Iwasawa theory of totally real fields for certain non-commutative $p$-extensions

IW ASA W A THEOR Y OF TOT ALL Y REAL FIELDS F OR CER T AIN NON-COMMUT A TI VE p -EXT ENSIONS T AK ASHI HARA Abstra ct. In this pap er, w e prov e the Iw asaw a main conjecture (in the sense of [CFKSV]) of totally real ﬁelds for certain sp eciﬁc non- comm utative p -adic Lie ex t ensions, using the in tegral logarithms intro - duced by Oliver and T a ylor. Our result gives certain g eneralization of Kato’s proof of the main conjecture for Galois exten sions of Heisen b erg type ([Kato1]). 0. Introduction 0.1. In tro duction. The Iw asa w a m ain conjecture, which describes myste- rious relation b et we en “arithmetic” charact eristic elemen ts and “analytic” p -adic zeta fu nctions, has b een pro v en under many situati ons for ab elian ex- tensions. Ho w ev er, for non-ab elian extensions, it took many y ea rs even to formulate the main conjecture. In 2005, Coates et al. form ulated it ([CFKSV]) by us ing algebraic K -theory (esp ecially the lo calization exact sequence), and Kazuy a Kato has prov en it for certain sp eciﬁc p -adic Lie extensions of tota lly real ﬁelds up to th e presen t ([Kato1]). Mahesh Kakd e generalized Kato’s pro of and pro v ed the main conjecture for other t yp es of extensions ([Kakde]). J ¨ urgen Ritter and Alfr ed W eiss also formulated the main conjecture in a littl e diﬀerent wa y (“Equiv arian t I wasa w a theory ,” see [R-W1].), also u sing alg ebraic K -theory . In this p ap er, we prov e the Iwasa w a m ain conjecture (in the sense of [CFKSV]) of totally real ﬁ elds for certain non-commutat ive p -extensions, using the metho d of Kato in [Kato1]. Let F b e a tota lly r eal n umber ﬁeld, and let p b e a prime n um b er. Let F ∞ /F b e a totally real Lie extension con taining the cycloto mic Z p -extension F cyc of F . W e assume t hat only ﬁnitely man y primes of F ramify in F ∞ . The aim of this pap er is to pro v e the follo wing theo rem. Theorem 0.1 (=Theorem 3.1) . Assume that the fol lowing c onditio ns ar e satisﬁe d . (1) G = Gal( F ∞ /F ) ∼ =     1 F p F p F p 0 1 F p F p 0 0 1 F p 0 0 0 1     × Γ , wher e Γ is a c ommutative p -adic Lie gr oup isomorphic to Z p . (2) p 6 = 2 , 3 . Date : Au gust 12, 2021. 1 2 T A KASHI HARA (3) Ther e exists a ﬁnite extension F ′ ⊆ F ∞ of F such that the µ -invariant of ( F ′ ) cyc /F ′ e quals to zer o, wher e ( F ′ ) cyc /F ′ is the cyclotomic Z p - extension of F ′ . Then the p -adic zeta function ξ F ∞ /F for F ∞ /F exists and the Iwasawa main c onje ctur e for F ∞ /F is true. Let us summarize ho w to pro v e this theorem. W e consider the follo wing family of su b groups of G . U 0 = G, V 0 =     1 0 F p F p 0 1 0 F p 0 0 1 0 0 0 0 1     × { 1 } , U 1 =     1 F p F p F p 0 1 0 F p 0 0 1 F p 0 0 0 1     × Γ , V 1 =     1 0 0 F p 0 1 0 0 0 0 1 0 0 0 0 1     × { 1 } , f U 2 =     1 0 F p F p 0 1 F p F p 0 0 1 F p 0 0 0 1     × Γ , f V 2 =     1 0 0 F p 0 1 0 F p 0 0 1 0 0 0 0 1     × { 1 } , U 2 =     1 0 0 F p 0 1 F p F p 0 0 1 F p 0 0 0 1     × Γ , V 2 =     1 0 0 0 0 1 0 F p 0 0 1 0 0 0 0 1     × { 1 } , U 3 =     1 0 0 F p 0 1 0 F p 0 0 1 F p 0 0 0 1     × Γ , V 3 = { I 4 } × { 1 } , where I 4 is the u n it mat rix of GL 4 ( F p ). In the f ollo win g, we u se the nota tion U i (resp. V i ) for o ne of the subgroups U 0 , U 1 , f U 2 , U 2 and U 3 (resp. V 0 , V 1 , f V 2 , V 2 and V 3 ). Note that eac h quotien t group U i /V i is a b elian. No w w e ha ve a homomorp h ism θ i : K 1 (Λ( G )) Nr − → K 1 (Λ( U i )) → K 1 (Λ( U i /V i )) = Λ( U i /V i ) × where Λ( G ) denotes th e Iw asa wa algebra of G . Here the ﬁrst map is the norm map of K -theory and the second one is the natural map induced b y Λ( U i ) → Λ( U i /V i ). On the other han d , John Coates et al. introdu ced the canonical Øre set S of Λ( G ) (See § 2 and [CFKSV]) and considered the Ø r e lo calization Λ( G ) S to formulate th e main conjecture. F or this lo calized Iwasa wa algebra, we also ha ve a homomorp h ism θ S,i : K 1 (Λ( G ) S ) Nr − → K 1 (Λ( U i ) S ) → K 1 (Λ( U i /V i ) S ) = Λ( U i /V i ) × S IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 3 b y the same constru ction as θ i . W e denote θ = ( θ i ) i : K 1 (Λ( G )) → Y i Λ( U i /V i ) × and θ S = ( θ S,i ) i : K 1 (Λ( G ) S ) → Y i Λ( U i /V i ) × S . Then w e ha ve the follo wing prop osition. Prop osition 0.2 (Prop osition 5.4, Prop osition 6.3 and Prop osition 6.4) . Ther e exi st sub gr oups Ψ ⊆ Y i Λ( U i /V i ) × and Ψ S ⊆ Y i Λ( U i /V i ) × S which satisfy the fol lowing c onditions : (1) Image( θ S ) ⊆ Ψ S . (2) Image( θ ) = Ψ . (3) Ψ S ∩ Y i Λ( U i /V i ) × = Ψ . W e can c haracterize b oth Ψ and Ψ S as the subgroups consisting of all elemen ts whic h satisfy certain n orm relations and certain congruences (for details, see Deﬁnition 5.3 and P rop osition 6.2.). In the f ollo wing, w e denote the induced h omomorphisms b y the same symbols θ : K 1 (Λ( G )) → Ψ and θ S : K 1 (Λ( G ) S ) → Ψ S . W e call the su rjectiv e homomorphism θ the theta map f or G and the homomorphism θ S the lo c alize d theta map for G . Then we obtain the follo wing outstanding theorem whic h was ﬁr s t ob- serv ed b y Da vid Bur ns. Theorem 0.3 (=Th eorem 3.4, Burns) . L et ξ i b e the p -adic zeta function for the ab elian p -adic Lie extension F V i /F U i wher e F U i ( r esp. F V i ) is the maximal interme diate ﬁeld of F ∞ /F ﬁxe d by U i ( r esp. V i ) . If ( ξ i ) i is c ontaine d in Ψ S , the p - adic zeta function ξ for F ∞ /F exists as an element of K 1 (Λ( G ) S ) and satisﬁes the main c onje c tur e. Note that the p -adic zeta function (pseudomeasure) ξ i for F V i /F U i has b een constructed b y u sing the theory of Pierre R. Delig ne and K en neth A. Rib et ([De-Ri]), and the Iwasa wa main conjecture for F V i /F U i has b een pro ve n by And rew Wiles ([Wiles]). The cond ition for ( ξ i ) i to b e conta ined in Ψ S is essen tially giv en b y the congruences among ξ i ’s, so w e m ay reduce th e non-comm utativ e Iw asa wa main conjecture to the congruences among the ab elian p -adic zeta pseu- domeasures via the theta map. In order to study the congruences whic h ab elian p -adic zeta pseudomea- sures sh ou ld satisfy , w e u se the theory of Hilb ert mo dular forms of Del igne- Rib et ([De-Ri ]), and derive the congruences of p -adic zet a pseud omeasures (that is, the congruences of constan t terms of certain Λ-adic Hilbert mo dular 4 T A KASHI HARA forms) fr om those of the co eﬃcient s of non-constan t terms of Λ-adic Hilb er t mo dular forms (See § 7 ). Kato and Rit ter-W eiss ﬁrst used th is tec hniqu e in [Kato1] and [R-W3], and obtained many kinds of congruences among abelian p -adic zet a pseu d omeasures. Actually it is diﬃcult to prov e all the desired congruences b y usin g only this tec hnique, therefore we u se the existence o f th e p -adic z eta function for a certain quotien t group G of G , pro ven b y Kato in [Kat o1]. W e pro v e our main theorem (Th eorem 0 .1 ) b y usin g an inductive tec hniqu e. 0.2. Ov erview . T h e deta iled con ten t of this pap er is as follo ws. In § 1, w e review basic r esults of (c lassical) algebraic K -theory . In particu- lar, we summarize prop erties of in tegral logarithmic homomorphism s, whic h w ere ﬁrs t introd uced b y Rob ert O liv er and Laurence R. T ayl or to study the structure of the K 1 -group of a group ring R [ G ] where G is a ﬁn ite group and R is th e in teger rin g of a ﬁ nite extension of Q p . The in tegral logarithmic homomorphisms are maps from m ultiplicativ e K 1 -groups to certain additiv e groups, whic h are muc h ea sier to treat (Prop osition-Deﬁnition 1.29). In § 2, we review the theory of Coates et al. ([CFKSV]), esp ecially h o w to form u late the main conjecture. W e state our main theorem precisely in § 3, a nd introd u ce Burns’ tec hn iqu e under more general situations than Theorem 0.1. W e construct the theta map for our case from § 4 to § 6. In § 4, w e construct the additiv e v ersion of th e theta map (Prop osition-Deﬁnition 4.3) by using linear represen tation theory of ﬁnite grou p s. In § 5, w e tran s late the results on the additiv e theta map p ro ve n in § 4 into the (multiplicat ive) t heta map, using the inte gral logarithmic homomorph isms. In § 6, we construct the lo calized v ersion of the theta map θ S . F or this purp ose, w e tak e the p -adic completion \ Λ(Γ) ( p ) of Λ(Γ) ( p ) , and apply the argumen ts of § 4 and § 5 to \ Λ(Γ) ( p ) [ G f ]. The condition for ab elian p -adic zeta pseud omeasures to b e cont ained in Ψ S is essen tially giv en as the congruences among them. Hence in § 7, w e study c ongruen ces whic h ab elian p -adic zeta pseudomeasur es satisfy (Pr op o- sition 7.2). W e us e the theory of D eligne and Ribet o n Hilb ert mo du lar forms ([De-Ri]). Since the congruences obtained in § 7 are n ot su ﬃ cien t to conclude that ab elian p -adic zeta pseud omeasures are con tained in Ψ S , we in tro d u ce Kato’s p -adic zeta fu nction for a certain sub p -adic Lie extension F N /F of F ∞ /F (Theorem 8. 1 ), and p ro v e Theorem 0.1 b y an inductiv e tec hnique. Our main theorem giv es a new example whic h is not dedu ced fr om pre- vious results ([Kato1], [R-W2], [R-W3], and [R-W4]). Contents 0. In tro d u ction 1 1. Preliminaries on algebraic K -theory 6 2. Basic results of non-commuta tive I w asa wa t heory 18 3. The main theorem and Burns’ tec hn ique 23 4. The additiv e theta map 28 5. T ranslation in to the multiplicativ e th eta map 39 IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 5 6. Lo calized theta map 56 7. Congruences among ab elian p -adic zeta pseud omeasur es 60 8. Pro of of t he main theorem 72 References 76 0.3. Notation. In this paper, p alwa ys d en otes a p ositiv e prime n u mb er. W e denote by N the set of n atural num b er s (the set of strictly p ositiv e in tegers), denote by Z (resp. Z p ) t he ring of in tegers (resp. p -adic inte gers), and also denote b y Q (resp. Q p ) the rational n u m b er ﬁeld (resp. t he p -adic n umb er ﬁ eld). F or a n arbitrary grou p G , we d enote by Conj( G ) the set of all conjugacy classes of G . F or ev ery pro-ﬁnite group P , we alw a ys denote by Λ( P ) = Z p [[ P ]] its Iw asa wa a lgebra (that is, its completed group ring o ver Z p ). W e d en ote by Γ a comm utativ e p -adic Lie group whic h is isomorphic to Z p . Thr ou gh ou t this pap er, we ﬁx a top ologica l generator t of Γ. In other w ords, w e ﬁx an isomorphism Λ(Γ) ≃ − → Z p [[ T ]] t 7→ 1 + T where Z p [[ T ]] is the formal p ow er series ring o v er Z p . W e alwa ys assu me that ev ery asso ciativ e ring has 1. W e also us e the follo w ing notation: M n ( R ) = { the ring of n × n -mat rices with en tries in R } , GL n ( R ) = { M ∈ M n ( R ) | M is an in v ertible matrix } , where R is an associativ e ring. I f R is a comm utativ e d omain, w e denote by F rac( R ) its fractional ﬁeld. W e alwa ys regard K 0 -groups as additiv e groups, and K 1 -groups as m ul- tiplicativ e groups. Finally , w e ﬁx an algebraic closure Q of Q and ﬁx embb edings Q ֒ → C , Q ֒ → Q p where C d en otes the complex n umber ﬁeld and Q p the algebraic closur e of Q p . 0.4. Ac kno wledgment. The author w ould lik e to thank ev ery one with whom he h as b een concerned. He w ould lik e to express his sin cere grati- tude to Professor Shuji Saito, Professor Kazuy a Kato and Pr ofessor T ake sh i Tsuji among them; Professor Sh uj i Saito h as in vited the author t o the mys- terious and in teresting w orld of num b er theory thr ough his lectures and seminars; Professor Kazuy a K ato has shown the author th e m ystic asp ect of non-comm utativ e Iw asa wa theory and motiv ated the author to study it through his in tensive lect ur es at the Unive rsity of T oky o in 200 6; and Pro- fessor T ak eshi Tsu ji, whom the author is most grateful to, has giv en the author a lot of useful advice through his seminars and read the m an uscript v ery c arefully . Esp ecially , the main idea of the inductive tec hnique (used in 6 T A KASHI HARA § 8) is d u e to him. This pap er co uld nev er ha ve existe d without their direct and indirect co op eration. Mahesh Kakde has recent ly generalize d the Kato’s metho d u s ed in [Kato1] and pro ve n the main conjecture for other cases in the diﬀeren t w a y f r om this pap er (see [Kakde]) . The author would like to express his sincere grat itude to Ma hesh Kakde for sending th e latest v ers ion of his pap er. 1. Preliminaries on a lgebraic K -theor y In this section, we sum marize basic results of algebraic K -theo ry w hic h w e need to form ulate th e n on-comm utativ e Iw asaw a main conjecture and to construct the theta map. 1.1. Deﬁnitions and ﬁrst prop e rt ies. First, we review the deﬁnition of K -g roup s and their prop erties. F or more details, s ee [Ba ss1]. Deﬁnition 1.1 (Grothendiec k groups, K 0 -groups) . L et C b e a category with a pro d uct ⊥ (rec all that a category C w ith a prod uct ⊥ is a category equipp ed with a fu n ctor ⊥ : C × C → C ). Then w e deﬁne the Gr othendie ck gr oup of C K 0 ( C ) as a n ab elian group equipp ed w ith a map [ · ] : Ob C − → K 0 ( C ) satisfying the fol lo wing universal p rop erties. ( K 0 -1) F or ev ery X , Y ∈ O b C satisfying X ∼ = Y , [ X ] = [ Y ]. ( K 0 -2) F or ev ery X , Y ∈ O b C , [ X ⊥ Y ] = [ X ] + [ Y ]. Namely , if a map f : O b C − → A ( A : an ab elian group) satisﬁes the prop erties ( K 0 -1) and ( K 0 -2), th er e exists a u nique group homomorp hism ψ : K 0 ( C ) → A whic h mak es th e f ollo w in g diagram comm u te: Ob C [ · ] / / f $ $ J J J J J J J J J J K 0 ( C ) ∃ ! ψ   A F or ev ery associativ e ring R , w e deﬁne K 0 ( R ) as the Grothendiec k g roup of t he category of ﬁnitely generated pro jectiv e left R -mo du les. Deﬁnition 1.2 (Whitehead groups, K 1 -groups) . Let C b e a catego ry with a pro duct ⊥ . Let Aut( C ) b e the category of automorphism s of ob jects of C . Na mely , an ob ject of Aut( C ) is a pair ( X , σ ) w here X ∈ Ob C and σ : X → X is an automorphism of X . A m orphism f : ( X, σ ) − → ( Y , τ ) is a morphism f : X → Y in C whic h satisﬁes f ◦ σ = τ ◦ f . Then w e d eﬁne the Whitehe ad gr oup of C K 1 ( C ) as an ab elian grou p equipp ed with a map [ · ] : Ob Aut( C ) − → K 1 ( C ) satisfying the fol lo wing universal p rop erties. ( K 1 -1) F or ev ery ( X , σ ) , ( Y , τ ) ∈ Ob Aut( C ) satisfying ( X, σ ) ∼ = ( Y , τ ), [( X, σ )] = [( Y , τ )]. ( K 1 -2) F or ev ery ( X , σ ) , ( Y , τ ) ∈ Ob Aut( C ), [( X , σ ) ⊥ ( Y , τ )] = [( X, σ )] · [( Y , τ )]. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 7 ( K 1 -3) F or ev ery ( X, σ ) , ( X , σ ′ ) ∈ Ob Au t( C ), [( X , σ ◦ σ ′ )] = [( X , σ )] · [( X, σ ′ )]. F or ev ery associativ e ring R , we deﬁne K 1 ( R ) as th e Whitehead g roup of the c ategory of ﬁnitely generat ed pro j ectiv e left R -mo d ules. Remark 1.3. When R is the group ring O K [ G ] of a ﬁn ite group G o v er the in teger r in g of a n umb er ﬁeld K , t he terminology “Whitehead group ” is usually used fo r the g roup Wh( O K [ G ]) = K 1 ( O K [ G ]) / ( µ ( O K ) × G ab ) , where µ ( O K ) is the m ultiplicativ e group consisting of all ro ots of 1 cont ained in O K . W e ha ve another inte rp retation of the Wh itehead group of a n a sso ciativ e ring R (Whitehead’s constru ction): let E n ( R ) b e a su bgroup of GL n ( R ) generated b y all “elemen tary m atrices,” that is, E n ( R ) = h I n + r E ij | 1 ≤ i 6 = j ≤ n, r ∈ R i where E ij =        j 0 0 · · · 0 i 0 . . . 1 . . . . . . . . . . . . 0 · · · · · · 0        . Here w e denote the unit matrix of GL n ( R ) b y I n . Note that E n ( R ) is normal in GL n ( R ). Let GL( R ) = lim − → n GL n ( R ) and E( R ) = lim − → n E n ( R ) , then w e hav e K 1 ( R ) = GL( R ) / E( R ) . F or the equiv alence of Deﬁnition 1.2 and Whitehead constru ction, see [Bass1], Chapter I X. Deﬁnition 1.4 (Relativ e Whitehead group s) . Let R b e an associativ e ring and a ⊆ R an arb itrary (t wo- sided) id eal. Set GL n ( R, a ) = Ker(GL n ( R ) − → GL n ( R/ a )) , E n ( R, a ) = Th e m inimal normal su bgroup of GL n ( R, a ) con taining { I n + r E ij | 0 ≤ i 6 = j ≤ n, r ∈ a } , and GL( R, a ) = lim − → n GL n ( R, a ) , E( R, a ) = lim − → n E n ( R, a ) . Then w e deﬁne K 1 ( R, a ) = GL( R, a ) / E( R, a ) . 8 T A KASHI HARA Prop osition 1.5 (Whitehead’s lemma) . L et R and a b e as ab ove. Then E( R ) = [GL( R ) , GL( R )] , E( R, a ) = [E( R ) , E( R, a )] = [GL( R ) , GL( R, a )] . Pr o of. See [Milnor], Lemma 3.1, Lemma 4.3.  The follo win g tw o pr op ositions are w ell kno wn and we often use them later. Prop osition 1.6. L et R b e an asso ciative ring and a b e a two-side d ide al c ontaine d in its Jac obson r adic al. Then the c anonic al homomorphism K 1 ( R ) − → K 1 ( R/ a ) induc e d by R → R/ a i s surje ctive. Pr o of. See [Bass1 ], Chapter IX, Prop osition (1.3) .  Prop osition 1.7. L et R b e a semi-lo c al asso ciative ring ( r e c al l that R is semi-lo c al if R/J is semi-simple wher e J is the Jac obson r adic al of R ) . Then the fol lowing pr op erties hold. (1) The gr oup homo morphism R × − → K 1 ( R ); u 7→ [( R, − · u )] is surje ctive, wher e − · u i s the automorp hism of R deﬁne d by multipli- c ation by u fr om the right ( her e we r e gar d R as a left R -mo dule ) . If R is semi-lo c al and also c ommutative, the homo morphism ab ove is an isomorphism. (2) ( stability pr op erty ) F or e ach d ≥ 2 , we have the c anonic al isomorp hism K 1 ( R ) ∼ = GL d ( R ) / E d ( R ) . Pr o of. F or (1), see [Bass1] Chapter IX, Prop osition (1.4). When R is com- m utativ e, the determinan t map giv es the in v erse map of R × → K 1 ( R ). F or (2), see [Bass1] Chapter V, Th eorem (9.1).  No w let us stud y the pro j ect limit of K 1 -groups for semi-lo cal rings. Prop osition 1.8. L et R b e a semi-lo c al ring and { R ( n ) } n ∈ N a pr oje ctive system of semi-lo c al rings such that R ( n ) → R ( m ) is surje ctive f or e ach n > m ≥ 1 and lim ← − n R ( n ) ∼ = R . Then the homomorphism K 1 ( R ) → lim ← − n K 1 ( R ( n ) ) induc e d by the c anonic al homo morphisms K 1 ( R ) → K 1 ( R ( n ) ) is an isomo r- phism. Pr o of. By the stabilit y prop ert y (Prop osition 1.7 (2)), w e ha v e K 1 ( R ) ∼ = GL d ( R ) / E d ( R ) and K 1 ( R ( n ) ) ∼ = GL d ( R ( n ) ) / E d ( R ( n ) ) for ev ery d ≥ 2. Fix d ≥ 2. Consider the exact sequence (1.1) 1 − − − − → E d ( R ( n ) ) − − − − → GL d ( R ( n ) ) − − − − → K 1 ( R ( n ) ) − − − − → 1 IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 9 for ea ch n ≥ 1. Then for eve ry n > m ≥ 1, the natural homomorph ism E d ( R ( n ) ) → E d ( R ( m ) ) induced b y R ( n ) → R ( m ) is cl early sur jectiv e. Therefore { E d ( R ( n ) ) } n ∈ N sat- isﬁes the Mittag-Leﬄer condition, and we hav e the follo wing exact sequence 1 / / lim ← − n E d ( R ( n ) ) / / lim ← − n GL d ( R ( n ) ) / / lim ← − n K 1 ( R ( n ) ) / / 1 b y taking t he pro j ective limit of (1.1). No w consider the follo wing diagram: 1 / / E d ( R )   / / GL d ( R )   / / K 1 ( R )   / / 1 1 / / lim ← − n E d ( R ( n ) ) / / lim ← − n GL d ( R ( n ) ) / / lim ← − n K 1 ( R ( n ) ) / / 1 here the ve rtical homomorphisms are ind uced by the canonical h omomor- phism R → R ( n ) . W e ca n easily chec k that eac h square diag ram comm utes. Then the le ft and middle v er tical h omomorphisms are ca nonically isomor- phic. Hence the righ t v ertical homomorph ism is also an isomorphism by the snak e lemma.  1.2. Norm maps in K -theory. Th e theta map, wh ic h we will construct in the follo wing sections, is essen tially a family of n orm maps in algebraic K -t heory . 1 So let us review the construction and prop erties of norm maps of K -groups. Let R ′ /R b e an extension of a rin g R . Supp ose that R ′ is a ﬁnitely generated pro jective mo dule as a left R -mo dule. Then we m a y r egard R ′ as left R - r igh t R ′ -bimo dule R R ′ R ′ . W e deﬁne Nr R ′ /R :=  R R ′ R ′ ⊗ R ′ −  : K i ( R ′ ) − → K i ( R ) ( i = 0 , 1) . W e often u se norm maps in the follo win g situation: let G b e a group and H b e its subgroup of ﬁnite ind ex. Th en w e h a v e an inclusion of group rings Z p [ H ] → Z p [ G ]. Hence w e obtain a norm map Nr Z p [ G ] / Z p [ H ] : K i ( Z p [ G ]) − → K i ( Z p [ H ]) ( i = 0 , 1) . If G is a pro-ﬁnite group and H is its op en subgroup, w e also obtain Nr Λ( G ) / Λ( H ) b y the same constru ction. No w w e ca lculate Nr Z p [ G ] / Z p [ H ] : K 1 ( Z p [ G ]) − → K 1 ( Z p [ H ]) ∼ = Z p [ H ] × under the sp eciﬁc condition that G is a group and H is a comm u tativ e subgroup of ﬁn ite index. Note that b oth Z p [ G ] a nd Z p [ H ] are local rings. By Prop osition 1.7 (1), w e can iden tify K 1 ( Z p [ H ]) with Z p [ H ] × . T ak e a system of represen tativ es { u 1 , . . . , u r } of the coset d ecomp osition H \ G . Then Z p [ G ] is rega rded as a le ft free Z p [ H ]-modu le w ith basis { u 1 , . . . , u r } . 1 In some b o oks and pap ers, n orm maps are also called “transfer homomorphisms.” 10 T A KASHI HARA Let φ b e an element of K 1 ( Z p [ G ]). By Prop osition 1.7 (1) ag ain, we obtain x ∈ Z p [ G ] × suc h that [ x ] = φ . Let u j x = r X i =1 x ij u i ( x ij ∈ Z p [ H ]) for ea ch j . Then w e can calculat e Nr Z p [ G ] / Z p [ H ] φ as Nr Z p [ G ] / Z p [ H ] φ = d et(( x ij ) 1 ≤ i,j ≤ r ) ∈ Z p [ H ] × . Pr o of. By the calc ulation ab ov e, we ma y iden tify Nr Z p [ G ] / Z p [ H ] φ w ith the im- age of ( x ij ) i,j in K 1 ( Z p [ H ]) = GL( Z p [ H ]) / E( Z p [ H ]). Since Z p [ H ] is comm u- tativ e, the surjection Z p [ H ] × → K 1 ( Z p [ H ]) is an isomorphism b y Prop osi- tion 1.7 (1) . The determinan t map giv es the in verse map of the isomorphism ab o ve , and it maps Nr Z p [ G ] / Z p [ H ] φ = [( x ij ) i,j ] t o det( x ij ) i,j .  When G is a pro-ﬁnite group and H is its commutati ve op en subgroup, w e ma y calculat e Nr Λ( G ) / Λ( H ) explicitly in th e sa me wa y . W e end this sub section with a certain compatibilit y prop erty of norm maps. Prop osition 1.9. L et G b e a gr oup and H b e its sub gr oup of ﬁnite index. L et N b e a sub gr oup of H normal in G and H . Then for i = 0 , 1 , the fol lowing diagr am c ommutes : K i ( Z p [ G ]) Nr Z p [ G ] / Z p [ H ] − − − − − − − − → K i ( Z p [ H ]) π G   y   y π H K i ( Z p [ G/ N ]) − − − − − − − − − − − → Nr Z p [ G/ N ] / Z p [ H/N ] K i ( Z p [ H / N ]) wher e π G and π H ar e c anonic al hom omorphisms induc e d by π G : Z p [ G ] − → Z p [ G/ N ] and π H : Z p [ H ] − → Z p [ H / N ] . When G is a pr o-ﬁnite gr oup, H is its op en sub gr oup and N is a close d sub gr oup of H normal in G and H , the same statement holds for Nr Λ( G ) / Λ( H ) and Nr Λ( G/ N ) / Λ( H/ N ) . Pr o of. Let { u 1 , . . . , u r } b e a system of represen tativ es of H \ G . Then it is clear that { u 1 , . . . , u r } is that of ( H / N ) \ ( G/ N ) w here u i = π G ( u i ). Then w e ha v e π H ◦ Nr Z p [ G ] / Z p [ H ] = [ Z p [ H / N ] ⊗ Z p [ H ]  Z p [ H ] Z p [ G ] Z p [ G ]  ⊗ Z p [ G ] − ] = " Z p [ H / N ] ⊗ Z p [ H ] r M i =1 Z p [ H ] u i ! ⊗ Z p [ G ] − # = " r M i =1 Z p [ H / N ] u i ! ⊗ Z p [ G ] − # =  Z p [ H/ N ] Z p [ G/ N ] Z p [ G/ N ]  ⊗ Z p [ G/ N ] ( Z p [ G/ N ]) ⊗ Z p [ G ] −  = Nr Z p [ G/ N ] / Z p [ H/ N ] ◦ π G . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 11 The p ro-ﬁnite v ersion can b e v eriﬁ ed in the same manner.  1.3. The lo calization exact sequenc es. Let R b e an a sso ciativ e ring and S b e a multiplicat ivel y closed subset of R con taining 1. In 1968, Hyman Bass constructed “the K 1 - K 0 lo calizat ion sequence” ([Ba ss1 ]): K 1 ( R ) → K 1 ( S − 1 R ) ∂ − → K 0 ( R, S ) → K 0 ( R ) → K 0 ( S − 1 R ) for ce ntral S (recal l that S is c entr al if S con tained in the c enter of R ). In this su bsection, we d eﬁne Øre lo calizations (certain “go o d” lo caliza- tions of non-co mmutativ e asso ciativ e rings) and introdu ce the K 1 - K 0 lo cal- ization exac t sequence for a non-cen tral Øre set S which is the generalization of B ass’ exact sequence for cent ral S . Deﬁnition 1.10 (Øre sets) . Let R b e an a sso ciativ e ring, and S ⊆ R b e a m ultiplicativ ely closed subs et con taining 1. S is called a left ( r esp. right ) Ør e set if S satisﬁes foll owing tw o conditions. (Øre-1) F or ev ery r ∈ R and s ∈ S , S r ∩ Rs 6 = ∅ ( resp . r S ∩ sR 6 = ∅ ). (Øre-2) If r ∈ R satisﬁes r s = 0 (resp. sr = 0) for a certain elemen t s ∈ S , there exi sts s ′ ∈ S s u c h that s ′ r = 0 (resp. rs ′ = 0). Example 1.11. Let R b e an asso ciativ e r ing and S b e a m ultiplicativ ely closed subset con taining 1. Su pp ose that S is con tained in the cen ter of R . Then S is a left and right Øre set: for an arbitrary r ∈ R and s ∈ S , there exist r ′ ∈ R and s ′ ∈ S su c h that s ′ r = r ′ s and r s ′ = sr ′ since w e ma y c ho ose s ′ = s and r ′ = r . Prop osition 1.12. L et R b e an asso ciative ring and S a left ( r e sp. right ) Ør e set. L et k ∈ N . Then the fol lowing pr op erty holds : ( Ør e- 1 ′ ) F or arbitr ary s 1 , . . . , s k ∈ S , ther e exist r 1 , . . . , r k ∈ R which satisfy r 1 s 1 = · · · = r k s k ∈ S ( r esp. s 1 r 1 = · · · = s k r k ∈ S ) . Pr o of. Directl y b y (Øre-1) for k = 2. T h en w e can sho w (Øre-1 ′ ) for k ≥ 3 b y induction.  Prop osition-Deﬁnition 1.13 (Øre lo calizatio n) . L et R b e an asso ci ative ring and S b e a left ( r esp. right ) Ør e set of R . (1) ( Existenc e of the Ør e lo c alization ) Ther e exi st a ring [ S − 1 ] R ( r esp. R [ S − 1 ]) and a c anonic al ring homo- morphism ι : R − → [ S − 1 ] R ( r esp. R [ S − 1 ]) which satisfy the fol lowing c onditions : (lo c-1) F or ev e ry s ∈ S , ι ( s ) is inve rtible. (lo c-2) Eve ry e lement in [ S − 1 ] R ( r esp. R [ S − 1 ]) c an b e describ e d in the form ι ( s ) − 1 ι ( r ) ( r esp. ι ( r ) ι ( s ) − 1 ) for c ertain r ∈ R and s ∈ S . (lo c-3) F or r ∈ R , ι ( r ) = 0 if and only if ther e exists s ∈ S such that sr = 0 ( r esp. r s = 0) in R . (2) ( The universality pr op erty ) Supp ose that a ring homo morphism f : R − → e R maps al l elements of S to invertible elements of e R . Then ther e exists a uni q ue ring homo - morphism ψ which makes the fol lowing diagr am c ommute : 12 T A KASHI HARA [ S − 1 ] R ( r esp. R [ S − 1 ]) ∃ ! ψ   R ι 6 6 n n n n n n n n n n n n n n n f / / e R W e call [ S − 1 ] R (resp. R [ S − 1 ]) the left ( r esp. right ) Ør e lo c alization of R with r esp e ct to S . F or an arbitrary left (resp. righ t) R -mo dule M , we deﬁne the left ( r esp. right ) Ør e lo c alization of M with r esp e ct to S to b e the mo dule [ S − 1 ] M = [ S − 1 ] R ⊗ R M ( r esp. M [ S − 1 ] = M ⊗ R R [ S − 1 ]) . Sketch of the pr o of. W e only giv e the construction of the left Øre lo calization [ S − 1 ] R . S et [ S − 1 ] R = S × R / ∼ where ∼ is an equiv alence relation deﬁned by ( s, r ) ∼ ( s ′ , r ′ ) if and on ly if there exi st a, b ∈ R wh ich satisfy ar = br ′ and as = bs ′ ∈ S . Then w e ma y deﬁn e the additiv e la w and the m u ltiplicativ e law as follo ws: ( s, r ) + ( s ′ , r ′ ) = ( t, ar + br ′ ) where t = as = bs ′ ∈ S, ( s, r ) · ( s ′ , r ′ ) = ( ts, ar ′ ) where tr = as ′ , t ∈ S . F or the w ell-deﬁnedness of ∼ , +, · , and for the univ ersalit y prop erty , w e use the Ø re co nd itions ( Ør e-1) and (Ø r e-2). F or details, see [Sten], Chapter II.  Corollary 1.14. L et R b e an asso c iative ring. If a multiplic atively close d subset S satisﬁes b oth left and right Ør e c onditions, then the c anonic al iso- morphism [ S − 1 ] R ≃ − → R [ S − 1 ] exists. Pr o of. Directl y fr om the universalit y pr op ert y .  Prop osition 1.15. L et R b e an asso ciative ring and S a left ( r e sp. right ) Ør e set. Then the left ( r esp. right ) Ør e lo c alization [ S − 1 ] R ( r esp. R [ S − 1 ]) i s ﬂat as a right ( r esp. left ) R -mo dule. In other wor ds, the left ( r esp. right ) Ør e lo c alization deﬁnes the e xact func- tor fr om the c ate gory of lef t ( r esp. rig ht ) R -mo dules to that of left [ S − 1 ] R - mo dules ( r esp. right R [ S − 1 ] -mo dules ) . Pr o of. See [Sten], Chapter I I, Prop osition 3 .5.  Deﬁnition 1.16 ( S - torsion mo dules) . Let R b e an asso ciativ e ring and S b e a left Øre set. W e deﬁne M S ( R ) to b e the category of ﬁ nitely g enerated S - torsion left R -mo dules, that is, an ob ject M of M S ( R ) is a ﬁnitely generat ed left R -mo dule satisfying [ S − 1 ] M = 0. W e ma y easily sho w that for an arbitrary ob ject M of M S ( R ), there exists an el ement of S s uc h that sM = 0 if S is cen tral. Bass constru cted the foll owing lo calization exact sequence for ce ntral S . Deﬁnition 1.17. Let R b e an asso ciativ e ring and S its multiplicati vely closed subset. Let H S ( R ) b e the category of ﬁnitely generated S -torsion left R -mo dules with p ro jectiv e resolutions of ﬁ nite length. Then we p ut K 0 ( R, S ) = K 0 ( H S ( R )). IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 13 Remark 1.18. Note that w e m ay i dentify K 0 ( R, S ) with the rela tiv e Gro- thendiec k group asso ciated to the canonical ring homomorphism R − → [ S − 1 ] R . F or the deﬁnition of relativ e Grothendiec k groups, see [Bass1], Chapter IX, § 1. W e can also id en tify this group with the Grothendieck group of the cat- egory of b ounded co mp lexes of ﬁnitely g enerated pr o jectiv e left R -mo du les whose co homologies are of S -torsion. Theorem 1.19 (Bass, th e lo calization exact sequence for cent ral S ) . L et R b e an asso ciative ring with 1 and let S b e its multiplic atively close d sub- set c ontaine d in the c e nter of R . Supp ose that for every element s of S , multiplic ation by s in R induc es an inje ction R s − → R . Then ther e exists an exact se q uenc e of K -gr oups : K 1 ( R ) → K 1 ( S − 1 R ) ∂ − → K 0 ( R, S ) → K 0 ( R ) → K 0 ( S − 1 R ) . Pr o of. See [Bass1 ], Chapter IX, Theorem (6.3 ).  Prop osition 1.20. If R has ﬁnite glob al dimension ( in other wor ds, if every ﬁnitely gener ate d lef t R -mo dule has ﬁnite pr oje ctive dimension ) , we have K i ( R, S ) = K i ( M S ( R )) ( i = 0 , 1) . Pr o of. It is clear since H S ( R ) = M S ( R ) in this case.  A. J. Berric k and M. E. Keating constr u cted th e lo calization sequence for Øre localizations by generalizing th e Bass’ construction of the lo calizat ion sequence for central S : Theorem 1.21 (Berric k-Keating) . L e t R b e an asso ciative ring w ith 1 and S a left Ør e set. Supp ose that no e lements of S ar e zer o divisors in R . L et H S, 1 ( R ) b e the sub c ate gory of M S ( R ) c onsisting of ﬁnitely pr esente d S -torsion left R -mo dules with pr oje ctive dimension 1 . Then ther e exists an exact se q u enc e of K - gr oups : (1.2) K 1 ( R ) → K 1 ([ S − 1 ] R ) ∂ − → K 0 ( H S, 1 ( R )) → K 0 ( R ) → K 0 ([ S − 1 ] R ) . Pr o of. See [Ber-Keat].  W e remark th at Daniel R. Grayson also constructed the exact sequence (1.2) b y using Qu illen’s higher K - theory ([Gra yson ]). In th e follo win g, we u se the mod iﬁed v ers ion of Berrick-Kea ting’s lo cal- ization sequence. Theorem 1.22 (The localization exact sequence for Øre lo calization) . L et R b e an asso ciative ring with 1 and S a left Ør e set. Supp ose that no elements o f S ar e zer o divisors in R . Then ther e exists an exact se quenc e of K -gr oups : K 1 ( R ) → K 1 ([ S − 1 ] R ) ∂ − → K 0 ( R, S ) → K 0 ( R ) → K 0 ([ S − 1 ] R ) . Pr o of. It su ﬃ ces to sh o w that (1.3) K 0 ( H S ( R )) = K 0 ( H S, 1 ( R )) . 14 T A KASHI HARA First, we s ho w that for eve ry M ∈ Ob H S ( R ), M has a ﬁn ite resolution b y ob jects of H S, 1 ( R ). Let { m 1 , . . . , m k } b e e lements of M which generate M , and d the pro jectiv e dimen sion of M . Then w e ha v e the canonical s u rjection π : ⊕ k i =1 Re i → M ; e i 7→ m i (1 ≤ i ≤ k ) , where { e i } 1 ≤ i ≤ k is th e free basis. W e set L = Ker( π ). Note that L is a left free m o dule. Since M is a S -to rsion mo d u le, there exists s i ∈ S w hic h annihilates m i for eac h i . Then w e obta in an elemen t s ∈ S whic h annihilates all m i b y (Øre-1 ′ ). F or this s , w e ha v e π  ⊕ k i =1 Rse i  = 0, therefore ⊕ k i =1 Rse i is a left free R -submo d ule of L . Consider 0 → L/ ⊕ k i =1 Rse i → ⊕ k i =1 Re i / ⊕ k i =1 Rse i → M → 0 , then ⊕ k i =1 Re i / ⊕ k i =1 Rse i is free and of S -torsion (Let x = P k i =1 r i e i b e an arbitrary elemen t of ⊕ k i =1 Re i / ⊕ k i =1 Rse i . F or eac h i there exist s ′ i ∈ S and r ′ i ∈ R suc h that s ′ i r i = r ′ i s . W e obtain r ′′ i ∈ R (1 ≤ i ≤ k ) whic h satisﬁes r ′′ 1 s ′ 1 = · · · = r ′′ k s ′ k ∈ S by (Ør e-1 ′ ). Th en the e lement e s = r ′′ 1 s ′ 1 = · · · = r ′′ k s ′ k annihilates x ), therefore ⊕ k i =1 Re i / ⊕ k i =1 Rse i is an ob jec t of H S, 1 ( R ). Note that the pro j ective dimension of L/ ⊕ k i =1 Rse i is d − 1. Hence w e can construct a resolution of M by ob j ects of H S, 1 ( R ) by ind uction on the pro jectiv e dimension d . Then (1.3) redu ces to Grothendiec k’s resolution theorem (S ee Theorem 1.23). Note th at b oth H S ( R ) and H S, 1 ( R ) are admissible sub catego ries of M ( R ) (that is, H S ( R ) and H S, 1 ( R ) are fu ll additiv e sub catego ries of M S ( R ) whic h hav e at most sets of isomorphism classes of ob jects, and if 0 → M ′ → M → M ′′ → 0 is an a rb itrary exa ct s equence in M ( R ) for wh ic h M an d M ′ are ob jects of H S ( R ) (resp . H S, 1 ( R )), M ′′ is also an ob ject of H S ( R ) (resp . H S, 1 ( R ))). S ee [Ba ss2 ], Corollary (8.5).  Theorem 1.23 (Grothendiec k’s resolution theorem) . L et M b e an ab elian c ate gory and P ⊆ P ′ admissible sub c ate gories of M . Assume an arbitr ary obje ct P ′ of P ′ has a ﬁnite r esolution by obje cts of P . Then the inclusion P ⊆ P ′ induc es an isomorph ism K 0 ( P ) ∼ = K 0 ( P ′ ) . Pr o of. See [Bass2 ], Theorem (7.1).  Remark 1.24. In our case, the canonical Øre set S for Λ( G ) (See § 2) is essen tially con tained in the ce nter Λ(Γ) (See § 6, Lemma 6.1), therefore Bass’ lo calizat ion sequence is suﬃcient f or the pro of of our main theorem. 1.4. Theory of the integral logarithm. Integral logarithmic homomor- phisms w ere u sed by Ro b ert Oliv er and Laurence R. T a ylor to stu dy struc- ture of Whitehead groups of g roup rin gs o f ﬁnite groups ([Oliver], [Oli-T a y ]). W e use these homomorphisms to translate “the add itiv e theta map” into “the (m ultiplicativ e) theta ma p” (See § 5). Ritter and W eiss also used them to form u late their “equiv arian t Iwa saw a theory” ([R-W1]). Let R b e a complete discrete v aluation ring with mixed characte ristics (0 , p ), K its f r actional ﬁeld, π a uniformizer of R and k the resid ue ﬁeld of R . Let A b e an “ R -order,” that is, an R -algebra whic h is free as a left R -mo dule (though Oliv er and T a ylor treate d only Z p -orders in [Oliv er] and [Oli-T a y ], w e n eed int egral logarithms for the \ Λ(Γ) ( p ) -order \ Λ(Γ) ( p ) [ G f ] to construct IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 15 the lo calized ve rsion of the th eta map (see § 6), th er efore we in tro duce here the g eneralized ve rsion of in tegral logarithms for general R -ord ers). Lemma 1.25 ([Oliv er], Lemma 2.7 for Z p -orders) . L et R , K , π , k a nd A b e as ab ove, and let J b e the Jac obson r adic al of A . Then for every two-side d ide al a ⊆ J , 1 + a is a multiplic ative gr oup. Mor e over, for an arbitr ary element x ∈ a , the p ower series (1.4) log(1 + x ) = ∞ X i =1 ( − 1) i − 1 x i i c onver ges p -adic al ly in a Q = Q ⊗ Z a , and satisﬁes (1.5) log ((1 + x )(1 + y )) ≡ log (1 + x ) + log(1 + y ) mod [ A Q , a Q ] for every x, y ∈ a wher e A Q = Q ⊗ Z A . In p articular, log induc es a homo morphism of gr oups log : 1 + a − → a Q / [ A Q , a Q ] . Here for arbitrary t wo -sided ideals a and b of R , [ a , b ] denotes the tw o- sided ideal generated by [ a, b ] = ab − ba where a ∈ a and b ∈ b . Pr o of. Since A / A π is ﬁ nite o ver k = R/Rπ , it is a left Ar tinian k -algebra. Note t hat J / A π is the Jacobson radical of A / A π , and b y c ommutati ve ring theory , the Jacobson radical of an arbitrary (left) Artinian rin g is nilp otent. Hence for an arbitrary x ∈ a ⊆ J , there exists a certain natural num b er m suc h that (1.6) x m ∈ A π . Let e b e the absolute ramiﬁcation index of K . T hen w e hav e x n ∈ A p [ n/em ] for eac h n ≥ em wh ere [ r ] denotes the greatest intege r not greater than r for ev ery r ∈ R . A p [ n/em ] con v erges to zero as n → ∞ , therefore w e obtain lim n →∞ | x n | p = 0 , whic h implies the p o we r series (1 + x ) − 1 = ∞ X i =0 ( − 1) i x i con v erges in A . Since a is closed in p -adic top ology , w e may conclude that (1 + x ) − 1 ∈ 1 + a . It is c lear that 1 + a is closed under multiplicati on (note that a is a t wo-sided idea l), therefore 1 + a is a m ultiplicativ e group. No w w e also ha v e x n /n ∈ A · ( p [ n/em ] /n ) for eac h n ≥ em b y (1.6), therefore w e obtain (1.7) lim n →∞     x n n     p = 0 , whic h implies that the p o we r series (1.4 ) con v erges in A Q . Th en we can sho w that (1 .4 ) co nv erges in a Q b y the same argum ent a s ab o v e. W e ma y also sho w the equation (1.5) b y direct calculation, but this cal- culation is quite complicated b ecause of th e n on-comm utativit y of A . S ee [Oliv er], Chapter 2, Lemma 2.7.  16 T A KASHI HARA Remark 1.26. Note that though Olive r constructed log only for Z p -orders in [Oliv er], Lemma 2.7 of Chapter 2, his pro of do es not use p eculiarities of Z p (esp ecially the ﬁ niteness of the residue ﬁeld of R ). S o w e can apply his pro of to t he case of general R -order s . Prop osition 1.27 ([Oliv er], Theorem 2.8 for Z p -orders) . L e t A and a b e as ab ove. Then the lo garithmic homomorphism of L emma 1.25 induc es the fol lowing homomorphism of ab elian gr oups : log a : K 1 ( A , a ) − → a Q / [ A Q , a Q ] . Sketch of the pr o of. F or every n ≥ 1 and for an ideal M n ( a ) ⊆ M n ( A ), we ha v e a loga rithmic h omomorp hism ( Lemma 1.25) log ( n ) : G L n ( A , a ) = I n + M n ( a ) log − − → M n ( a Q ) / [M n ( A Q ) , M n ( a Q )] trace − − − → a Q / [ A Q , a Q ] where I n is the u n it mat rix of GL n ( A ). F or ev ery X ∈ GL n ( A ) and A ∈ GL n ( A , a ) = I n + M n ( a ), w e ha ve log ( n ) ([ X, A ]) = log ( n ) ( X AX − 1 ) − log ( n ) ( A ) (b y (1.5)) = T r( X log( A ) X − 1 ) − T r(log ( A )) = 0 . Hence log ( n ) induces log ( n ) : GL n ( A ) / [GL n ( A ) , GL n ( A , a )] − → a Q / [ A Q , a Q ] , and b y ta king the pro jectiv e limit, w e obtain log a : K 1 ( A , a ) − → a Q / [ A Q , a Q ] , using Whitehead’s lemma (Prop osition 1 .5). (See [Oliv er], Theorem 2.8.)  Remark 1.28. Let R b e the in teger r ing of a ﬁnite extension of Q p , and tak e a to b e the Jacobson radical J of A = R [ G ] where G is a ﬁnite group. Then we ma y extend the domain of th e log arithmic homomorphism obtained in Prop osition 1.27 to K 1 ( R [ G ]) uniquely . The f ollo win g exact sequence o f K - groups is w ell kno wn: K 2 ( R [ G ] /J ) → K 1 ( R [ G ] , J ) → K 1 ( R [ G ]) → K 1 ( R [ G ] /J ) → 1 (See [Milnor] Lemma 4.1 and Theorem 6.2 ). Note that the homomorphism K 1 ( R [ G ]) → K 1 ( R [ G ] /J ) is surjectiv e by Prop osition 1.6. Since R [ G ] /J is a ﬁ n ite semi-simple algebra with p -p o wer order, K 2 ( R [ G ] /J ) = 1 and p ∤ a = ♯K 1 ( R [ G ] /J ) ([Oliv er], T heorem 1.16). Therefore, we ma y regard K 1 ( R [ G ] , J ) as a subgroup of K 1 ( R [ G ]), and ma y extend the domain of log J to K 1 ( R [ G ]) b y setting log R [ G ] φ = 1 a log J φ a for eve ry φ ∈ K 1 ( R [ G ]) (note that φ a ∈ K 1 ( R [ G ] , J )). T he un iqueness of the e xtension is trivial from this construction. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 17 No w w e deﬁne inte gr al lo garithmic homom orphisms . Let R b e a complete discrete v aluation ring whic h is absolutely unramiﬁed, a nd let G b e a ﬁnite p -group. F or simp licit y , w e assu me that p 6 = 2. W e consider a group r ing R [ G ]. Let J b e t he Jacobson radical of R [ G ]. Let R [Conj( G )] = R [ G ] / [ R [ G ] , R [ G ]] (resp. K [Conj( G )] = K [ G ] / [ K [ G ] , K [ G ]]) b e the free R -mo dule (resp. the K -v ector space) with basis C on j( G ) . In th e follo w ing, w e ﬁ x the F rob enius automorph ism e ϕ : K → K when k = R/pR is not p erfect. Then w e ha v e e ϕ ( r ) ≡ r p mo d pR (1.8) for ev ery r ∈ R . 2 Let ϕ : K [Conj ( G )] − → K [ G ] b e the homomorphism deﬁned by ϕ X g k g [ g ] ! = X g e ϕ ( k g )[ g p ] ( k g ∈ K, [ g ] ∈ Conj( G )) . Prop osition-Deﬁnition 1.29 (The in tegral logarithms) . Se t Γ G,J ( u ) = log J ( u ) − 1 p ϕ (log J ( u )) ∈ K [Conj( G )] , u ∈ K 1 ( R [ G ] , J ) . Then Γ G,J induc es a homomo rphism Γ G,J : K 1 ( R [ G ] , J ) − → R [Conj( G )] , whic h w e call the inte gr al lo garithmic homomorp hism for R [ G ] . When K is a ﬁnite unr amiﬁe d extension of Q p and R is its i nte ger ring, then Γ G ( u ) = log R [ G ] ( u ) − 1 p ϕ  log R [ G ] ( u )  ∈ K [Conj( G )] , u ∈ K 1 ( R [ G ]) also induc es a homomorphism Γ G : K 1 ( R [ G ]) − → R [Conj( G )] . Sketch of the pr o of. T ak e an arbitrary x ∈ J . Since Γ G,J (1 − x ) = −  x + x 2 2 + · · ·  + 1 p ϕ  x + x 2 2 · · ·  ≡ − ∞ X i =1 1 pi  x pi − ϕ ( x i )  mo d R [C on j( G )] , it is suﬃcien t to sho w that pi | ( x pi − ϕ ( x i )) for eve ry i ≥ 1, or p n | ( x p n − ϕ ( x p n − 1 )) for ev ery n ≥ 1 (note that all primes other than p are inv ertible in R ). T o do this, we should c h ec k eac h term of th e expan s ion of x p n and ϕ ( x p n − 1 ) carefully . S ee [Oliver], Theorem 6.2 for details. The second part follo ws immediately by the ﬁ rst p art and th e uniqueness of t he extension of log (Re mark 1.28).  2 If R is the integer rin g of a ﬁnite unramiﬁed ex t ension of Q p , e ϕ is the ordinary F rob eniu s en domorphism (determined uniq uely u p to th e in ertia sub group ) . I f R is the p - adic completion of Λ(Γ) ( p ) (w e use t h is ring in § 6), we ma y choose e ϕ as the endomorphism induced by t 7→ t p . 18 T A KASHI HARA No w assum e th at R is the p -adic in teger rin g Z p and G is a ﬁnite p -group. In th is case we can derive fur ther inf ormation ab out the kernel and image of t he inte gral logarithms. Theorem 1.30. L et G b e a ﬁnite p -gr oup and K 1 ( Z p [ G ]) tors the torsion p art of K 1 ( Z p [ G ]) . Then ther e exists an exact se quenc e 1 → K 1 ( Z p [ G ]) /K 1 ( Z p [ G ]) tors Γ G − − → Z p [Conj( G )] ω G − − → G ab → 1 . Her e ω G is deﬁne d by ω G X g a g [ g ] ! = Y g g a g ( a g ∈ Z p , [ g ] ∈ Conj ( G )) and G ab is the ab elization of G . We denote by g the image of [ g ] in G ab . Pr o of. See [Olive r ], Theorem 6. 6.  F or the torsion part of K 1 ( Z p [ G ]), the fol lo wing prop erties a re known. Deﬁnition 1.31 ( S K 1 ( Z p [ G ])) . Let G b e a ﬁn ite group . Then w e put S K 1 ( Z p [ G ]) = Ker ( K 1 ( Z p [ G ]) − → K 1 ( Q p [ G ])) . Remark 1.32. W e ma y deﬁne the S K 1 -group for an arbitrary asso ciativ e ring by Whitehead construction, but w e omit this sin ce w e only use S K 1 - groups for group rings of Z p [ G ]-t yp e. Prop osition 1.33. L et G b e a ﬁnite gr oup. (1) K 1 ( Z p [ G ]) tors ∼ = µ p − 1 × G ab × S K 1 ( Z p [ G ]) . (2) S K 1 ( Z p [ G ]) is ﬁnite. (3) If G is c ommutative, S K 1 ( Z p [ G ]) = 1 . Pr o of. (1) See [ Oliver], Theorem 7.4. (2) See [ W all ], Theorem 2.5. (3) This follo ws from the d eﬁnition of S K 1 -groups: sin ce Z p [ G ] an d Q p [ G ] are lo cal and commuta tive , w e hav e K 1 ( Z p [ G ]) ∼ = Z p [ G ] × and K 1 ( Q p [ G ]) ∼ = Q p [ G ] × b y Prop osition 1.7 (1), then Z p [ G ] × → Q p [ G ] × is o bviously injecti ve.  2. Basic resul ts of n on-commut a t ive Iw asa w a theor y In t his section, w e review basic results of [CFKSV]. Let G b e a compact p -adic Lie group con taining a normal closed sub- group H whic h satisﬁes G/H ∼ = Z p , and let Λ( G ) b e its Iwa saw a algebra. In the non-commuta tive Iwasa wa theory of [CFKSV], c haracteristic elements (the “arithmetic” p -adic zeta fun ctions) are deﬁn ed as elemen ts of White- head groups of certain Ør e lo calization of Λ( G ) by using the lo calization exact sequence. W e ﬁr st d eﬁne the c anonic al Ør e set S for the group G in tro du ced in [CFKSV] § 2. Then for ev ery elemen t [ C ] of the relativ e K 0 - group K 0 (Λ( G ) , Λ( G ) S ) (in the follo wing, we regard K 0 (Λ( G ) , Λ( G ) S ) as the Grothendiec k group of the category of b ounded complexes of ﬁ nitely gen- erated pro jectiv e left Λ( G )-mod ules wh ose cohomolog ies are of S -torsion), w e deﬁn e a char acteristic element of [ C ] as an element of K 1 (Λ( G ) S ). Next, IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 19 w e c haracterize the p -adic zeta function (the “analytic” elemen t) as an el- emen t of K 1 (Λ( G ) S ) interp olating sp ecial v alues of Artin L -functions, and form ulate the non-comm u tativ e Iw asa wa m ain c onjecture. 2.1. The canonical Øre set a nd c haracteristic elemen ts. Let G b e a compact p -adic Lie group and Λ( G ) = lim ← − U : open normal Z p [ G/U ] b e its Iwa saw a a lgebra. S upp ose that there exists a normal closed su bgroup H of G wh ic h satisﬁes G/H ∼ = Γ, wher e Γ is a comm u tativ e p -adic Lie group isomorphic to Z p (See § 0. 3). In the follo wing, we ﬁx suc h a s u bgroup H . Prop osition-Deﬁnition 2.1 (Th e canonical Øre set) . L e t G and H b e as ab ove. Then S = { f ∈ Λ( G ) | Λ( G ) / Λ( G ) f is ﬁnitely gener ate d as a left Λ( H ) -mo dule } is a left and right Ør e set ( Se e Deﬁnition 1.10 ) . W e ca ll this S the c anonic al Ør e set for the gr oup G . 3 Since S is a left and right Ør e set, the left lo calization and the right lo calizat ion of Λ( G ) with r esp ect to S are canonically isomorp hic to eac h other b y Corollary 1.1 4. Th erefore w e ma y identify these tw o lo calizations. Because of this reason, w e denote b y Λ( G ) S the left (or righ t) lo calization of Λ( G ) with resp ect to S . The canonical Øre sets and th eir p rop erties are discus s ed w ell in [CFKSV ], § 2. Here w e use the follo wing t wo pr op erties: Prop osition 2.2. L et G b e a c omp act p -adic Lie gr oup and S the c anonic al Ør e set for it. (1) The lo c alize d Iwasawa algebr a Λ( G ) S is semi-lo c al. (2) The elements of S ar e non-zer o divisors in Λ( G ) . Pr o of. (1) See [CFKSV], Prop osition 4.2. (2) See [ CFKS V], Theorem 2.4.  Note that we ha ve the canonical surjection Λ( G ) × S → K 1 (Λ( G ) S ) by Prop osition 1.7 (1) and Prop osition 2.2 (1). Also n ote that b y Prop osition 2.2 (2), w e ma y consider the lo calizatio n sequen ce for the Øre lo calizati on Λ( G ) → Λ( G ) S (Theorem 1. 22 ). Remark 2.3. Assume that G has no p -torsion element s. In this case we often use S ∗ = [ n ≥ 0 p n S in place of S , as is remarked in [CFKSV]. W e may iden tify K 0 (Λ( G ) , Λ( G ) S ∗ ) with K 0 ( M S ∗ (Λ( G ))) where M S ∗ (Λ( G )) is the categ ory of ﬁnitely generated left S ∗ -torsion Λ( G )-modu les. 3 Since the deﬁn ition of S dep ends on th e subgroup H , we should call S the c anonic al Ør e set for ( G, H ). But by abuse of notation, w e often call S the canonical Ø re set for G when the su b group H is ﬁxed . 20 T A KASHI HARA But in our ca se (See § 3), G has man y p -torsion elemen ts. Therefore w e ha v e to treat derived categories of co mp lexes of ﬁnitely generated mo dules. No w let us consider the lo calization exact sequence (T h eorem 1.22) for Λ( G ) → Λ( G ) S : (2.1) K 1 (Λ( G )) → K 1 (Λ( G ) S ) ∂ − → K 0 (Λ( G ) , Λ( G ) S ) → K 0 (Λ( G )) → K 0 (Λ( G ) S ) . Prop osition 2.4. The c onne cting homom orphism ∂ in ( 2.1 ) is surje ctiv e. Therefore, for an arbitrary elemen t [ C ] ∈ K 0 (Λ( G ) , Λ( G ) S ), there exists an el ement f ∈ K 1 (Λ( G ) S ) whic h satisﬁes ∂ ( f ) = − [ C ]. Deﬁnition 2.5 (Characteristic elemen t) . Let [ C ] ∈ K 0 (Λ( G ) , Λ( G ) S ). W e call an element f of K 1 (Λ( G ) S ) a char acteristic element of [ C ] if f satisﬁes ∂ ( f ) = − [ C ]. Remark 2 .6. By the lo calization exact sequence, c haracteristic elemen ts of [ C ] a re determined up t o multiplica tion by elemen ts of K 1 (Λ( G )). Pr o of of Pr op osition 2.4. W e ma y pro ve this prop osition by the almost same argumen t as t he pro of o f Prop osition 3.4 i n [CFKSV]. Though w e assume that G h as no p -torsion el ements in t he pro of of Prop o- sition 3.4 in [CFKS V], this assumption is used only to a v oid treating com- plexes directl y (this p oint is also remarked in [CFKSV]). F or eac h complex C = C · , its canonical image in K 0 (Λ( G )) is X i ∈ Z ( − 1) i [ C i ]. By using this fact and translating λ, τ , ε in [CFKSV] appr opriately , w e may apply their proof of Prop osition 3.4 in [CFKSV] to the case w here G has p -torsion elemen ts.  Example 2.7. Let us consider the classical Iw asa wa Z p -extension case: G = Γ , H = { 1 } and Λ(Γ) ∼ = Z p [[ T ]]; t 7→ 1 + T . In t his case, b y p -adic W eierstrass’ pr eparation theorem, we ha ve f ( T ) = u ( T ) p n g ( T ) for an arbitrary non-zero elemen t f ( T ) ∈ Z p [[ T ]], where u ( T ) ∈ Z p [[ T ]] × , n ∈ Z ≥ 0 , and g ∈ Z p [ T ] is a distinguished p olynomial. Hence Λ(Γ) / Λ(Γ) f is ﬁ nitely generated o v er Z p if and only if p ∤ f (or equiv alent ly n = 0). Therefore the c anonical Øre set S is Λ(Γ) \ p Λ(Γ). Since Γ has no p -torsion elemen ts, w e ma y consider th e Ø r e set S ∗ as in Remark 2.3. It is easy to see that in this case S ∗ is the subset of Λ(Γ) consisting of all non-zero elemen ts. Therefore Λ(Γ) S ∗ coincides with th e fractional ﬁeld F rac(Λ(Γ)). F urthermore, since Λ( Γ) is local and comm utativ e, w e ha ve K 1 (Λ(Γ)) = Λ(Γ) × and K 1 (Λ(Γ) S ∗ ) = F rac(Λ(Γ)) × . Hence the lo calization exact sequence is d escrib ed a s follo ws: Λ(Γ) × → F rac(Λ(Γ)) × ∂ − → K 0 ( M tor (Λ(Γ))) → 0 , where M tor (Λ(Γ)) is the category of all ﬁ n itely generated torsion Λ(Γ)- mo dules. Th e connecting homomorp hism ∂ is characte rized by ∂ ( f ) = [Λ(Γ) / Λ(Γ) f ] for f ∈ Λ(Γ) \ { 0 } in this (ab elian) ca se. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 21 Let M b e an arbitrary ﬁnitely generate d torsion Λ(Γ)-mo du le. Th en b y the famous structure theorem of ﬁn itely generated torsion Λ(Γ)-mo dules, there exi st f i ∈ Λ(Γ) \ { 0 } (1 ≤ i ≤ N ) and satisfy M ∼ N M i =1 Λ(Γ) / Λ(Γ) f i (pseudo-isomorphic) where the image of eac h f i in Z p [[ T ]] is a non-inv ertible p olynomial. Sin ce the image of ev ery pseudo-null Λ(Γ)-mo du le in K 0 ( M tor (Λ(Γ))) v anishes (See [Sc h -V en]), w e ha v e [ M ] = N X i =1 [Λ(Γ) / Λ(Γ) f i ] . Hence, b y the explicit description of ∂ , w e ha v e ∂ ( f M ) = [ M ] where f M = Q N i =1 f i ( the char acteristic p olynomial of M ). f M is determined up to multiplica tion b y a n elemen t of Λ(Γ) × . The calculation ab o v e shows that the charact eristic elemen ts in [CFKS V] are generaliz ed n otion of the classical c haracteristic p olynomials. 2.2. The non-comm utative Iwasa w a main conjecture for totally real ﬁelds. Now let us review the form ulation of th e non-comm utativ e Iw asa wa mai n conjecture in th e sense of [CFKSV]. Fix a prime num b er p 6 = 2. Let F b e a to tally real n u mb er ﬁeld and F ∞ /F a Galo is extension of inﬁnite degree satisfying the follo wing conditions: 4 (1) The Gal ois group G = Gal( F ∞ /F ) is a compact p -adic Lie group . (2) Only ﬁnitely many pr imes of F ramify in F ∞ . (3) F ∞ is to tally real and cont ains the cycloto mic Z p -extension F cyc of F . Fix a ﬁn ite set Σ of primes of F con taining all p rimes whic h ramify in F ∞ . Deﬁnition 2.8. Und er the conditions ab o ve, w e deﬁne the complex C by C = C F ∞ /F = R Hom( R Γ ´ et (Sp ec O F ∞ [1 / Σ] , Q p / Z p ) , Q p / Z p ) . Here Γ ´ et is the global section functor for ´ etale top ology . Note that H 0 ( C ) = Z p , H − 1 ( C ) = Gal( M Σ /F ∞ ) where M Σ is the max- imal ab elian p ro- p extension of F ∞ unramiﬁed outside Σ, and H n ( C ) = 0 for n 6 = 0 , − 1. W e denote Ga l( M Σ /F ∞ ) b y X Σ ( F ∞ /F ). No w set H = Ga l( F ∞ /F cyc ) and Γ = Gal( F cyc /F ) ∼ = Z p . Then if [ C ] is an element of K 0 (Λ( G ) , Λ( G ) S ), we can apply the results of § 2.1 to [ C ] an d obtain a char acteristic element for F ∞ /F as a charact eristic e lement of [ C ]. Conjecture 2.9. [ C ] is always an element of K 0 (Λ( G ) , Λ( G ) S ) . In other wor ds, X Σ ( F ∞ /F ) is of S -torsion. 4 Since the Galois extension we wi ll consider has p -torsion elements , w e need to weak en the conditions in [CFKSV]. See [Kato1] § 2 and [F u-Ka] § 4.3. 22 T A KASHI HARA Prop osition 2.10. L et G ′ ⊆ G b e a pr o- p sub gr oup of G and let F ′ b e the maximal interme diate ﬁeld of F ∞ /F ﬁxe d by G ′ , th en the fol lowings ar e e quivalent : (1) X Σ ( F ∞ /F ) is of S -torsion. (2) µ ( F ′ cyc /F ′ ) = 0 wher e µ is the µ -invariant. In p articular, if the fol lowing c ondition ( ∗ ) is satisﬁe d, X Σ ( F ∞ /F ) is of S -torsion : ( ∗ ) Ther e exists a ﬁnite sub extension F ′ of F ∞ such that Gal( F ∞ /F ′ ) is pr o- p and µ ( F ′ cyc /F ′ ) = 0 . Pr o of. This pr op osition is a v ariant of [Ha-Sh], Lemma 3 .4.  F or the µ -in v arian ts of cyclot omic Z p -extensions, Kenkic hi I w asa wa con- jectured: Conjecture 2.11 (Iw asa w a’s µ = 0 conjecture) . F or every nu mb er ﬁeld K , µ ( K cyc /K ) = 0 . Corollary 2.12. Assume that Iwasawa ’s µ = 0 c onje ctur e is true, then X Σ ( F ∞ /F ) is always of S - torsion. Corollary 2.13. L et K/ Q b e a ﬁnite ab elian extension. Then X Σ ( K ∞ /K ) is of S -tor sion. Pr o of. µ ( K cyc /K ) = 0 b y F errero-W ashington’s theorem ([FW]).  In the foll o win g, w e alw a ys assume the condition ( ∗ ) in Prop osition 2 .10. No w we deﬁne the “ p -adic zeta f unction” as an element of K 1 (Λ( G ) S ). Let ρ : G − → GL d ( Q ) → GL d ( Q p ) b e an arbitrary Ar tin represent ation (that is, ρ ( G ) ⊆ GL d ( Q ) is a ﬁ nite subgroup). Th en their exists a ﬁnite extension E of Q p suc h that GL d ( E ) con tains the imag e of ρ , and ρ induces a ring homomorphism ρ : Λ( G ) − → M d ( E ) This a lso induces a homomo rp hism of K -groups ev ρ : K 1 (Λ( G )) − → K 1 (M d ( E )) ≃ − → K 1 ( E ) ∼ = E × where the isomorphism K 1 ( M d ( E )) ≃ − → K 1 ( E ) is giv en by the Morita equiv- alence b et we en M d ( E ) and E . Comp osing this with the natural inclusion E × → Q × p , w e obtain t he map ev ρ : K 1 (Λ( G )) − → Q × p . As is d iscussed in [CFKSV] § 2, this map can b e exte nd ed to ev ρ : K 1 (Λ( G ) S ) − → Q p ∪ {∞} . W e call ev ρ the evaluation map at ρ . W e den ote ev ρ ( f ) b y f ( ρ ) for ev ery f ∈ K 1 (Λ( G ) S ). Let κ : Gal( F ( µ p ∞ ) /F ) − → Z p × IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 23 b e the p -adic cyclotomic c haracter. Then for ev ery p ositiv e integ er r divisible b y p − 1, κ r factors Γ = Gal( F cyc /F ). Therefore w e ma y extend the d omain of κ r to G by G = Gal( F ∞ /F ) → Gal( F cyc /F ) κ r − → Z p × where the ﬁr st map is the ca nonical sur jection. Deﬁnition 2.14 ( p -adic zet a fu n ction) . If ξ F ∞ /F ∈ K 1 (Λ( G ) S ) satisﬁes (2.2) ξ F ∞ /F ( ρ ⊗ κ r ) = L Σ (1 − r ; F ∞ /F , ρ ) for ev ery p ositiv e integ er r divisible b y p − 1 and for an arbitrary Artin represent ation ρ of G , w e call ξ F ∞ /F the p -adic zeta function for F ∞ /F . Here L Σ ( s ; F ∞ /F , ρ ) is the complex Artin L -function of ρ in which the Euler facto rs at Σ a re remo ved. Conjecture 2.15. L et F ∞ /F b e as ab ove. (1) ( The existenc e and the unique ne ss of the p -adic zeta function ) The p -adic zeta function ξ F ∞ /F for F ∞ /F exists uniquely. (2) ( The non-c ommutative Iwasawa main c onje ctur e ) The p -adic zeta function ξ F ∞ /F satisﬁes ∂ ( ξ F ∞ /F ) = − [ C F ∞ /F ] . Remark 2.16 (The ab elian case) . Let G = Gal( F ∞ /F ) b e an ab elian p - adic Lie group. In this case, Coates ob s erv ed that if certain congru en ces among the sp ecial v alues of the partial zeta functions w ere pro ven, we could construct the p -adic L -fu nction for F ∞ /F (S ee [Coates], Hyp otheses ( H n ) and ( C 0 )). These co ngru ences w ere pr ov en b y Deligne and Rib et using the deep result ab out Hi lb ert-Blumen thal mo du lar v arieties ([De- Ri ]). Using the Deligne-Rib et’s congruences, Serr e constructed the elemen t ξ F ∞ /F of F rac(Λ( G )) whic h satisﬁed the follo wing t wo pr op erties (Serre’s p -adic zet a pseu d omeasure [Ser r e2] for F ∞ /F , also s ee § 7). (1) F or an arbitrary elemen t g of G , (1 − g ) ξ F ∞ /F is co nta ined in Λ( G ). (2) ξ F ∞ /F satisﬁes the interp olation prop ert y (2.2). Remark 2.17. The non-commutati ve Iwasa wa main conjecture whic h we in tro du ced h ere was established ﬁrst by Coates, F uk ay a, Kato, Sujatha and V enj ak ob for elliptic curv es without complex multiplic ation (the GL 2 - conjecture) in [CFKS V]. F or more precise statemen ts, s ee [CFKSV]. In [F u-Ka], F uk a y a an d Kato established the main conjecture for gen- eral cases and show ed the compatibilit y of the main conjecture with the equiv ariant T amaga w a n umber conject ur e. 3. The main theorem a nd Burns’ technique 3.1. The main theorem. Fix a p rime num b er p . W e consid er the on e dimensional p -adic Lie group G = G f × Γ where G f =     1 F p F p F p 0 1 F p F p 0 0 1 F p 0 0 0 1     (ﬁnite part of G ) and Γ is a co mmutativ e p -adic L ie group isomorphic to Z p (See § 0.3). 24 T A KASHI HARA In t he follo wing, we ﬁx generato rs of G f and d enote them b y α =     1 1 0 0 0 1 0 0 0 0 1 0 0 0 0 1     , β =     1 0 0 0 0 1 1 0 0 0 1 0 0 0 0 1     , γ =     1 0 0 0 0 1 0 0 0 0 1 1 0 0 0 1     , δ =     1 0 1 0 0 1 0 0 0 0 1 0 0 0 0 1     , ε =     1 0 0 0 0 1 0 1 0 0 1 0 0 0 0 1     , ζ =     1 0 0 1 0 1 0 0 0 0 1 0 0 0 0 1     . Then the cent er of G f is h ζ i and there are four non-trivial fundamenta l relations of G f : [ α, β ] = δ, [ β , γ ] = ε, [ α, ε ] = ζ , [ δ , γ ] = ζ , where [ x, y ] = xy x − 1 y − 1 is the comm utator of x a nd y . Other comm utators among α, β , . . . , ζ are 1. W e alwa ys denote the ind ices of α, β , . . . , ζ by a, b, . . . , f . Note that G satisﬁes the conditions in § 2 for H = G f . W e also assume that p 6 = 2 , 3. Und er this assu mption, the exp onen t of the group G f is p . Let F b e a totally r eal n umb er ﬁeld and F ∞ a Galois extension of F satisfying Gal( F ∞ /F ) ∼ = G and the cond ition ( ∗ ) in Prop osition 2.10, that is, t here exists an in termediate ﬁeld F ∞ /F ′ /F suc h that F ′ /F is ﬁnite and the µ -in v arian t µ (( F ′ ) cyc /F ′ ) equals t o zero. Theorem 3.1. U nder the notation and the assumptions as ab ove, the p -adic zeta function ξ F ∞ /F for F ∞ /F exi sts and the main c onje ctur e ( Conje ctur e 2.15 (2)) is true. Remark 3.2. In this p ap er w e do not discuss the uniqueness of ξ F ∞ /F . Also see Remark 5.15. 3.2. Burns’ tec hnique. I n this su b section, let F ∞ /F b e a general p -adic Lie extension of a tot ally real ﬁeld F satisfying the conditions in § 2.2. Put G = Gal( F ∞ /F ). Let F b e a family of pairs ( U, V ) where U is an op en subgroup of G and V is an op en subgrou p of H such that V is normal in U and U /V is comm utativ e. F or F , we a ssum e t he follo win g h yp othesis: ( ♭ ) F or an arbitrary Artin representat ion ρ of G , ρ is a Z - linear com bination of in duced represent ations In d G U j ( χ j ), as a virtual r epresen tation, w here ( U j , V j ) is an elemen t of F and χ j is a c haracter of U j /V j of ﬁnite ord er. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 25 In the follo wing, we ﬁx a family F s atisfying the hyp othesis ( ♭ ). F or ev ery ( U, V ) ∈ F , w e ha ve a homomorp h ism θ U,V : K 1 (Λ( G )) − → Λ( U /V ) × whic h is t he comp osite of the norm map of K -groups (se e § 1.2) Nr Λ( G ) / Λ( U ) : K 1 (Λ( G )) − → K 1 (Λ( U )) and the canonical homomorphism K 1 (Λ( U )) − → K 1 (Λ( U /V )) = Λ( U /V ) × . Let S b e the canonical Ør e set for G . Then similarly we h a v e a homo- morphism θ S,U,V : K 1 (Λ( G ) S ) − → Λ( U /V ) × S . Here we also denote the canonical Øre set for U /V by the same sy mb ol S b y abuse of n otation. Set θ = ( θ U,V ) ( U,V ) ∈ F : K 1 (Λ( G )) → Y ( U,V ) ∈ F Λ( U /V ) × and θ S = ( θ S,U,V ) ( U,V ) ∈ F : K 1 (Λ( G ) S ) → Y ( U,V ) ∈ F Λ( U /V ) × S . Let Ψ S b e a subgroup of Q ( U,V ) ∈ F Λ( U /V ) × S and let Ψ = Ψ S ∩ Y ( U,V ) ∈ F Λ( U /V ) × . Deﬁnition 3.3 (Th e theta map, [ Kato1] § 2.4) . Let G , F , θ S , θ , Ψ S and Ψ b e as ab o v e. If θ and θ S satisfy ( θ -1) I mage( θ S ) ⊆ Ψ S , ( θ -2) I mage( θ ) = Ψ, w e call t he indu ced surjectiv e homomorphism θ : K 1 (Λ( G )) − → Ψ the theta map for the gr oup G , and c all the induced homomorphism θ S : K 1 (Λ( G ) S ) − → Ψ S the lo c alize d theta map for the gr oup G . F or ( U, V ) ∈ F , let F U (resp. F V ) b e the maximal su bgroup of F ∞ ﬁxed by U (resp. V ). Sin ce Gal( F V /F U ) ∼ = U /V is abelian, the p -adic zeta f unction (pseudomeasure) ξ U,V ∈ F rac(Λ( U /V )) for F V /F U (see Remark 2.16) exists. Theorem 3.4 (Burns, [Kato1] Prop osition 2.5) . L et G b e a c omp act p -adic Lie gr oup. Assume that the theta map θ and its lo c alize d version θ S for G exist, and also assume that ( ξ U,V ) ( U,V ) ∈ F is c ontaine d i n Ψ S . Then the p -adic zeta fu nction ξ F ∞ /F for F ∞ /F exists and satisﬁes the main c onje ctur e. Mor e over, if θ is inje ctive, ξ F ∞ /F is determine d uniquely. 26 T A KASHI HARA Pr o of. Let f ∈ K 1 (Λ( G ) S ) b e an arbitrary c haracteristic elemen t for F ∞ /F . Signiﬁcan tly , f is an elemen t o f K 1 (Λ( G ) S ) whic h satisﬁes ∂ ( f ) = − [ C F ∞ /F ]. Put f U,V = θ S,U,V ( f ) and [ C U,V ] = θ U,V ([ C ] ) for eac h ( U, V ) ∈ F . Set u U,V = ξ U,V f − 1 U,V . Since the Iw asa wa main conjecture is tr u e for ab elian extensions of totall y real ﬁelds ([Wiles ]), w e hav e ∂ ( ξ U,V ) = − [ C U,V ]. Hence w e obtain ∂ ( u U,V ) = 0. By the localizatio n exact sequ ence (Th eorem 1 .22 ), w e ha v e u U,V ∈ Λ( U /V ) × . On the other hand, since ( f U,V ) ( U,V ) ∈ F ∈ Im age( θ S ) ⊆ Ψ S b y the condi- tion ( θ -1) and ( ξ U,V ) ( U,V ) ∈ F ∈ Ψ S b y a ssu mption, ( u U,V ) ( U,V ) ∈ F is an element of Ψ S . Therefore b y the deﬁnition of Ψ, w e ha ve ( u U,V ) ( U,V ) ∈ F ∈ Ψ = Ψ S ∩ Y ( U,V ) ∈ F Λ( U /V ) × . By the condition ( θ -2), there exists an elemen t u of K 1 (Λ( G )) whic h satisﬁes θ U,V ( u ) = u U,V . W e denote the image of u in K 1 (Λ( G ) S ) by the same sym b ol u . Put ξ F ∞ /F = uf . Since u is an elemen t of K 1 (Λ( G )), w e ha v e ∂ ( u ) = 0. Therefore ∂ ( ξ F ∞ /F ) = ∂ ( uf ) = ∂ ( u ) + ∂ ( f ) = ∂ ( f ) = − [ C ] . This implies that ξ F ∞ /F is a lso a c h aracteristic element o f [ C ]. By t he construction of ξ F ∞ /F , it is clear that (3.1) θ S,U,V ( ξ F ∞ /F ) = ξ U,V . Finally , w e p ro v e that ξ F ∞ /F satisﬁes the in terp olation p rop erty (2.2). Since ξ U,V is th e p -adic zeta function for F V /F U , it satisﬁes th e follo wing in terp olating prop ert y: (3.2) ξ U,V ( χ ⊗ κ r ) = L Σ (1 − r ; F V /F U , χ ) for r ∈ N , ( p − 1) | r where χ is an arbitrary c haracter of ﬁnite index of U /V . By the hyp othesis ( ♭ ), an arbitrary Artin representa tion ρ is w r itten in the form ρ = X ( U,V ) ∈ F X i ∈ I U,V a ( i ) U,V Ind G U ( χ ( i ) U,V ) where I U,V is a in dex set, a ( i ) U,V is an in teger (assume all but ﬁnitely man y a ( i ) U,V are zero), and χ ( i ) U,V is a characte r of ﬁnite ind ex of U /V . Then we ha v e IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 27 ξ F ∞ /F ( ρ ⊗ κ r ) = Y ( U,V ) ∈ F Y i ∈ I U,V ξ (Ind G U ( χ ( i ) U,V ) ⊗ κ r ) a ( i ) U,V = Y ( U,V ) ∈ F Y i ∈ I U,V θ S,U,V ( ξ )( χ ( i ) U,V ⊗ κ r ) a ( i ) U,V = Y ( U,V ) ∈ F Y i ∈ I U,V ξ U,V ( χ ( i ) U,V ⊗ κ r ) a ( i ) U,V (b y (3.1)) = Y ( U,V ) ∈ F Y i ∈ I U,V L Σ (1 − r ; F V /F U , χ ( i ) U,V ) a ( i ) U,V (b y (3.2)) = L Σ   1 − r ; F ∞ /F , X ( U,V ) ∈ F X i a ( i ) U,V Ind G U ( χ ( i ) U,V )   = L Σ (1 − r ; F ∞ /F , ρ ) , here the ﬁr s t equalit y follo ws from the d eﬁnition of the ev aluation maps, and the ﬁf th equalit y follo ws from the compatibilit y of Artin L -fun ctions with direct s u m o f representat ions. The s econd equalit y follo ws from the next lemma (Lemma 3.5). Note that if θ is injectiv e, the element u a b o v e is determined uniquely , so is ξ F ∞ /F .  Lemma 3.5. L et ( U, V ) i s an e lement of F . Then for an arbitr ary cha r acter χ of U /V , the fol lowing diagr am c ommutes. K 1 (Λ( G ) S ) ev Ind G U ( χ )   θ S,U,V / / K 1 (Λ( U /V ) S ) = Λ( U /V ) × S ev χ u u k k k k k k k k k k k k k k k Q p ∪ {∞} Pr o of. W e will sh o w the in tegral v ersion of this lemma. In other w ords, we will pro v e the c ommutat ivit y of the f ollo win g diagram: K 1 (Λ( G )) ev Ind G U ( χ )   θ U,V / / K 1 (Λ( U /V )) = Λ( U /V ) × ev χ u u k k k k k k k k k k k k k k k k k k Q × p Let W b e the represen tation space of Ind G U ( χ ) o ve r Q and let W ′ b e that of χ . Sup p ose that End Q ( W ) (resp. E nd Q ( W ′ )) acts on W (resp. W ′ ) from the righ t. Then b y the deﬁnition of ev aluation map, w e obtain ev Ind G U ( χ ) = [ W ⊗ Λ( G ) − ] . On the other hand, w e ha ve W ∼ = W ′ ⊗ Λ( U /V ) Λ( G ) 28 T A KASHI HARA b y the deﬁnition of induced represen tations, therefore w e ha ve ev Ind G U ( χ ) =  W ′ ⊗ Λ( U /V )  Λ( U /V ) Λ( G ) Λ( G )  ⊗ Λ( G ) −  , and the right h and side is nothing but the deﬁn ition of ev χ ◦ θ U,V . It is not diﬃcult to g eneralize this result to the localized v ersion.  Kazuy a Kato has constru cted the theta m ap s for p -adic Lie groups of Heisen b erg type ([Kato1], see also § 8.1) and for certain op en subgroups of Z p × ⋉ Z p ([Kato2]). Kakde also constr u cted the th eta map for the p -adic Lie group H ⋊ Γ of “sp ecial type” wh ere H is a compact pro- p ab elian p -adic Lie group. He used the metho d of Kato in [Kato1] (See [Kakde]). In our case G = G f × Γ, it is diﬃcult to sho w that ( ξ U,V ) ( U,V ) ∈ F ∈ Ψ S (See § 7 and § 8), so w e should mo dify th e pro of of this pr op osition to show our main theorem (Theorem 3.1). But this tec hn iqu e giv es u s the ideas how to r ed uce the diﬃculties come from n on-comm utativit y to t he co nd itions of comm utativ e cases. 4. The additive thet a ma p In the follo wing thr ee sections, we construct the theta map (and its lo- calized v ersion) for the group G = G f × Γ. First, we construct a family F = { ( U i , V i ) } whic h satisﬁes the h yp othesis ( ♭ ) of § 3.2. Then we deﬁn e a Z p -mo dule homomorphism θ + : Z p [[Conj( G )]] − → Y i Z p [[ U i /V i ]] and c haracterize its image Ω. W e sho w th at θ + induces an isomorphism from Z p [[Conj( G )]] to Ω, which w e cal l the add itive theta map for the g r oup G . 4.1. Construction of t he family F . First, w e deﬁne su b groups H and N of G f as fo llo ws: H =     1 F p F p 0 0 1 F p 0 0 0 1 0 0 0 0 1     , N =     1 0 0 F p 0 1 0 F p 0 0 1 F p 0 0 0 1     . Note that N is a b elian and normal in G . Ther e a re canonical isomo rp hisms H ≃ − →   1 F p F p 0 1 F p 0 0 1   ;     1 a d 0 0 1 b 0 0 0 1 0 0 0 0 1     7→   1 a d 0 1 b 0 0 1   , N ≃ − →   F p F p F p   ;     1 0 0 f 0 1 0 e 0 0 1 c 0 0 0 1     7→   f e c   . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 29 By the isomorphisms ab o ve , w e identify H with   1 F p F p 0 1 F p 0 0 1   and N with   F p F p F p   . Then G f is represen ted as a semi-direct pro du ct G f ∼ = H ⋉ N =   1 F p F p 0 1 F p 0 0 1   ⋉   F p F p F p   , where H acts on N f rom the left as ordinary prod u ct of matrices. F rom represen tation th eory of s emi-dir ect pro ducts of ﬁnite group s (See [Serre1], Ch apitre 8.2), all ir r educible representat ions of G f are obtained from represen tations of H and those o f N . Let us review this construction. First, let X ( N ) be the c haracter group of the ab elian group N : X ( N ) =  χ ij k     0 ≤ i, j, k ≤ p − 1  where χ ij k     f e c     = exp  2 π √ − 1 p ( f i + ej + ck )  ( c, e, f ∈ F p ). The group H acts on X ( N ) from the righ t by ( χ ∗ h )( n ) = χ ( hn ) f or al l n ∈ N where h ∈ H and χ ∈ X ( N ). S p eciﬁcally , χ ij k ∗   1 a d 0 1 b 0 0 1   = χ i,ai + j,di + bj + k for ev ery   1 a d 0 1 b 0 0 1   ∈ H where χ ij k ∈ X ( N ). Then w e ha v e χ 00 k (0 ≤ k ≤ p − 1) , χ 0 j 0 (1 ≤ j ≤ p − 1) , χ i 00 (1 ≤ i ≤ p − 1) as a system of represen tativ es for the orbital decomp osition X ( N ) /H . Next, let H ij k b e the isotropic subgroup of H at eac h χ ij k . Then for eac h represent ativ e ab ov e, we h a v e H 00 k = H 0 = H (0 ≤ k ≤ p − 1) , H 0 j 0 = H 1 =   1 F p F p 0 1 0 0 0 1   (1 ≤ j ≤ p − 1) , H i 00 = H 2 =   1 0 0 0 1 F p 0 0 1   (1 ≤ i ≤ p − 1) . Let G f ℓ = H ℓ ⋉ N for ℓ = 0 , 1 , 2. Then w e ma y extend the domai n of the c haracter χ 00 k (resp. χ 0 j 0 , χ i 00 ) t o G f 0 (resp. G f 1 , G f 2 ) b y sett ing χ 00 k ( h 0 n ) = χ 00 k ( n ) , χ 0 j 0 ( h 1 n ) = χ 0 j 0 ( n ) , χ i 00 ( h 3 n ) = χ i 00 ( n ) 30 T A KASHI HARA where h ℓ ∈ H ℓ ( ℓ = 0 , 1 , 2) and n ∈ N . Note that eac h χ 00 k (resp. χ 0 j 0 , χ i 00 ) is a c haracter of degree 1 of the group G f 0 (resp. G f 1 , G f 2 ). No w let ρ ℓ ( ℓ = 0 , 1 , 2) b e an arbitrary irreducible repr esen tation of H ℓ . Then w e obtain an irr educible representa tion of G f ℓ b y comp osin g ρ ℓ with the canonical pro jection G f ℓ → H ℓ , which w e also denote by ρ ℓ . Consider the tensor pro ducts of repr esen tations χ 00 k ⊗ ρ 0 (resp. χ 0 j 0 ⊗ ρ 1 , χ i 00 ⊗ ρ 2 ), and let θ k ,ρ 0 = Ind G f G f 0 ( χ 00 k ⊗ ρ 0 ) = χ 00 k ⊗ ρ 0 , θ j,ρ 1 = Ind G f G f 1 ( χ 0 j 0 ⊗ ρ 1 ) , θ i,ρ 2 = Ind G f G f 2 ( χ i 00 ⊗ ρ 2 ) . Prop osition 4.1. Supp ose 0 ≤ k ≤ p − 1 , 1 ≤ j ≤ p − 1 and 1 ≤ i ≤ p − 1 . L et ρ ℓ b e an irr e ducible r epr esentation of H ℓ . Then e ach θ k ,ρ 0 , θ j,ρ 1 , θ i,ρ 2 is an irr e ducible r epr esentation of G f . Mor e over, an arbitr ary irr e ducib le r epr esentation of G f is isomorphic to one of the θ k ,ρ 0 , θ j,ρ 1 , θ i,ρ 2 . Pr o of. See [Serr e1], Ch ap itre 8.2 (for irreducibilit y of eac h θ , w e use Mac k ey’s irreducibilit y criterion).  Note t hat H 1 and H 2 are ab elian groups, so if w e set U f 1 = H 1 ⋉ N V ′ 1 = { I 3 } ⋉   F p 0 F p   =     1 F p F p F p 0 1 0 F p 0 0 1 F p 0 0 0 1     , =     1 0 0 F p 0 1 0 0 0 0 1 F p 0 0 0 1     , U f 2 = H 2 ⋉ N V ′ 2 = { I 3 } ⋉   0 F p F p   =     1 0 0 F p 0 1 F p F p 0 0 1 F p 0 0 0 1     , =     1 0 0 0 0 1 0 F p 0 0 1 F p 0 0 0 1     , eac h irred ucible representa tion θ j,ρ 1 (resp. θ i,ρ 2 ) is obtained as the induced represent ation Ind G f U f 1 ( χ ′ 1 ) (resp. In d G f U f 2 ( χ ′ 2 )) where χ ′ 1 (resp. χ ′ 2 ) i s a certain c haracter of U f 1 /V ′ 1 (resp. U f 2 /V ′ 2 ) o f ﬁnite order. On the other hand, w e ma y regard H 0 (= H ) a s a semi-direct prod uct H ∼ =  1 F p 0 1  ⋉  F p F p  ;   1 a d 0 1 b 0 0 1   7→  1 b 0 1  ⋉  a d  IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 31 where  1 b 0 1  ∈  1 F p 0 1  acts on  a d  ∈  F p F p  from the l eft b y  1 b 0 1  ∗  a d  =  a d   1 b 0 1  − 1 (ordinary pro duct o f matrices) . Let U H 0 = H V H 0 = { I 2 } ⋉  0 F p  =   1 F p F p 0 1 F p 0 0 1   , =   1 0 F p 0 1 0 0 0 1   , U H 1 = { I 2 } ⋉  F p F p  V H 1 = { I 2 } ⋉  0 0  =   1 F p F p 0 1 0 0 0 1   , = { I 3 } , then by an argumen t similar to ab ov e, e very irredu cible representat ion of H is obtained as the indu ced represen tation Ind H U H ℓ ( χ ℓ )( ℓ = 0 or 1) where χ ℓ is a certa in c haracter of U H ℓ /V H ℓ . Therefore if w e set U f 0 = U H 0 ⋉ N V f 0 = V H 0 ⋉   F p F p 0   = G, =     1 0 F p F p 0 1 0 F p 0 0 1 0 0 0 0 1     , U f 1 = U H 1 ⋉ N V ′′ 1 = V H 1 ⋉   F p F p 0   =     1 F p F p F p 0 1 0 F p 0 0 1 F p 0 0 0 1     , =     1 0 0 F p 0 1 0 F p 0 0 1 0 0 0 0 1     , eac h irred ucible repr esen tation θ k ,ρ 0 is obtained as the induced representa- tion Ind G f U f 0 ( χ 0 ) (resp. Ind G f U f 1 ( χ ′′ 1 )) where χ 0 (resp. χ ′′ 1 ) is a certain c haracter of U f 0 /V f 0 (resp. U f 1 /V ′′ 1 ) of ﬁn ite ord er. Let V f 1 = V ′ 1 ∩ V ′′ 1 = h ζ i . Then b y the argument ab o v e, ev ery irreducible represent ation of G f is obtained as the induced r epresent ation of a certain c haracter χ i ∈ X ( U f i /V f i ) wh ere X ( U f i /V f i ) is the c haracter group of the ab elian group U f i /V f i . Hence F f = { ( U f 0 , V f 0 ) , ( U f 1 , V f 1 ) , ( U f 2 , V ′ 2 ) } satisﬁes the h yp othesis ( ♭ ) for the group G f . 32 T A KASHI HARA F or certai n tec hnical reasons, w e replace V ′ 2 b y V f 2 = { I 3 } ⋉   0 F p 0   =     1 0 0 0 0 1 0 F p 0 0 1 0 0 0 0 1     and add follo wing sub groups to our family F f f U 2 f =   1 0 F p 0 1 F p 0 0 1   ⋉ N f V 2 f = { I 3 } ⋉   F p F p 0   =     1 0 F p F p 0 1 F p F p 0 0 1 F p 0 0 0 1     , =     1 0 0 F p 0 1 0 F p 0 0 1 0 0 0 0 1     , U f 3 = { I 3 } ⋉ N V f 3 = { I 3 } ⋉ { 0 } =     1 0 0 F p 0 1 0 F p 0 0 1 F p 0 0 0 1     , = { I 4 } . Note that V f 2 (resp. f V 2 f , V f 3 ) is normal in U f 2 (resp. f U 2 f , U f 3 ) and U f 2 /V f 2 (resp. f U 2 f / f V 2 f , U f 3 /V f 3 ) is ab elian. F or eac h i (including f U 2 and f V 2 ), let U i = U f i × Γ , V i = V f i × { 1 } and let F = { ( U i , V i ) } i . It is clear that F satisﬁes the h yp othesis ( ♭ ) for the group G . 4.2. Construction of the isomorphism θ + . In the f ollo w in g, w e denote Y i Z p [[ U i /V i ]] = Z p [[ U 0 /V 0 ]] × Z p [[ U 1 /V 1 ]] × Z p [[ f U 2 / f V 2 ]] × Z p [[ U 2 /V 2 ]] × Z p [[ U 3 /V 3 ]] and we u se the n otatio n U i (resp. V i ) for one of the subgroup s U 0 , U 1 , f U 2 , U 2 and U 3 (resp. V 0 , V 1 , f V 2 , V 2 and V 3 ). F or an arb itrary ﬁn ite group F , w e d en ote by Z p [Conj( F )] the fr ee Z p - mo dule of ﬁn ite rank with free bases Conj ( F ). F or an arbitrary p r o-ﬁnite group P , let Z p [[Conj( P )]] b e the pro jectiv e limit of the free Z p -mo dules Z p [Conj( P λ )] o ver ﬁ nite quotien t groups P λ of P . Let { u 1 , u 2 , . . . , u r i } ⊆ G b e a system of representat ive s of the coset decomp osition G/U i for eac h i . T hen for an arbitrary conjugacy class [ g ] ∈ C onj( G ), set T r i ([ g ]) = r i X j =1 τ j ([ g ]) where τ j ([ g ]) = [ u − 1 j g u j ] if u − 1 j g u j is con tained in U i , and τ j ([ g ]) = 0 oth- erwise. It is easy to see that T r i ([ g ]) is indep endent of the choic e of the IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 33 represent ativ es { u j } r i j =1 , t herefore T r i induces a well -deﬁn ed Z p -mo dule ho- momorphism T r i : Z p [[Conj( G )]] − → Z p [[Conj( U i )]] , whic h we call the tr ac e homomorphism fr om Z p [[Conj( G )]] to Z p [[Conj( U i )]]. W e deﬁ n e the homomorphism θ + i as the comp osition of the trace map T r i and the natural surj ection Z p [[Conj( U i )]] − → Z p [[ U i /V i ]] . No w let us calc ulate θ + i ([ g 0 ]) for [ g 0 ] ∈ Conj( G f ). Here w e regard [ g 0 ] as an el ement ([ g 0 ] , 0) con tained in Conj( G ) = C on j( G f ) × Conj(Γ). Here w e only compute θ + 2 ([ g 0 ]). W e may tak e { α j δ k | 0 ≤ j, k ≤ p − 1 } as a system of representa tive s of G/U 2 . If g 0 =     1 a d f 0 1 b e 0 0 1 c 0 0 0 1     , w e ha ve ( α j δ k ) − 1 g 0 ( α j δ k ) =     1 a d − bj f − ej − ck 0 1 b e 0 0 1 c 0 0 0 1     . This im p lies that ( α j δ k ) − 1 g 0 ( α j δ k ) is con tained in U 2 if and only if a = 0 and d − bj = 0. First, supp ose that a = 0 and b 6 = 0. Then there exists a unique j satisfying d − bj = 0. W e denote this j by d b . Then w e ha ve θ + 2 ([ g 0 ]) = p − 1 X k =0     1 0 0 f − e · d b − ck 0 1 b e 0 0 1 c 0 0 0 1     mo d h ε i = ( β b γ c (1 + ζ + · · · + ζ p − 1 ) if c 6 = 0, pβ b ζ f − e · d b if c = 0. If a = b = 0 and d 6 = 0, ( α j δ k ) − 1 g 0 ( α j δ k ) is not co ntai ned in U 2 for eac h 0 ≤ j, k ≤ p − 1, hence θ + 2 ([ g 0 ]) = 0. If a = b = d = 0, we ha ve θ + 2 ([ g 0 ]) = p − 1 X j,k =0     1 0 0 f − ej − ck 0 1 0 e 0 0 1 c 0 0 0 1     mo d h ε i =      p 2 ζ f if c = e = 0, pγ c (1 + ζ + . . . + ζ p − 1 ) if c 6 = 0, p (1 + ζ + . . . + ζ p − 1 ) if c = 0 , e 6 = 0. 34 T A KASHI HARA In t his w a y , we h a v e θ + 2 ([ g 0 ]) =                    0 if a 6 = 0 or a = b = 0 , d 6 = 0, β b γ c (1 + ζ + . . . + ζ p − 1 ) if a = 0 , b 6 = 0 , c 6 = 0, pβ b ζ f − e · d b if a = c = 0 , b 6 = 0, pγ c (1 + ζ + . . . + ζ p − 1 ) if a = b = d = 0 , c 6 = 0, p (1 + ζ + . . . + ζ p − 1 ) if a = b = c = d = 0 , e 6 = 0, p 2 ζ f if a = b = c = d = e = 0. Similarly , we ma y tak e { β j | 0 ≤ j ≤ p − 1 } for a system of represen tativ es of G/U 1 , { α j | 0 ≤ j ≤ p − 1 } for that of G/ f U 2 , and { α j β k δ ℓ | 0 ≤ j, k , ℓ ≤ p − 1 } for that of G/U 3 . Using these representa tiv es, we m a y calc ulate θ + ([ g 0 ]) as follo ws: θ + : Z p [[Conj( G )]] − → Y i Z p [[ U i /V i ]] c I ( a, b, c ) = { α a β b γ c δ d ε e ζ f | 0 ≤ d, e, f ≤ p − 1 } ( a 6 = 0 , b 6 = 0) 7→ ( α a β b γ c , 0 , 0 , 0 , 0) , c II ( a, c, d, e ) = { α a γ c δ d ε e ζ f | 0 ≤ f ≤ p − 1 } ( a 6 = 0 , c 6 = 0) 7→ ( α a γ c , α a γ c δ d ε e h ε c δ − a , 0 , 0 , 0) , c II I ( a, e ) = { α a ε e δ d ζ f | 0 ≤ d, f ≤ p − 1 } ( a 6 = 0) 7→ ( α a , α a ε e h δ , 0 , 0 , 0) , c IV ( b, c ) = { β b γ c δ d ε e ζ f | 0 ≤ d, e, f ≤ p − 1 } ( b 6 = 0 , c 6 = 0) 7→ ( β b γ c , 0 , β b γ c h δ , β b γ c h ζ , 0) , c V ( b, f ) = { β b δ d ε e ζ f ′ | f ′ − d b e ≡ f mo d p } ( b 6 = 0) 7→ ( β b , 0 , β b h δ , pβ b ζ f , 0) , c VI ( c, d ) = { γ c δ d ε e ζ f | 0 ≤ e, f ≤ p − 1 } ( c 6 = 0 , d 6 = 0) 7→ ( γ c , γ c δ d h ε , pγ c δ d , 0 , 0) , c VII ( c ) = { γ c ε e ζ f | 0 ≤ e, f ≤ p − 1 } ( c 6 = 0) 7→ ( γ c , γ c h ε , pγ c , pγ c h ζ , pγ c h ε h ζ ) , c VII I ( d, e ) = { δ d ε e ζ f | 0 ≤ f ≤ p − 1 } ( d 6 = 0) 7→ (1 , pδ d ε e , pδ d , 0 , 0) c IX ( e ) = { ε e ζ f | 0 ≤ f ≤ p − 1 } ( e 6 = 0) 7→ (1 , pε e , p, ph ζ , p 2 ε e h ζ ) , c X ( f ) = { ζ f } 7→ (1 , p, p, p 2 ζ f , p 3 ζ f ) , where for an element g 0 ∈ G f , h g 0 denotes an elemen t 1 + g 0 + g 2 0 + . . . + g p − 1 0 of Z p [[Conj( G )]]. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 35 No w let u s consider the ﬁnite quotien ts G ( n ) = G f × Γ / Γ p n and U ( n ) i = U f n × Γ / Γ p n . Then w e ha v e Z p [ G ( n ) ] ∼ = Z p [ G f ] ⊗ Z p Z p [Γ / Γ p n ] , Z p [ U ( n ) i /V i ] ∼ = Z p [ U f i /V f i ] ⊗ Z p Z p [Γ / Γ p n ] . θ + i induces a homomo rp hism θ + , ( n ) i : Z p [ G ( n ) ] → Z p [ U ( n ) i /V i ] whic h sends [ g ] = ([ g 0 ] , t z ) ∈ Conj( G ( n ) ) = Conj( G f ) × Γ / Γ p n to ( θ + i ([ g 0 ]) , t z ). F or example, we ha ve θ + , ( n ) (( c I ( a, b, c ) , t z )) = (( α a β b γ c , t z ) , (0 , t z ) , (0 , t z ) , (0 , t z ) , (0 , t z )) , and so on. W e w ould lik e to c h aracterize the image of θ + , ( n ) . Let I f i b e the submo dule of Z p [ U f i /V f i ] deﬁned as follo w s: I f 1 = [ p δ d ε e , α a ε e h δ ( a 6 = 0) , γ c δ d h ε ( c 6 = 0) , α a γ c δ d ε e h ε c δ − a ( a 6 = 0 , c 6 = 0)] Z p , e I 2 f = [ β b γ c h δ ( b 6 = 0) , pγ c δ d ] Z p , I f 2 = [ p 2 ζ f , pγ c h ζ , β b γ c h ζ ( b 6 = 0 , c 6 = 0) , pβ b ζ f ( b 6 = 0)] Z p , I f 3 = [ p 3 ζ f , p 2 ε e h ζ ( e 6 = 0) , pγ c h ε h ζ ( c 6 = 0)] Z p . Here we denote by [ S ] R the R -submo dule of M generated by S o v er R where R is a comm utativ e ring, M is an R -mo dule and S is a ﬁnite subset of M . Eac h I f i is a Z p -submo d ule of Z p [ U f i /V f i ] generated by { θ + i ([ g 0 ]) | g 0 ∈ G f } . Let I ( n ) i = I f i ⊗ Z p Z p [Γ / Γ p n ]  ⊆ Z p [ U ( n ) i /V i ]  and I i = lim ← − n I ( n ) i ( ⊆ Z p [[ U i /V i ]]) . F or eac h i , I i is the image of the homomorph ism θ + i . Note that generators of I ( n ) 1 (resp. e I 2 ( n ) , I ( n ) 3 ) ab ov e are Z p [Γ / Γ p n ]-linearly indep en d en t, but those of I ( n ) 2 are n ot Z p [Γ / Γ p n ]-linearly ind ep endent, which ha v e a relatio n (4.1) p − 1 X f =0 p 2 ζ f = p · ph ζ . Deﬁnition 4.2 . W e d eﬁne Ω to b e the Z p -submo d ule of Y i Z p [[ U i /V i ]] con- sisting of all elemen ts ( y 0 , y 1 , e y 2 , y 2 , y 3 ) satisfying the follo wing t w o condi- tions: (1) (trace relati ons) (rel-1) T r Z p [[ U 0 /V 0 ]] / Z p [[ U 1 /V 0 ]] y 0 ≡ y 1 , (rel-2) T r Z p [[ U 0 /V 0 ]] / Z p [[ f U 2 /V 0 ]] y 0 ≡ e y 2 , (rel-3) T r Z p [[ f U 2 / f V 2 ]] / Z p [[ U 2 / f V 2 ]] e y 2 ≡ y 2 , 36 T A KASHI HARA Λ( U 0 /V 0 ) rel-1 w w o o o o o o o o o o o o rel-2 ' ' O O O O O O O O O O O Λ( U 1 /V 1 ) rel-5   5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 ' ' rel-4 Λ( f U 2 / f V 2 ) rel-3   w w Λ( U 1 ∩ f U 2 / f V 2 ) Λ( U 2 /V 2 ) rel-6 w w n n n n n n n n n n n n Λ( U 3 /V 3 ) Figure 1. T race and n orm relati ons. (rel-4) T r Z p [[ U 1 / f V 2 ]] / Z p [[ U 1 ∩ f U 2 / f V 2 ]] y 1 ≡ T r Z p [[ f U 2 / f V 2 ]] / Z p [[ U 1 ∩ f U 2 / f V 2 ]] e y 2 , (rel-5) T r Z p [[ U 1 /V 1 ]] / Z p [[ U 3 /V 1 ]] y 1 ≡ y 3 , (rel-6) T r Z p [[ U 2 /V 2 ]] / Z p [[ U 3 /V 2 ]] y 2 ≡ y 3 . (These trac e relations are describ ed in Figure 1.) (2) y i ∈ I i for ea ch i . Prop osition-Deﬁnition 4.3. The homomo rphism θ + = ( θ + i ) induc es an isomorph ism θ + : Z p [[Conj( G )]] ≃ − → Ω . W e ca ll this induced iso morp hism θ + the additive theta map for G . Pr o of. It is clear from the calculation ab o ve that Ω con tains th e image of θ + . Hence it is suﬃcient to pro ve its injectivit y and surjectivit y . W e no w show that θ + induces an isomorph ism on the ﬁnite quotien ts θ + , ( n ) : Z p [Conj( G ( n ) )] ≃ − → Ω ( n ) , for ev ery n ≥ 1, where Ω ( n ) is deﬁned to b e t he subgroup o f Y i Z p [ U ( n ) i /V i ] b y the same conditions in Deﬁnition 4.2. Then by taking th e pro j ectiv e limit, w e obtain the desired isomorph ism . T o simplify th e notation, w e d enote the ind uced homomorph ism θ + , ( n ) b y θ + in the f ollo win g. • In jectivit y . Let y ∈ Z p [Conj( G ( n ) )] b e an elemen t satisfying θ + ( y ) = 0. Then from the hyp othesis ( ♭ ) for the f amily { ( U ( n ) i , V i ) } i , 5 an arbitrary 5 Recall th at an arbitrary irreducible rep resentation τ of G ( n ) = G f × Γ / Γ p n is iso- morphic to ρ ⊗ ω n where ρ is an irreducible represe ntation o f G f and ω n is a c haracter of Γ / Γ p n . By the constru ction of U i and V i , an arbitrary i rreducible represen tation ρ of G f is described as P i,j a ( j ) i Ind G f U f i ( χ ( j ) i ) where a ( j ) i is an integer and χ ( j ) i is a character IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 37 irreducible r epresen tation of G ( n ) is isomorp hic to a Z -linear com- bination (as a virtu al r epresen tation) of Ind G ( n ) U ( n ) i ( χ i )’s w here χ i is a c haracter of U ( n ) i /V i . W e den ote b y e χ i the c h aracter of the induced represent ation Ind G ( n ) U ( n ) i ( χ i ). Then w e ha ve e χ i ( y ) = X [ g ] ∈ Conj( G ( n ) ) m [ g ] X j,u − 1 j g u j ∈ U ( n ) i χ i ( u − 1 j g u j ) = χ i ◦ θ + i ( y ) b y the prop erty of induced c haracters, where { u 1 , . . . , u r i } is a system of represen tativ es of G ( n ) /U ( n ) i and y = X [ g ] ∈ Conj( G ( n ) ) m [ g ] [ g ] ( m g ∈ Z p ) . On the other hand, θ + i ( y ) v anishes by assumption, h ence e χ i ( y ) = 0 This implies that χ ( y ) = 0 for an arb itrary irreducible charact er χ of G ( n ) . Since all ir reducible c h aracters of G ( n ) form a d ual basis of the Q p -v ector space Q p ⊗ Z p Z p [Conj( G ( n ) )], we ma y conclude that y = 0. • S urjectivit y . Let ( y 0 , y 1 , e y 2 , y 2 , y 3 ) b e an arbitrary elemen t of Ω ( n ) , th en eac h y i is a Z p [Γ / Γ p n ]-linear co mbination of generators of I ( n ) i : y 0 = X a,b,c κ abc α a β b γ c , y 1 = X d,e λ (1) de δ d ε e + X a 6 =0 ,e λ (2) ae α a ε e h δ + X c 6 =0 ,d λ (3) cd γ c δ d h ε + X a 6 =0 ,c 6 =0 ,d,e λ (4) acde α a γ c δ d ε e h ε c δ − a , e y 2 = X b 6 =0 ,c µ (1) bc β b γ c h δ + X c,d pµ (2) cd γ c δ d , y 2 = X f p 2 ν (1) f ζ f + X c pν (2) c γ c h ζ + X b 6 =0 ,c 6 =0 ν (3) bc β b γ c h ζ + X b 6 =0 ,f pν (4) bf β b ζ f , y 3 = X f p 3 σ (1) f ζ f + X e 6 =0 p 2 σ (2) e ε e h ζ + X c 6 =0 pσ (3) c γ c h ε h ζ . Note th at by the r elation (4.1), ν (1) f (0 ≤ f ≤ p − 1) and ν (2) 0 are not dete rmin ed uniquely . of U f i /V f i of ﬁnite index. Then w e hav e τ = ρ ⊗ ω n = “ P i,j a ( j ) i Ind G f U f i ( χ ( j ) i ) ” ⊗ ω n = P i,j a ( j ) i Ind G ( n ) U ( n ) i ( χ ( j ) i ⊗ ω n ). χ i,j ⊗ ω n is a chara cter of U ( n ) i /V i of ﬁnite index, h ence { U ( n ) i , V i } i also satisﬁes the hypothesis ( ♭ ) for the group G ( n ) . 38 T A KASHI HARA The trace relations in Deﬁnition 4.2 (1) giv e linear relati ons among these c o eﬃcient s. F or example, let us d escrib e th e (rel-6) explicitl y . T ak e { β j | 0 ≤ j ≤ p − 1 } as a system of represent ativ es of U ( n ) 2 /U ( n ) 3 . Then for u 2 ∈ U f 2 /V f 2 , β − j u 2 β j is cont ained in U ( n ) 3 /V 2 if and only if b ( u 2 ) = 0 where u 2 ≡ β b ( u 2 ) γ c ( u 2 ) ζ f ( u 2 ) mo d h ε i . It is obvious that β − j u 2 β j = u 2 when b ( u 2 ) = 0. T herefore, T r Z p [ U ( n ) 2 /V 2 ] / Z p [ U ( n ) 3 /V 2 ] y 2 = X f p 2 ( pν (1) f + ν (2) 0 ) ζ f + X c 6 =0 p 2 ν (2) c γ c h ζ . On the other hand, y 3 ≡ p 2 X e 6 =0 ,f ( pσ (1) f + σ (2) e ) ζ f + X c 6 =0 p 2 σ (3) c γ c h ζ mo d V 2 . By c omparing these coeﬃcients, w e ha ve pν (1) f + ν (2) 0 = p σ (1) f + X e 6 =0 σ (2) e (0 ≤ f ≤ p − 1) , (4.2) ν (2) c = σ (3) c (1 ≤ c ≤ p − 1) . (4.3) Ho w ev er, b ecause of the relation (4.1), we ma y c hange ν (1) f → ν (1) f + z and ν (2) 0 → ν (2) 0 − pz for ev ery z ∈ Z p [Γ / Γ p n ]. Therefore in the e quation (4.2), w e ma y c ho ose ν (1) f and ν (2) 0 as they sat isfy ν (1) f = σ (1) f (0 ≤ f ≤ p − 1) , ν (2) 0 = X e 6 =0 σ (2) e . Similarly , w e can describ e the other trace r elations explicitly as follo w s: (rel-1) κ 000 = X d,e λ (1) de , κ a 00 = X e λ (2) ae ( a 6 = 0), κ 00 c = X d λ (3) cd ( c 6 = 0), κ a 0 c = X d,e λ (4) acde ( a 6 = 0 , c 6 = 0), (rel-2) κ 00 c = X d µ (2) cd , κ 0 bc = µ (1) bc ( b 6 = 0), (rel-3) µ (1) bc = ν (3) bc ( b 6 = 0 , c 6 = 0), µ (1) b 0 = X f ν (4) bf ( b 6 = 0), µ (2) c 0 = ν (2) c ( c 6 = 0), µ (2) 00 = X f ν (1) f + ν (2) 0 , IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 39 (rel-4) X e λ (1) de = µ (2) 0 d , λ (3) cd = µ (2) cd ( c 6 = 0) (rel-5) λ (1) 00 = X f σ (1) f , λ (1) 0 e = σ (2) e ( e 6 = 0), λ (3) c 0 = σ (3) c ( c 6 = 0), (rel-6) ν (1) f = σ (1) f , ν (2) 0 = X e 6 =0 σ (2) e , ν (2) c = σ (3) c ( c 6 = 0). Then w e may sho w by direct calcula tion, using the explicit trace relations abov e, that an e lement o f Z p [Conj( G ( n ) )] y = X a 6 =0 ,b 6 =0 ,c κ abc c I ( a, b, c ) + X a 6 =0 ,c 6 =0 ,d,e λ (4) acde c II ( a, c, d, e ) + X a 6 =0 ,e λ (2) ae c II I ( a, e ) + X b 6 =0 ,c 6 =0 µ (1) bc c IV ( b, c ) + X b 6 =0 ,f ν (4) bf c V ( b, f ) + X c 6 =0 ,d 6 =0 µ (2) cd c VI ( c, d ) + X c 6 =0 ν (2) c c VII ( c ) + X d 6 =0 ,e λ (1) de c VII I ( d, e ) + X e 6 =0 σ (2) e c IX ( e ) + X f ν (1) f c X ( f ) satisﬁes θ + ( y ) = ( y 0 , y 1 , e y 2 , y 2 , y 3 ). Therefore θ + induces an isomorphism b et we en Z p [Conj( G ( n ) )] and Ω ( n ) .  5. Transla tion into the mul t iplica tive thet a map In th e previous section, w e constru ct th e a dd itive theta map θ + . No w we shall translat e it into the multiplic ative theta map θ . The main to ol fo r this translation is t he in tegral logarithmic homomorphism in tro duced in § 1.3. Let θ i : K 1 (Λ( G )) − → Λ( U i /V i ) × b e the comp osition of the norm homo- morphism of K -groups Nr i : K 1 (Λ( G )) − → K 1 (Λ( U i )) and the natural h omomorp hism K 1 (Λ( U i )) − → K 1 (Λ( U i /V i )) = Λ( U i /V i ) × induced b y Λ( U i ) → Λ( U i /V i ). Set θ = ( θ i ) i : K 1 (Λ( G )) − → Y i Λ( U i /V i ) × . Prop osition-Deﬁnition 5.1 (Th e F rob enius homomorph ism) . F or an ar- bitr ary element g ∈ G , set (5.1) ϕ ( g ) = g p . Then ϕ induc es a gr oup homomorph ism ϕ : G → Γ . W e call this homo- morphism the F r ob eniu s hom omorphism. W e denote th e indu ced ring homomorphism Λ( G ) → Λ(Γ) by the same sym b ol ϕ , and also call it the F r ob eniu s homomorph ism. Pr o of. Since the exp onent of G f is exactly equal to p , w e m ay d ecomp ose ϕ as follo ws: 40 T A KASHI HARA G ϕ / / π G   ? ? ? ? ? ? ? Γ Γ e ϕ ? ?        where e ϕ : Γ → Γ is the F rob enius endomorphism of Γ deﬁ ned by t 7→ t p , and π G is the canonical sur jection. Hence ϕ is clearly a group homomorphism.  Remark 5.2. In general, the corresp ondence (5.1) do es not induce a group homomorphism ϕ : G − → G . In the follo win g, w e d enote by the same s ym b ol ϕ the induced ring ho- momorphism Λ( U i /V i ) → Λ(Γ). Deﬁnition 5.3. W e deﬁne Ψ to b e the subgroup of Y i Λ( U i /V i ) × consisting of a ll elemen ts ( η 0 , η 1 , e η 2 , η 2 , η 3 ) satisfying t he follo wing t w o conditions: (1) (norm relat ions) (rel-1) Nr Λ( U 0 /V 0 ) / Λ( U 1 /V 0 ) η 0 ≡ η 1 , (rel-2) Nr Λ( U 0 /V 0 ) / Λ( f U 2 /V 0 ) η 0 ≡ e η 2 , (rel-3) Nr Λ( f U 2 / f V 2 ) / Λ( U 2 / f V 2 ) e η 2 ≡ η 2 , (rel-4) Nr Λ( U 1 / f V 2 ) / Λ( U 1 ∩ f U 2 / f V 2 ) η 1 ≡ Nr Λ( f U 2 / f V 2 ) / Λ( U 1 ∩ f U 2 / f V 2 ) e η 2 , (rel-5) Nr Λ( U 1 /V 1 ) / Λ( U 3 /V 1 ) η 1 ≡ η 3 , (rel-6) Nr Λ( U 2 /V 2 ) / Λ( U 3 /V 2 ) η 2 ≡ η 3 , (See Figure 1). (2) (congruences) η 1 ≡ ϕ ( η 0 ) mod I 1 , e η 2 ≡ ϕ ( η 0 ) mod e I 2 , η 2 ≡ ϕ ( η 0 ) p mo d I 2 , η 3 ≡ ϕ ( η 0 ) p 2 mo d I 3 . No w let u s recall the deﬁnition of norm maps of commutativ e r in gs: let R b e a commutat ive r ing and R ′ a commutati ve R -algebra. Assume th at R ′ is free and ﬁnitely g enerated as a n R -mo d u le. Then w e deﬁne Nr R ′ /R ( y ) to b e the d etermin ant o f the m ultiplication- y homomorphism. Prop osition 5.4. The homomorph ism θ induc es a su rje ction (5.2) θ : K 1 (Λ( G )) → Ψ . In other wo rd s , θ induces the ( multiplic ative ) theta map for G ( in the sense of Deﬁnition 3.3 ) . In the rest of this section, w e prov e Prop osition 5.4 by using th e additive theta map θ + and the int egral logarithmic homomorph ism s. Since the in- tegral logarithmic homomo rp hisms are deﬁned only for group rings of ﬁ nite groups, w e ﬁx n ≥ 1 and construct an isomorph ism f or ﬁnite q u otien ts θ ( n ) : K 1 ( Z p [ G ( n ) ]) /S K 1 ( Z p [ G ( n ) ]) ≃ − → Ψ ( n ) ⊆ Y i Z p [ U ( n ) i /V i ] ! IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 41 where G ( n ) = G f × Γ / Γ p n , U ( n ) i = U f i × Γ / Γ p n , and Ψ ( n ) is the su bgroup of Q i Z p [ U ( n ) i /V i ] d eﬁned by the same conditions of Deﬁnition 5.3. Th en w e obtain the sur jection (5.2) by taking the pro jectiv e limit. T o simplify the nota tion, w e denote the theta map fo r G ( n ) b y θ in § 5 .1, § 5.2, and § 5.3. In § 5.4, where w e tak e the pro jectiv e limit of theta maps, w e denote the theta map for o f G ( n ) b y θ ( n ) again. 5.1. Logarithmic isomorphisms. In the follo wing three sub sections, w e ﬁx n ≥ 1. F or eac h i ≥ 1, we deﬁn e the ideal J ( n ) i of Z p [ U ( n ) i /V i ] a s follo ws: J ( n ) 1 = ( h ε j δ k (0 ≤ j, k ≤ p − 1)) , e J 2 ( n ) = ( p, h δ ) , J ( n ) 2 = J ( n ) 3 = ( p, h ζ ) . Note t hat J ( n ) i con tains I ( n ) i . Since h ε 0 δ 0 = p , J ( n ) 1 also co ntai ns p . Lemma 5.5. F or e ach i ≥ 1 , 1 + J ( n ) i is a multiplic ative gr oup and the p -adic lo garithmic homo morphism induc es an isomorph ism 1 + J ( n ) i ≃ − → J ( n ) i . Pr o of. Since J ( n ) i is an ideal, ( J ( n ) i ) 2 ⊆ J ( n ) i . Therefore 1 + J ( n ) i is closed under m ultiplication. It is easy to calc ulate that h 2 δ = ph δ and h 2 ζ = ph ζ . Then for e J 2 ( n ) , J ( n ) 2 and J ( n ) 3 , w e ha ve ( J ( n ) i ) 2 = pJ ( n ) i . F or J ( n ) 1 , w e ha ve h ε j δ k h ε j ′ δ k ′ = ( h ε,δ if ( j , k ) and ( j ′ , k ′ ) are Z -linearly indep endent , ph ε j δ k otherwise , where h ε,δ = X 0 ≤ ℓ,ℓ ′ ≤ p − 1 ε ℓ δ ℓ ′ . F or eac h 0 ≤ j, k ≤ p − 1, we h a v e h ε,δ h ε j δ k = ph ε,δ and h 2 ε,δ = p 2 h ε,δ (note that the exp onent of U f 1 /V f 1 is p ). Thus w e ha v e ( J ( n ) 1 ) 3 = p ( J ( n ) 1 ) 2 ⊆ pJ ( n ) 1 . Set N = 3 for i = 1 and N = 2 for other i . • Th e exi stence of in v erse elemen ts. By the argument ab o ve , we ha v e y m ∈ p m − N +1 J ( n ) i for an arbi- trary ele ment y ∈ J ( n ) i and ev ery m ≥ N . Th er efore, (5.3) (1 + y ) − 1 = X m ≥ 0 ( − 1) m y m con v erges in J ( n ) i p -adically . • Conv ergence of logarithms. It is obvious that y m /m i nJ ( n ) i for 1 ≤ m < N , and w e ha v e y m m ∈ p m − N +11 m · J ( n ) i ⊆ J ( n ) i 42 T A KASHI HARA for an arbitrary y ∈ J ( n ) i and every m ≥ N (here we use the fact that p 6 = 0). Th us the logarithm log (1 + y ) = P m ≥ 1 ( − 1) m − 1 ( y m /m ) con v erges p -adically in J ( n ) i . Since n ( J ( n ) i ) m o N = p m ( J ( n ) i ) m ( N − 1) ⊆ p m ( J ( n ) i ) m , we may a lso sho w that 1 +  J ( n ) i  m is a sub group o f 1 + J ( n ) i and log  1 +  J ( n ) i  m  ⊆  J ( n ) i  m for ev ery m ≥ 1 , b y the same argument as ab o ve. • Logarithmic isomorphisms. First note that p -adic top ology and J ( n ) i -adic top ology are th e same on J ( n ) i : this is b ecause p m +1 J ( n ) i ⊆ ( J ( n ) i ) m + N − 1 ⊆ p m J ( n ) i for eve ry m ≥ 1. Since J ( n ) i is p -adically complete, it is also J ( n ) i - adically c omplete. Therefore t o sho w that the logarithm induces an isomorphism 1 + J ( n ) i ≃ − → J ( n ) i , it i s suﬃcien t to s h o w t hat it induces an isomo rp hism log : (1 + ( J ( n ) i ) m ) / (1 + ( J ( n ) i ) m +1 ) ≃ − → ( J ( n ) i ) m / ( J ( n ) i ) m +1 ; 1 + y 7→ y for ea ch m ≥ 1, and to pro ve this, it suﬃces to show that y p k /p k ∈  J ( n ) i  m +1 for ev ery 1 + y ∈ 1 +  J ( n ) i  m and k ≥ 1 . (5.4) Ho w ev er, since y N +1 ∈ p  J ( n ) i  m +1 , w e obtain y p ∈ p  J ( n ) i  m +1 (recall p 6 = 2 , 3 by assumption). This implies (5.4).  Lemma 5.6. F or e ach i ≥ 1 , 1 + I ( n ) i is a multiplic ative gr oup and the p -adic lo garithmic homomor phism induc es an isomorphism 1 + I ( n ) i ≃ − → I ( n ) i . Remark 5.7. Since eac h I ( n ) i is not an ideal but on ly a Z p -submo d ule of Z p [ U ( n ) i /V i ], it is not tr ivial ev en that 1 + I ( n ) i is closed u nder m ultiplication. Pr o of. By direct calculation, w e ha v e ( I ( n ) 1 ) 2 = [ pα a γ c δ d ε e h ε c δ − a , pδ d ε e h ε c δ − a , α a γ c h ε,δ (( a, c ) 6 = (0 , 0))] Z p [Γ / Γ p n ] , ( e I 2 ( n ) ) 2 = [ pβ b γ c h δ , p 2 γ c δ d ] Z p [Γ / Γ p n ] , ( I ( n ) 2 ) 2 = [ pβ b γ c h ζ , p 2 β b ζ f ] Z p [Γ / Γ p n ] , ( I ( n ) 3 ) 2 = [ p 6 ζ f , p 5 ε e h ζ , p 4 γ c h ε h ζ ] Z p [Γ / Γ p n ] , These resu lts imply that ( I ( n ) i ) 2 ⊆ I ( n ) i . 6 Therefore, eac h 1 + I ( n ) i is stable under m ultiplication. 6 F or I ( n ) 1 , recall that I ( n ) 1 is also stable under the multiplica tion of δ and ε . Therefore we hav e pδ d ε e h ε c δ − a = p − 1 X ℓ =0 δ d − ℓa ε e + ℓc ! p ∈ I ( n ) 1 and α a γ c h ε,δ = IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 43 No w let I ′ 1 = I ( n ) 1 , e I 2 ′ = [ β b γ c h δ , pγ c δ d ] Z p [Γ / Γ p n ] , I ′ 2 = [ pβ b ζ f , β b γ c h ζ ] Z p [Γ / Γ p n ] , I ′ 3 = [ p 3 ζ f , p 2 ε e h ζ , pγ c h ε h ζ ] Z p [Γ / Γ p n ] . Note that I ′ i ⊇ I ( n ) i , and b y simp le cal culation w e ha ve ( I ( n ) 1 ) 3 = ( I ′ 1 ) 3 ⊆ p ( I ′ 1 ) 2 for i = 1 and ( I ( n ) i ) 2 = ( I ′ i ) 2 ⊆ pI ′ i for ot her i . Then we ma y sho w the existence of in verse elemen ts of 1 + I ′ i and the logarithmic isomorphism log : 1 + I ′ i ≃ − → I ′ i for eac h i by th e same argument as the pro of of Lemma 5.5 b y replacing J i b y I ′ i . Since ( I ′ i ) 2 ⊆ I ( n ) i ⊆ I ′ i and I ( n ) i is a closed subset of I ′ i , the p o we r series (1 + x ) − 1 and the p - adic logarithm log (1 + x ) on 1 + I ( n ) i con v erges in to I ( n ) i . Moreo ver since log : 1 + ( I ′ i ) k → ( I ′ i ) k is a n isomorphism for k = 1 , 2 an d log : 1 + I ( n ) i / 1 + ( I ′ i ) 2 − → I ( n ) i / ( I ′ i ) 2 ; 1 + y i 7→ y i mo d ( I ′ i ) 2 is also an isomorphism, w e ma y conclude that log : 1 + I ( n ) i → I ( n ) i is an isomorphism.  5.2. Ψ ( n ) con tains the image of θ . Lemma 5.8. The fol lowing diagr am c ommutes for e ach i and n ≥ 1 : K 1 ( Z p [ G ( n ) ]) log − − − − → Z p [Conj( G ( n ) )] Nr Z p [ G ( n ) ] / Z p [ U ( n ) i ]   y   y T r Z p [ G ( n ) ] / Z p [ U ( n ) i ] K 1 ( Z p [ U ( n ) i ]) − − − − → log Z p [Conj( U ( n ) i )] Pr o of. By the pro of of [Oli-T a y], Theorem 1.4., the fol lo wing d iagram com- m utes. K 1 ( Z p [ G ( n ) ]) log − − − − → Z p [Conj( G ( n ) )] Nr Z p [ G ( n ) ] / Z p [ U ( n ) i ]   y   y R ′ K 1 ( Z p [ U ( n ) i ]) − − − − → log Z p [Conj( U ( n ) i )] where R ′ : Z p [Conj( G ( n ) )] → Z p [Conj( U ( n ) i )] is deﬁned as follo ws: for an arbitrary x ∈ G ( n ) , let { u ′ 1 , . . . , u ′ s i } b e a set of represen tativ es of the double coset decomp osition h x i\ G ( n ) /U ( n ) i , and l et J = { j | 1 ≤ j ≤ s i , u ′ j − 1 xu ′ j ∈ U ( n ) i } . Then w e deﬁne R ′ ( x ) = X j ∈ J u ′ j − 1 xu ′ j . p − 1 X ℓ =0 ( ε c ′ δ − a ′ ) ℓ ! α a γ c h ε c δ − a ∈ I ( n ) 1 where ( a ′ , c ′ ) ∈ Z 2 is an arbitrary element Z -linearly indep endent of ( a, c ). Therefore we hav e ( I ( n ) 1 ) 2 ⊆ I ( n ) 1 . 44 T A KASHI HARA It suﬃces to sho w that (5.5) R ′ = T r Z p [ G ( n ) ] / Z p [ U ( n ) i ] , but this is not diﬃcult at all. Let { u ′ 1 , . . . , u ′ s i } b e as a b o v e. Then w e can write G ( n ) = G j h x i u ′ j U ( n ) i = p − 1 [ k =0 G j x k u ′ j U ( n ) i , therefore it is clear that { x k u ′ j } 0 ≤ k ≤ p − 1 ,j con tains a set of representat ive s of the righ t coset decomp osition G ( n ) /U ( n ) i . Moreo ver, since eac h u ′ j U ( n ) i is disjoin t, { u ′ j } j is a sub set o f representati ves of G ( n ) /U ( n ) i . (Case-1) G ( n ) = G j u ′ j U ( n ) i . In th is case, the desired equation (5.5) is ob vious b y deﬁnition of R ′ and T r. (Case-2) G ( n ) = p − 1 G k =0 G j x k u ′ j U ( n ) i . Then b y simple calculati on R ′ ( x ) = X j ∈ J u ′ j − 1 xu ′ j = 1 p p − 1 X k =0 X j ∈ J ( x k u ′ j ) − 1 x ( x k u ′ j ) = 1 p T r Z p [ G ( n ) ] / Z p [ U ( n ) i ] x. On the other hand, recall that j ∈ J imp lies that xu ′ j is an ele- men t o f u ′ j U ( n ) i . Ho w ev er, this is impossib le s in ce { x k u ′ j } 0 ≤ k ≤ p − 1 ,j is a set of repr esen tativ es of G ( n ) /U ( n ) i in this case. Therefore J = ∅ a nd R ′ ( x ) = T r Z p [ G ( n ) ] / Z p [ U ( n ) i ] ( x ) = 0.  Prop osition 5.9. F or an ar bitr ary element η ∈ K 1 ( Z p [ G ( n ) ]) , t he fol lowing e quations hold : θ + 1 ◦ Γ G ( n ) ( g ) = log θ 1 ( η ) ϕ ( θ 0 ( η )) , e θ + 2 ◦ Γ G ( n ) ( g ) = log e θ 2 ( η ) ϕ ( θ 0 ( η )) , θ + 2 ◦ Γ G ( n ) ( g ) = log θ 2 ( η ) ϕ ( θ 0 ( η )) p , θ + 3 ◦ Γ G ( n ) ( g ) = log θ 3 ( η ) ϕ ( θ 0 ( η )) p 2 , wher e Γ G ( n ) : K 1 ( Z p [ G ( n ) ]) → Z p [Conj( G ( n ) )] is the inte gr al lo garithm for G ( n ) ( Pr op osition-Deﬁnition 1.29 ) . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 45 Pr o of. This pr op osition follo ws from Lemma 5.8 and by follo wing commu- tativ e d iagrams: Z p [Conj( G ( n ) )] θ + 0 − − − − → Z p [ U ( n ) 0 /V 0 ] Z p [Conj( G ( n ) )] θ + 0 − − − − → Z p [ U ( n ) 0 /V 0 ] 1 p ϕ   y   y ϕ 1 p ϕ   y   y ϕ Z p [Conj( G ( n ) )] − − − − → θ + 1 Z p [ U ( n ) 1 /V 1 ] Z p [Conj( G ( n ) )] − − − − → e θ + 2 Z p [ f U 2 ( n ) / f V 2 ] Z p [Conj( G ( n ) )] θ + 0 − − − − → Z p [ U ( n ) 0 /V 0 ] Z p [Conj( G ( n ) )] θ + 0 − − − − → Z p [ U ( n ) 0 /V 0 ] 1 p ϕ   y   y pϕ 1 p ϕ   y   y p 2 ϕ Z p [Conj( G ( n ) )] − − − − → θ + 2 Z p [ U ( n ) 2 /V 2 ] Z p [Conj( G ( n ) )] − − − − → θ + 3 Z p [ U ( n ) 3 /V 3 ] These diagrams are easily c h ec k ed for an arb itrary elemen t of G ( n ) . Then w e ma y calculat e as θ + 2 ◦ Γ G ( n ) ( η ) = θ + 2  log η − 1 p ϕ (log( η ))  = θ + 2 (log( η )) − pϕ ( θ + 0 (log( η )) = log( θ 2 ( η )) − pϕ (log( θ 0 ( η ))) = log θ 2 ( η ) ϕ ( θ 0 ( η )) p , here w e use t he diagram ab o ve f or the seco nd equalit y , and use Lemm a 5.8 for the th ir d e qualit y . T he other equations ma y b e deriv ed similarly . Note t hat for an a rb itrary y ∈ Z p [ U ( n ) i /V i ], w e h a v e ϕ (log ( y )) = log ϕ ( y ) b y the deﬁn ition of the p -adic logarithms and the fact that the F rob en iu s homomorphism ϕ is a ring h omomorphism on eac h Z p [ U ( n ) i /V i ].  Lemma 5.10. F or an arbitr ary element η ∈ K 1 ( Z p [ G ( n ) ]) , the f ol lowing c ongruenc e s hold : θ 1 ( η ) ≡ ϕ ( θ 0 ( η )) mo d J ( n ) 1 , e θ 2 ( η ) ≡ ϕ ( θ 0 ( η )) mo d e J 2 ( n ) , θ 2 ( η ) ≡ ϕ ( θ 0 ( η )) p mo d J ( n ) 2 , θ 3 ( η ) ≡ ϕ ( θ 0 ( η )) p 2 mo d J ( n ) 3 . Pr o of. W e calculate the image o f the theta map as in § 1.2 . In the follo w ing p ro of, we use the same notation η for a lift of η to Z p [ G ( n ) ] × (Prop osition 1.7). • Th e co ngru en ces f or θ 1 , e θ 2 . W e only sho w the congruence for e θ 2 . W e ma y prov e the congruence for θ 1 in the s ame manner. 46 T A KASHI HARA Since { α i | 0 ≤ i ≤ p − 1 } is a Z p [ f U 2 ( n ) ]-basis of Z p [ G ( n ) ], η is describ ed as a Z p [ f U 2 ( n ) ]-linear com b ination as fol lo ws: η = p − 1 X i =0 η i α i , η i ∈ Z p [ f U 2 ( n ) ] . Then w e ha v e α j η = p − 1 X i =0 ( α j η i α − j ) · α i + j = p − 1 X i =0 ν j ( η i − j ) α i where ν j : Z p [ f U 2 ( n ) ] → Z p [ f U 2 ( n ) ]; x 7→ α j xα − j . Here w e consider the sub-index of η i as an elemen t of Z /p Z . By abu se of notation, we denote the image of η i in Z p [ f U 2 ( n ) / f V 2 ] b y th e sa me symb ol η i . Then w e ma y co mp u te e θ 2 ( η ) as follo ws: e θ 2 ( η ) = d et( ν j ( η i − j )) i,j = X σ ∈ S p sgn( σ ) p − 1 Y j =0 ν j ( η σ ( j ) − j ) = X σ ∈ S p P σ where S p is the p ermutatio n group of { 0 , 1 , . . . , p − 1 } and P σ = sgn( σ ) Q p − 1 j =0 ν j ( η σ ( j ) − j ). No w w e ha v e ν k ( P σ ) = sgn( σ ) Y j ν j + k ( η σ ( j ) − j ) = sgn( σ ) Y j ν j ( η ( σ ( j − k )+ k ) − j ) . First Supp ose that σ do es not satisfy (5.6) σ ( j − k ) + k = σ ( j ) for eac h k ∈ Z /p Z , then τ k ( j ) = σ ( j − k ) + k , k ∈ Z /p Z are d istinct elemen ts of S p and w e h a v e (5.7) ν k ( P σ ) = P τ k . Here w e use the f act s gn( σ ) = sgn( τ k ). T his follo ws fr om the equation τ k = c k ◦ σ ◦ c − 1 k where c k ( i ) = i + k . The equatio n (5.7) implies p − 1 X k =0 P τ k ∈ p − 1 X k =0 ν k  Z p [ f U 2 ( n ) / f V 2 ]  . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 47 Note that eac h ν k is a Z p [ f U ′ 2 ( n ) / f V 2 ]-linear map where f U ′ 2 ( n ) = h γ , δ, ε, ζ i × Γ / Γ p n and Z p [ f U 2 ( n ) / f V 2 ] is generated by { β b | b ∈ Z /p Z } o ver Z p [ f U ′ 2 ( n ) / f V 2 ]. Since p − 1 X k =0 ν k ( β b ) = ( p if b = 0 , β b (1 + δ + · · · δ p − 1 ) otherwise , w e ma y co nclud e that (5.8) p − 1 X k =0 ν k  Z p [ f U 2 ( n ) / f V 2 ]  ⊆ e J 2 ( n ) . This implies p − 1 X k =0 P τ k ∈ e J 2 ( n ) . Next assume σ satisﬁes (5.6 ). I t is easy to see that all p er muta- tions sa tisfying (5.6) are c h ( j ) = j + h (0 ≤ h ≤ p − 1) . F or σ = c h , w e ha ve P c h = sgn( c h ) p − 1 Y i =0 ν j ( η c h ( j ) − j ) = p − 1 Y i =0 ν j ( η h ) . Note that since c h ( h 6 = 0) is a cyclic p ermuta tion of degree p , c h is a n ev en p ermutatio n, therefore sgn( c h ) = 1 for 0 ≤ h ≤ p − 1. Claim. F or an a rb itrary el ement x ∈ Z p [ f U 2 ( n ) / f V 2 ], p − 1 Y j =0 ν j ( x ) ≡ ϕ ( x ) m o d e J 2 ( n ) . If this claim is true, w e ha ve e θ 2 ( η ) ≡ p − 1 X h =0 P c h mo d e J 2 ( n ) ≡ p − 1 X h =0 p − 1 Y j =0 ν j ( η h ) mo d e J 2 ( n ) ≡ p − 1 X h =0 ϕ ( η h ) mo d e J 2 ( n ) ≡ p − 1 X h =0 ϕ ( η h β h ) mo d e J 2 ( n ) ≡ ϕ ( η ) mo d e J 2 ( n ) , and th e congruence for e θ 2 holds. 48 T A KASHI HARA No w let x = N X ℓ =1 x ℓ where eac h x ℓ is a monomia l of Z p [ f U 2 ( n ) / f V 2 ]. Then (5.9) p − 1 Y j =0 ν j ( x ) = p − 1 Y j =0 N X ℓ =1 ν j ( x ℓ ) = X 1 ≤ ℓ 0 ,...,ℓ p − 1 ≤ N ν 0 ( x ℓ 0 ) ν 1 ( x ℓ 1 ) · · · · · ν p − 1 ( x ℓ p − 1 ) = X 1 ≤ ℓ 0 ,...,ℓ p − 1 ≤ N Q ℓ 0 ,...,ℓ p − 1 . Here w e set Q ℓ 0 ,...,ℓ p − 1 = ν 0 ( x ℓ 0 ) ν 1 ( x ℓ 1 ) · · · · · ν p − 1 ( x ℓ p − 1 ). If ℓ 0 = ℓ 1 = . . . = ℓ p − 1 = λ (constan t ), we obtain Q λ,...,λ = ϕ ( x λ ) by easy calculatio n (here we use that the exp onent of G f is p ). Otherwise, eac h Q ℓ k ,ℓ k +1 ,...,ℓ p − 1 ,ℓ 0 ,ℓ 1 ,...,ℓ k − 1 = ν p − k ( Q ℓ 0 ,...,ℓ p − 1 ) is a distinct term in the expansion (5.9), so we h a v e p − 1 X k =0 Q ℓ k ,ℓ k +1 ,...,ℓ p − 1 ,ℓ 0 ,ℓ 1 ,...,ℓ k − 1 ∈ p − 1 X k =0 ν k  Z p [ f U 2 ( n ) / f V 2 ]  , whic h is c onta ined in e J 2 ( n ) b y (5.8). Hence, p − 1 Y j =0 ν j ( x ) ≡ N X λ =1 ϕ ( x λ ) mo d e J 2 ( n ) ≡ ϕ ( x ) mo d e J 2 ( n ) , and C laim is pro ven. • Th e co ngru en ce for θ 2 . The b asic strategy is essen tially the same as th e congru en ce for e θ 2 , but since U 2 is n ot a normal sub group of G , the calculati on is m uch more complicated. Since { α i δ j | 0 ≤ i, j ≤ p − 1 } is a system of repr esentati ves of U ( n ) 2 \ G ( n ) and { β k | 0 ≤ k ≤ p − 1 } is that of U ( n ) 3 \ U ( n ) 2 , we can describ e η in the follo wing form: η = X 0 ≤ i,j,k ≤ p − 1 ( η ( k ) i,j β k ) α i δ j IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 49 where η ( k ) i,j is a n elemen t of Z p [ U ( n ) 3 ]. Then w e ha ve α ℓ δ m η = X 0 ≤ i,j,k ≤ p − 1 ν ℓ,m ( η ( k ) i,j β k ) α i + ℓ δ j + m = X 0 ≤ i,j,k ≤ p − 1 { ν ℓ,m ( η ( k ) i,j ) δ ℓk β k } α i + ℓ δ j + m = X 0 ≤ i,j ≤ p − 1 p − 1 X k =0 ν ℓ,m ( η ( k ) i − ℓ,j − m − ℓk ) β k ! α i δ j where ν ℓ,m ( x ) = ( α ℓ δ m ) x ( α ℓ δ m ) − 1 . W e use the fund amen tal rela- tions [ α , δ ] = 1 a nd [ α, β ] = δ for the s econd equalit y . Here w e regard the sub-index i, j of η ( k ) i,j as an elemen t of ( Z /p Z ) ⊕ 2 , and the upp er-index ( k ) of it as an e lement of Z /p Z . W e denote the image of η ( k ) i,j in Z p [ U ( n ) 3 /V ( n ) 2 ] by the same symb ol η ( k ) i,j . Then we obtain θ 2 ( η ) = d et p − 1 X k =0 ν ℓ,m ( η ( k ) ( i,j ) − ( ℓ,m ) − (0 ,ℓk ) ) β k ! ( i,j ) , ( ℓ,m ) = X σ ∈ S  sgn( σ ) Y ( ℓ,m ) ∈ ( Z /p Z ) ⊕ 2 p − 1 X k =0 ν ℓ,m ( η ( k ) σ ( ℓ,m ) − ( ℓ,m ) − (0 ,ℓk ) ) β k ! = X σ ∈ S  P σ where S  is the p ermuta tion group o f { ( ℓ, m ) | 0 ≤ ℓ, m ≤ p − 1 } . No w b y using [ β , δ ] = 1, w e ha v e ν 0 ,µ ( P σ ) = sgn( σ ) Y 0 ≤ ℓ,m ≤ p − 1 p − 1 X k =0 ν ℓ,m + µ ( η ( k ) σ ( ℓ,m ) − ( ℓ,m ) − (0 ,ℓk ) ) β k ! = sgn( σ ) Y 0 ≤ ℓ,m ≤ p − 1 p − 1 X k =0 ν ℓ,m ( η ( k ) ( σ ( ℓ,m − µ )+(0 ,µ )) − ( ℓ,m ) − (0 ,ℓk ) ) β k ! . Therefore if σ do es not satisfy (5.10) σ ( ℓ, m ) = σ ( ℓ, m − µ ) + (0 , µ ) for eac h µ ∈ Z /p Z , then τ µ ( ℓ, m ) = σ ( ℓ, m − µ ) + (0 , µ ) , µ ∈ Z /p Z are d istinct elemen ts of S  , a nd we obtain p − 1 X µ =0 P τ µ ∈ p − 1 X µ ν 0 ,µ  Z p [ U ( n ) 2 /V 2 ]  , here w e use the fact sgn( σ ) = sgn( τ µ ) whic h can b e p ro ve n by the same argu m en t as the case of e θ 2 . 50 T A KASHI HARA Note that eac h ν 0 ,µ is a Z p [ U ′ ( n ) 2 /V 2 ]-linear map wh ere U ′ 2 ( n ) = h β , ε, ζ i × Γ / Γ p n and Z p [ U ( n ) 2 /V 2 ] is generated by { γ c | c ∈ Z /p Z } o v er Z p [ U ′ 2 ( n ) /V 2 ]. Since p − 1 X µ =0 ν 0 ,µ ( γ c ) = ( p if c = 0 , γ c (1 + ζ + · · · + ζ p − 1 ) otherwise , w e ma y co nclud e that (5.11) p − 1 X µ =0 ν 0 ,µ  Z p [ U ( n ) 2 /V 2 ]  ⊆ J ( n ) 2 and this imp lies p − 1 X k =0 P τ k ∈ e J 2 ( n ) (note that the argument up to here is a lmost the same a s the case of e θ 2 ). No w supp ose that σ ∈ S  satisﬁes (5.10). If w e set σ ( ℓ, 0) = ( a ℓ , b ℓ ), then by (5. 10 ) w e ha ve σ ( ℓ, − µ ) = ( a ℓ , b ℓ − µ ) for eac h µ ∈ Z /p Z . This calc ulation implies that all p ermutatio ns satisfying (5.1 0 ) are describ ed in th e follo wing forms: (5.12) c s,h ( ℓ, m ) = ( s ( ℓ ) , h ℓ + m ) where s is a p erm utation of { 0 , 1 , . . . , p − 1 } and h = ( h ℓ ) ℓ is an elemen t of ( Z /p Z ) ⊕ p . Then w e ha v e P c s,h = sgn( c s,h ) Y 0 ≤ ℓ,m ≤ p − 1 p − 1 X k =0 ν ℓ,m ( η ( k ) c s,h ( ℓ,m ) − ( ℓ,m ) − (0 ,ℓk ) ) β k ! = sgn( s ) Y 0 ≤ ℓ,m ≤ p − 1 p − 1 X k =0 ν ℓ,m ( η ( k ) ( s ( ℓ ) − ℓ,h ℓ − ℓk ) ) β k ! = sgn( s ) p − 1 Y ℓ =0 ν ℓ, 0 p − 1 Y m =0 ν 0 ,m ( Q s,h ; ℓ ) ! , where Q s,h ; ℓ = p − 1 X k =0 η ( k ) ( s ( ℓ ) − ℓ,h ℓ − ℓk ) β k . Note that s gn ( c s,h ) = sgn( s ) p = sgn( s ) since p is odd . Since Q s,h ; ℓ is indep endent of m , we ma y pro ve p − 1 Y m =0 ν 0 ,m ( Q s,h ; ℓ ) ≡ ϕ ( Q s,h ; ℓ ) mo d J ( n ) 2 in the same w a y as Claim abov e. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 51 Therefore, X s,h P c s,h ≡ X s,h sgn( s ) p − 1 Y ℓ =0 ν ℓ, 0 ( ϕ ( Q s,h ; ℓ )) ! mo d J ( n ) 2 ≡ X s,h sgn( s )   X 0 ≤ k 0 ,...,k p − 1 ≤ p − 1 p − 1 Y ℓ =0 ϕ ( η ( k ℓ ) ( s ( ℓ ) − ℓ,h ℓ − ℓk ℓ ) )   mo d J ( n ) 2 = X s,h,k ℓ R s,h ; k 0 ,k 1 ,...,k p − 1 (5.13) where R s,h ; k 0 ,...,k p − 1 = sgn( s ) p − 1 Y ℓ =0 ϕ ( η ( k ℓ ) ( s ( ℓ ) − ℓ,h ℓ − ℓk ℓ ) ). W e use the fact that ν ℓ, 0 ◦ ϕ = ϕ (recall ϕ ( Z p [ U ( n ) 2 /V 2 ]) ⊆ Z p [Γ / Γ ( n ) ]) for the second congruence. If s ( ℓ ) − ℓ = λ (constan t ), k 0 = k 1 = . . . = k p − 1 and h ℓ − ℓk ℓ = µ (constan t), that is, if s ( ℓ ) = s λ ( ℓ ) = ℓ + λ , k ℓ = κ for eac h ℓ , a nd h = h µ,κ = ( ℓκ + µ ) ℓ for 0 ≤ κ, λ, µ ≤ p − 1, we h a v e R s λ ,h µ,κ ; κ,...,κ = ϕ ( η ( κ ) λ,µ ) p . Note that sgn( s λ ) = 1 b ecause s λ is a cyclic p ermutati on of deg ree p . Otherwise, set k ( w ) ℓ = k ℓ + w , s ( w ) ( ℓ ) = s ( ℓ + w ) − w, h ( w ) ℓ = h ℓ + w − wk ℓ + w , for ea ch 0 ≤ w ≤ p − 1. Then w e ha ve R s (0) ,h (0) ; k (0) 0 ,...,k (0) p − 1 = . . . = R s ( p − 1) ,h ( p − 1) ; k ( p − 1) 0 ,...,k ( p − 1) p − 1 and R s ( w ) ,h ( w ) ; k ( w ) 0 ,...,k ( w ) p − 1 (0 ≤ w ≤ p − 1) are distinct terms in the expansion (5. 13 ). Hence w e ha v e p − 1 X w =0 R s ( w ) ,h ( w ) ; k ( w ) 0 ,...,k ( w ) p − 1 ∈ p Z p [Γ ( n ) ] ⊆ J ( n ) 2 . 52 T A KASHI HARA Therefore w e ha ve θ 2 ( η ) ≡ X s,h,k w R s,h ; k 0 ,...,k p − 1 mo d J ( n ) 2 ≡ X 0 ≤ λ,µ,κ ≤ p − 1 R s λ ,h µ,κ ; κ,κ,. ..,κ mo d J ( n ) 2 ≡ X 0 ≤ λ,µ,κ ≤ p − 1 ϕ ( η ( κ ) λ,µ ) p mo d J ( n ) 2 ≡   X 0 ≤ λ,µ,κ ≤ p − 1 ϕ ( η ( κ ) λ,µ β κ α λ δ µ )   p mo d J ( n ) 2 ≡ ϕ   X 0 ≤ λ,µ,κ ≤ p − 1 η ( κ ) λ,µ β κ α λ δ µ   p mo d J ( n ) 2 ≡ ϕ ( η ) p mo d J ( n ) 2 . Note that s in ce J ( n ) 2 ⊇ p Z p [ U ( n ) 2 /V 2 ], w e ha ve ( x + y ) p ≡ x p + y p mo d J ( n ) 2 . • Th e co ngru en ce for θ 3 . W e ma y sh o w the congruence for θ 3 b y an argument similar to that for θ 2 . Moreo v er, the calculation for θ 3 is muc h simpler than the case for θ 2 since U 3 is normal in G . Therefore we omit the pro of.  By Le mma 5.10 and the fact that ϕ ( θ 0 ( η )) is in vertible, w e ha ve θ 1 ( η ) ϕ ( θ 0 ( η )) ∈ 1 + J ( n ) 1 , e θ 2 ( η ) ϕ ( θ 0 ( η )) ∈ 1 + e J 2 ( n ) , θ 2 ( η ) ϕ ( θ 0 ( η )) p ∈ 1 + J ( n ) 2 , θ 3 ( η ) ϕ ( θ 0 ( η )) p 2 ∈ 1 + J ( n ) 3 . Hence b y Lemma 5.5, we ha ve log θ 1 ( η ) ϕ ( θ 0 ( η )) ∈ J ( n ) 1 , log e θ 2 ( η ) ϕ ( θ 0 ( η )) ∈ e J 2 ( n ) , log θ 2 ( η ) ϕ ( θ 0 ( η )) p ∈ J ( n ) 2 , log θ 3 ( η ) ϕ ( θ 0 ( η )) p 2 ∈ J ( n ) 3 . But b y Prop osition 5 .9 , these elemen ts are con tained in the image of θ + i , and ther efore con tained in I ( n ) i (recall the deﬁn ition of I ( n ) i ). Moreo ve r, since the p -adic log arithmic homomorph isms induce isomorp h isms 1 + I ( n ) i ≃ − → I ( n ) i (b y Lemma 5. 6), we may conclude that (5.14) θ 1 ( η ) ϕ ( θ 0 ( η )) ∈ 1 + I ( n ) 1 , e θ 2 ( η ) ϕ ( θ 0 ( η )) ∈ 1 + e I 2 ( n ) θ 2 ( η ) ϕ ( θ 0 ( η )) p ∈ 1 + I ( n ) 2 , θ 3 ( η ) ϕ ( θ 0 ( η )) p 2 ∈ 1 + I ( n ) 3 . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 53 Since ea c h θ i is a norm map in K - theory , it is clear that ( θ i ( η )) i satisﬁes norm relatio ns. This and (5. 14 ) imply that θ ( η ) ∈ Ψ ( n ) . 5.3. Surjectivit y for ﬁnite quotients. Let ( η i ) i b e a n arbitrary elemen t of Ψ ( n ) . By the congruences in Deﬁnition 5.3 and by the fact that ϕ ( η 0 ) is in ve rtible, we h a v e η 1 ϕ ( η 0 ) ∈ 1 + I ( n ) 1 , e η 2 ϕ ( η 0 ) ∈ 1 + e I 2 ( n ) , η 2 ϕ ( η 0 ) p ∈ 1 + I ( n ) 2 , η 3 ϕ ( η 0 ) p 2 ∈ 1 + I ( n ) 3 . Then w e obtain log η 1 ϕ ( η 0 ) ∈ I ( n ) 1 , log e η 2 ϕ ( η 0 ) ∈ e I 2 ( n ) , log η 2 ϕ ( η 0 ) p ∈ I ( n ) 2 , log η 3 ϕ ( η 0 ) p 2 ∈ I ( n ) 3 via the p -adic logarithms (Lemma 5.6). Note that since ϕ ( η 0 ) is con tained in the cen ter of Z p [ U ( n ) i /V i ] × , it is easy to see that Nr Z p [ g U ( n ) 2 / f V 2 ] / Z p [ U ( n ) 2 / f V 2 ] ϕ ( η 0 ) = ϕ ( η 0 ) p , Nr Z p [ U ( n ) 1 / f V 2 ] / Z p [ U ( n ) 1 ∩ f U 2 ( n ) / f V 2 ] ϕ ( η 0 ) = Nr Z p [ f U 2 ( n ) / f V 2 ] / Z p [ U ( n ) 1 ∩ f U 2 ( n ) / f V 2 ] ϕ ( η 0 ) = ϕ ( η 0 ) p , Nr Z p [ U ( n ) 1 /V 1 ] / Z p [ U ( n ) 3 /V 1 ] ϕ ( η 0 ) = ϕ ( η 0 ) p 2 , Nr Z p [ U ( n ) 2 /V 2 ] / Z p [ U ( n ) 3 /V 2 ] ϕ ( η 0 ) p = ϕ ( η 0 ) p 2 . By using Lemma 5.8, ( y 1 , e y 2 , y 2 , y 3 ) =  log η 1 ϕ ( η 0 ) , log e η 2 ϕ ( η 0 ) , log η 2 ϕ ( η 0 ) p , log η 3 ϕ ( η 0 ) p 2  ∈ I ( n ) 1 × e I 2 ( n ) × I ( n ) 2 × I ( n ) 3 satisﬁes the trace relations in D eﬁnition 4.2 (except for Z p [ U ( n ) 0 /V 0 ]-part). No w set y 0 = Γ U ( n ) 0 /V 0 ( η 0 ) = (1 /p ) log ( η p 0 /ϕ ( η 0 )) ∈ Z p [ U ( n ) 0 /V 0 ]. Then w e may easily chec k that ( y 0 , y 1 , e y 2 , y 2 , y 3 ) satisﬁes the trace relatio ns in Deﬁnition 4.2. Hence w e ha ve ( y 0 , y 1 , e y 2 , y 2 , y 3 ) ∈ Ω ( n ) . By the add itiv e theta isomorphism (Prop osition-Deﬁnition 4.3), there ex- ists a un ique ele ment y ∈ Z p [Conj( G ( n ) )] satisfying (5.15) θ + ( y ) = ( y 0 , y 1 , e y 2 , y 2 , y 3 ) =  Γ U ( n ) 0 /V 0 ( η 0 ) , log η 1 ϕ ( η 0 ) , log e η 2 ϕ ( η 0 ) , log η 2 ϕ ( η 0 ) p , log η 3 ϕ ( η 0 ) p 2  . Prop osition 5.11. ω G ( n ) ( y ) = 1 , that is, y is in the image of the inte gr al lo garithmic homomorp hism Γ G ( n ) wher e ω G ( n ) is deﬁne d as in The or em 1 .30. 54 T A KASHI HARA Pr o of. By the deﬁnition, ω G ( n ) is deco mp osed as f ollo ws : Z p [Conj( G ( n ) )] ω G ( n ) / / π ( n )   G ab = U ( n ) 0 /V 0 Z p [ U ( n ) 0 /V 0 ] ω U ( n ) 0 /V 0 6 6 m m m m m m m m m m m m where π ( n ) is the canonical su rjection. Not e that π ( n ) is the same map as θ + 0 b y the deﬁnition of θ + . Therefore, w e ha v e ω G ( n ) ( y ) = ω U ( n ) 0 /V 0 ( θ + 0 ( y )) = ω U ( n ) 0 /V 0 (Γ U ( n ) 0 /V 0 ( η 0 )) by (5 .15 ) = 1 b y Theorem 1.30 .  Let η b e an elemen t of K 1 ( Z p [ G ( n ) ]) satisfying Γ G ( n ) ( η ) = y . Th en η is determined up to m ultiplication b y a torsion elemen t of K 1 ( Z p [ G ( n ) ]) (S ee Theorem 1.3 0 ). By the deﬁnition of η and the equation (5. 15 ), θ + ◦ Γ G ( n ) ( η ) = ( y 0 , y 1 , e y 2 , y 2 , y 3 ) . Com bining with Prop osition 5.9, we hav e y 0 = 1 p log η p 0 ϕ ( η 0 ) = 1 p log θ 0 ( η ) p ϕ ( θ 0 ( η )) , that is, (5.16) Γ U ( n ) 0 /V 0 ( η 0 ) = Γ U ( n ) 0 /V 0 ( θ 0 ( η )) . W e also hav e (5.17) log η 1 ϕ ( η 0 ) = log θ 1 ( η ) ϕ ( θ 0 ( η )) , log e η 2 ϕ ( η 0 ) = log e θ 2 ( η ) ϕ ( θ 0 ( η )) , log η 2 ϕ ( η 0 ) p = log θ 2 ( η ) ϕ ( θ 0 ( η )) p , log η 3 ϕ ( η 0 ) p 2 = log θ 3 ( η ) ϕ ( θ 0 ( η )) p 2 . Prop osition 5.12. Ther e exists an element τ ∈ µ p − 1 ( Z p ) × ( G ( n ) ) ab such that θ ( η τ ) = ( η 0 , η 1 , e η 2 , η 2 , η 3 ) wher e µ p − 1 ( Z p ) is the multiplic ative gr oup ge ne r ate d by al l ( p − 1) -th r o ots of 1 in Z p . In p articular, θ surje cts on Ψ ( n ) . Pr o of. F rom th e equation (5.16) and the fact that the inte gral logarithm homomorphism is injectiv e mo dulo torsion elemen ts, there exists an elemen t τ ∈ K 1 ( Z p [ U ( n ) 0 /V 0 ]) tors satisfying η 0 = τ θ 0 ( η ). Then b y Lemma 1.33 ( 3), τ is con tained in K 1 ( Z p [ U ( n ) 0 /V 0 ]) tors = µ p − 1 ( Z p ) × ( U ( n ) 0 /V 0 ) = µ p − 1 ( Z p ) × ( G ( n ) ) ab . Hence w e ha ve θ 0 ( η τ ) = θ 0 ( η ) θ 0 ( τ ) = θ 0 ( η ) τ = η 0 . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 55 Since τ ∈ Ker (Γ G ( n ) ), w e ha v e θ + ◦ Γ G ( n ) ( η τ ) = θ + ◦ Γ G ( n ) ( η ). Therefore w e ma y replace η b y η τ in the righ t hand sides of the equat ions (5.17). F rom this fact and Lemma 5.6, w e ha ve θ i ( η τ ) = η i for eac h i , using θ 0 ( η τ ) = η 0 .  5.4. T aking the pro jective limit. In this sub section, w e tak e the pr o jec- tiv e limit of the surjections θ ( n ) : K 1 ( Z p [ G ( n ) ]) − → Ψ ( n ) obtained b y the argument in the previous su bsections, and construct the surjection (5. 2 ). First, w e s tudy the kernel o f the surjection θ ( n ) precisely . Supp ose that τ ∈ Ker( θ ( n ) ). Then we hav e θ + , ( n ) ◦ Γ G ( n ) ( τ ) = 0 by Prop osition 5.9. Since θ + , ( n ) is an isomorphism b y Prop osition-Deﬁnition 4.3, w e ob tain Γ G ( n ) ( τ ) = 0. In ot her words, Ker( θ ( n ) ) ⊆ Ker(Γ G ( n ) ) = K 1 ( Z p [ G ( n ) ]) tors = µ p − 1 ( Z p ) × ( G ( n ) ) ab × S K 1 ( Z p [ G ( n ) ]) . By t he deﬁnition of S K 1 groups, norm ma ps of K 1 -groups induce h omo- morphisms on S K 1 , so we h a v e θ ( n ) ( S K 1 ( Z p [ G ( n ) ])) ⊆ Y i S K 1 ( Z p [ U ( n ) i /V i ]) . Ho w ev er, S K 1 ( Z p [ U ( n ) i /V i ]) = 1 s in ce eac h U ( n ) i /V i is ab elian (Proposition 1.33 (3) ). Hence w e obtain S K 1 ( Z p [ G ( n ) ]) ⊆ Ker( θ ( n ) ). On the o ther hand, f or ev ery elemen t τ ∈ µ p − 1 ( Z p ) × ( G ( n ) ) ab not equal to 1, w e ha v e θ ( n ) 0 ( τ ) = τ 6 = 1. This implies  µ p − 1 ( Z p ) × ( G ( n ) ) ab  ∩ Ker( θ ( n ) ) = { 1 } . Therefore we ma y conclude that K er( θ ( n ) ) = S K 1 ( Z p [ G ( n ) ]), and we ob- tain an exact sequence of p ro jectiv e systems of a b elian groups (5.18) 1 − − − − → S K 1 ( Z p [ G ( n ) ]) − − − − → K 1 ( Z p [ G ( n ) ]) θ ( n ) − − − − → Ψ ( n ) − − − − → 1 . By Prop osition 1.33 (2), S K 1 ( Z p [ G ( n ) ]) is a ﬁnite ab elian group for ev- ery n ≥ 1. Hence the pr o jec tiv e system { S K 1 ( Z p [ G ( n ) ]) } n ∈ N satisﬁes the Mittag- Leﬄer condition, wh ic h implies lim ← − 1 n S K 1 ( Z p [ G n ]) = 0. By taking the pro jectiv e limit of (5.1 8 ), w e obtain the follo wing exact sequence (5.19) 1 → lim ← − n S K 1 ( Z p [ G ( n ) ]) → lim ← − n K 1 ( Z p [ G ( n ) ]) ( θ ( n ) ) n − − − − → Ψ → 1 . Therefore, the surjectivit y of (5.2) is reduced to the follo w in g prop osition. Prop osition 5.13. The homomorp hism K 1 (Λ( G )) → lim ← − n K 1 ( Z p [ G ( n ) ]) induc e d by c anonic al homomorphisms K 1 (Λ( G )) → K 1 ( Z p [ G ( n ) ]) is an iso- morphism . Pr o of. Apply Proposition 1.8 for Λ( G ) and { Z p [ G ( n ) ] } n ∈ N .  56 T A KASHI HARA Remark 5.14. I n the c ase of Kato’s Heisen b erg t yp e ([Kato1]), that is, in the c ase that Gal( F ∞ /F ) ∼ = G where G =   1 F p F p 0 1 F p 0 0 1   × Γ = G f × Γ , it is known ([Oliver]) that S K 1 ( Z p [ G ( n ) ]) = S K 1 ( Z p [ G f ]) ⊕ S K 1 ( Z p [Γ / Γ p n ]) = 1 , therefore w e ma y show that the thet a map K 1 (Λ( G )) ≃ − → Ψ ⊆ Y i Λ( U i /V i ) ! is a n isomorph ism , and ξ F ∞ /F is dete rm in ed uniquely . Remark 5.15. By the calculatio n ab o ve , we kno w that the k ernel of th e theta map tends to b e S K 1 (Λ( G )). I t is conjectured that S K 1 (Λ( G )) v an- ishes for general compact p -adic L ie groups. F or our sp ecial case G = G f × Γ, Otmar V enjak ob announced to the author that he an d Pete r Sc hneider ha ve recen tly prov en the v anishin g of S K 1 (Λ( G )). If we admit their r esu lt, we m a y pr o v e the un iqueness of the p -adic zet a fun ction ξ F ∞ /F whic h w e w ould construct in § 8. 6. Localized thet a map In t his section, w e construct th e lo c alize d theta map θ S : K 1 (Λ( G ) S ) − → Ψ S . Let Λ(Γ) = Z p [[Γ]] b e the Iw asa wa algebra f or Γ. Then sin ce G = G f × Γ, w e ha v e Λ( G ) = Z p [[ G f × Γ]] ∼ = Z p [ G f ] ⊗ Z p Z p [[Γ]] ∼ = Λ(Γ)[ G f ] , and w e also hav e Λ( U i /V i ) ∼ = Λ(Γ)[ U f i /V f i ] Let S b e the canonical Øre set (see § 2.1) for the group G and let S 0 b e that for Γ. Note that S 0 = Λ(Γ) \ p Λ(Γ) ( see Example 2.7). Lemma 6.1. Λ( G ) S ∼ = Λ(Γ) S 0 [ G f ] and Λ( U i /V i ) ∼ = Λ(Γ) S 0 [ U f i /V f i ] . Pr o of. W e sh o w this lemma only for the group G . First, we sho w that Λ( G ) S = Λ( G ) S 0 (b y ab u se of notation, w e a lso d enote the im age of S 0 under the canonical map Λ(Γ) − → Λ( G ) by the same symb ol S 0 ). F or an arbitrary f 0 ∈ S 0 , Λ( G ) / Λ( G ) f 0 ∼ = (Λ(Γ) / Λ(Γ) f 0 ) [ G f ] is Z p - ﬁnitely generated, so S 0 ⊆ S . Th en by the universalit y of Øre lo calizati ons (Prop osition-Deﬁnition 1.13), w e obtain the c anonical homomorphism (6.1) Λ( G ) S 0 − → Λ( G ) S . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 57 Let f b e an elemen t of S . By the deﬁ n ition of the canonical Ør e set, Λ( G ) / Λ( G ) f is a ﬁ nitely generated left Z p -mo dule. 7 Let z b e a kernel of the follo w ing comp osition of h omomorphisms: (6.2) Λ(Γ) − − − − → Λ( G ) − − − − → Λ( G ) / Λ( G ) f , where the ﬁr st map is the ca nonical map Λ(Γ) − → Λ( G ) ∼ = Λ(Γ) ⊗ Z p Z p [ G f ] ; f 0 7→ f 0 ⊗ 1 . T ak e an a rb itrary non -zero el ement g 0 ∈ z . Since Λ(Γ) is a u nique factor- ization domain, g 0 has a decomp osition g 0 = up n π n 1 1 · · · π n k k , where u ∈ Λ(Γ) × and π i is a pr ime e lement o f Λ(Γ). Supp ose that for every n on-zero g 0 ∈ z , the index of the prime p ( n in the d ecomp osition ab o v e) is n ot zero. Then w e hav e z ⊆ p Λ(Γ), and this induces a s urjection Λ(Γ) / z → Λ(Γ) /p Λ(Γ). This is con tradiction since Λ(Γ) / z ⊆ Λ( G ) / Λ( G ) f is ﬁnitely generated o v er Z p b y the deﬁnition of S , but Λ(Γ) /p Λ(Γ) ∼ = F p [[Γ]] is not s o. Hence there exists g 0 ∈ z such that p ∤ g 0 , whic h implies g 0 ∈ S 0 . Via the homomorphism ( 6.2 ), g 0 is con tained in Λ( G ) f , hen ce there exists φ ∈ Λ( G ) satisfying g 0 = φf . Th is means ( g − 1 0 φ ) · f = 1 in Λ( G ) S 0 , in other w ords , f has i ts (left) inv erse elemen t g − 1 0 φ in Λ( G ) S 0 . Hence, b y th e unive rsalit y of Øre localizations again, w e obtain (6.3) Λ( G ) S − → Λ( G ) S 0 . It is easy to sho w that th e homomorphisms (6.1) and (6.3) are the inv erse maps of eac h other, therefore Λ( G ) S ∼ = Λ( G ) S 0 . On the other hand, the canonical homomorphisms Λ( G ) S 0 − → Λ(Γ) S 0 [ G f ] obtained b y the un iversalit y of Øre localizations and Λ(Γ) S 0 [ G f ] = Λ(Γ) S 0 ⊗ Z p Z p [ G f ] − → Λ( G ) S 0 obtained b y the un iversalit y of tensor pro ducts from the map Λ(Γ) S 0 × Z p [ G f ] → Λ( G ) S 0 ; ( φ 0 , h ) 7→ φ 0 h are in v erse maps of eac h other. T his imp lies Λ( G ) S ∼ = Λ(Γ) S 0 [ G f ]. The r esult for e ac h Λ( U i /V i ) S is obtained in the same w ay .  By this lemma, w e ma y r eplace the co eﬃcien t ring Z p b y Λ(Γ) S 0 and apply the same discussion in § 4 and § 5. Since we need to use p -adic logarithms to tr an s late the additiv e theta map, w e take the p -adic completion \ Λ(Γ) S 0 of Λ(Γ) S 0 . Set R = \ Λ(Γ) S 0 . Then the add itiv e theta map b θ + : R [Conj( G f )] ≃ − → b Ω 7 Since H is ﬁnite, a left Z p [ H ]-mo du le is ﬁnitely generated if and only i f it is ﬁnitely generated as a left Z p -mo d ule. 58 T A KASHI HARA is obtained by the completely same argument as in § 4, w h ere b Ω is the R - submo d ule of Y i R [ U f i /V f i ] deﬁned by the same conditions as in Deﬁnition 4.2. More pr ecisely , b Ω is an R -sub mo dule consisting of all elemen ts ( b y i ) i satisfying the trac e relations and b y i ∈ b I i for ea ch i where b I i = I f i ⊗ Z p R . W e use the generalized in tegral p -adic lo garithm (See § 1.4) Γ R [ G f ] ,J : K 1 ( R [ G f ] , J ) − → R [Conj( G f )] , where J is the Jacobson radical of R [ G f ]. In this case, J is the k ernel of th e natural surjection R [ G f ] → R /pR . By the same argumen t as in § 5, we ha v e b θ + ◦ Γ R [ G f ] ,J ( ˆ η ′ ) ∈ b Ω , for an arbitrary ˆ η ′ ∈ K 1 ( R [ G f ] , J ), and w e may conclude that b θ ( ˆ η ′ ) is con- tained in b Ψ, where b Ψ is the subgroup of Y i R [ U f i /V f i ] consisting of all ele- men ts ( ˆ η ′ i ) i whic h satisfy th e norm relations and th e follo w ing congruences: (6.4) ˆ η ′ 1 ≡ ϕ ( ˆ η ′ 0 ) mo d b I 1 , ˆ e η ′ 2 ≡ ϕ ( ˆ η ′ 0 ) mo d b e I 2 , ˆ η ′ 2 ≡ ϕ ( ˆ η ′ 0 ) p mo d b I 2 , ˆ η ′ 3 ≡ ϕ ( ˆ η ′ 0 ) p 2 mo d b I 3 . Let ˆ η b e an arbitrary elemen t of K 1 ( R [ G f ]). W e use the same notation ˆ η for its lift to R [ G f ] × . Let F ⊆ R [ G f ] b e a sub s et of repr esentati ves of π × : R [ G f ] × → ( R [ G f ] /J ) × ∼ = F p (( T )) × , t hat is, F b e a subset of R [ G f ] for whic h π × induces a bijection of sets F ≃ − → ( R [ G f ] /J ) × . Note that we can tak e F as a subset of R × . Then there exists φ ∈ F and ˆ η ′ ∈ 1 + J su c h that e η = φ ˆ η ′ . ˆ η ′ deﬁnes an elemen t of K 1 ( R [ G f ] , J ), and we ha ve already kn o wn that b θ ( ˆ η ′ ) is contai ned in b Ψ. T herefore, if w e pro v e that b θ ( φ ) ∈ b Ψ for every φ ∈ F , we may conclud e that b θ ( K 1 ( R [ G f ])) ⊆ b Ψ. It suﬃces to s ho w that b θ ( r ) ∈ b Ψ for ev ery r ∈ R × . F or an arbitrary r ∈ R × , w e ha ve b θ 0 ( r ) = r, b θ 1 ( r ) = b e θ 2 ( r ) = r p , b θ 2 ( r ) = r p 2 , b θ 3 ( r ) = r p 3 , b y dir ect calculat ion. 8 This implies that ( b θ i ( r )) i satisﬁes the n orm relations. The congru ences for ( θ i ( r )) i are d eriv ed from the follo wing r elation r p ≡ ϕ ( r ) m o d pR whic h follo w s from the deﬁn ition of the F rob enius au tomorp h ism (S ee § 1.4). No w ( b θ i ( r )) i ∈ b Ψ holds. Set Ψ S = b Ψ ∩ Y i Λ(Γ) S 0 [ U f i /V f i ]. 8 Since R is in the center of R [ G f ], the images of norm maps of r ∈ R are easily calculated by using the metho d in § 1.3. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 59 Prop osition 6.2. Ψ S is char acterize d as the sub gr oup of Y i Λ( U i /V i ) × S c on- sisting of al l elements ( η 0 , η 1 , e η 2 , η 2 , η 3 ) satisfying the fol lowing two c ondi- tions. (1) ( norm r elations ) (rel-1) Nr Λ( U 0 /V 0 ) S / Λ( U 1 /V 0 ) S η 0 ≡ η 1 , (rel-2) Nr Λ( U 0 /V 0 ) S / Λ( f U 2 /V 0 ) S η 0 ≡ e η 2 , (rel-3) Nr Λ( f U 2 / f V 2 ) S / Λ( U 2 / f V 2 ) S e η 2 ≡ η 2 , (rel-4) Nr Λ( U 1 / f V 2 ) S / Λ( U 1 ∩ f U 2 / f V 2 ) S η 1 ≡ Nr Λ( f U 2 / f V 2 ) S / Λ( U 1 ∩ f U 2 / f V 2 ) S e η 2 , (rel-5) Nr Λ( U 1 /V 1 ) S / Λ( U 3 /V 1 ) S η 1 ≡ η 3 , (rel-6) Nr Λ( U 2 /V 2 ) S / Λ( U 3 /V 2 ) S η 2 ≡ η 3 . ( These ar e the same as those of Ψ . Se e Figur e 1 ) (2) ( c ongruenc es ) η 1 ≡ ϕ ( η 0 ) mo d I S, 1 , e η 2 ≡ ϕ ( η 0 ) mo d g I S, 2 , η 2 ≡ ϕ ( η 0 ) p mo d I S, 2 , η 3 ≡ ϕ ( η 0 ) p 2 mo d I S, 3 , wher e I S,i = I f i ⊗ Z p Λ(Γ) S 0 . Pr o of. An arbitrary element ( η i ) i of Ψ S satisﬁes the norm relations b y the deﬁnition of Ψ S . Since ( η i ) i is con tained in b Ψ, it satisﬁes (6.4). Then the desired c ongruen ces hold (note that b I i ∩ Λ(Γ) S 0 [ U f i /V f i ] = I f i ⊗ Z p Λ(Γ) S 0 = I S,i ).  No w let θ S = ( θ S,i ) i b e the family of the homomorphisms θ S,i where eac h θ S,i is deﬁned a s the composite of the norm map Nr Λ( G ) S / Λ( U i ) S : K 1 (Λ( G ) S ) → K 1 (Λ( U i ) S ) and the canonical homomorphism K 1 (Λ( U i ) S ) → K 1 (Λ( U i /V i ) S ) = Λ( U i /V i ) × S induced b y Λ( U i ) S → Λ( U i /V i ) S . Prop osition 6.3. Ψ S c ontains the image of θ S . Pr o of. By the comm utativ e d iagram K 1 (Λ( G ) S ) θ S − − − − → Q i Λ( U i /V i ) × S   y   y K 1 ( R [ G f ]) − − − − → b θ Q i R [ U f i /V f i ] × , w e ha ve Im age( θ S ) ⊆ Image( b θ ) ⊆ b Ψ (note that the r ight v ertical map is injectiv e since the canonical m ap Λ(Γ) S 0 → R is injectiv e, using th e fact that Λ(Γ) S 0 is separate d with resp ect to p -adic top ology). T herefore, Image( θ S ) ⊆ b Ψ ∩ Y i Λ( U i /V i ) × S ! = Ψ S .  60 T A KASHI HARA Prop osition 6.4. Ψ S ∩ Y i Λ( U i /V i ) × = Ψ . Pr o of. This follo ws easily from the fac t I S,i ∩ Λ( U i /V i ) = I i .  W e ca n sum m arize the t wo pr op ositions abov e as follo ws: Theorem 6.5. θ S : K 1 (Λ( G ) S ) − → Ψ S is the lo c alize d theta map ( deﬁne d in Deﬁnition 3.3 ) for G . 7. Congruences amo ng abelian p -ad ic zet a pseudo measure s 7.1. Norm relations. Let F U i (resp. F V i ) b e the maximal su bﬁeld of F ∞ ﬁxed b y U i (resp. V i ). In th e p r evious sections we hav e constructed the theta map θ and its lo calized v ersion θ S . If the Iw asa w a main conjecture is true in o ur case, the assumption of Theorem 3.4 should b e true, that is, the family of th e ab elian p -adic zeta pseudomeasures ( ξ 0 , ξ 1 , e ξ 2 , ξ 2 , ξ 3 ) (eac h ξ i ∈ F rac(Λ( U i /V i )) is the p -adic zeta pseudomeasure for F V i /F U i ) should b e con tained in Ψ S . F or ( ξ i ) i to b e an elemen t of Ψ S , it is n ecessary that ξ i ’s satisfy the norm relations and the congruences in Prop osition 6.2. Prop osition 7.1. ξ i ’s satisfy the norm r elations in Pr op osition 6.2. Pr o of. W e ma y sho w th is pr op osition b y formal calculation, us ing only the in terp olatio n prop erties (See Deﬁnition 2.14 (2 .2 )). Here w e only s h o w Nr Λ( U 0 /V 0 ) S / Λ( U 1 /V 0 ) S ξ 0 ≡ ξ 1 in Λ( U 1 /V 0 ) S . Let ρ b e an arbitrary Ar tin repr esentati on of U 1 /V 0 . T hen by Lemma 3.5, w e h a v e (Nr ξ 0 ) ( ρ ⊗ κ r ) = ξ 0 (Ind U 0 U 1 ( ρ ⊗ κ r )) = ξ 0 (Ind U 0 U 1 ( ρ ) ⊗ κ r ) = L Σ (1 − r ; F V 0 /F U 0 , Ind U 0 U 1 ( ρ )) = L Σ (1 − r ; F V 0 /F U 1 , ρ ) = ξ 1 ( ρ ⊗ κ r ) for ev ery p ositive in teger r divisible by p − 1. W e ma y also sho w the other norm relatio ns b y the same argument.  7.2. Kato’s observ ation. No w let us study the congruences among ξ i ’s. Let I ′′ 2 b e the imag e of the comp osition Z p [[Conj( f U 2 )]] T r − → Z p [[Conj( U 2 )]] → Z p [[ U 2 /V 2 ]] . W e ha ve the follo wing explicit description of I ′′ 2 : I ′′ 2 = lim ← − n  [ β b γ c h ζ ( c 6 = 0) , pβ b ζ f ] Z p [Γ / Γ p n ]  ⊆ Λ( U 2 /V 2 ) . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 61 Prop osition 7.2 (Congruences among abelian p -adic z eta functions) . The p -adic zeta functions ξ i for F V i /F U i satisfy the fol lowing c ongruenc es : (1) ξ 1 ≡ ϕ ( ξ 0 ) mo d I S, 1 , f (2) e ξ 2 ≡ ϕ ( ξ 0 ) mo d g I S, 2 , (2) ξ 2 ≡ c 2 mo d I ′′ S, 2 , (3) ξ 3 ≡ c 3 mo d I S, 3 , wher e c 2 and c 3 ar e c ertain e lements of Λ(Γ) S 0 . Remark 7.3. Th ese congruences are not suﬃcien t to sh ow that ( ξ i ) i ∈ Ψ S . Hence w e s hould mo dify Bur ns’ tec h nique (Prop osition 3.4) to pro ve the Iw asa wa main conjecture in our case. This mo diﬁcation is discussed in the next sec tion. Kato observ ed in [Kato1] that these kinds of c ongruen ces among ab elian p -adic zeta pseudomeasures w ere derived from Deligne-Rib et’s metho d of p - adic Hilb ert mo dular forms. T he fundamenta l philosophy of p -adic (Λ-adic) Hilb ert mod ular forms is as fol lo ws (quite r ough): ( ♯ ) Congruence b et w een constan t te rms of tw o p -adic (Λ-adic) Hilb ert mo d u lar forms is d eriv ed from congru ences b et w een co eﬃcien ts of al l p ositiv e degree term s of them. Before the precise pr o of of Prop osition 7. 2, let us review ho w to deriv e the congruence b et w een ab elian p -adic z eta pseudomeasures from Hilbert mo du - lar forms, follo wing [Kato1] § 4. W e tak e the congru ence ξ 1 ≡ ϕ ( ξ 0 )mo d I S, 1 as an example. The Λ( U i /V i )-adic F U i -Hilb ert Eisenstein series f i = ξ i 2 [ F U i : Q ] + X ( a ,x ) ∈ P i  F V i /F U i a  q x i = 0 , 1 w as essen tially constructed b y Del igne and Rib et, where P i = { ( a , x ) | a ⊆ O F U i : non-ze ro ideal prime to Σ , x ∈ a : totally p ositive } and  L/K −  denotes the Ar tin symbol in Gal( L/K ) for an ab elian extension L/K . By r estricting f 1 to the Hilb ert mo d ular v ariet y of F em b edded as the diagonal in th e Hilb ert mo dular v ariet y of F U 1 , we obtain a Λ( U 1 /V 1 )-adic F -Hilb ert mo du lar form (7.1) g 1 = ξ 1 2 r p + X ( a ,x ) ∈ P 1  F V 1 /F U 1 a  q T r F U 1 /F ( x ) where r is [ F : Q ]. Note that [ F U 1 : Q ] is equ al to r p . The Galo is group Gal( F U 1 /F ) = h β i acts on P 1 from the left as β j ∗ ( a , x ) = ( β j a , β j x ) 62 T A KASHI HARA for ea ch 0 ≤ j ≤ p − 1. Then we ma y divide p ositiv e degree terms of g 1 b y isotropic sub groups Gal( F U 1 /F ) ( a ,x ) as follo ws: g 1 = ξ 1 2 r p + X ( a ,x ) ∈ P ′ 1  F V 1 /F U 1 a  q T r F U 1 /F ( x ) + X ( a ,x ) ∈ P ′′ 1  F V 1 /F U 1 a  q T r F U 1 /F ( x ) where P ′ 1 = { ( a , x ) ∈ P 1 | Gal( F U 1 /F ) ( a ,x ) = Gal( F U 1 /F ) } , P ′′ 1 = { ( a , x ) ∈ P 1 | Gal( F U 1 /F ) ( a ,x ) = { id F U 1 }} . One observes that there exits a bijectio n P 0 ∼ − → P ′ 1 ; ( b , y ) 7→ ( a , x ) where a = b O F U 1 and x = y . By the functorialit y of the Artin s y mb ol, w e ha ve g 1 = ξ 1 2 r p + X ( b , y ) ∈ P 0 V er  F V 0 /F U 0 b  q py + X ( a ,x ) ∈ P ′′ 1  F V 1 /F U 1 a  q T r F U 1 /F ( x ) , where V er : Gal( F V 0 /F U 0 ) → Gal( F V 1 /F U 1 ) is the V erlagerung (transfer) ho- momorphism. Lemma 7.4. L et H f 1 and H f 2 b e arbitr ary sub gr oups of G f . Se t H i = H f i × Γ . If H 1 c ontains H 2 , the V erlagerung homomorphism V er H 1 H 2 : H ab 1 → H ab 2 c oincides with the c omp osition of the F r ob enius hom omorphism ϕ : H ab 1 → Γ and the c anonic al inje ction Γ → H ab 2 . Pr o of. By the transitivit y of the V erlagerung homomorph ism, it su ﬃces to pro ve th e prop osition w h en ( H 1 : H 2 ) = p . In th is case it is well kno wn that H f 2 is n ormal in H f 1 . L et σ b e an elemen t of H f 1 whic h generate s H f 1 /H f 2 . Then { σ j } p − 1 j =0 is a set of represen tativ es of H 1 /H 2 . Sup p ose that h ∈ H f 1 satisﬁes h ≡ σ j ( h ) mo d H f 2 . By the deﬁ nition of the V erlagerung homomorphisms, V er H 1 H 2 ( h ) = p − 1 Y j =0 σ − ( j + j ( h )) hσ j = h p = 1 .  Hence in our sp ecial case (the case when the exp onen t of G f is p ), we ma y identify ev ery V erlagerung h omomorphism with the F rob eniu s homo- morphism ϕ . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 63 If we set ϕ ( q y ) = q py , w e ha ve g 1 = ξ 1 2 r p + X ( b , y ) ∈ P 0 ϕ  F V 0 /F b  q y  + X ( a ,x ) ∈ P ′′ 1  F V 1 /F U 1 a  q T r F U 1 /F ( x ) and the ﬁr st summation is the same as p ositiv e degree terms o f ϕ ( f 0 ). Next, le t ˜ P 1 ′′ b e a system of representati ves of the o rb ital decomp osition Gal( F U 1 /F U 0 ) \ P ′′ 1 . By the prop ert y of the Artin sym b ol, w e ha ve p − 1 X j =0  F V 1 /F U 1 β j c  = p − 1 X j =0 β j  F V 1 /F U 1 c  β − j , therefore the righ t hand side of the equation ab ov e is con tained in I 1 since I 1 is the image of θ + 1 (trace map). Hence, w e obtain g 1 − ϕ ( f 0 ) =  ξ 1 2 r p − ϕ ( ξ 0 ) 2 r  + X ( c ,z ) ∈ ˜ P 1 ′′    p − 1 X j =0 β j  F V 1 /F U 1 c  β − j    q T r F U 1 /F ( z ) and all p ositiv e degree terms of g 1 − ϕ ( f 0 ) are conta ined in I 1 , w hic h implies ξ 1 2 r p ≡ ϕ ( ξ 0 ) 2 r mo d I S, 1 b y the philosoph y ( ♯ ) . Note that 2 p ≡ 2 mo d I S, 1 since p ∈ I S, 1 . Hence we hav e the desired congruence. In the rest of this section, w e make this argument precise and p ro v e the congruences in Prop osition 7.2. W e adopt the metho d of Ritter and W eiss ([R-W3]). 7.3. Appro ximation of ab elian p -adic zeta pseudomeasures. First, follo w ing [R-W3], w e app ro ximate the ab elian p -adic zeta pseudomeasures and r educe the congru ences a mong pseudomeasures to those among sp ecial v alues of partia l zeta functions. Let W i b e U i /V i , and let W f i b e U f i /V i . F or an arb itrary open set U ⊆ W i , w e deﬁne a non-negativ e inte ger m ( U ) by κ p − 1 ( U ) = 1 + p m ( U ) Z p where κ is the p -adic cyclotomic c h aracter. Let ω = m (Γ). Then one can easily see that m ( U f × Γ p j ) = ω + j for an arbitrary U f ⊆ W f i and j ∈ Z ≥ 0 . Lemma 7.5. The c anonic al surje ction Z p [[ W i ]] → Z p [ W i / U ] /p m ( U ) Z p [ W i / U ] induc es the isomorph ism Z p [[ W i ]] ∼ − → lim ← − U ⊆ W i : op en Z p [ W i / U ] /p m ( U ) Z p [ W i / U ] . 64 T A KASHI HARA Pr o of. The injectivit y is ob vious since m ( U ) is not b ounded. The su rjectivit y is also not diﬃcu lt, for w e can construct the lift of ( x U ) U to Z p [[ W i ]]. See [Kakde] § 4.2 .1.  It is clear that { Γ p j } j ∈ Z ≥ 0 is the coﬁnal system of the inv erse limit ab o ve. Let ε = ε i is a C -v alued lo cally constan t fun ction on W i . Th en there exists an op en sub set U ⊆ W i suc h that ε is constan t on eac h coset of W i / U . Therefore w e ca n write ε = X x ∈ W i / U ε ( x ) δ ( x ) where δ ( x ) is the charact eristic f u nction with resp ect t o a coset x , that is, δ ( x ) ( w ) = ( 1 if w ∈ x , 0 otherwise . Deﬁnition 7.6 (partial zeta function) . Let x b e an arb itrary co set in W i / U . Then w e call ζ F V i /F U i ( s, δ ( x ) ) = X 0 6 = a ⊆O F U i δ ( x )  F V i /F U i a  ( N a ) s the p artial zeta fu nc tion for δ ( x ) . Here  F V i /F U i −  is th e Artin symb ol and N a is the absolute norm of an ideal a . This function has an alytic con tin uation to the wh ole complex plane except for a simple pole at s = 1, and for every k ∈ N , ζ F V i /F U i (1 − k, δ ( x ) ) is a rational n umber . F or an arbitrary lo cal constan t function ε on W i , w e deﬁn e the partial zeta function f or ε as ζ F V i /F U i ( s, ε ) = X x ∈ W i / U ε ( x ) ζ F V i /F U i ( s, δ ( x ) ) where ε = P x ∈ W i / U ε ( x ) δ ( x ) is the d ecomp osition as ab o v e. F or an arbitrary element w ∈ W i , Q p -v alued lo cally constan t fun ction ε on W i , and positiv e in teger k divisib le b y p − 1, w e deﬁne ∆ w i (1 − k, ε ) = ζ F V i /F U i (1 − k, ε ) − κ ( w ) k ζ F V i /F U i (1 − k, ε w ) ∈ Q p where ε w ( w ′ ) = ε ( w w ′ ) for ev ery w ∈ W i . Deligne and Rib et sho wed the in - tegralit y of ∆ w i (1 − k , δ ( x ) ), that is, for an arbitrary w ∈ W i and an arbitrary coset x ∈ W i / U , ∆ w i (1 − k , δ ( x ) ) ∈ Z p ([De-Ri] Theoreme (0.4), see also Hyp othesis ( H n − 1 ) in [Coates]). Prop osition 7.7 (Appro ximation lemma, Ritter-W eiss) . L e t U ⊆ W i b e an arbitr ary op en set. Then f or every p ositive inte ge r k divisible by p − 1 and w ∈ W i , (1 − w ) ξ i maps to (7.2) X x ∈ W i / U ∆ w i (1 − k, δ ( x ) ) κ ( x ) − k x mod p m ( U ) IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 65 under the c anonic al surje ction Z p [[ W i ]] → Z p [ W i / U ] /p m ( U ) Z p [ W i / U ] . ( Note that κ ( x ) − k is wel l-deﬁne d by the deﬁnition of m ( U ) . ) Sketch of the pr o of. Let ξ w i b e the limit elemen t of (7.2). This limit ele- men t is indep end en t of k due to Deligne-Rib et’s congru ence ([De-Ri] The- oreme (0.4) and [Coates] Hyp othesis ( C 0 )). One sees that ξ w i is con tained in Z p [[ W i ]] (that is, ξ w i is a p - adic me asur e on W i ) by the integ ralit y of ∆ w i (1 − k, δ ( x ) ) and L emma 7 .5. Then one sees that Z W i εκ k dξ w i = ∆ w i (1 − k, ε ) for eac h lo cally constan t f u nction ε on W i and p ositive in teger k divisible b y p − 1, bu t this p rop erty c haracterizes th e p -adic measure (1 − w ) ξ i where ξ i is Serre’s p -adic zeta p seudomeasure for F V i /F U i ([Serre2]). Thus ξ w i = (1 − w ) ξ i . See [R- W3 ] Prop osition 2 for the more precise p ro of.  Let N U i b e th e normalizer of U i in G . Th en we ha v e N U 1 = N f U 2 = N U 3 = G and N U 2 = f U 2 . The quotien t group N U i /U i acts on the set of lo cally constan t fun ctions o n W i b y ε σ ( w ) = ε ( σ − 1 wσ ) where ε is a locally constan t functions on W i and σ ∈ N U i /U i . Prop osition 7.8 (Suﬃ cient conditions) . Suﬃcient c onditions for the c on- gruenc es (1) , f (2) , (2) , (3) in Pr op osition 7.2 to hold ar e the fol lowing (1) ′ , f (2) ′ , (2) ′ , (3) ′ r esp e ctively : (1) ′ ∆ ϕ ( g ) 1 (1 − k, ε ) ≡ ∆ g 0 (1 − pk, ε ◦ ϕ ) mo d p Z p for al l g ∈ W 0 and lo c al ly c onstant Z ( p ) -value d function ε on W 1 ﬁxe d by N U 1 /U 1 (= G/U 1 ) . f (2) ′ e ∆ ϕ ( g ) 2 (1 − k, ε ) ≡ ∆ g 0 (1 − pk, ε ◦ ϕ ) mo d p Z p for al l g ∈ W 0 and lo c al ly c onstant Z ( p ) -value d function ε on f W 2 ﬁxe d by N f U 2 / f U 2 (= G/ f U 2 ) . (2) ′ ∆ w 2 (1 − k, δ ( y ) ) ≡ 0 mo d p Z p for al l w ∈ Γ and for every c oset y ∈ W 2 / Γ p j ( j ∈ N ) which is not c ontaine d in Γ and ﬁxe d by N U 2 /U 2 (= f U 2 /U 2 ) . (3) ′ ∆ w 3 (1 − k, δ ( y ) ) ≡ 0 mo d p m y Z p for al l w ∈ Γ and for every c oset y ∈ W 3 / Γ p j ( j ∈ N ) which is not c ontaine d in Γ . Her e p m y is the or der of ( N U 3 /U 3 ) y = ( G/U 3 ) y , the isotr opic sub gr oup of G/U 3 at y . Pr o of. The pro of of the su ﬃciency of f (2) ′ (resp. (2) ′ ) for f (2) (resp. (2)) is completely th e sa me as that of (1) ′ (resp. (3) ′ ) for (1) (resp. (3)). Therefore w e only pro ve the latter t w o cases. • Th e suﬃciency of (1) ′ for (1 ). W e use simila r metho d to [R-W3], Pr op osition 4. 66 T A KASHI HARA The congruence (1) is equiv alen t to (1 − ϕ ( g )) ξ 1 ≡ ϕ ((1 − g ) ξ 0 ) mo d I 1 for an arbitrary g ∈ W 0 . Ap plying the appro ximation lemma (Prop o- sition 7.7) to them, w e obtai n (1 − ϕ ( g )) ξ 1 ≡ X y ∈ W 1 / Γ p j +1 ∆ ϕ ( g ) 1 (1 − k, δ ( y ) ) κ ( y ) − k y (7.3) in Z p [ W 1 / Γ p j +1 ] /p ω + j +1 ϕ ((1 − g ) ξ 0 ) ≡ X x ∈ W 0 / ( W f 0 × Γ p j ) ∆ g 0 (1 − pk, δ ( x ) ) κ ( x ) − pk ϕ ( x ) (7.4) in Z p [ W 0 / ( W f 0 × Γ p j )] /p ω + j Note that ϕ ((1 − g ) ξ 0 ) is ﬁxed by the conjugate action of G/U 1 since it is contai ned in Λ(Γ). ϕ ( g ) is also ﬁ xed b y G/U 1 , whic h implies that (1 − ϕ ( g )) ξ 1 is ﬁxed b y G/U 1 (see [R-W3 ] Lemma 3.2 ). Therefore we see that (1 − ϕ ( g )) ξ 1 − ϕ ((1 − g ) ξ 0 ) is con tained in ( Z p [ W 1 / Γ p j +1 ] /p ω + j +1 ) G/U 1 (the G/U 1 -ﬁxed part). (Case-1) y is ﬁxed by G/U 1 . Then δ ( y ) is also ﬁxed b y G/U 1 , therefore w e ma y apply (1) ′ for δ ( y ) : ∆ ϕ ( g ) 1 (1 − k, δ ( y ) ) ≡ ∆ g 0 (1 − pk, δ ( y ) ◦ ϕ ) mo d p Z p . If y = ϕ ( x )(= V er( x )), x is determined u niquely and κ ( y ) − k = κ (V er( x )) − k = κ ( x ) − pk . Moreo ve r, δ ( x ) coincides with δ ( y ) ◦ ϕ . This implies that ∆ ϕ ( g ) 1 (1 − k, δ ( y ) ) κ ( y ) − k y − ∆ g 0 (1 − pk, δ ( x ) ) κ ( x ) − pk ϕ ( x ) ∈ p ( Z p [ W 1 / Γ p j +1 ] /p ω + j +1 ) G/U 1 ⊆ I ( j +1) 1 /p ω + j +1 . If y / ∈ Image( ϕ ), w e h a v e δ ( y ) ◦ ϕ = 0, hence the y -summand v anishes mo dulo I ( j +1) 1 . (Case-2) y is not ﬁxed b y G/U 1 . Since ∆ ϕ ( g ) 1 (1 − k, δ ( y ) ) = ∆ ϕ ( g ) 1 (1 − k, δ ( y σ ) ) for ev ery σ ∈ G/U 1 (see [R-W 3 ] Lemma 3 .2), w e ha v e X σ ∈ G/U 1 (∆ ϕ ( g ) 1 (1 − k, δ ( y ) ) κ ( y ) − k y ) σ = ∆ ϕ ( g ) 1 (1 − k, δ ( y ) ) κ ( y ) − k X σ ∈ G/U 1 y σ ∈ I ( j +1) 1 /p ω + j +1 . Therefore (7.3) and (7.4) are congruent mo dulo I ( j +1) 1 /p ω + j +1 . T ak- ing the pro jectiv e limit, w e obtain the congruence (1) in P rop osition 7.2 (Le mma 7.5). • Th e suﬃciency of (3) ′ for (3 ). Apply the approximat ion lemma (Prop osition 7.7) to (1 − w ) ξ 3 . Then w e ha ve IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 67 (1 − w ) ξ 3 ≡ X y ∈ W 3 / Γ p j ∆ w 3 (1 − k, δ ( y ) ) κ ( y ) − k y (7.5) in Z p [ W 3 / Γ p j ] /p ω + j . Then b y (3) ′ , X σ ∈ ( G/U 3 ) / ( G/U 3 ) y ∆ w 3 (1 − k, δ ( y σ ) ) κ ( y σ ) − k y σ = ∆ w 3 (1 − k, δ ( y ) ) κ ( y ) − k X σ ∈ ( G/U 3 ) / ( G/U 3 ) y y σ ≡ p m y X σ ∈ ( G/U 3 ) / ( G/U 3 ) y σ − 1 y σ (b y (3) ′ ) ∈ I ( j ) 3 /p ω + j for eve ry y ∈ W 3 / Γ p j whic h is not con tained in Γ. Therefore it is ob vious th at the right hand side of the congruence (7.5) is con tained in ( Z p [Γ / Γ p j ] + I ( j ) 3 ) /p ω + j . By taking the pro j ectiv e limit (Lemma 7.5), w e ha ve (1 − w ) ξ 3 ≡ c w mo d I 3 for a certain elemen t c w ∈ Λ(Γ). Since 1 − w ∈ S 0 , we can obtain the c ongruen ce (3 ) in Prop osition 7.2.  7.4. Hilb ert mo dular forms and H ilb ert-Eisenstein series. In this subsection, we review Deligne-Rib et’s th eory of Hilb ert m o dular forms. See [De-Ri] and [R-W3] Section 3 for more information. Let K b e a totally real n umb er ﬁeld of degree r , K ∞ /K a totally r eal p -adic Lie extension, and Σ a ﬁxed ﬁn ite set of primes of K con taining all primes whic h ramify i n K ∞ . Let f b e an int egral ideal of O K with all prime factors in Σ. W e denote h = { τ ∈ K ⊗ C | Im( τ ) ≫ 0 } the Hilb ert upp er-half plane. F or an eve n p ositiv e in teger k , we deﬁ ne the action of GL (2 , K ) + = { g ∈ GL (2 , K ) | d et( g ) ≫ 0 } on functions F : h → C b y ( F | k  a b c d  )( τ ) = N ( ad − bc ) k / 2 N ( cτ + d ) − k F ( aτ + b cτ + d ) where N : K ⊗ C → C is the norm map. Deﬁnition 7.9 (Hilb ert mo d ular forms) . Let Γ 00 ( f ) = {  a b c d  ∈ S L (2 , K ) | a, d ∈ 1 + f , b ∈ D − 1 , c ∈ fD } 68 T A KASHI HARA where D is the diﬀeren tial of K . A Hilb ert mo dular form F of weight k on Γ 00 ( f ) is a holomorphic function 9 F : h → C satisfying F | k M = F for all M ∈ Γ 00 ( f ). W e denote the space of Hilb ert mo dular forms of w eigh t k on Γ 00 ( f ) b y M k (Γ 00 ( f ) , C ). A Hilb ert mo du lar form F h as a F ourier series expans ion (standard q - expansion) c (0) + X µ ∈O K ,µ ≫ 0 c ( µ ) q µ K where q µ K = exp(2 π √ − 1T r K/ Q ( µτ )). Deligne and Rib et constru cted the Hilb ert-Eisenstein series attac hed to ev ery lo cally constan t function ε on G = Gal( K ∞ /K ) ([De-Ri ] Theorem (6.1)). Theorem-Deﬁnition 7.10 (Hilb ert-Eisenstein series) . L et ε b e a lo c al ly c onstant fu nction on G = Gal( K ∞ /K ) and k a p ositive even inte g e r. Then ther e exists an inte gr al ide al f ⊆ O K with its prime f actors in Σ and a Hilb ert mo dular form G k ,ε ∈ M k (Γ 00 ( f ) , C ) whose standar d q - exp ansion is 2 − r ζ K ∞ /K (1 − k, ε ) + X µ ∈O K ,µ ≫ 0   X µ ∈ a ⊆O K , prime to Σ ε ( a ) κ ( a ) k − 1   Her e we denote ε ( a ) = ε  K ∞ /K a  and κ ( a ) = κ  K ∞ /K a  wher e  K ∞ /K −  is the Artin symb ol. We c al l G k ,ε the Hi lb ert-Eisenstein series of weigh t k attac hed to ε . Pr o of. See [De-Ri] Theorem (6.1).  Next let us d iscuss the q -expansion of Hilb ert mo dular form s at cusps. Let A f K b e the ﬁnite ad` ele rin g of K . By the strong appr o ximation theorem S L (2 , A f K ) = ˆ Γ 00 ( f ) · S L (2 , K ) , ev ery M ∈ S L (2 , A f K ) is decomp osed as M = M 1 M 2 where M 1 ∈ ˆ Γ 00 ( f ) (the closure of Γ 00 ( f ) in S L (2 , A f K )) and M 2 ∈ S L (2 , K ). W e deﬁne F | k M to be F | k M 2 . F or ev ery ﬁn ite id` ele α ∈ ( A f K ) × , set F α = F | k  α 0 0 α − 1  . F α also has a F ourier series expansion F α = c (0 , α ) + X µ ∈O K ,µ ≫ 0 c ( µ, α ) q µ K , and w e cal l it the q -e xp ansion of F at the cusp determine d by α . 9 If K = Q , we also assume th at F is holomorphic at ∞ . IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 69 Prop osition 7.11. L et k b e a p ositive even inte g er a nd ε a lo c al ly c onstant function on G . Then the q -exp ansion of G k ,ε at the cu sp determine d by α ∈ ( A f K ) × is given by N (( α )) k ( 2 − r ζ K ∞ /K (1 − k, ε a ) + X µ ∈O K ,µ ≫ 0   X µ ∈ a ⊆O K , prime to Σ ε a ( a ) κ ( a ) k − 1   ) wher e ( α ) is the ide al gener ate d by α and a =  K ∞ /K ( α ) α − 1  . Pr o of. See [De-Ri] Theorem (6.1).  No w w e in tro duce the Deligne-Rib et’s deep p rinciple. Theorem 7.12 (Deligne-Ribet) . L et F k ( k ≥ 0) b e r ational Hilb ert mo d- ular forms of weight k on Γ 00 ( f ) ( tha t is, al l c o eﬃcie nts of the q -exp ansion of F k at every cusp ar e r ational numb ers ) , and F k = 0 for al l but ﬁnitely many k . L et α ∈ ( A f K ) × . We denote by α p the p -th c omp onent of α . Set S ( α ) = X k ≥ 0 N α − k p F k ,α . If S ( α ) has al l c o eﬃcients in p j Z p ( j ∈ Z ) for one α ∈ ( A f K ) × , S ( α ) has al l c o eﬃcients in p j Z p for ev ery α ∈ ( A f K ) × . Pr o of. See [De-Ri] Theorem (0.2).  Corollary 7.13 ([De-Ri] C orollary (0.3)) . L et S ( α ) as in The or em 7.12. Supp ose that ther e exists α ∈ ( A f K ) × such tha t al l non-c onstant c o eﬃcients of S ( α ) ar e c ontaine d in p j Z ( p ) ( j ∈ Z ) . Then for arbitr ary two distinct elements β , β ′ ∈ ( A f K ) × , the diﬀer enc e b etwe en the c onstant terms of the q -exp ansions of S ( β ) and S ( β ′ ) is an element of p j Z ( p ) . Pr o of. Let c (0 , α ) b e the constan t term of S ( α ). W e may interpret c (0 , α ) as an element o f M 0 (Γ 00 ( f ) , Q ). Set S ( β ) c (0 ,α ) = S ( β ) − c (0 , α ). Then S ( α ) c (0 ,α ) has all co eﬃcients in p j Z ( p ) , so do es S ( β ) c (0 ,α ) b y Theorem 7.12. Esp ecially the constant term of S ( β ) c (0 ,α ) is also an elemen t of p j Z ( p ) , b ut it is no other than c (0 , β ) − c (0 , α ).  7.5. Pro of of t he suﬃcien t conditions . No w we p ro v e the suﬃcient con- ditions in Proposition 7.8. W e only pro v e the conditions (1) ′ and (3) ′ . O ne can pro ve the conditions f (2) ′ (resp. (2) ′ ) in the same manner as (1) ′ (resp. (3) ′ ). Condition (1) ′ . Let k b e a p ositiv e ev en integ er and ε a locally constant Z ( p ) -v alued fun ction on W 1 ﬁxed b y the action of G/U 1 . Let G k ,ε (resp. G pk ,ε ◦ ϕ ) b e the Hilb ert-Eisenstein series of w eigh t k (resp . w eigh t pk ) at- tac h ed to ε (resp. ε ◦ ϕ ) ( Th eorem-Deﬁnition 7. 10 ). 70 T A KASHI HARA The natural in clusion F → F U 1 induces the r estriction of the Hilb ert- Eisenstein series res G k ,ε on h F . It is easy to see that res G k ,ε is a Hilb ert mo dular form of weig ht pk and its stand ard q -expansion is giv en b y ([R-W3] Lemma 7 ) res G k ,ε = 2 − [ F U 1 : Q ] ζ F V 1 /F U 1 (1 − k, ε ) + X µ ∈O F ,µ ≫ 0   X ( b , ν ) ∈ P µ 1 ε ( b ) κ ( b ) k − 1   q µ F where q µ F = exp(2 π √ − 1T r F / Q ( µτ )) and P µ 1 = { ( b , ν ) | ν ∈ b ⊆ O F U 1 , b is p rime to Σ , ν ≫ 0 , T r F U 1 /F ( ν ) = µ } . F or λ ∈ O F , we m a y construct the Hec ke op erator U λ asso ciated to λ (See [R-W3] Lemm a 6), wh ic h “shifts the co eﬃcien ts of q -expansion b y λ .” Therefore w e ha ve (res G k ,ε ) | pk U p = 2 − [ F U 1 : Q ] ζ F V 1 /F U 1 (1 − k, ε ) + X µ ∈O F ,µ ≫ 0   X ( b , ν ) ∈ P pµ 1 ε ( b ) κ ( b ) k − 1   q µ F On the other hand, the standard q -expan s ion o f G pk ,ε ◦ ϕ is G pk ,ε ◦ ϕ = 2 − [ F : Q ] ζ F V 0 /F U 0 (1 − pk, ε ◦ ϕ ) + X µ ∈O F ,µ ≫ 0   X µ ∈ a ⊆O F , prime to Σ ε ◦ ϕ  F V 0 /F U 0 a  κ ( a ) pk − 1   q µ F . and note that ϕ  F V 1 /F U 1 a  = V er  F V 1 /F U 1 a  = F V 0 /F U 0 a O F U 1 ! b y the comm u tativit y of the Artin symbol and the V erlagerung homomor- phism (see Lemma 7.4). Set S = (res G k ,ε ) | pk U p − G pk ,ε ◦ ϕ , t hen the µ -th c o eﬃcient o f S is X ( b , ν ) ∈ P pµ 1 ε ( b ) κ ( b ) k − 1 − X µ ∈ a ⊆O F ε ( a O F U 1 ) κ ( a ) pk − 1 Let ( G/U 1 ) ( b , ν ) b e the isotropic sub group of G/U 1 at ( b , ν ) ∈ P pµ 1 . (Case-1) ( G/U 1 ) ( b , ν ) = { id } . Note that ε  F V 1 /F U 1 b σ  = ε  σ − 1  F V 1 /F U 1 b  σ  = ε σ  F V 1 /F U 1 b  = ε  F V 1 /F U 1 b  IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 71 for eve ry σ ∈ G/U 1 . Similarly we hav e κ ( b σ ) k − 1 = κ ( b ) k − 1 . There- fore the su m of ( b , ν )- orbit is giv en b y X σ ∈ G/U 1 ε ( b σ ) κ ( b σ ) k − 1 = pε ( b ) κ ( b ) k − 1 ∈ p Z ( p ) . (Case-2) ( G/U 1 ) ( b , ν ) = G/U 1 . In this case G/U 1 ﬁxes ( b , ν ), therefore ν ∈ F and b = a O F U 1 for unique in tegral ideal a of F p rime to Σ. Since T r F U 1 /F ( ν ) = pµ , w e hav e ν = µ . Therefore ε ( b ) κ ( b ) k − 1 = ε ( a O F U 1 ) κ ( a O F U 1 ) k − 1 = ε ( a O F U 1 ) κ ( a ) p ( k − 1) ≡ ε ( a O F U 1 ) κ ( a ) pk − 1 mo d p and so th e ( a O F , µ )-term v anish es modu lo p . Therefore S has a ll t he n on-constan t co eﬃcient s in p Z ( p ) , and we can ap- ply Deligne- Rib et’s p rinciple (Corollary 7.13) to S = S (1). There exists a ﬁnite id ` ele γ suc h that  F V 0 /F U 0 (( γ )) γ − 1  = g (see [De-Ri] (2.23)). T hen b y Corol- lary 7.13 S (1) − S ( γ ) h as its constan t term in p Z ( p ) . By easy calculation, the c onstant te rm of E − E ( γ ) is 2 − pr ζ F V 1 /F U 1 (1 − k, ε ) − 2 − r ζ F V 0 /F U 0 (1 − pk, ε ◦ ϕ ) − N ( γ p ) − pk N (( γ )) pk × { 2 − pr ζ F V 1 /F U 1 (1 − k, ε ϕ ( g ) ) − 2 − r ζ F V 0 /F U 0 (1 − pk, ( ε ◦ ϕ ) g ) } =2 − pr { ζ F V 1 /F U 1 (1 − k, ε ) − κ ( ϕ ( g )) k ζ F V 1 /F U 1 (1 − k, ε ϕ ( g ) ) } − 2 − r { ζ F V 0 /F U 0 (1 − pk, ε ◦ ϕ ) − κ ( g ) pk ζ F V 0 /F U 0 (1 − pk, ( ε ◦ ϕ ) g ) } =2 − pr ∆ ϕ ( g ) 1 (1 − k , ε ) − 2 − r ∆ g 0 (1 − pk, ε ◦ ϕ ) ≡ 2 − r { ∆ ϕ ( g ) 1 (1 − k, ε ) − ∆ g 0 (1 − pk, ε ◦ ϕ ) } mo d p. Here we set r = [ F : Q ] and use N (( γ )) = N ( γ p ) κ ( g ) ([R-W3] Lemma 9), κ ( ϕ ( g )) = κ ( g ) p . F or pr ecise calculation, s ee [R-W3], the pro of of the Theorem. Condition (3) ′ . Let j b e a suﬃcien tly large int eger and y ∈ W 3 / Γ p j is a coset whic h is not con tained in Γ. Let S = G k ,δ ( y ) b e the Hilb ert-Eisenstein series of w eigh t k attac hed to δ ( y ) , then th e st and ard q -expansion o f S is giv en b y S = 2 − [ F U 3 : Q ] ζ F V 3 /F U 3 (1 − k, δ ( y ) ) + X ν ∈O F U 3 ,µ ≫ 0    X ν ∈ b ⊆O F U 3 , prime t o Σ δ ( y ) ( b ) κ ( b ) k − 1    q ν F U 3 where q ν F U 3 = exp(2 π √ − 1T r F U 3 / Q ( ν τ )). (Case-1) ( G/U 3 ) ( b , ν ) = { id } . I n this case, we can easily calculate th e su m of ( b , ν )-orbit : X σ ∈ G/U 3 δ ( y ) ( b σ ) κ ( b σ ) k − 1 = p m y X σ ∈ ( G/U 3 ) / ( G/U 3 ) y δ ( y σ − 1 ) ( b ) κ ( b ) k − 1 . 72 T A KASHI HARA (Case-2) ( G/U 3 ) ( b , ν ) 6 = { id } . Let F ( b , ν ) b e the ﬁxed ﬁeld of ( G/U 3 ) ( b , ν ) . Then b y the same argument as the pro of of the condition (1) ′ (Case-2), there exists µ ∈ F ( b , ν ) and a ⊆ O F ( b ,ν ) uniquely suc h that ( b , ν ) = ( a O F U 3 , µ ). F or such ( a , µ ), δ ( y ) ( b ) = δ ( y ) F U 3 /F V 3 a O F U 3 !! = δ ( y ) ◦ V er  F V 3 /F ( b , ν ) a  = 0 b ecause Im(V er) = Im( ϕ ) ⊆ Γ. Therefore S has all the non-constan t co eﬃcien ts in p m y Z ( p ) . T ake a ﬁn ite id ` ele γ ′ suc h that  F V 3 /F U 3 ( γ ′ ) γ ′ − 1  = w . Then by Corollary 7.13, the constant term of S − S ( γ ′ ) i s also in p m y Z ( p ) , and it is equal to 2 − p 3 r ∆ w 3 (1 − k, δ ( y ) ) . Th us w e ha v e ﬁnished the p ro of o f Prop osition 7.8. 8. Proof of the main theorem Unfortunately , we cannot conclude that ( ξ i ) i is contai ned in Ψ S b y the congruences obtained in the previous sectio n, so w e ma y n ot app ly Burns’ tec h nique (Theorem 3.4) d irectly to ( ξ i ) i . In this section, we mod ify the pro of of Theorem 3.4 and pro v e our main theorem (Theorem 3.1) using an i nd uctiv e tec hn ique. 8.1. Kato’s p -adic zeta function for F N /F . Let N =     1 0 0 F p 0 1 0 F p 0 0 1 F p 0 0 0 1     × { 1 } b e a closed n ormal subgroup of G and set G = G/ N , U i = U i / N and V i = V i · N/ N . Then w e ha ve the splitting exact sequence 1 − − − − → N − − − − → G π − − − − → G − − − − → 1 . Let s : G → G ;     1 a d 0 1 b 0 0 1   , t z   7→         1 a d 0 0 1 b 0 0 0 1 0 0 0 0 1     , t z     b e a section of π . The p -adic Lie group G =   1 F p F p 0 1 F p 0 0 1   × Γ IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 73 is a group “of He isenberg t yp e,” for wh ic h Kazuya Kato has already prov en the e xistence of the p -a dic zeta function in [Kato1]. Theorem 8.1 ( Kato) . The p - adic zeta function ξ for F N /F exists uniqu e ly and it satisﬁes the Iwasawa main c onje ctur e ( Conje ctur e 2.15 (2)) . Sketch of the pr o of. This theorem is the sp ecial case of [Kato1], Theorem 4.1. First, he constru cted th e theta map (and its l o calized v ersion) θ : K 1 (Λ( G )) → Ψ ⊆ Λ( U 0 /V 0 ) × × Λ( U 1 /V 1 ) × θ S : K 1 (Λ( G ) S ) → Ψ S ⊆ Λ( U 0 /V 0 ) × S × Λ( U 1 /V 1 ) × S where Ψ (resp. Ψ S ) w as deﬁned to b e th e su bgroup consisting of all elemen ts ( η 0 , η 1 ) whic h satisﬁed the norm relation Nr Λ( U 0 /V 0 ) / Λ( U 1 /V 1 ) η 0 ≡ η 1 (resp. Nr Λ( U 0 /V 0 ) S / Λ( U 1 /V 1 ) S η 0 ≡ η 1 ) and the congruence η 1 ≡ ϕ ( η 0 ) mo d I 1 (resp. mod I S, 1 ) . In t his case, w e could sh o w the c ongruen ce ξ 1 ≡ ϕ ( ξ 0 ) mo d I S, 1 directly in the same manner as in § 7. Hence, by using Burns’ tec hnique (Theorem 3. 4), we migh t sho w the existence (and uniqueness) o f the p -a dic zeta function ξ for F N /F .  Let C = C F ∞ /F b e the complex deﬁned in Deﬁn ition 2.8. Since we alw a ys assum e the condition ( ∗ ) in Prop osition 2.10, C is con tained in K 0 (Λ( G ) , Λ( G ) S ). Let [ C ] b e its image in K 0 (Λ( G ) , Λ( G ) S ). Then Kato’s p -adic zet a fun ction ξ satisﬁes the main conjecture ∂ ( ξ ) = − [ C ]. Prop osition 8.2. Ther e exists a char acteristic element f ∈ K 1 (Λ( G ) S ) fo r F ∞ /F whose image in K 1 (Λ( G ) S ) c oincides with ξ . Pr o of. In the follo wing, w e den ote b y x the image of x in K 1 (Λ( G ) S ) for an elemen t x ∈ K 1 (Λ( G ) S ). Let f ′ ∈ K 1 (Λ( G ) S ) b e a charact eristic elemen t of [ C ]. Then by using the functorialit y of the connecting homomorphism ∂ , w e ha v e ∂ ( ξ · f ′ − 1 ) = − [ C ] + [ C ] = 0. By the lo calization exact sequence (Th eorem 1.22 ), u = ξ · f ′ − 1 is the image of an e lement o f K 1 (Λ( G )), whic h w e also d en ote u by abuse of notation. Then the element f = f ′ s ( u ) satisﬁes the assertion of the prop osition wh ere s denotes the h omomorp hism K 1 (Λ( G )) → K 1 (Λ( G )) induced b y s : G → G .  8.2. Completion of the pro of. Let f ∈ K 1 (Λ( G ) S ) b e a c haracteristic elemen t for F ∞ /F satisfying Prop osition 8.2, t hat is, π ( f ) = ξ . Let f i = θ S,i ( f ) ∈ Λ( U i /V i ) × S and let u i = ξ i f − 1 i . Th en ∂ ( u i ) = 0, so w e ha v e u i ∈ Λ( U i /V i ) × b y the localization sequence (Theorem 1.22 ). Then it is suﬃ cien t to sho w that ( u i ) i is con tained in Ψ: if ( u i ) i is con- tained in Ψ , there exists u ∈ K 1 (Λ( G )) such that θ i ( u ) = u i b y the surjec- tivit y o f the theta map (Proposition 5.4 ). One can easily c h ec k that ξ = uf 74 T A KASHI HARA satisﬁes the interp olating prop erties of the p -adic zeta function for F ∞ /F (Deﬁnition 2.14) and also satisﬁes ∂ ( ξ ) = − [ C ]. Namely , ξ is the desired p -adic zet a fun ction. Note that i t w as easy to sho w t hat ( u i ) i ∈ Ψ in the pro of of Theorem 3.4 b ecause of th e a ssu m ption ( ξ i ) i ∈ Ψ S . Prop osition 8.3. u i ’s satisfy the norm r elations in Deﬁnition 5.3, and satisfy fol lowing c ongruenc es : u 1 ≡ ϕ ( u 0 ) mo d I 1 , f u 2 ≡ ϕ ( u 0 ) mo d e I 2 , u 2 ≡ d 2 mo d I ′′ 2 , u 3 ≡ d 3 mo d I 3 , wher e d 2 and d 3 ar e c ertain e lements of Λ(Γ) . Pr o of. f i ’s a nd ξ i ’s sat isfy the norm rela tions, so do u i ’s. Since ( f i ) i is cont ained in Ψ S , f i ’s sat isfy the congruences in Prop osition 6.2. On the other hand, ξ i ’s satisfy the congruences in Prop osition 7.2. Hence, w e can easily sho w that u i ’s sat isfy the desired c ongru en ces. 10  Lemma 8.4. ( u i ) i ∈ Ker π × : Y i Λ( U i /V i ) × − → Y i Λ( U i /V i ) × ! . Pr o of. This lemma follo ws from the construction of f and the commutati vity of π and norm map s of K -group s (Pr op osition 1.9).  Lemma 8.5. s ◦ π ( I i ) ⊆ I i . Pr o of. Just simple calculation.  Lemma 8.6. F or e ach i , ϕ ( u i ) = 1 . Pr o of. The F r ob enius homomorphism ϕ : Λ( U i /V i ) → Λ(Γ) factors as Λ( U i /V i ) ϕ − − − − → Λ(Γ) π   y k x   s Λ( U i /V i ) − − − − → ϕ Λ(Γ) where ϕ : Λ( U i /V i ) − → Λ(Γ) is the F rob enius homomorphism (mo d N ). Then this lemma holds since ( u i ) i ∈ Ker( π × ) b y Lemma 8.4.  Pr o of of The or e m 3.1 . By op erating s ◦ π to the congruences of Prop osition 8.3, w e h a v e s ◦ π ( u 2 ) ≡ d 2 mo d s ◦ π ( I ′′ 2 ) , s ◦ π ( u 3 ) ≡ d 3 mo d s ◦ π ( I 3 ) . By Lemma 8.4, b oth of s ◦ π ( u 2 ) and s ◦ π ( u 3 ) are 1. Therefore w e obtain u 2 ≡ d 2 ≡ 1 = ϕ ( u 2 ) mo d I ′′ 2 , (8.1) u 3 ≡ d 3 ≡ 1 = ϕ ( u 3 ) mo d I 3 , (8.2) 10 Note that we may replace c 2 and c 3 in Prop osition 7.2 by in vertible elemen ts. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 75 b y Lemma 8.5 and 8.6. Hence if we sho w that u 2 ≡ 1(= ϕ ( u 2 )) mo d I 2 , (8.3) w e ha v e ( u i ) i ∈ Ψ, whic h is the desired result. No w we s ho w the congruence (8.3). W e kno w that u 2 ∈ 1 + I ′′ 2 and u 3 ∈ 1 + I 3 b y (8.1) and (8.2). Note that the p -adic logarithmic homomorph ism induces an iso morp hism of ab elian groups 1+ I ′′ 2 ≃ − → I ′′ 2 b y the same argumen t as t he pro of of Lemma 5.6. By the logarithmic isomorphisms (Lemma 5.6), w e ha v e log u 2 ∈ I ′′ 2 and log u 3 ∈ I 3 . W e ma y describ e log u 2 and log u 3 as Λ(Γ)-linear com binations of gener- ators of I ′′ 2 and I 3 : log u 2 = X b,c 6 =0 e ν (3) bc β b γ c h ζ + X b,f p e ν (4) bf β b ζ f , log u 3 = X f p 3 σ (1) f ζ f + X e 6 =0 p 2 σ (2) e ε e h ζ + X c 6 =0 pσ (3) c γ c h ε h ζ . Then w e ha ve T r Λ( U 2 /V 2 ) / Λ( U 3 /V 2 ) log u 2 = X c 6 =0 p e ν (3) 0 c γ c h ζ + X f p 2 e ν (4) 0 f h ζ and log u 3 ≡ X f p 2   pσ (1) f + X e 6 =0 σ (2) e   ζ f + X c 6 =0 p 2 σ (3) c γ c h ζ (mo d V 2 ) . By c omparing the coeﬃcien ts, w e ha ve e ν (4) 0 f = pσ (1) f + X e 6 =0 σ (2) e , e ν (3) 0 c = pσ (3) c ( c 6 = 0) . Therefore if we set ν (1) f = σ (1) f , ν (2) c = ( P e 6 =0 σ (2) e if c = 0 , σ (3) c if c 6 = 0 , ν (3) bc = e ν (3) bc ( b 6 = 0 , c 6 = 0) , ν (4) bf = e ν (4) bf ( b 6 = 0) , w e ha v e log u 2 = X f p 2 ν (1) f ζ f + X c pν (2) c γ c h ζ + X b 6 =0 ,c 6 =0 ν (3) bc β b γ c h ζ + X b 6 =0 ,f pν (4) bf β b ζ f whic h implies log u 2 ∈ I 2 . Hence by the logarithmic isomorphism (Prop osi- tion 5.6), u 2 is co nta ined in 1 + I 2 , whic h implies the congruence (8.3).  76 T A KASHI HARA Remark 8.7. By the c onstru ction of ξ = uf , w e ha v e θ S ( ξ ) = ( ξ i ) i . Since Ψ S con tains the image of θ S , we esp ecially obtain the follo wing non-trivial congruences among ab elian p -adic pseu domeasures: ξ 2 ≡ ϕ ( ξ 0 ) p mo d I 2 , ξ 3 ≡ ϕ ( ξ 0 ) p 2 mo d I 3 . It seems to b e imp ossible to sh o w th ese congruences dir ectly b y using only the theory of Delig ne-Rib et. Referen ces [Bass1] Bass, H., Algebr aic K-the ory , Benjamin (1968). [Bass2] Bass, H ., Intr o duction to some metho ds of algebr aic K -the ory , C onference Board of the Mathematical Sciences Regio nal Conference Series in Mathematics, No. 20 . American Mathematical So ciet y , Providence, R. I . (1974). [Ber-Keat] Berrick, A. J., and Keating, M. E., The l o c alization se quenc e in K -t he ory , K -Theory 9 (1995), 577-589. [Coates] Coates, J., p -adic L -functions and Iwasawa’s the ory , in Algebr aic Numb er Fiel ds : L -functions and Galois pr op erties , Academic Press, London (1977). [CFKSV] Coates, J., F uka ya, T., Kato, K ., Sujatha, R., and V en jak ob, O ., The GL 2 main c onje ctur e for el liptic curves without c omplex multipl ic ation , Publ. Ma th. Inst. Hautes ´ Etudes Sci. 101 (2005), 163-208. [De-Ri] Deligne, P ., and Ribet, K . A., V alues of a b elian L -functions at ne gative inte gers over total ly r e al ﬁelds , I nven t. Math. 59 (1980), 227-286. [F u- Ka] F uk ay a, T., and Kato, K., A f ormulation of c onje ctur es on p -adic zeta functions in nonc ommutative Iwasawa the ory , Amer. Math. So c. T ransl. Ser. 2, 219 (2006) [FW] F errero, B., and W ashington, L. C., The Iwasawa i nvariant µ p vanishes for ab elian numb er ﬁelds , Ann. of Math. 109 (1979), 377-395. [Gra yson] Gra yson, D . R., K -the ory and lo c alization of nonc ommutative rings , J. Pure Appl. Algebra 18 (1980), 125-127. 18F25 (57R65) [Ha-Sh] H achimori, Y., and Shariﬁ, R. T., On the failur e of pseudo- nul lity of Iwasawa mo dul es , J. A lgebraic Geom. 14 (2005), 567-591. [Kakde] Kakde, M., Pr o of of the main c onje ctur e of nonc ommutative Iwasawa the ory for total ly r e al numb er ﬁelds in c ertain c ases , arXiv:0802.227 2 v2 [math.NT] . [Kato1] Kato, K., Iwasawa the ory of total ly r e al ﬁelds for Galois extensions of Heisenb er g typ e , preprint. [Kato2] Kato, K., K 1 of some non-c omm utative c omplete d gr oup rings , K -Theory , 34 (2005), 99-140. [Milnor] Milnor, J. , Intr o duction to Algebr aic K -the ory , Princeton Univ. Press, Princeton, N.J., (1971). [Oliv er] Oliver, R., Whitehe ad gr oups of ﬁnite gr oups , London Mathematical Society Lec- ture Note S eries 132 (1988), Cam bridge Univ. Press. [Oli-T ay] Oliver, R., and T a ylor, L. R., L o garithmic descriptions of Whitehe ad gr oups and class gr oups for p -gr oups , Mem. Amer. Math. S oc. 76 (1988), no. 392. [R-W1] Ritter, J ., and W eiss, A., T owar d e quivariant Iwasawa the ory , Part I , Man uscripta Math. 109 ( 2002), 131 -146, P art I I, Indag. Math. (N.S.) 15 (2004), 549-572, PartII I , Math. Ann . 336 (2006), 27-49, Part IV, Homology , Homotop y Appl. 7 (2005), 155- 171(electronic). [R-W2] Ritter, J., and W eiss, A., Non-ab elian pseudome asur es and c ongruenc es b etwe en ab eli an Iwasawa L-functions , Pure Appl. Math. Q. 4 (2008), no. 4, part 1, 1085-1106. [R-W3] Ritter, J., and W eiss, A., Congruenc es b etwe en ab elian pseudome asur es , Math. Res. Lett. 15 (2008), no. 4, 715-725. [R-W4] Ritter, J., and W eiss, A., Equivariant Iwasawa the ory: an example , Doc. Math. 13 (2008), 117-129. IW ASA W A THEOR Y FOR NON-COMMUT A TIVE p -EXTENSIONS 77 [Sch-V en] Schneider, P ., and V enjakob, O., On the c o dimension of mo dules over skew p ower series rings with applic ations to Iwasawa algebr as , J. Pure Appl. Algebra 204 (2006), 349-367. [Serre1] Serre, J.-P ., R epr ´ esentations lin´ eair es des gr oup es ﬁn is , Hermann (1967). [Serre2] Serre, J.-P ., Sur l e r´ esidu de l a fonction z ˆ eta p -adique d’un c orps de nombr es , C. R. Acad. S ci., Pari s 287 (1978), s´ erie A, 183-188. [Sten] Stenstr¨ om, B., Rings of Q uotients , Springer-V erlag, New Y ork-Heidelb erg (1975). [Sw an] Swa n, R. G., Alge br aic K -the ory , Lecture Notes in Mathematics, 76 , Springer- V erlag, Berlin-New Y ork (1968). [W all] W all, C. T. C., Norms of units in gr oup rings , Proc. London Ma th. Soc. (3), 29 (1974), 593-632. [Wiles] Wiles, A., The Iwasawa c onje ctur e for total ly r e al ﬁelds , Ann. of Math. Second Ser., 131 (1990), no.3, 493-540. Gradua te School of Ma thema ti cal S ciences, The Un iversity of Tokyo, 8-1 Ko maba 3-chome, Meguro -ku, Tok yo, 15 3-8914, Jap an E-mail addr ess : thara@ms.u-t okyo.ac.jp

Iwasawa theory of totally real fields for certain non-commutative $p$-extensions

Original Paper

Comments & Academic Discussion

Leave a Comment

Original Paper

Related Papers

Comments & Academic Discussion

Leave a Comment