Journal of the Mathematical Society of Japan Vol. 7, Supplement, December,1955
Class formations III
By Yukiyosi KAWADA
(Received Nov. 25, 1955)
This paper is a continuation of our former papers (Kawada [7], Kawada-rate [8]) concerning class formations and their application in algebraic function fields in one variable. After some preliminaries in § 1 we shall consider in § 2 the system of groups { W(K/k)} in a class formation, which was investigated by A. Weil [12] in case of number fields. We shall arrange the formulas for { W(KK/k)} so that we are able to consider their inverse limit groups in § 3. The sections § 4, § 5 are devoted to the application of the results in § 2, § 3 to the case of algebraic function fields. There we shall find the explicit structure of these groups { W(K/k) } and their limit groups using the formulations in [8]. The results of A. Weil [12] were treated cohomology-theoreti- cally first by T. Nakayama and G. Hochschild [5], [10]. Though the results in § 2, § 3 of this paper are not published hitherto in the literature, they would be known by mathematicians working in this field. The author does not claim any priority on these results. It should be mentioned that there are unpublished investigations of E. Artin and J. Tate concerning the structure of the inverse limit groups of { W(K/k) } in case of number fields. Also the explicit structure of { W(k) } in Theorem 6 (~ 5) was suggested by J. Tate. The author wishes to express his hearty thanks to Professor John Tate for his discussions during the preparation of this paper.
§ I. Preliminaries I. We repeat here some necessary preparations from Part I (Kawada [7]) which we need later. Let ko be a fixed ground field and 12 be a fixed infinite separable normal algebraic extension of k0. Let S be the set of all finite extensions of ko which are contained 454 Y. KAWABA in 2. In the following we denote the fields which belong to S by k, 1,K,Letc. We assume further that a certain abelian group A(K) is as- sociated with each K E S with the following properties : F 1. Let kCK then there exists an isomorphism qiK ,k of A(k) into A(K). F 2. Let kcl cK then the transitivity relation qK,l°glk=q'K,k holds. F 3. Let K/k be a normal extension with the Galois group G=G(K/k)'~ then G acts on A(K) as an operator group such that qK,kA(k)=A(K)G holds.2~ F 4. In CASE I : k ci Kc L, K/k and L/k are both normal with the Galois groins G = G(L/k), H= G(L/K) and F= G(K/k) the relation o- o qi ,x(a) = q'L,x ° (*F,G°-)(a)3) holds for every o E G and a A(K). Furthermore, for every normal extension K/k with G= G(K/k) we assume C 1: H' (G, A(K)) = 0 and C 2 : H2(G, A(K)) Z/[G :1] • Z. We call then {S, A(K) } a class formation. Let {, A(K) } be a class formation. Then we can choose for each normal extension K/k a 2-cocycle fKI/ of G over A(K) with the following properties: (i) the cohomology class of fK/k is a generic class of H2(G, A(K)), (ii) in case I we have inflG, F fKlk^-fLr'k(m= [L: K]), (iii) in CASE II: k cit c K, K/k is normal with G = G(K/k) and H= G(K/l ) we have resH,GfKlk^'fKll •1) We call then that {fK/k} is a system of fundamental (canonical) 2-cocycles in this class formation. Let {K/k} be another system of 2-cocycles. Then {K/k} is also a system of fundamental 2-cocycles if and only if (i) gxlk"'fKlklk~ rKlk and [K: k] are relatively prime; (ii) in case I rLlk=rKlk (mod [K: k]); (iii) In case II rKil=rK/k (mod [K: l]). Now let us consider the group pairing (Z, A)-~A by (n, a)-4a2.
1) We use the notation G(K/k) to denote the Galois group of a normal extension K/k. 2) In a G-group A we assume alz=(ar )d. By AG we mean the set of all elements a e A which are invariant by all o~E G. 3) For a subgroup H of G we mean by 'G,H the injection mapping H-~G and for a normal subgroup H of G and F=G/H we mean by 'F, G the canonical homomorphism G->F. As usual we mean by Z, R and C the modules of all integers, all real numbers and all complex numbers respectively. 4) By res H,G we mean the restriction mapping: H"(G, A)->H"(H, A) for an abelian G-group A in case II, and by inflG, F we mean the inflation mapping (lift) : H'(F, AH)-> II''r(G, A) in case I. Class formations III. 455
Then the cup-multiplication g'-g' U fK/k gives the isomorphism: Hr(G, Z)~Wr+2(G, A(K)) for all rEZ (Theorem of Tate [11]). In particular, for r= -2 112(G, Z)~G/Gc is isomorphic to H°(G, A(K)) A(K)G/NGA(K)5) by the mapping a- (mod G) -~ fK,k(G, a-) (mod NGA(K))6~(Isomorphism theorem). We define the norm-residue symbol (a, K/k)=o- (mod Gc) for qi ,k(a)=fKlk(G, a-) (mod NGA(K)). Let Ak be the maximal abelian extension of k in 2 and we denote by I7(k) the compact Galois group G(Ak/k). The generalized norm-residue symbol (a, k) (a EEA(k)) which takes value in F(k) is defined as the limit of (a, KA/k) for k C K,, C Ak. The definition (a, k) depends on the choice of a system of fundamental 2-cocycles {fKlk}. But if we define (a, k)' by menas of another system {gKlk},then there exists an isomorphism qi of I'(k) onto itself such that (a, k)'= ji o(a, k) holds for all a A(k). The following two axioms are satisfied in case of class field theory and in several other cases of class formations. P. For any extension K/k (k, KE S) there exists a cyclic extension Z/k such that [Z: k] is a multiple of [K: k]. P'. There exists an increasing sequence of fields K1, K2,..., Ku,... in S such that 2= U K,2 holds. n=1 LEMMA1. Suppose that the axiom P is satisfied in S. I f the generalized norm-residue symbols (a, k) (a A(k)) defined by two systems {fKIk} and {gKIk} of fundamental 2-cocycles coincide, then fKIk-gKJk holds for every normal K/k. (PROOF) Let gKlk-fx klk then we have (a, K/k)"CI k= (a, K/k) by our assumption. Especially this implies rZIk-1 (mod [Z: k]) for cyclic extension Z/k. Now for an arbitrary extension K/k take a cyclic extension Z/k such that [Z: k] is a multiple of [K: k]. Then from the relations rKZIk-rZIk (mod [Z: k]) and rKZlk=rKlk (mod [K: k]) follows that rKlk-1 (mod [K: k]). This means fK/k-gK/k,-q. e. d. 2. We remark here some elementary properties of group ex- tensions. Let A, E, G be groups, c, Jt be homomorphisms such that
c (1) 1--- A---*E-- *G--*1 5) Gc means the commutator group of G. NG means the norm with respect to G. 6) fKIk(G,o")=HreG1Klk(T, d) 456 Y. KAWADA is an exact sequence. Then we say that E is a group extension of A by G and we denote E(A, G t, v). By a homomorphism µ(X, v) of E(A, G, t, fr) into E'(A', G', ~', fr') we mean a set of homomorphisms x : A -A',A', ,u: E--~ E', v : G- ~ G' such that t 'fr 1-- ~A --~E -*G - --~1 (2) x t 1->A'---E'----G'-----+i is a commutative diagram. In the rest of this section we assume that the group A is abelian. Then G can be considered as an operator group of A by a~ = t-' (u~• t(a) • uc') (o G, a ETA) where {u~} is a system of representatives of { Jr-' (o-)c E; o-E G }. (3) f(o-, T)= t-'(uQuzzc) (a-, T E G) is a 2-cocycle of G over the G-group A. We call such a pair 2-({u}, {f(o-, T)}) a frame of E(A. G, t, tJr). Any two pairs ({u~}, {f(a-, T)}) and ({v6}, {g(a-,T) }) are related by suitable b;s (bQE A) by vQ= t(b6)u6, with g(o-,T) = f(a-, T) bdbTb;V'. For a fixed 2-cocycle {f(o-, T)} there are as many different pairs ({u~}, {f(o-,T)}) as sets {b~; aEG(b6E A)} which satisfies bUb°b6r'=1. The cohomology class of {f (a-, T)} is uniquely determined by E(A, G, c, ijc) and is inde- pendent of the choice of the representatives {u6}. So we call it the 2-cohomology class associated with E. Suppose that two group extensions E(A, G, ~, J.r) and E'(A', G', t', J'), a homomorphism A : A-~A' and an isomorphism v: G-~G' (into) are given such that X(a6)_ x(a)) (a A, o- G) holds. Then there exists a homomorphism µ(x, v) : E(A, G, t, fr)-~E'(A', G', t', fr') if and only if the with E and E' associated 2-cocycles {f(o-, T)} and {f'(6', T') } satisfy x(f(o-, T))' 'f'(v(a•), v(T)) (o-,T E G). If we fix the frames 2-({u}, ~ {f (a-, T)}) and ~' = ({u~' }, {f'(o-', T') }) of E and E' respectively such that A(f(o-, T))= f'(v(6), y(T)) holds, then a homomorphism ./ is uniquely determined by µ: U--U). It should be remarked that usually there are many such choices of µ's according to that of 2's. Finally, for a given group G, an abelian G-group A and a 2-cocycle {f(a-, T)} of G over A there exists a group extension E(A, G, t,) such that Class formations III. 457
E= J A*•uQ (A*~A), u~~u~=f(Q-,T)*UQT,u~•a*=(ad)*•u6 (4) c(a)=a*, f'(a*u6)=o~ .
§ 2. System of Well groups I. Let {S, A(K)} be a class formation. Let us assume that a group W(K/k) is attached to each normal extension K/k (k, KE S) with the following properties: A 1 There exist homomorphisms (1) XK,k . A(K)--W(K/k) (2) k : W(K/k)-~G(K/k) such that XK, k ~K k (3) 1-OA(K)--~ W(K/k)--*G(K/k)--*1 is exact. Moreover, let us take an arbi(rary representative ud in each ~Kk(°) then (4) uQ• (Xa)• u61=x(ac) o-E G, a A(K) holds. A2. Let (5) TK,k . W(K/k)-+XK,kA(K) be the transfer mapping and let us put
(6) /'K k_'PK, k°XK k°T K, k : W (K/k)-A(k) . Then
~ µK , h (7) 1~W(K/k)c_ >W(K/k) >A(k)--1 is exact. B1. In case II there exists an into-isomorphism
(8) UK,1, k : W (K/l )-~ W(K/k) such that 458 Y. KAWADA
K ,l 1--OA(K)---W(K/l) ---p G(K/l) --~1 (exact) (9) vK,l,k~. c 1-->A(K)~W K k -p G K k --~1 (exact) is a commutative diagram. We remark that from (9) follows
(10) vK,l,kW(K/1)=7'K~k(G(K/i)) C1. In case I there exists an onto-homomorphism (11) PL,K,k: W(L/k)-W(K/k) such that
1-*-W(L/k)~ ----* W(L/k) -* A(k)--->1 (exact) (12) PL,K,k PL,K,kµ 1 1 _ *W(Klk)c --* W(Kk) --* A(k)--*1 (exact) and
vL ,K,k 1--~ W(L/K) -- * W(L/k) --~ G(K/k)-- *1 (exact) (13) µL,KXPL,K,k 1- A(K) -*W(K/k) -* G(K/k)-~1 (exact) are commutative diagrams. We remark here that from (7) and (13) follows (14) Kernel pL,K,k=Kernel µL,K= W(L/K)c. A1'. Let the homomorphism
(15) ~K: A(K)-- F(K) be defined by the generalized norm-residue symbol 1K(a)= (a, K) (aEA(K)). Let 1'(K/k) = G(AK/k) for normal extension K/k. Th e n there exists a homomorphism (15)' K,/: W(K/k)-~r(K/k) such" that Class formations III. 459
XK,k ~K>k 1--OA(K)----W(K/k) -~G(K/k)--*1 (exact) (16) '~KK,k 1 1--~r(K)-- F(K/k) ---~G(K/k)---1 (exact) is a commutative diagram. A2'. Let F(K/k)c be the topological commutator groin of 1'(K/k) then 1 K,k 1--~ W(K/k)~--~ W(K/k) ---~A(k)-->1 (exact) (17)K>k K,k ~k 1--*I'(K/k)c--->I7(K/k) --*r(k)--*1 (exact) is a commutative diagram. B'. In case II we can chooseK,l and K,k such that UK,l,k W(K/l) ---* W(K/k) (18) K,lK,k r(K/l) ----*F(K/k) is a commutative diagram. C'. In case I we can choose L,k and K,k such that PL,K,k W(L/k) ---~W(K/k) (19)L,kK,k r(L/k) ---~I'(K/k) is a commutative diagram. If all these properties are satisfied we call the system {W(K/k) } a system of Weil groups associated with the class formation (cf. Weil [12]) 2. Now we can prove the following existence theorem: THEOREM1. For any given class formation {S, A(K) } there exists always a system of Weil groups {W(K/k) } associated with it. (PROOF) (i) Let {fKfk} be a system of fundamental 2-cocycles and (a, k) be the generalized norm-residue symbol defined by them. Since fK/k(°-,T) is a 2-cocycle of G(K/k) over A(K) we can define a
I a'eG 6
460 Y. KAWADA
group extension W(K/k) of A(K) by G(K/k) defined by fx/k(°-,T) : W(K/k) = U A(K)*•u
(20) ud' uz =fK/k(0, T)* . u6r, uQ° a* = (a°) *`u6
~'K,ka=a, -K,k(a `uQ)= o
where A(K)* is an isomorphic replica of A(K) as abelian G-group. Thus we can construct W(K/k) satisfying (3). In the following we fix a frame 2JKIk= ({u6 }, {fKlk(° , T)} ). Clearly XK,k and 7rK,k are inde- pendent of the choice of a frame. (ii) The transfer mapping TKk : W(K/k)-*A(K)* = XK,kA(K) is de- fined by
(21) TK,k(a*) = (NGa) *, TK,k(u6) = 1fll uTu6u;1 =fKl k(G, o')* TEG
By the isomorphism theorem we have A(K)G= f l f (G, o-)• (NGA(K)) 6eG which implies TK,kW(K/k)=XK,kA(K)G. Next consider the kernel of TK,k. So let TK,k(a*• uC)= T(a*) • T(u6)=1. It follows from the same theorem that o.EGC. Let o-=II TI(o-Jp~o1pT 1)E'T9 1, then we have u~ = ar.• c, c=1.1UT,(ud. up• u;~ ic.1)E1 zz and a1EA. Hence 1t follows
1 that a* . uQ= a2 • c where c+ W(K/k)c and a2EA(K) with TKka2 =1. To prove Kernel T = W(K/k)c it is enough to see that (NGa)*=1 implies a*EW(K/k)c. Since H-'(G, A(K))~A(K)NG~1/I(A(K)) (1(A) =fjA 1)is 6eG isomorphic to H-3(G, Z) =H2(G, Z) by the Theorem of Tate, we have
A(K)NG~1_ {g-3UfKlk}'I(A(K))
Here g-3 is a 2-cocycle of G over Z, i. e. g-3= ~, zap, z(a-,T) (a6,. EZ) with 6g-3= ~6 , za6,T {(T) - (6T) + (o-)} -0. By the explicit formula of cup-product (see e. g. Artin-rate [3]) we have
(~aa, T(Q.T)) Uf= L1 {f (o-7T)6-1T- 1} -ac, T= L 1 (i0 uTuQ)-aQ,C. Q>T J>'C
This term belongs to W(K/k)c since it becomes the unity element by taking mod W(K/k)c by means of ~g-3=0. Also (a*)6-1=uQa*ic a-1~ Classformations III. 461
W(K/k)c implies I(A(K)) c W(K/k)c. Hence we have proved Kernel T = W(K/k)c. Since Kernel 1 K,k=Kernel TK,k=W(K/k)c we get the exactness of (7) (cf. Kawada [6], p. 92). Evidently the homomorphism ' X,kis independent of the choice of a frame 2Klk of W(K/k). (iii) B1. Firstly, let us normalize fK/l=resH,GfK/k and let us choose '~JKlk= ({u6 } , {fKl k(°~ T) }) and ~'KIl= ({vp } , {fKll(°~ T) ) of W(K/k) and W(K/i) respestively : (22) W(H/k)= U A(K) • u~, W(K/i) = U A(K)' up QeG peH Then we can define an into-isomorphism vK,l,k(1,c) of W(K/i) (A(K), H) into W(K/k) (A(K), G) by (23) v(a')=a*(aEA(K)), v(up)=up (pEH) It is then easy to verify (9). It should be remarked that vK,l,k depends on the choice of the frames 2K/k and Z'Kll. (iv) C1. Let us normalize fL/K=resH,GfL/k and let us choose ~'LIh=({u~}, {fLlh}) and ~'LIK=({up}, {fLIK}) of W(L/k) and W(L/K) respectively: (24) W(L/k)= U A(L)*. uQ, W(L/K) = U A(L)' •up . QeG peH Let us fix for a moment a system of representatives {~-} of cosets : G=UHF and put g(o-, T) =fLik(°", T)m• (~h) (a-, T) (m= [L: K]) (25) h (a-)=fLlk(H, fLIk(a-~H)-1 Then it follows that g(o-,T)=g(~, T) and so g(Q-,T)=inflG,Fg(?, T) where denotes the class of a- mod H. Hence we can normalize
(26) fKIk(~~T) =g(I, T) By a known formula (see e. g. Hochschild-Nakayama [5], Lemma 7) we have (27) infG,F°PL,K(fKlk(°,T)) _X L,K°TL,K°vL,K,k(u6u~uQr). We choose a frame 2KIk= ({v~}, {fKIk}) of W(K/k) : W(K/k) = U A(K)* • v6 . REF 462 Y. KAWADA
By B1 we can decompose W(L/k) = JvL ,K,k(W(L/K)) • u6. As in the considerations in § 1, 2 we can define a homomorphism of W(L/k) into W,(K/k) as an extension of (28) P -µL,K°UL'K,k: UL,K,k(W(L/K))--OA(K) by means of (29) p(u)=v~ (-EG) because using (27) we can verify the corresponding conditions: P*(u~, uz' u6r) -fKlk(?, ?)*, P*(ua • (a* • up)' u61) =v_ • p*(a* aup) • v:l for o-, TC G. Then from the definition of P = PL,K,k follows the com- mutativity of (13). Next we shall verify (2). By a known formula of Akizuki-Witt [1] g(6, T) in (25) can be expressed in another form: g(o', T) =fLJk(H, ~) °fLlk(H, aT)_1.fLlk(H~, T) . Hence we have (30) fLIk(G, ~)=-(F, a-)=fKl k(F, i) by the normalization (26). Now compare the two mappings µL,k and µK,k ° PL,K,k• On the subgroup vL,K,k(W(L/K)) we have by (28) PK,k°PL,K,k-µK,k°µL,K°vLiK,j-µL,k using the transitivity of the trans- fer mappings. On the set {u6} we have by (21) and (29) µx,k°PL,x,k (u6)=µK,k(v~)=fKlk(F, ?) and p(u) =fLlk(G, ~) which are the same by (30). Thus we have proved (12). We should remark here that the definition of PL,K,k depends on the choice of the frames ~'Lik, EL/K and EKlk but does not depend on the choice of representatives {} of cosets G mod H. (v) To show the existence of K,k we need a generalized theorem v of Savarevic (see, e. g. Hochschild-Nakayama [5], Theorem 3.2 or Artin-rate [3]). Namely let fKk(,,~7) (o, 7EG) be a fundamental 2- cocycle of G over A(K), then
(31) fKlk(°' T)-(fKlk(a~ T), K)EF(K) is a 2-cocycle of G over 11(K) associated with the group extension f(K/k) (F(K), G, t, -vi). Since we need the proof itself we repeat the known proof. By a known formula of Akizuki-Witt [1] g(o-, T) in Class formations III. 463
(25) can be expressed in another form (32) g(o-, T)=fL,k(H, y)' (fL,k(6~T)/fLlk(y, vT))H where ~T-'EH. Now take k c Kc L c AK and denote here G= G(K/k). Then from (32) follows (g(?, T), L/K) = • 7 o=T-i~H. Since fKIk('~)=g(, 7) • cU• ci • c G (c6EEA(K)) we have (33) (fKIk(~'T), UK) =(p~ • i)• (pi•T)• (p~~•~T)-,EH where we put p~= (ce, L/K)EH. Take the limit with respect to all {L; Kc L c AK} we have (34) (fKlk(?, 7), K)=~-*•T*•(oT)*-IEP(K) where o-*~f(K/k) means some element with (o*) = o ~G(K/k). This proves our proposition. Let us put in (34) U~=o *(o EEG), then (35) P(K/k)-Ur(K)•U6, U~'U =fKlk(o~T)`U6z. ~eG By the general method shown in § 1, 2 we can define the homomor- phism SK,k: W(K/k)-~F(K/k) as an extension of 11K by K,k(u~)-J (?EG). Then the commutativity of (16) holds. We should remark here that K,kdepends on the choice of frames 2Klk- ({u6}, {fKlk}) of W(K/k) and 2'Klk=({ Uo}, {f Klk}) of 1'(K/k) where fKlkis defined by (31). If we fix ~Klk there are as many possibilities of eK,k as different frames ({BU}, {fKlk})of F(K/k) where B~Er(K) and (36) BuBiB;i 1 (?, TEG) (vi) LEMMA2. I f a homomorphisrn .K,k : W(K/k)--l7(K/k) satisfies A1' then the property A2' is satisfied automatically for this K,k. (PROOF) To prove the commutativity of (17) it is enough to see (37) r(a, K) = (NGa, k) (aEA(K)) (38) f'U6= (f Klk(G(K/k), 6), k) (o EG(K/k)) where Jr=fr(I'(k), r(K/k))7~. Here (37) is a known property of the norm-residue symbol. To prove (38) let kcKcLEAK, L/k be normal and put L'=Akf L. Then (38) is equivalent to 7) i(F, G) means1'F, c' i. e. the cannonicalhomomorphism of G on F. 464 Y. KAWADA
(39) f''U~ _ (fA,k(G(K/k), a), L'/k) for all such L where 1'=fr(G(L'/k), [(K/k)). From the choice of U~_ O* in (v) follows fr'U~=*"(p~ • o-) in (33). Put F= G(K/k). Then from f=g.6cK,kit follows that fKIk(F,~)=g(F, U)•NFc~. On the other hand we have J"(p~) _ V'(c~, L/K) _ (NFc~, L'/k) and Jr"(6) _ (fL/k(G,~), L'/k) by definition. Hence using these relations and (30) we have i) _ (g(F, ~) Fc~ L'/k) _ (fK/k(F, ~), L'/k), q. e. d. (vii) B'. Let us fix frames 2K/k and as in (iii) for the normalization fKI~=resx ,cfKlk, and choose the frames 2KIk of [(K/k) and 2of (K/l) as in (v). If we define and K 1,k by (40) 7"K,k(up ) = u1, K ~(uA) = Up K,k(UQ)= U6 then we have t(U) = U6 and B' holds. C'. Let us fix the frames 2'Llk, ~'LIK and 21KIkas in (iv) for fL/K =resx ,cfL/k and for fK/k as in (26). Let us choose the frames ~'L/k _ ({ U~}, {f Llk}) and 'KIk= ({ V~}, {f Klk}) of F(L/k) and [(K/k) respec- tively such that (41) Jr(U~)= V~ (BEG, ='4'(r(K/k), F(L/k)) holds. Then we can verify the commutativity of (19) immediately. That we can choose 2'K/k and LIk satisfying (41) can be shown as follows. Let us take £L/k as in (v) from £LIk of W(L/k) and define L,k as there. Next put V~= r(Uff). Then from (U6U~U6t)- (~ L,K(u6u~uQT),K) ` (fK/k(~, T ), K) follows that ~u`Vz-~(U~`U~)^ l (Uo`U~`Ua'T`U6T)-(fKlk(~~~)~K)`V~z• This shows that ({V~}, {f K/k}) gives a desired frame of [(K/k). q.e.d. Let {W(K/k)} be a system of Weil groups. If we can choose the homomorphisms v, p once and for all such that the conditions B2. Let k c iiic j c K and K/k be normal. Then (42) PK,l,k °UK, j, l = UK,j, k holds. C2. Let k c KG L GM and M/k, L/k, K/k are all normal. Then Class formations III. 465
(43) PL,K,k°PM,L,k=PM,K,k . C3. Let k c lc KcL and L/k, K/k be normal. Then (44) PL,K,k°UL,1>k-vK,l>k°PL,x,I holds. (In case K= l C3 is contained in C1). are satisfied then we call {W(K/k)} a strong system of Weil groups. REMARK. In a strong system of Weil groups we can choose the homomorphisms & ,k once and for all such that B' and C' hold. For, what we have proved is that if we fix the frames 'Klk of W(K/k) for all K/k then we can choose (i) 'K/k and 2'K/ of F(K/k) and 1'(L/t) respectively so as to satisfy B' and (ii) 2'Llk and 2'Klk of F(L/k) and F(K/k) so as to satify C'. What we want to prove is that for each individual {k, 1, K} or {k, K, L} we can choose the frame 2TK/k for each K/k once for all such that B' and C' holds for these frames. For that purpose let us consider the set E'K,k of all homomor- phisms K,k satisfying A'. If we fix a frame ~'Klk of W(K/k), K,k depends on the choice among different frames 2'Kfk in F(K/k) with a fixed 2-cocycle fKfk as in (31) and this depends on 1-cocycle {B} of G in 17(K). Since 17(K) is compact we can introduce a natural topology in `~K,k such that 8K,k is compact. Let 2= {KA}be the set of all finite normal extesnions KA/ko in and put H = KA,ko . If K,kc K, (E2) then we can define the continuous mapping '/''a,u of u , into L d by (41). It is easy to see that {E,, r ~} makes an inverse mapping system. Since each A is compact the limit space is not empty. Take an element in its limit space. Then we have a set {K A,ko; KAE3} such that (41) holds for each pair {& ~,ko,eKA,ko ; KKAEK~}. Thus we have chosenK ~,ko. For a general normal extension K/k we can take koc k c Kc L (LE 2) and put
(45) 2K,k= kL,K°res ~'L,ko where ''L,K means the mapping 8L,k--~LK,k defined by (41) and res means the restriction from I'(L/ko) on F(L/k). We can then prove in the usual way that the definition (45) does not depend on the choice of LE2 and thus defined family {K,k} satisfies B', C', q. e, d. 466 Y. KAWADA
3. Let A(K)} be a class formation and ijK(a)_ (a, K) be a generalized norm-residue symbol defined there. Suppose that two systems of Weil groups {W(K/k)} and {W'(K/k)} associated with this class formation be given. Let us denote the groups and homomor- phisms in A, B, C, A', B', C' with respect to W'(K/k) with prime. We call {W(K/k)} and {W'(K/k)} are isomorphic systems if there exists an isomorphism
(46) c'K,k: W(K/k)-+W'(K/k) (onto) for each K/k such that the following conditions A*, B*, C, A'* hold: ~K,k°XK,k_ K,k, ~K,k-~K,k°K,k, I K,k-IK,k°K,k B PK,k°PL,K,k- PL,K,k °L,K A*: K,k- K,k°`PK,k (K= K) THEOREM2. Let us assume that the axioms P and P' are satisfied in S. Then any two strong systems of Weil groups {W(K/k)} and {W'(K/k)} are isomorphic. (PROOF) (i) We may assume that the system {W(K/k)} is the one defined in the proof of Theorem 1. Let us choose an arbitrary frame z"K,k=({u6}, {gKlk(°-, T)}) in each W'(K/k) :
(47) W'(K/k) =veG U A(K)' • u6, u~' u~= gKlk(6fT )' • UQT• From the conditions B1, C1 follows that vK,~,kinduces gKl~~resH,GgKlk and PL,K,k induces 111flG,TgLIh~gKlk (m= EL : K]). Moreover, gKlk is a generic 2-cocycle in H2(G, A(K)). For, if it were not so g-1 would hold on some cyclic subgroup H=G(K/l) (by the above rela- tion). On the other hand we have A(K)H= U g / (H, p) • NHA(K) by peH A2. Since gKlx~1 we would have gKl (H, p)ENHA(K) which would imply A(K)H=NHA(K). But this contradicts with the isomorphism theorem A(K)H/NHA(K)~H. We shall prove next that gKik-fKlk holds for every K/k. Let us first assume that G= G(K/k) is cyclic and so put G= {1, o, cr2,.• . , crn-1}. Then we can normalize Class formations III. 467
fxIk(°,-z6') =1 (i +1 C n) ; = XK,k(a)(i + j >_n) (a+A(k)) gKlk(°, 6J) =1 (i +j W (K/k) = U A(K)' • uQ, u6 • uT=fK k(~' T)' • u6T QeG we define then PK,k(a*) ~ a' (aEI A(K)), cPK,k(u~)= u . It is evident that A*, B*, C* hold for these TPK,k. (iii) That these ~K,k satisfy A'* follows from the following Lemma: LEMMA 3. I f a system of groups {W(K/k) } satisfies the conditions A, B, C, A', B', C' then the homomorphisms K,k are determined uniquely by other homomorphisms i, x, µ, v, (PROOF) Let K,k andK,k be two homomorphisms satisfying the conditions A', B', C', while the other homomorphisms X,'ir, µ, v, p,' are the same for . and e'. If we use the same notations as in the proof of Theorem 1 we have K,k(uQ)= K,k(uQ)' BK(cr) BK(o")EF(K ) BK(o)• BK(T)6• BK(oT)-1=1. By the considerations and notations in (vii) of the proof of Theorem 1 it follows from the condition C' for k c Kc L that (48) ~frBL(o)=Bx(a) where -tfr_'tfr(I'(K), I'(L/K)). In particular, if we take k c Kc L c AK then F(L/K) = I'(K) and 'Jr is the identity. Hence (48) implies that BK(~) (~EG(K/k)) belong to G(AKIL). Since L is an arbitrary inter- mediate field in AK/K we have BK(~)=1 (~EG(K/k)), namely x,k= K,k' q. e. d. THEOREM 3. Let B(K) be the kernel of ?K: A(K)-+P(K) and let us assume that (49) H1(G, B(K)) = 0. Then the isomorphism !K,k in Theorem 2 is determined up to the inner automorphism by an element of XK,kB(K). (PROOF) Let us assume that x_X', 'ir=ir', etc. in A*, B*, C*, A'*. Let us fix a frame £K,k= ({u6},{fK k}) of W(K/k) : W (K/k) = U A(K) * • ui,, u~• uT=fK,k(°' T)* • u6T 6eG Class formations III. 469 Then ~K ,k is determined by PK,k(a*)=a* and cPK,k(u~)=u6. By condi- tions A, A' we have uv' uT- K/k(cT,T) *U, ~K,k(ui) _ ~K,la(ua)' Hence we have uQ=b~•u6, (bdEA(K)), b.b.b;=1, and K(b) 1. Therefore, we have bdEB(K). Since H'(G, B(K))=0 we can find an element cEB(K) such thet b~= c'-~(o-EG) hold. Then we have u6 = c'-'u~ and hence (50) PK,k(a*u6)= a* (cl-d)*, u~= c*(a* , u~)c*-1 Conversely the mapping (50) satisfies all the conditions A*, B*, C*, A/*, q. e. d. REMARK. We omit the discussion on the uniqueness property of a (not necessarily strong) system of Weil groups. § 3. Strong system of generalized Weil groups 1. Let {S, A(K)} be a class formation and let {W(K/k)} be a strong system of Weil groups associated with it. We shall prove the following existence theorem of generalized Weil groups: THEOREM4. Let us assume the axiom F': (1) S2=UKn (kocK1cK2c... ciKnc... cQ) in S. Then there exists a system of groups {W(k) ; kc S} with the following properties : (A1) Let G(k) be the compact Galois group of S2/k (i. e. G(k) _ G(S2/k)). There exists an into-homomorphism (2) 7T"k:W(k)-G(k) whose image 7rkW(k) is dense in G(k), and an onto-homomorphism (3) ''k: W(k)--~A(k). We denote its kernel by W(k)cc such that ILk (4) 1--~ W(k) - >W(k)-A(k)--->1 h 470 Y. KAWADA is an exact sequence. (A2). The diagram W(k) -- p G(k) (5) µk ? ' A(k) ----* F(k) is commutative. (B,) For k c t (k, t E S) there exists an into-isomorphism (6) k : W()-4.W(k) such that 7rl W(t) ---* G(1) (7) Ul,k ~. G W(k) --* G(k) is a commutative diagram. We have also (8) vj, kW(t) _ 7k'G(t) (B2) For k c t cii) the transitivity relation (9) v~ , h °Uj,~ _ Uj, k hold s. (C1) For a normal extension K/k (k, KES) there exists an onto- home mor phism (10) PK,k : W (k)- W(K/k) such that W(k) -G(k) (11) PK,k ,k W(Klk)-P(K 1)k 1-- * W(K) --- * W(k) --~G(K/k)---*1 (exact) (12) ~K PK,k ~. 1 1---*A(K)--~ W K k)- G K k ---*1 (exact) Class formations III. 471 are commutative diagrams. Here f'=t(G(K/k), G(k)). We remark here that from (12) follows (13) Kernel PK,k=Kernel 1K = W(K). (C2) In the same case Pk 1-~W(k)~~ ----*W(k) ---*A(k)--*1 (exact) (14) PK,k PK,k K> k 1- >W(K/k)c___ W(K/k) --- *A(k)-- *1 (exact) is a commutative diagram. We remark here that from (11), (14) and § 2, (17) follows (5). (C3) Let k c Kc L and K/k, L/k be both normal. Then the transi- tivity relation (15) PL,K,k°PL,k-PK,k holds. (C4) Let kc1EK and K/k be normal. Then (16) PK,k01l,k-1)K,l,k°PK,I. We call this system {W(k) ; kE~} a strong system of generalized Weil groups associated with {S, A(K), W(K/k)}. (PROOF) (i) Let kE h. By (1) k c Ky for a certain r. In the fol- lowing we assume that the fields {K12}in (1) are all normal over k0. Then we consider a sequence of groups {W(K,1/k) ; n = r, r+1, . .} and a set of onto-homomorphisms IIn=P(Kn, Kn-1, k): W(Kn/k)-~W(Kn-1/k) (n=r+1, r+2,..-). These determine the inverse limit group W(k) : (17) W(k)=lim {W(Kn/k), II}. Since all III are onto-homomorphisms we can define naturally the onto-homomorphism (18) p(Kn, k) : W(k)--aW(Kn/k) (kcKn)° such that p(Kn_1,Kn, k) p(Kn_1fk) = p(Kn, k) holds. On the other hand 8) We also use the notationsp(K, k) insteadof pK,ketc. 472 Y. KAWADA the Galois group G(k) = G(12/k) can be considered also as the inverse limit group of the system {G(K/k), Jt(G(K7/k), G(K12~/k))}. Since the following diagram is commutative : P P W(K~/k)~--W(K1/k) *--...... ~-W(k) K /k 7rKn ~1/k G(Kn/k) ~--G(K2 ,_1/k) F-...... F ---G(k) we can define the homomorphism 7rk: W(k)-G(k) by (19) 7rk(u) = lira (7r(K,2,k) ° p(Kn, k) (u)) uE W(k). Since each 7r(K2/k) is an onto-homomorphism the image 7rkW(k) is dense in G(k). (ii) Consider the following commutative diagram: 1 1 P P W(KJk)c< W(K~_,-,/k)~E F- yV(k)~~ G G P P (20) W(K~2/k) W(Kn f1/k) E---...... --W(k) µ 1 ~` 1 A(k) F-- A(k) E -...... F---A(k) 1 1 where each column is exact. Then we can define the homomorphism /k: W(k)-A(k) by (21) µk(u)= lim (µ(K,2,k) °p(K2, k) (u))EA(k) uE W(k). Since p(K~, k) is an onto-homomorphism µk is also an onto-homomor- phism. Here the kernel of µk is the limit group of {W(KK/k)c} and contains the commutator group of W(k). (iii) Let kcl and W(~)= lim {W(Kn/1), p(Kn+1, K, 1)}, W(k) = lim {W(K~1/k),p(K 1,Kn, k)} . Then from the following commutative diagram Classformations III. 473 W P P (K~/1)~--W(Kn 1/i)f--...... F--yV( ) H) v P P W(Kn/k)~--W(Kf1/k)E--...... --W(k) it follows that we can define the homomorphism v.k: W(t)-W(k) by (22) v~;k(u)=1im (7)(K, 1, k)op(Kn, l) (u)) uEW(l) . That vi; k is an into-isomorphism follows from the fact that each v(Kn, 1, k) is an into-isomorphism. The commutativity of (7) follows from the following commutative diagram and the definition of ?Tk and v~;k W(KJ l) P W(K~+1/1) < - P - - - - <- v v __ l l W Klk)-- n P W(K n +t'k) ~-- P - - - E---- W(k) G(Kailk) G(Kn+1fJk)f ------E--- G(k) (B2) can be proved similarly. (iv) Let k c Kc Kn (n = r, r+1, ... ). Then using the commutative diagram: P P W(Kn/k)F--W(Kn+1/k)-----...... --W(k) P 1 P 1 W(K/k) E---- W(K/k) F-...... F____yV(K/k) we define the homomorphism PK;k: W(k)-- W(K/k) by (23) PK,k(u)= P(Kn, K, k) op(Kn, k) (u) uE W(k) Since p(Kn, K, k) and p(Kn, k) are onto-homomorphisms PK,k is an onto- homomorphism. Now we can verify (C1), (C2), (C3) (C4) as above using corresponding diagrams. It is evident that the definition of W(k), p, ir, v, does not depend on the choice of the sequence {Kn},q.e.d. 2. Let B(K) be the kernel of ilK: A(K)-- I'(K). Let {Kn} be the sequence in (1) and let us put 474 Y. KAWADA (24) H, 1=q'(K~, Kn+1)_1`NG(Kn+1IKn):B(Kn+1)`~B(Kn) (n1, 2,...). Then we can define the inverse limit group B by (25) B=lim {B(Kn), H}. Let us denote the canonical homomorphism by 'tJrn: B--B(K) so that (25) Jn = Hn+1°cn+1. THEOREM5. Let us assume, furthermore, that (i) for each kES . (26) ii (A(k)) = I'(k) holds, and (ii) for each normal K/k with G=G(K/k) (27) c'K,kB(k)= NGB(K) holds. Then we have stronger results than in Theorem 4: (A1) There exists (28) an into-isomorphism ?k: B-- W(k) such that (29) 1----*B--* W(k)---*G(k)-----i1 is exact. (A2) Let C(k) be the kernel of i' k: B--B(k). Instead of (5) we have the following commutative diagram where each column and each row are exact sequences : 1 1 1 1----~C(k)--- k (30) 1--~ B ---~ W(k) -G(k) ---~1 Y' k ~'Ikya 1--- *B(k)-->A(k) ---k ir(k) -p1 1 1 1 where J' k is defined naturally by the definition of B in (25). G *- Class formations III 475 (C1)' Instead of (11) we have the foll owing commutative diagram, where each column and each row are exac sequences t 1 1 1 ->G Xk UK,k 71'k (31) 1--* B --* W(k) ----* G(k) -*1 11k ~ 'K ,k PK,k cK,k i-- B(k) W(K k) ->F * 1 (K/k) l 1 1. (PROOF) By our assumption (27) II' (n =1, 2, ..•) are onto-homo- morphisms and hence ~~ : B--B(Kn) are also onto-homomor p hisms. Since the following is a commutative diagram II' II' E-- B(K,1) P P E--W(Kn/k) we can define Xk: B--~ W(k) by (32) Xk(b)= lim (B(Kn, k) °~n(b)) bEB. Since ~,(Kn, k) : B(Kn) -~W(K nik) are all into-isomorphisms, ~,h is also an into-isomorphism. Next consider the following commutative diagram : H' H' F- B(Kn) E- B(K1) E-- ...... F---B P P F--W(K k) (33) ~r(K k) F--G(Kn k) 476 Y. KAWADA Here we have G(k) = lira (P(K/k), '4')= lira (G(K/k), fr). For o-EG(k) let us denote o- = '4r(G(K/k), G(k)) (o-)EG(K/k) and p,~= '4r(F(K~/k), G(k))(o-) so that o-= lira o-n= lim pn. Now take arbitrarily anE W(K/k) such that (Kn, k) (c;) = pn holds. Then by means of II +1B(K+1)= B(K) and Kernel (K/k)=XB(Kz) we can find an element an+~e W(K+1/k) such that (34) an = p(Kn+1,Kn)art+1, Pn+~= (Kn {1, k)an+1 holds. (Precisely, take ~n+1~ W(K+1lk) such that (K2_F1,k)$1= p1 holds. Then its image ,9 = p(K+1, Kn, k)$n+1satsifies (K/k)c 1 • $n i =1 by the commutativity of (33). Hence a?2•$n'EX(K, k)B(Kl). So take 7n+1EB(Kn I-1) such that c n. 9;1=X(K,2, k) o1In(y~+1)holds. Let us put c ~ 1=h(K+1, k) (7+)$n+ Then this element an+1 satisfies (34)). Continuing this process we can find a sequence {cnE W(K/k)} such that a =1im anE W(k) satisfies 7-k(a)= a-. Hence we have 7rkW(k)= G(k). Finally the whole exactness of the sequence (29) follows also from the commutativity of the diagram (33). The commutativity of the diagrams (30), (31) can be proved in similar manner, q. e. d. REMARK1. As to the uniqueness theorem for generalized Weil groups satisfying all the formaulas (A1), (A2), (B1), (B2), (C1), (C2), (C3), (C4), (A,)', (A2)', (C1)' we can prove the analogous theorem as Theorem 2 under the assumption P, P' and (i), (ii) in Theorem 5. REMARK2. It is an open question whether every system of Weil groups {W(K/k)} is a strong system or not. REMARK3. We don't discuss here the problem to construct a system of generalized Weil groups from a (not necessarily strong) system of Weil groups {W(K/k)}. § 4. System of Weil groups in algebraic function fields 1. Let ko be an algebraic function field in one variable over the complex number field C and 12 be the maximal unramified ex- tension over k0. As we have proved (Kawada-rate [8j) we can define a class formation {, A(K)} as follows. Let KES, D(K) be the group of all divisors of K, E(K) be the group of all divisor classes of K. Let us introduce a locally compact topology in E(K) such that the Class formations III. 477 subgroup E0(K) of all divisor classes of degree 0 is compact and E(K)/E0(K) is discrete. Consider the character group E(K)A (in the sense of Pontrjagin). Put A(K)=E(K)'. For kcK(k, KES) we define ~K;k: E(k)A--~E(K)A by conorm. Then we can prove that {S, A(K)} is a class formation. Let R(K) be the Riemann surface of K, F(K) be the fundamental group of R(K), H(K) be the one-dimensional homology group of R(K) with integral coefficients. We can identify H(K) _ F(K)/F(K)c. We denote by I' a closed curve on R(K) (with initial point by r its homotopy class (so an element of F(K)), by P its homology class. We use the additive notation for chains and the multiplicative nota- tion in F(K). Let ?=fl ' be a divisor of degree 0 on R(K) and d log '?1be the differential of the 3rd kind on R(K) such that (i) at each point j3t d log ?I has a simple pole with residue vi, (ii) all periods of d log 2i are pure imaginary. If B is a 1-chain on R(K) we define (1) [B, I]= d log I, (B, !X)=exp [B, ?I]. Let us fix a prime divisor 3. xEE(K)'' induces a character on D(K) for which we use the same notation: x(2I) = x(2i). Here by 1 we mean the divisor class containing a divisor ?I. Then x can be ex- pressed in the form (2) x(13') _ (I', ) • A5 tEE0(K) by a 1-cycle I' and a complex number A with absolute value 1. Here F is uniquely determined up to boundaries. We denote then ti x=x(P, A)• Let K/k be a finite unramified extension. We denote be ~r= ~xrh the projection of R(K) onto R(k) and by V=VKIk the transfer map- ping : H(k)-+H(K). Then we have (3) ~K,kxk(Y,A) = XK(V'Y,A) 7EH(k). Let K/k be normal with the Galois group G=G(K/k). We can verify (4) x(I, A)6= x(~ , A• (F, ~-~1)) In the following we use the notation 478 Y. KAWADA (5) Q[cr, B] = s[B, Q(~, B) = exp Q[r, B] for o-~G and for 1-chain B, where denote the imaginary part of a complex number. Let F(k)=J 7rF(K) `R~ and RQ• 7rF `f9 ' = 7r(o-['). 6EG Choose a curve B6 on R(K) such that B6 connects 43owith 4, and 7rBQbelongs to f3. Let then I'(o-, T) = Ba + (OBT)- B6T and (6) X(o , T)= Q(0, BT). Then (7) fKIk(°' T) - x(P(0, T), h(cr, T)) satisfies all the properties of a fundamental 2-cocycle. We use later the following relations Q[T T, I'] = Q[o-,I'] + Q[T, UI'] ($) Q [p, I'(o, T)]= Q[P, Bd]- Q[o, BT]+ Q[Po, BT]- Q[P, B6T] which can be verified easily. 2. We shall consider next a system of Weil groups in algebraic function fields. For normal K/k (k, KES) we can express W(K/k) by W(K/k) = U A(K)*. u6 (A(K) = E(K)A) yeG (9) uQ`x(r, X)=x(o-P, ~' `QKIk(°-,r))* `uQ u6 `u T= x(P(o-! T), 7(o', T))*' uvQT From (9) follows that belongs to the centre of W(K/k). We identify x(0, X) with ? and we denote T= {x ; XEC, Jx =1}. Let (11) J(K/k) = F(k)/7rF(K)c Then we have J(K/k) = U H(K). R~ (12) ti ieG Hence if we put Class formations III. 479 (13) ~Klk(ui)=RQ, xlk(x(r, X))_ ~'Krk we have an exact sequence * 'ir, (14) 1---~T-~W(K/k)- ~J(K/k)--~1. For aUF(k) let us denote by a* the class of a mod irF(K)c (i. e. a*UJ(Klk)), and let us choose a fixed representative path [a*] on R(K) starting from 430such that ir[a*] belongs to a. In particular, we choose (15) [('irf•RQ)]=[(7r1'(o,T))*]=BQ•o-B.•BQZ. Then for new representatives Ua* = x(r,1) • u~ for a* = 7TT• 3 EJ(K/k) we have W(K/k) = U T * * Ua* a*EJ(KI k) (16) Ua*•?,*_? *•Ua* (XET) U*. Ua*= aKlk(a* , R*) • Ua* p* where we put for a* ='irt • $, and /9*= 7rd•,&T (17) aKlk(a*,$*) = X (o-,T)QKIk(0, d) = QKIk(0,[l~*]) Next we shall choose another system of representatives mod T*, namely we put (18) Va*= exp( =- QKIk[G,[a*]]/n) • Ua • Then we have W(K/k) = U T* • Va* a*EJ(KI k) (19) X* • Va* = Va* • X* (XET) Va*V=bKlk(a*, $*)*. Va*(3* Using (8) b(a*, $*)UT can be determined as follows : 480 Y. KAWADA b(a*, ,&*)= a(a*, $*). exp( - {Q[G, [a*]] + Q[G, [$*]] - Q[G, [a*R*]]}/n) = exp(Q[o, 4] + Q[o-,BT] - {Q[G, F] + Q[G, Bd]}/n - {Q[G, 4] + Q[G, Bz]}/n + {Q[G, F + o4 + F(o, T)]+ Q[G, B]}/n) = exp( - {Q[G, F] + Q[G, o4] + Q[G, F(c, T)]- Q[G, F + o4 + F(o, T)]}/n) = exp([ - A, NG43/4n]/n) where A = F +B d + a-4 + ~Bz - Bdr _ (r + o4 + F(r, T)) _ [a*] + O[$*] - [a*$*] . On R(K) A belongs to F(K)c. Hence 1-chain A is homolo- gous 0 on R(K). Let us assume, for simplicity, a triangulation of R(k) and R(K) such that (i)~ is a vertex and 3 is an inner point of a 2-simplex, (ii) r and Bd are all simplicial 1-chains, (iii) 7rXikand are all simplicial mappings. Since A is homologous 0 we can take a simplicial 2-chain c2 such that ace = [a*] + o[$*] - [a*,3*]. Let us denote by KI(c2, C) the Kronecker index of c2 with respect to C (i. e. the algebraic sum of the number of coverings of C by each simplex in c2) and by LK(F1, C) = KI(c2, C) (ace= I'1) the linking coefficient. By the residue formula we have [ace, 43/C]- 27ri(KI(c2, 3)- KI(c2 C)). Hence we have b(a*, $*) = exp - 27rz {(n-1) KI(c2, 3) _ KI(c2, T-~ 3)} n =$1 = exp 27r2 KI(c2, T13) n vec So finally (20) bKIk(a*,/3*)=exp (2iri LK([a*] +(f'a*)[$*] --[a*/3*], T) n Tec where -4r='1t(G,f(K/k)). By (20) we see that bKJk(a*,$*)n =1 (n = [K: k]). We call the expression of W(K/k) by (19) and (20) the standard ex- pression of W(K/k). 2. Here we shall find the explicit form of the homomorphisms x, µ, v, p, 7r in Theorem 1 by means of the standard expression. A. By the formaulas in proof (i) of Theorem 1 we have Class formations III. 481 XK,kx(I', X)=X . exp(Q[G, r]/n)* • Vr (21) 7rK,k(X*• Va*) =ira*EG (n= [K: k], Jr=J'(G, J(K/k)) Next we have TK,k(UQ)=fKIk(G, o)*=x(F(G, cr),X(G, CT))* TK,k(x(r, X))= NGX(F,X) = x(NcI ~n • Q(G, fl)*, Then using (4) and V,8J=F(G, CT), V('ir )=NGP (V=VKIk) we have µu6= x,~(Ra,x(G, C-)),µxK(r, X) = xk(lrr, ? • Q(G, F)) for Let *=-tjr(H(k), J(K/k)) then for a*= (~F•R3)EJ(K/k) we have µ(X* • Ua*) = xk(*a*, ~n • Q(G, [c*])). Finally from (18) follows that (22) µK,k(?'*• Vas`)= xk(*a*, 7n). Thus the homomorphisms XK,k, ~K,k and µK,k are expressed by (21) and (22). B. Let G= Uo-H, and for PEH let us take the same path on R(K) for the extension K/k and for K/i. Now put W(K/k) = U A(K)* T* • Va* a E G a*~j(K/k) W(K/i) = U A(K)' u1 = U T V Q*. A e H R*EI(K/x) Then v = vKx ,k is determined in the proof of Theorem 1. In the standard expression we have (23) VKj,k(V(3"~)=bK,~,k($*)VQ* (R*E.l(K/j))~ Then for m= [K: t] and n= [1: k] we get b(R*)= exp( - QK/~[H,[R*]]/m + QK/k[G, [R*]]/mn) . 482 Y. KAWADA Using the formula (24) [K/J7 (K/1)']l-~~ -[Iv, NH S -1]K we have b($*) = exp([[3*], (NGJ3)(Nf43-r~)]Klmn) namely, (25) bK,h(i *) - exp 1 - Q~lk[i, 7rK/U3 IJ mn 6 E G/H Thus the homomorphism vK,j,k is determened by (23) and (25) explicitly. C. Let m= EL: K] and n= [K: k]. Let G= UHF. Using the results in A we can compute the term (25) in § 2 fL/k(°-.T)m(6h)(0', T) = XL1K°TL,K°vLiK,k(u, u~?) x(Nh~Llk(~~T), X L/k(o, T )rrr, QL/k(H, "L/k(°' ?)) where we put I'L/k(~,T) = Bo• ~B~ • B6z. On the other hand, let BQ (oEEG) be a fixed path from o to o-30 on R(L) and we take 7rB6= B6 on R(K) (7r= 7T/). Then "K/k(' ~) on R(K) is given by 7r(BBz)B61). Let us denote by V the transfer mapping of H(K) into H(L) then we can see that V rK/k(~, T ) = NHrL/k(-,T ) holds. As in B, (24) we have -1 -1 XK/k(0,T) = QK/k[7rB-,(~ 3 -1]= QLIk[B~~NH 3 -1] =XLlk(UH,T)XLlk(H, T)-1 . Furthermore, we have by (8) ~'Klk(°'T) `XLlk(o-, T)-m` QL/k(H, r'L/k(, T))-1 = {X(UH,T)Hj ?)^m•{X(U,T)rn•x(H~,T)-1•x(Hj 6T)•X(H, Q)-1} = (6g1)(o,T)-1 (X=~L/k) where we put (26) gKlh(~)=A'Llk(H~~) . Then comparing these equalities we get (27) inflG,F°~L,Kfxlk(~~T)(8gK/k)(~, T)=fLlk(°~ T)m`(8h)(o-, T) Class formations III. 483 for the above normalization of I'K/k(~, T). Now let W(L/k) = U {J A(L)up}u6 6 ,peH W (K/k) = U A(K)*vG- 6E~+ so that v~vi-f /k(~,7) • v z, f/k(~~ I) =fKlk(6,T)' (~gklk)(~,T) Then we can apply the results in (iv) in the proof of Theorem 1 for these expressions. Namely, p=PL,K,kis given by p-XK,koIL,K°vL,K,k on UL,K,kW(L/K) p(u6)= v6 Then we shall use the standard expressions of W(L/k) and W(K/k): (28) W(L/k)= J T • Va*, W(K/k)= U T' • V.1..a' a*EJ(Llk) a EJ(K/k) We can see first that X*ET* is mapped by p to (xn1)I. For $*=$ Vas= exp( - QL/k[G,B6]/mn)* • uct is mapped by p to exp( - QL/k[G, Bs]/n) •vG = exp({QK/k[F,7rBj - QLIk[G,B6]}/n)' ' XL/k(H,Q) • V~. Here the component in T'' is 1 which can be proved similarly as in B. Hence we have p(Va*)=V * for $*= 9 (where Jr=*(J(K/k), J(L/k)). By similar computations we also get that p(Va*)= V~;a*holds for a = 7J* .$. Therefore the homomorphism PL,K,kis given for the standard expression by PL,K,k(" ) _ (Am)! (XET ) (27) PL,K,k(Va*)= V ~a* (a*+J(K/k)) with the above normalization of I'K/k(?,T ). REMARK.If we normalize [a*] on R(L) and [fra*] on R(K) to be [1fra*]=7rL/K[a*]then the 2-cocycle of the group extensions are bLk(a*, R*) = exp 27ra LK([a*] + (J,*) [R*]- [a*/3*, ? mnreG bKlk(f'a*,f,$*) = eXp _27ri LK(7r[a*] + (1J,'*a*)7r[$*~ ] n REF 484 Y. KAWADA 7r[a*$*], ?,~) By (27) these must satisfy the relation (28) bL/k(a*, $* )m- bK/k(/, *, fr 9* ) But we can also verify (28) dirsctly from the above expressions of b J /k(a*, $f) and bK/k(tJ,ca*,*$*) using the formula LK(7rL/Kh, ~L/K )K = LK(P, P )L. peH A'. Now we have to check the formulas in A'. As we have seen already (Kawada-rate [8], or Kawada [7]) the generalized norm- residue symbol (x, k) for x=x(y, x)EA(k) can be identified with of). Here 6(y) means the automorphism of Ak/k which is attached canoni- cally to y. Also we can identify each ry~F(k) with the automorphism of S2/k which is induced canonically by ry in the classical theory. Then we can verify that the homomorphism K,k is given by (29) K,kP''' Va*) = a* (EJ(K/ k)) for which all the formulas in A' are satisfied. § 5. System of generalized Weil groups in algebraic function fields I. We shall consider in this section the system of generalized Weil groups in algebraic function field. If we could apply the general method in § 3 we should have rather too big groups {W(k)} (kEn). Hence we shall choose a suitable subgroup of each W(k) which satisfy the properties in Theorems 4, 5 in § 3 with suitable modifications. THEOREM 6. In the class formation {S, A(K)} defined in § 4,1 in algebraic function fields there exists a system of generalized Weil groups {W(k)} (kES) which satisfy the following properties : (A1)* There exist an onto-homomorphism Irk : W(k)-F(k) k : R---~W(k)into-isomorphism an ) an onto-homomorphism µk : W(k)--~E(k)A such that Class formations III. 485 Ak ~k (1) O- *R---* W(k)-*F(k)--*O L ~'k (2) 1_* W(k)- > W(k)___>E(k)A_0 are exact sequences. Here XAR is contained in the centre o f W(k). Let : a--~exp(27ria) (aER). The following diagram is also commutative, where each column and each row are exact sequences : 0 1 1 Ak ~h 0----~ Z ----* W(ky 1 L A L G k ~h (3) O- *R ---*W(k) --- F(k) ---* 1 * µIk ~k O---*T --*E(k)" ---*H(k) -- * 0 1 l U (B1)* Let k c l (k, l E S) and m = [I: k]. Then there exists an into-isomorphism v~,k W (l)- + W (k) such that Al 7Tl 0--*R -~ W(i) -* F(l )- * 1 (exact) (4) Im~k Ul,h~k L O--*R --* W(k) -~F(k)--- * 1 (exact) is a commutative diagram, where m denotes the homomorphism a--ma (aER). (C1)* For normal K/k there exists (5) an onto-homomorphism PK,k: W(k)-+W(K/k) such that 486 Y. KAWADA 1 UK,k J °7rk --* W(K) --~ W(k) --- * G(K/k)-* 1 (exact) (6) 1 1 k XK,k 4,PK, k K,k _ _*E(K)A ---pW(K/k)---~ G(K/k)--> 1 (exact) 1----* W(ky --- * W(k) __ *E(k)A__ 1 (exact) (7) 4, PK,la L PK,k 1K,k 1 1--~W(K/k)~--*W(K/k)---~E(kY----~ 1 (exact) are commutative diagrams. Moreover, the following diagram is also commutative, where each column and each row are exact : 0 1 1 0 ---~ Z --- * W(K)~ ----~F(K)~--~ 1 n k 7'K, k ~.k L (8) 0----*R -- * W(k) -- *F(k) ----> 1 Eln PK,k 1---*T --~ W(Klk)----j(K/k)--~ 1 1 1 1 where n= [K: k]. (PROOF) (i) We shall define W(k) directly as a group extension of R by F(k). Let {al, • ••, ag, /9g} be a system of generators of F(k) and cc' .$' be the fundamental relation among them, where g denotes the genus of R(k). Let k be the free group generated by the free generators {A,,..., Ag, R„• • •, leg}, Put C= j;AJR;A;1RJ 1 and let J (C) be the normal subgroup of k generated by C. Then by the mapping R,--~$1 we have k/J (C)^~F(k). Let us fix a representative W(a)E k for each aEF(k) in the cor- responding class. In particular, we choose W(a) = A,, W(f3) = Bi (i=1, 2,•••, g). For each pair (a,/3) we have (9) W(a)• W(/3)• W(c4& '=f iTi•CEi • T11EJ(C) (Ei= ±1) Then we define Class formations III. 487 (10) ak(a, 3) EZ (ER) Clearly ak(a, ,3) is a 2-cocycle of F(k) over Z (or over R). Hence we can define the group W(k) by W(k)= U R* `Va (R* _ {x* ; XER}) (11) aFFk) Va•Vf=ak(a,/3) •Va where R* belongs to the centre of W(k). Let us define then ak(~,)=~, (XER) (12) Va)=a (aEF(k)) (13) µkG • Va) = x(a, exp(27riX))EE(k)n where a means the class of aEF(k) mod F(k)c. Then (14) Kernel Itk= U Z*•Vr r 6 F(k)c It is evident that W(k)c is included in Kernel µk. To see the converse it is enoughto prove Z~c W(k)6. This follows from 1*= f Vaz,• Va.. V1. Vaz1~W(k)a, which is a consequence of f; W(a,) • W($1).W(c)1. W($;)-1=C in The commutativity of the diagram (3) follows from the above results. (ii) B. Let us identify F(l) with the subgroup 7rF(l)cciF(k). Take the same element W(/3) for $EF(l) both in s and in k, and [i3] be the path corresponding to W(f ). Let F(k) = Uo-• F(1) and for $EF(1) let us put (15) b~/k($)_ [[/3],[J3' • Then for (16) W(k)= aeF,k) U R* • Va, W(1)= aEF~~)U R' •V1 let us define = (mX)* (XER) (17) v~k(Vp) = b~/k($)• Va . From the relations b(/3./3') = b($) • b(/3') and ak(,3,y) = ma~($,y) (/3,(3', 488 Y. KAWADA yEF(1)) follows that v is a homomorphism and the diagram (4) is commutative. We can also verify (5) easily. (iii) For normal K/k let W(k)= U R*•Va, W(K/k)= U T*.Va* a E F(k) a EJ(K/k) To each a*EE j(K/k) let us fix a representative aEF(k) (i. e. a* = a mod F(K)c) and let [a*] be a closed path on R(k) which corresponds to W(a)U k. Then we define the homomorphism p by PK,k(~*)= exp(27riX/n) (XER) (18) PK,k(Va) _ V a* Then from the definition of bK/k in 4 follows that (19) bK/k(a*,$*) = exp(27riak(a, f3)/n) We can verify the commutativity of diagrams (6), (7), (8) easily, q.e.d. 2. Here we shall view the group W(k) from another point of view. For that purpose let us define the groups W*(k) and W*(K/k) by W*(k) = U Z* • Va a E F(k) (20) Va • Va = ak(% a) • Vaa, X* • V a - Va • X* (X+Z) ak(a, /') as in (11) ( W*(K/k)= U (Z/nZ)*.Va* a'EJ(K/k) (21) ) Va*`Va*=CK/k(a*,/*)*`Va*fl cK/k(a*, 3*) = LK([a*] + (J''a*) [R*]- [a*,9*], T3) (mod nZ) reG Let us denote (22) M*(K/k) = U (Z/nZ)* • V a* (c W*(KIk)) . a*EH(K) Since cK/k(a*,,9*)=o (mod n) for a*, R*UH(K) M*(K/k) is a split ex- tension of (Z/nZ) by H(K). Now let us fix a point on the Riemann surface R(k) and let R* (k) = R(k) - {p}. Let C be a small circle around p in the negative Class }formations III. 489 sense. Then the fundamental group (~ of R*(K) is a free group with 2 g free generators {A1,..., Ag. B1,• • •, Bg} such that C=171A9B1 A; 1B71. Consider (C)cc J1(C)C C$Jthen from the definition of the 2-cocycle ak(a,$) and from (20) follows that (23) W*(k)/1(C)c For, ak(a, 9) is none other than the 2-cocycle of the group extension of(C)/1(C)c^__'Z by F(k). Let K/k be a finite unramified extension with G= G(K/k) and let R* (K/k) = R(K) - {T3; TE G} where '7r~3= ~. Since R* (K/k) is an un- ramified regular covering of R*(k) there exists a normal subgroup flK/kof i such that (24) /JK/AG(K/k).~" Let us identify F(K) with 7rF(K) C F(k) and for aEF(K) let us take the same element W(a) both in UK and in In the isomorphism (23) we have (25) IC/k/ (CY=' U Z• Vr rEF(K) and we may identify this group with W*(K). Now for a*EJ(K/k) let us fix a representative aEF(k) in its class and take [a*]= W(a) on R*(K/k). Then let us define the homomorphism PK,k: W*(k)-- W * (K/k) by PKkP ) _ (h mod nZ) (XEZ) (26) PK,k(Va)= V a (aEF(k)) PK,kis an onto-homomorphism and (27) Kernel PK/k= U (nZ)* • Vr= W*(K)c. reF(K)c Thus we have proved: THEOREM7. (i) Let ( be a free group with 2 g free generators. There exists a normal subgroup 1k(_ Y(C)) such that ~/1Y1kF(k), /J1W*(k),k^ k/Jlk~Z. (ii) For normal K/k there exists a normal subgroup K/k of C such 490 Y. KAWADA that ~I1t/k^ G(KJk), ~IK/kW * (,KJk), J1KIkIKik _ M* (K/ k) (iii) If we consider C as the fundamental group of R*(k) then 1k and 1 KIkare the to unramified extensions 12 and K corresponding subgroups respectively. The homomorphism PK,k is defined by the canonical homomorphism between these groups. University of Tokyo Bibliography [1] Y. Akizuki, Eine homomorphe Zuordnung der Elemente der Galoisschen Gruppe zu den Elementen einer Untergruppe der Normklassengruppe, Math. Ann., 112 (1936), 566-571. [2] E. Artin, Algebraic numbers and algebraic functions I, Mimeographed Notes, New York University, (1951). [3] E. Artin and J. Tate, Algebraic numbers and algebraic functions II, Mimeographed Notes (in preparation). [4] C. Chevalley, Class field theory, Nagoya University (1953-54). [5] G. Hochschild and T. Nakayama, Cohomology in class field theory, Ann. of Math., 55 (1952), 348-366. [6] Y. Kawada, On the structure of the Galois group of some infinite extensions II, J Fac. Sci. 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