion. Proposition 4.1.5.
The map E is bijective.Proof. Injectivity: Suppose E ( ϕ ) = E ( ϕ ′ ) for ϕ, ϕ ′ ∈ IA( b U ). Then we23ave ϕ | H = ϕ ′ | H . Since an F p -algebra endomorphism of ˆ U is determined byits restriction on H , we have ϕ = ϕ ′ .Surjectivity: Take any η ∈ Hom F p ( H, b U ). We can extend η + id H : H → b U uniquely to a F p -algebra endomorphism ϕ of b U . Then we have obviously ϕ ( b U n ) ⊂ b U n for all n ≥
0. Since b U / b U = H and we see that[ ϕ ]( h mod b U ) = ϕ ( h ) mod b U = h + η ( h ) mod b U = h mod b U , we have [ ϕ ] = id H . By Kawazumi’s Lemma 2.1, we have ϕ ∈ IA( b U ) and E ( ϕ ) = η . (cid:3) By Lemma 4.1.4 and Proposition 4.1.5, we have the following
Corollary 4.1.6.
We have a bijection ˆ E : Aut fil ( b U ) ≃ Hom F p ( H, b U ) × GL( H )given by ˆ E ( ϕ ) = ( E ( ϕ ◦ [ ϕ ] − ) , [ ϕ ]) . The Magnus embedding θ : F ֒ → b U × in (4.1.2) is extended to an F p -algebra isomorphism, denoted by the same θ ,(4 . . θ : F p [[ F ]] ∼ −→ b U , which satisfies(4 . . θ ( I nF ) = b U n for m ≥ . For m ≥
0, let θ m denote the component of θ in H ⊗ m as in (4.1.1): θ ( α ) = ∞ X m =0 θ m ( α ) , θ m ( α ) ∈ H ⊗ m ( α ∈ F p [[ F ]]) . Note that θ ( f ) = 1 and θ ( f ) = [ f ] for f ∈ F . Further we can write θ m ( α )as(4 . . θ m ( α ) = X ≤ i ,...,i m ≤ r ǫ ( i · · · i m ; α ) X i · · · X i m , ǫ ( i · · · i m ; α ) is given in terms of the pro- p Fox freederivative ∂/∂x j : Z p [[ F ]] → Z p [[ F ]] ([Ih], [Ms2, 8.3]) ǫ ( i · · · i m ; α ) = ǫ Z p [[ F ]] (cid:18) ∂ m ˜ α∂x i · · · ∂x i m (cid:19) mod p, where ǫ Z p [[ F ]] : Z p [[ F ]] → Z p is the augmentation map and ˜ α ∈ Z p [[ F ]] suchthat ˜ α mod p = α .An F p -algebra automorphism ϕ of F p [[ F ]] is said to be filtration-preserving if ϕ ( I nF ) = I nF for all n ≥ fil ( F p [[ F ]]) the group offiltration-preserving automorphisms of F p [[ F ]]. By (4.1.7) and (4.1.8), wehave an isomorphism(4 . .
10) Aut fil ( F p [[ F ]]) ≃ Aut fil ( b U ); ϕ θ ◦ ϕ ◦ θ − . Now, let φ ∈ Aut( F ). Then φ induces a filtration-preserving F p -algebraautomorphism ˆ φ of F p [[ F ]]). In fact, φ induces an automorphism [ φ ] n of afinite p -group F/F n and hence an F p -algebra automorphism, denoted by thesame [ φ ] n , of a finite group ring F p [ F/F n ][ φ ] n : F p [ F/F n ] ∼ −→ F p [ F/F n ]for each n ≥
1, which sends the augmentation ideal of F p [ F/F p ] onto it-self. Taking the inverse limit with respect to n , we obtain an F p -algebraautomorphism b φ := lim ←− n [ φ ] n : F p [[ F ]] ∼ −→ F p [[ F ]]such that b φ ( I F ) = I F . Thus we have an injective homomorphismAut( F ) −→ Aut fil ( F p [[ F ]]); φ b φ. By composing with the isomorphism (4.1.10), we obtain an injective homo-morphism b κ θ : Aut( F ) −→ Aut fil ( b U ); φ θ ◦ b φ ◦ θ − . Lemma 4.1.11.
Let [ φ ] denote the F p -linear automorphism of H induced by φ ∈ Aut( F ) . Then we have [ b κ θ ( φ )] = [ φ ] in GL( H ) . roof. We have, for X j ∈ H (1 ≤ j ≤ r ) , b κ θ ( φ ) = ( θ ◦ b φ ◦ θ − )( X j )= ( θ ◦ b φ ◦ θ − )( θ ( x j ) − θ ◦ b φ )( x j − θ ( φ ( x j )) − ≡ [ φ ( x j )] mod b U = [ φ ]( X j ) mod b U . Hence we have [ b κ θ ( φ )] = [ φ ]. (cid:3) By Lemma 4.1.11, we have, for φ ∈ Aut( F ), b κ θ ( φ ) = ( b κ θ ( φ ) ◦ [ φ ] − , [ φ ])under the semi-direct decomposition Aut fil ( b U ) = IA( b U ) ⋊ GL( H ) of Lemma4.1.4. We set(4 . . κ θ ( φ ) := b κ θ ( φ ) ◦ [ φ ] − = θ ◦ b φ ◦ θ − ◦ [ φ ] − ( φ ∈ Aut( F )) . Now, we define the extended p -Johnson map b τ θ : Aut( F ) −→ Hom F p ( H, b U ) ⋊ GL( H )by composing b κ θ with ˆ E of Corollary 4.1.6, and we define the p -Johnson map τ θ : Aut( F ) −→ Hom F p ( H, b U )by the composing b τ θ with the projection on Hom F p ( H, b U ), namely, for φ ∈ Aut( F ),(4 . . τ θ ( φ ) := E ( κ θ ( φ )) = κ θ ( φ ) | H − id H . For m ≥
1, we define the m -th p -Johnson map τ θm : Aut( F ) −→ Hom F p ( H, H ⊗ ( m +1) )by the m -th component of τ K :(4 . . τ θ ( φ ) := X m ≥ τ θm ( φ ) ( φ ∈ Aut( G )) . p -Johnson homomorphisms (Theorem 2.2.8), the p -Johnsonmap τ θ = E ◦ κ θ : Aut( F ) → Hom( H, b U ) is no longer a homomorphism. Infact, we have the following Proposition 4.1.15.
We have κ θ ( φ ◦ φ ) = κ θ ( φ ) ◦ [ φ ] ◦ κ θ ( φ ) ◦ [ φ ] − . Proof.
By (4.1.12), we have κ θ ( φ φ ) = θ ◦ ( d φ φ ) ◦ θ − ◦ [ φ φ ] − = θ ◦ b φ ◦ b φ ◦ θ − ◦ [ φ ] − ◦ [ φ ] − = θ ◦ b φ ◦ θ − ◦ [ φ ] − ◦ [ φ ] ◦ θ ◦ b φ ◦ θ − ◦ [ φ ] − ◦ [ φ ] − = κ θ ( φ ) ◦ [ φ ] ◦ κ θ ( φ ) ◦ [ φ ] − . (cid:3) Proposition 4.1.15 yields an infinite sequence coboundary relations whichJohnson maps τ θm satisfies. Here we give the formulas for τ θ and τ θ . Proposition 4.1.16.
We have τ θ ( φ φ ) = τ θ ( φ ) + [ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − ,τ θ ( φ φ ) = τ θ ( φ ) + ( τ θ ( φ ) ⊗ id H + id H ⊗ τ θ ( φ )) ◦ [ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − +[ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − . Proof.
By definition (4.1.14), we have(4 . . . τ θ ( φ φ ) = X m ≥ τ θm ( φ φ ) .
27n the other hand, by Proposition 4.1.15 and (4.1.13), we have, for h ∈ H , τ θ ( φ φ ) = − h + κ θ ( φ φ )( h )= − h + ( κ θ ( φ ) ◦ [ φ ] ◦ κ θ ( φ ) ◦ [ φ ] − )( h )= − h + ( κ θ ( φ ) ◦ [ φ ] ◦ (id H + τ θ ( φ )))([ φ ] − ( h ))= − h + ( κ θ ( φ ) ◦ [ φ ]) [ φ ] − ( h ) + X m ≥ ( τ θm ( φ ) ◦ [ φ ] − )( h ) ! = − h + κ θ ( φ ) h + X m ≥ ([ φ ] ⊗ m ◦ τ θm ( φ ) ◦ [ φ ] − )( h ) ! = − h + κ θ ( φ )( h )+ κ θ ( φ )(([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h ))+ κ θ ( φ )(([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h )) mod b U . We note that κ θ ( φ ) | H ⊗ m = (id H + τ θ ( φ )) ⊗ m : H ⊗ m −→ H × b U m for any φ ∈ Aut( F ) and so we have the following congruences mod b U : κ θ ( φ )( h ) ≡ h + τ θ ( φ )( h ) + τ θ ( φ )( h ) ,κ θ ( φ )(([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h )) ≡ ([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h )+(( τ θ ( φ ) ⊗ id H + id H ⊗ τ θ ( φ )) ◦ [ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h ) ,κ θ ( φ )(([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h )) ≡ ([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h ) . Therefore we have(4 . . . τ θ ( φ φ )( h ) ≡ τ θ ( φ )( h ) + τ θ ( φ )( h )+([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h )+(( τ θ ( φ ) ⊗ id H + id H ⊗ τ θ ( φ )) ◦ [ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h )+([ φ ] ⊗ ◦ τ θ ( φ ) ◦ [ φ ] − )( h ) mod b U . Comparing (4.1.16.1) and (4.1.16.2), we obtain the assertions. (cid:3)
Next, we compute the p -Johnson maps for inner automorphisms of F . Proposition 4.1.17.
Let f ∈ F and h ∈ H . For m ≥ , we have τ θm (Inn( f ))( h ) = θ m ( f ) h + m X j =1 X q + ··· + q j = m q ≥ ,q ,...,qj ≥ ( − j θ q ( f ) hθ q ( f ) · · · θ q j ( f ) . n particular, we have, for m = 1 , , τ θ (Inn( f ))( h ) = [ f ] h − h [ f ] ,τ θ (Inn( f ))( h ) = θ ( f ) h − hθ ( f ) + h [ f ][ f ] − [ f ] h [ f ] . Proof.
Since [Im( f )] = id H , by (4.1.12), we have κ θ (Inn( f ))( z ) = ( θ ◦ \ Inn( f ) ◦ θ − )( z )= θ ( f ) zθ ( f − )= X m ≥ θ m ( f ) ! z X j ≥ ( − j ( X q ≥ θ q ( f )) j ! for z ∈ b U . Therefore, by (4.1.13), we have τ θ (Inn( f ))( h ) = κ θ (Im( f ))( h ) − h = X m ≥ θ m ( f ) h + X j ≥ X q o ≥ .q ,...,q j ≥ ( − j θ q ( f ) hθ q ( f ) · · · θ q j ( f )for h ∈ H . Taking the component in H ⊗ ( m +1) , we obtain the assertion. (cid:3) Finally we give the relation between the p -Johnson maps and the p -Johnsonhomomorphisms in Section 2. Proposition 4.1.18.
The restriction of τ θm to A F ( m ) coincides with θ m +1 ◦ τ m for each m ≥ τ θm | A F ( m ) = θ m +1 ◦ τ m : A F ( m ) −→ Hom(
H, H ⊗ ( m +1) ) , where θ m +1 is the injection gr m +1 ( F ) ֒ → H ⊗ ( m +1) in (4.1.1). Proof.
It suffices to show that for φ ∈ A F ( m ), τ θm ( φ )( X j ) = θ m +1 ( τ m ( φ )( X j )) 1 ≤ j ≤ r. By (4.1.13) and [ φ ] = id H , we have τ θ ( φ )( X j ) = ( κ θ ( φ ) | H − id H )( X j )= ( θ ◦ b φ ◦ θ − )( θ ( x j ) − − ( θ ( x j ) − θ ( φ ( x j )) − θ ( x j ) . . . . τ θm ( φ )( X j ) = the component in H ⊗ ( m +1) of θ ( φ ( x j )) − θ ( x j ) . On the other hand, since φ ( x j ) x − j ∈ F m +1 , we have θ ( φ ( x j ) x − j ) ≡ θ m +1 ( φ ( x j ) x − j ) = 1 + θ m +1 ( τ m ( φ )( X j )) mod b U m +2 . Multiplying the above equation by θ ( x j ) from right, we have(4 . . . θ ( φ ( x j )) ≡ θ ( x j ) + θ m +1 ( τ m ( φ )( X j )) mod b U m +2 . By (4.1.18.1) and (4.1.18.2), we obtain the assertion. (cid:3)
Let us come backto the arithmetic situation set up in 3.1 and keep the same notations. So, asin (3.1.1) and (3.1.2), we have an exact sequence of pro- p Galois groups1 −→ G −→ G −→ Γ −→ , where G = Gal( M/k ∞ ) , G = Gal( M/k ) and G = Gal( k ∞ /k ) . In order to apply the materials in 4.1, we assume that(F) G = Gal( M/k ∞ ) is a free pro- p group F on x , . . . , x r . This condition (F) is satisfied for the following cases.
Example 4.2.1 ([Iw2], [W1]). Suppose that(1) k is totally real,(2) M := ˜ k S ,(3) the Iwasawa µ -invariant of H ∞ = G/ [ G, G ] is zero.Then the condition (F) is satisfied where the generator rank r is equal to theIwasawa λ -invariant of H ∞ .To give the following example, we introduce the notation. For a field K , K ( p ) denotes the maximal pro- p extension of K . Example 4.2.2 ([Sc], [W2]). Suppose that301) k is a CM-field containing µ p i.e., k = k + ( µ p ) where k + is the maximaltotally real subfield of k ,(2) the completions k + p of k + with respect to any prime p lying over p do notcontain µ p ,(3) k ∞ is the cyclotomic Z p -extension of k , and(4) the Iwasawa µ -invariant of the maximal Abelian unramified pro- p Galoisgroup over k ∞ is zero.The condition (4) is known to be true if k is Abelian over Q ([FW]). So, theabove four conditions are satisfied for the p -th cyclotomic field k = Q ( µ p ),for instance.A finite p -extension L/k is called positively ramified over S p if L p ⊂ k + p ( p )( µ p ) for any prime p over p . Since the composite of positively rami-fied p -extensions is positively ramified again, the maximal positively ramifiedpro- p extension of k exists, and it contains the cyclotomic Z p -extension k ∞ .We then let M := the maximal pro- p extension of k which is unramified outside S and positively ramified over S p .Then the condition (F) is satisfied with r = 2 λ − + S ( k ∞ ) \ S p ( k ∞ )) − , where λ − denotes the Iwasawa λ − -invariant of k , and S ( k ∞ ) (resp. S p ( k ∞ ))denotes the set of primes of k ∞ lying over S (resp. S p ). The pro- p Galoisgroup F = G = Gal( M/k ∞ ) has the following presentation F = h a , b , . . . , a λ − , b λ − , c v ( v ∈ S ( k ∞ ) \ S p ( k ∞ )) | Y v ∈ S ( k ∞ ) \ S p ( k ∞ ) c v λ − Y i =1 [ a i , b i ] = 1 i . We may take S to be S p ∪ { q } such that there is only one prime of k ∞ lyingover q (there are infinitely many such q ). Then F and G may be seen asarithmetic analogues of the fundamental groups of a one-boundary surfaceand a surface bundle over a circle (a fibered knot complement), respectively.We fix a lift ˜ γ ∈ G of a topological generator γ of Γ and consider theautomorphism φ ˜ γ := Inn(˜ γ ) ∈ Aut( F ) as in (3.1.3). The p -power iteratedaction of [ φ ˜ γ ] m on F/F m +1 is described by the m -th p -Johnson map τ θm : Aut( F ) −→ Hom F p ( H, H ⊗ ( m +1) ) ( m ≥ . d ≥
0, we can write τ θm ( φ p d ˜ γ )([ f ]) = X ≤ i ,...,i m +1 ≤ r τ θ ( φ p d ˜ γ )( i · · · i m +1 ; [ f ]) X i · · · X i m +1 . Suppose that φ p d ˜ γ ∈ A F ( m ). Then we can also write(4 . . θ m +1 ◦ τ m ( φ p d ˜ γ )([ f ]) = X ≤ i ,...,i m +1 ≤ r τ ( φ p d ˜ γ )( i · · · i m +1 ; [ f ]) X i · · · X i m +1 and, by Proposition 4.1.18, we have τ θ ( φ p d ˜ γ )( i · · · i m +1 ; X j ) = τ ( φ p d ˜ γ )( i · · · i m +1 ; X j ) ∈ F p . These coefficients are numerical datum encoded in the Johnson maps/ ho-momorphisms. In Section 5, we express these coefficients in terms of Masseyproducts in Galois cohomology.
5. Massey products
In this section, we give a cohomological interpretation of p -Johnson ho-momorphisms in terms of Massey products in Galois cohomology.A fixed prime number p is arbitrary in 5.1 and assumed to be odd in 5.2. Firstly, we re-call some general materials on Massey products. For the sign convention,we follow [Dw]. Let G be a pro- p group and let α , . . . , α m ∈ H ( G , F p ). A Massey products h α , . . . , α m i is said to be defined if there is an array A = { a ij ∈ C ( G , F p ) | ≤ i ≤ m + 1 , ( i, j ) = (1 , m + 1) } such that [ a i,i +1 ] = α i (1 ≤ i ≤ m ) ,da ij = j − X l =1 a l ∪ a lj ( j = i + 1) , where d denotes the differential on cochains and ∪ denotes the cup product.An array A is called a defining system for h α , . . . , α m i . Then we define h α , . . . , α m i A by the cohomology class represented by the 2-cocycle m X l =2 a l ∪ a l,m +1 .
32 Massey product of α , . . . , α m is then defined by h α , . . . , α m i := {h α , . . . , α m i A ∈ H ( G , F p ) | A ranges over defining systems } . We recall some basic properties of Massey products, which will be usedin 5.2.5.1.1. One has h α , α i = α ∪ α . For m ≥ h α , . . . , α m i is definedand consists of a single element if h α i , . . . , α i l i = 0 for all proper subsets { i , . . . , i l } of { , . . . , m } .5.1.2. Let Ψ : G → G ′ be a continuous homomorphism of pro- p groups. Thenif h α , . . . , α m i is defined for α i ∈ H ( G ′ , F p ) with defining system A = ( a ij ),then so is h Ψ ∗ ( α ) , . . . , Ψ ∗ ( α m ) i with defining system A ∗ = (Ψ ∗ ( a ij )) and wehave Ψ ∗ ( h α , . . . , α m i ) ⊂ h Ψ ∗ ( α ) , . . . , Ψ ∗ ( α m ) i .Next, we recall a relation between Massey products and the Magnus ex-pansion. Let G be a finitely generated pro- p group with a minimal presenta-tion 1 −→ N −→ F π −→ G −→ , where F is a free pro- p group on x , . . . , x s with s = dim F p H ( G , F p ). We set g i := π ( x i ) (1 ≤ i ≤ s ). Note that π induces the isomorphism H ( G , F p ) ≃ H ( F , F p ) . We let tg : H ( N, F p ) G → H ( G , F p ) be the transgression mapdefined as follows. For a ∈ H ( N, F p ) G , choose a 1-cochain b ∈ C ( F , F p )such that b | N = a . Since the value db ( f , f ), f i ∈ F , depends only on thecosets f i mod N , there is a 2-cocyle c ∈ Z ( G , F p ) such that π ∗ ( c ) = db . Thenwe define tg( a ) by the class of c . By Hochschild-Serre spectral sequence, tgis an isomorphism and so we have the dual isomorphism, called the Hopfisomorphism,(5 . .
3) tg ∨ : H ( G , F p ) ∼ → H ( N, F p ) G = N/N p [ N, F ] . Then we have the following Proposition. The proof goes in the same manneras in [Ms1, Theorem 2.2.2].
Proposition 5.1.4.
Notations being as above, let α , . . . , α m ∈ H ( G , F p ) and A = ( a ij ) a defining system for the Massey product h α , . . . , α m i . Let ∈ N and set β := (tg ∨ ) − ( f mod N p [ N, F ]) . Then we have h α , . . . , α m i A ( β )= m X j =1 ( − j +1 X c + ··· + c j = m X ≤ i ,...,i j ≤ s a , c ( g i ) · · · a m +1 − c j ,m +1 ( g i j ) ǫ ( i · · · i j ; f ) , where c , . . . , c j run over positive integers satisfying c + · · · + c j = m and g i := π ( x i ) (1 ≤ i ≤ s ) and ǫ ( i · · · i j ; f ) is the Magnus coefficient defined in (4 . . . p -Johnson homomorphisms. We comeback to the arithmetic situation in 4.2 and keep the same notations. So wehave an exact sequence of pro- p Galois groups1 −→ F −→ G −→ Γ −→ , where F = Gal( M/k ∞ ) , G = Gal( M/k ) and Γ = Gal( k ∞ /k ) , and F is a free pro- p group on x , . . . , x r . We fix a lift ˜ γ ∈ G of a topologicalgenerator γ of Γ and let φ ˜ γ := Inn(˜ γ ) ∈ Aut( F ).Let d (1) be the p -period of [ φ γ ] on H as in (3.2.3) so that φ p d (1) ˜ γ ∈ IA( F ).If necessary, we replace the base field k by the subextension k d (1) of k ∞ withdegree [ k d (1) : k ] = p d (1) and γ p d (1) with γ so that we may suppose that φ ˜ γ ∈ IA( F ) , namely, φ ˜ γ acts trivially on H .For each integer d ≥
0, let k d be the subextension of k ∞ with [ k d : k ] = p d and let G d := Gal( M/k d ) . Then the pro- p group G d has the presentation1 −→ N d −→ F π d −→ G d −→ F is the free pro- p group on x , . . . , x r , x r +1 with π d ( x r +1 ) = γ p d and N d is the closed subgroup of F generated normally by R j,d := φ p d ˜ γ ( x j )( x r +1 x j x − r +1 ) − (1 ≤ j ≤ r ) . emma 5.2.1. For each integer d ≥ , the homomorphism π d : F → G d induces the isomorphism of cohomology groups H ( G d , F p ) ∼ −→ H ( F , F p ) . Proof.
Since G d = F /N d , we have(5 . . . H ( G d , F p ) = Hom c ( G d / G pd [ G d , G d ] , F p ) ≃ Hom c ( F /N d F p [ F , F ] , F p ) , where Hom c stands for the group of continuous homomorphisms. Since φ p d ˜ γ acts trivially on H = F/F p [ F, F ], φ p d ˜ γ ( x j ) x − j ∈ F p [ F, F ] and so R j,d = φ p d ˜ γ ( x j ) x − j [ x j , x r +1 ] ∈ F p [ F , F ] (1 ≤ j ≤ r ). Therefore we have(5 . . . N d ⊂ F p [ F , F ] . By (5.2.1.1) and (5.2.1.2), we have H ( G d , F p ) ≃ Hom c ( F / F p [ F , F ] , F p ) = H ( F , F p ) . (cid:3) By Lemma 5.2.1, Hochschild-Serre spectral sequence yields the Hopf isomor-phism as in (5.1.3)tg ∨ : H ( G d , F p ) ∼ −→ H ( N d , F p ) G d = N d /N pd [ N d , F ] , and we define ξ j,d ∈ H ( G d , F p ) by ξ j,d := (tg ∨ ) − ( R j,d mod N pd [ N d , F ]) (1 ≤ j ≤ r ) . We set g j := π d ( x i ) (1 ≤ j ≤ r + 1) and let g ∗ i ∈ H ( G d , F p ) denote theKronecker dual to g j , namely g ∗ i ( g j ) = δ ij .For d ≥
0, let m ( d ) be the integer defined in (3.2.5). Since φ ˜ γ ∈ IA( F ), m ( d ) ≥
1. Let τ m ( d ) ( φ p d ˜ γ )( i · · · i m ( d ) ; X j ) be the coefficients of the m ( d )-th p -Johnson homomorphism defined in (4.2.3). The following theorem gives aninterpretation of τ m ( d ) ( φ p d ˜ γ )( i · · · i m ( d ) ; X j ) in terms of the Massey product inthe cohomology of G d . Theorem 5.2.2.
Notations being as above, let i , . . . , i m ( d )+1 ∈ { , . . . , r } . hen the Massey product h g ∗ i , · · · , g ∗ i m ( d )+1 i is uniquely defined and we have,for each d ≥ , τ m ( d ) ( φ p d ˜ γ )( i · · · i m ( d )+1 ; X j ) = ( − m ( d )+1 h g ∗ i , · · · , g ∗ i m ( d )+1 i ( ξ j,d ) . Proof.
Let G ′ d be the pro- p group given by the presentation1 −→ N ′ d −→ F π ′ d −→ G ′ d −→ , where N ′ is the closed subgroup of F generated normally by R ′ j,d := φ p d ˜ γ ( x j ) x − j (1 ≤ j ≤ r ) . We set g ′ j := π ′ ( x j ) (1 ≤ j ≤ r ) and let g ′ i ∗ be the Kronecker dual to g ′ j . Asin Lemma 5.2.1, π ′ d induces the isomorphism tg : H ( G ′ d , F p ) ∼ → H ( N ′ d , F p ) G ′ and so we have the Hopf isomorphism tg ∨ : H ( G ′ d , F p ) ∼ → H ( N ′ d , F p ).We define ξ ′ j,d ∈ H ( G ′ d , F p ) by (tg ∨ ) − ( R ′ j,d mod N ′ pd [ N ′ d , F ]). Since φ p d ˜ γ ∈ A F ( m ( d )), we note R ′ j,d ∈ F m ( d )+1 (1 ≤ j ≤ r ).Suppose m ( d ) ≥
2. By Proposition 5.1.4, if l ≤ m ( d ), we have h g ′ i ∗ , . . . , g ′ i l ∗ i A ′ ( ξ ′ j,d ) = 0for any i , . . . , i l ∈ { , . . . r } , 1 ≤ j ≤ r , and any defining system A ′ ,because we have ǫ ( i · · · i l ; R ′ j,d ) = 0. Since R ′ j,d ’s generate H ( N ′ d , F p ) G ′ d , h g ′ i ∗ , . . . , g ′ i l ∗ i = 0 for any i , . . . , i l ∈ { , . . . r } . Therefore, by 5.1.1, theMassey product h g ′ i ∗ , . . . , g ′∗ i m ( d )+1 i is uniquely defined and, by Proposition5.1.4 again, we have(5 . . . h g ′ i ∗ , . . . , g ′∗ i m ( d )+1 i ( ξ ′ j,d ) = ( − m ( d )+1 ǫ ( i · · · i m ( d )+1 ; R ′ j,d )= ( − m ( d )+1 τ m ( d ) ( φ p d ˜ γ )( i · · · i m ( d )+1 ; X j ) . We define the homomorphismΨ : G d −→ G ′ d by Ψ( g j ) := g ′ j (1 ≤ j ≤ r ) , Ψ( g r +1 ) := 1 . so that we have ξ ′ j,d = Ψ ∗ ( ξ j,d ) g ′ i ∗ = Ψ ∗ ( g ∗ i ) (1 ≤ i, j ≤ r ) . h g ∗ i , . . . , g ∗ i m ( d )+1 i ( ξ j,d ) = h Ψ ∗ ( g ′ i ∗ ) , . . . , Ψ ∗ ( g ′∗ i m ( d )+1 ) i (Ψ ∗ ( ξ ′ j,d ))= Ψ ∗ ( h g ′∗ i , . . . , g ′∗ i m ( d )+1 i )(Ψ ∗ ( ξ ′ j,d ))= h g ′∗ i , . . . , g ′∗ i m ( d )+1 i ( ξ ′ j,d )= ( − m ( d )+1 τ m ( d ) ( φ p d ˜ γ )( i · · · i m ( d )+1 ; X j ) . (cid:3) Remark 5.2.3. (1) Theorem 5.2.2 may be regarded as an arithmetic ana-logue in non-Abelian Iwasawa theory of Kitano’s result ([Ki, Theorem 4.1]).(2) For Massey products in cohomology of a pro- p group, we also refer to[G], [MT1] and [MT2]. Acknowledgement.
We would like to thank Yasushi Mizusawa, ManabuOzaki, Takuya Sakasai and Takao Satoh for helpful communication. Wewould also like to thank the referee for useful comments.
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