Burns , On integral Stark conjectures 159

Hence , converting now to additive notation, by multiplying (3.25) by w(K„)f„ and then using equations (3.2) and (3.5) (with n replaced by v throughout) one has

12w ( KM ■ ^orm^i^X°^v.mm) = HK) ■ NormK,^„„K^6>(l; J^„ ^)^-')(т„

= (Norm^,^„„^„((pjïJ*^")))a„ (N^ - g^S^))

which reduces to give an equality

( 3 . 26 ) ^(К„)-Ш^^^1^Х'^^^^{(0')) = w^^{K) ■ 8;a„(N^ -g^^^)) in the group K^,

Now by a standard lemma of class field theory one has

g . c . d . {NJ^ - 1 : i^ coprime to 6J^', i;(g^,(Jf)) = 1} = w(^:j.

Hence there exists a finite set {J^J^ of integral (^-ideals, each coprime to 6^' and such that v{g^^(^i)) = 1, and a set {rj; of rational integers such that

Xr , ( N^ , - l ) = w(ü:,).

i

We define a^, = ^ t^ • a^^^ g %j^. Adding the equations obtained by replacing a^,^ with each

i

t^ • (x^^^^ in (3.26), and then applying n^e^^ to the resulting equality gives

n , • Norm^^/j,^a,((9))e,^ = w^^{K)v^{a,) • z,.

Now i;^ ((7^) is non-zero and any prime divisor of the ideal it generates lies above «^, and so

( 3 . 27 ) [z,ZK] : Norm^/^Xa,((P))e,^ZK]]^f„^3 A Z^] .

At this stage, for any rational prime pJ['w{Hfr)w{K^)n^, we have constructed an Euler system which gives rise to the '/7-part' of the units z^ which occur in the expression (3.18)'. In terms of this approach we shall now consider the equalities (3.18). Well, taking into account (3.27), for any pvimQ pJf w(Hf^)w{K^)n^ the equalities (3.18) are equivalent to equalities

A ( T V) Д. [(^^ ®z^W)" : Norm,,^ДaДg)).,^Z[^]]^,,,

for each embedding; : O' c^ Op. But by definition of the integer r{v, ol^J о v^) (c. f. (3.21)), which we now abbreviate to r{j), one hag

Norm , / ^^a , ( ( P ) ep ; E (К^Г'^' МК^Г'^'^'.