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ON A FUNCTORIAL PROPERTY OF POWER RESIDUE SYMBOLS Erratum to: Solution of the congruence subgroup problem for SL, (n>~ 3) and Sp2 . (n~> 2), by Hyman Bass, John MILNOR and Jean-Pierre SEm~E (Publ. Math. LH.E.S., 33, I967, p. 59-I37). L Statement of results This concerns part (A.23) of the Appendix of the above paper (p. 9o-92). Let k 1Dk be a finite extension of number fields, of degree d= [k I : k]. Denote by ~k (resp. ~k~) the group of all roots of unity in k (resp. kl) , and by m (resp. rnl) the order of ~k (resp. ~k,)- We have Nk~(~k, ) C ~kC ~, and m divides m 1. It is easy to see (cf. (A. 23, a)) that there is a unique endomorphism q~ of~k such that q~(z "'/"~) -- Nk,/k(z ) for all ze~k ,. Since Exk is cyclic of order m, there is a well-defined element e of Z/mZ such that q)(z) =z~ for all ze~k. Two assertions about e are made in (A.23): (A.23), b) We have e=(I-Fm/2+ml/2 ) dm/mt; this makes sense because dm/m 1 has denominator prime to m. (A. 23), c) Let a be an algebraic integer of k, and let b be an ideal of k prime to ml a; identify b with the corresponding ideal of h t. Then (;) kj rn I k m where the left subscript denotes the field in which the symbol is defined. Both assertions are proved in (A. 23) by a "d~vissage" argument which is incorrect (the mistake occurs on p. 91 where it is wrongly claimed that one can break up the extension k(~k,)/k into layers such that the order of ~k increases by a prime factor in each one). The actual situation is: Theorem 1. -- Assertion (A. 23), b) is false and assertion (A. 23) , c) is true. To get a counter-example to (A.23) , b), take for h~ the field Q(%/-2, ~v/~) of 8th-roots of unity, and for k either Q(~/?) or Q(v' -2). In both cases, we have 31 ON A FUNCTORIAL PROPERTY OF POWER RESIDUE SYMBOLS ,~4,~ ra = 2, m t = 8, d= 2 ; this shows that the denominator of dm/m t need not be prime to m. Moreover, a simple calculation shows that eeZ/2Z is equal to o in the first case and to I in the second case; hence, there is no formula for e inwfiving only d, m and mt. The truth of (A.23) , c) will be proved in w 3 below. Remark. -- The reader can check that (A.23) , b) was not used at any place in the original paper, except for a harmless quotation on p. 8I. 2. A transfer property of Knmmer theory We generalize the notations of w I as follows: ki/k is a finite separable extension of commutative fields, d----[k 1 :k], (resp. ~q) is a finite subgroup of k" (resp. k~), m:=[~:I] and mt=[~t :I]. We make the following assumption: (,) c c a. As in w I, this implies that m divides m t and that there is a well-defined element eaZ/mZ such that Nk,/k(Z)---Z ''~'/'' for all zc~t t. Let now ]~ be a separable closure of k~, and put G, = Gal(k/kl) and G = Gal(k/k), so that G t is an open subgroup of index d of G. Denote by G ab (resp. G~ h) the quotmnt of G (resp. Gt) by the closure of its commutator group; this group is the Galois group of the maximal abelian extension k ab (resp. k~ ~') of k (resp. kt) in k-. The transfer map (Verlagerung) is a continuous homomorphism Ver : G ab ~ G~ ~. Lct aek'. Kummer theory attaches to a thc continuous charactcr Z~,,. : G at'-~ defined by : ),~,,,(s)=s(e)a -t for seG a" and affk ab with s Similarly, every element b of k~ defines a character ~. ..... : G~b-+~t, and this applies in particular when b-a. Theorem 2. -- If a belongs to k', the map a f'l~b t"~a') )(,., ,.~ o Ver : u -+ ut -+ ,at takes values in ~, and is equal to the e-th-power of ~, m. 24:? ON A FUNCTORIAL PROPERTY OF POWER RESIDUE SYMBOLS ~43 Proof. -- [In what follows, we write y~ (resp. ~a) instead of L~.,~ (resp. Z~,.m,); we view it indifferently as a character of G or G a~' (resp. of G 1 or G~b).] Let (si) iel be a system of representatives of the left cosets of G mod. Gx; we have G= ilsiG 1. If seG and ieI, we write ss i as ssi--:-s~ti, with jeI, /iEGi and Vet(s) il is the image of iPx ti in ~c"~' . Let now w : G-,~tt be the i-cocycle defined by w(s) = s(?,)X- 1, where )."1' = a. The restriction of w to Gj is t~,. Hence we have +~(Ver (s)) = i~I qb"(ti) = iet[[ w(ti). Since t i=s~ssi and w is a cocycle, we get: w( h) = w(s; . sj- ' (w(s) ) . s; '4w(,,)), hence +~(Ver(s)) = hxhzlh, with ht--Hw(si~), Ih--~is;t(w(s) ) and l~--,~sats(w(s,)). When i runs through I, the same is true for j, hence h 1 can be rewritten as [Iw(s~ 1) ; on the other hand, since t i acts trivially on [zl, we have s~ls(z)=tis~-l(z)=s~-l(z) for all ze~q, hence k~=Hs[-l(w(s~))=Ilw(si) -t since w is a cocycle. This shows that hlh3:=I , hence +~(Ver(s)) = h2 = Nk./,,(w(s) ) = w(s) "'/". Put now ~=X ''/m. We have ~"~-a, hence ~(s)=s(~)a-a~w(s) "~"' for all seG. This shows that ~(Ver(s)) = ~(s) ~, q.e.d. Remark.-- When m=ml, we have e=d and th. ~ reduces toaspecial case of the well-known formula Z~ .... o Ver- - Z~. ,,,, valid for b~k" 1 and a=Nk~/k(b)ek*. 3- The number field case We keep the notations of w 2, and assume that k is a number field. If b is an id~le of k, we denote by s~ the element of G *b attached to b by class field theory; for every aek', we define an element k(b),, of~ by: -- Lk, m k"k )" k m Similar definitions apply to k I and ml. :?43 244 ON A FUNCTORIAL PROPERTY OF POWER RESIDUE SYMBOLS Theorem 3. -- If a (resp. b) is an element of k* (resp. an idkle of k), we have kl mt k This follows from th. 2 and the known fact that Sklb _.Ver(s k~). Proof of (A. 23) , c). -- Assume now a to be an integer of k, and let b be an ideal ofk prime to mla. Choose for b an id61e with the following properties: (i) the v-th component of b is I if the place v is archimedean, or is ultrametric and divides mta; (ii) the ideal associated to b is b. It is then easy to check that and ka mt mt Hence (A.23) , c) follows from th. 3. 4. The local case We keep the notations of w 2, and assume that k is a localfield, i.e. is complete with respect to a discrete valuation with finite residue field. If b~k*, we denote by s~ the element of G ~b attached to b by local class field theory; if a~k*, the Hilbert symbol k ],E~ is defined by Theorem 4. -- If a, b are elements of k', we have: k\kM" This follows from th. 2 and the known fact that s~1 =Ver(s~). Remark. -- It would have been possible to prove th. 4 first, and deduce th. 3 and (A.23), c) from it. Manuscrit re~u le 7 mai 1974.
Publications mathématiques de l'IHÉS – Springer Journals
Published: Aug 10, 2007
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