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By Baruch Z. Moroz

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Ikp)' i = (I 2) we need the following lemma. p' : W ( K ~ Ik~) + GL(Z,~), £ EIN, be a representation and let W(K~ [kp) P~ = Indw(K~ Ik~) p' Then £p(p@ ,t) = &@(p; ,t f) , where Proof. (14) N k , i k ~ : pf. Let W ( K ~ Ik~) t (~) = I (~) n W ( K ~ Ik~) be the inertia subgroup of and let f W(K~IkP) n (15) = eEeU n=IU W ( K ~ Ik~)eT , where T E qp and e is a (finite) set of representatives modulo t(~). 2), f X(~) = 7. ~ (16) 25 a = m Since u + TnuT -n ~ X (Tmu) d~ (u) . I (p) is an a u t o m o r p h i s m of I (p), one o b t a i n s from (16) an equation: am = f Changing the v a r i a b l e u + TmuT -m Z ~ X' [e Tmue-1 ) d~ (u) .

When j > rI I ! J ! m, follows from (30). The function defined by (23) for Res > I whole complex plane ~. Since j <_r I Xj = (XIQ Nkj/k)~ j , s~ Proof. when L(s,Xj), L(s,x) (31) can be meromorphically I ! J ! m, continued to the is meromorphic in ~, this statement P is irreducible follows from (26). Conjecture (Artin-Weil)° Let p 6 R(k), X = tr P. If 30 and ~ I, t h e n the f u n c t i o n Generalised Riemann Hypothesis. L(s,x) Suppose (31) is h o l o m o r p h i c that L(s,x) Let ~ O p 6 gr(k) in ~.

Write, for brevity, A = B(f) (1+ It I) . log f(s)! < ~ I and say (1o) q I > O. c(o I ) If (9) log A, (9) follows. Corollary 2. 32). (1+Itl)n)ne) Then (L 11) 53 I Re s > ~+e, for Proof. phic It f o l l o w s Relation called (11), (9) and doesn't vanish implied of the Im s, X 6 gr(k). since Lindel~f X = tr p a n d functions n(x) that L(s,xj), I < (cf. [89], by assumption. Ch. m a y be XIII). 26); write m 2 Z n. j=1 3 = satisfies (11), then 1-s g(x) ]~ (24n(x)£ n(x) e n(x) e) [L(s,x) (I-~) = 0a (1+Itl) (a(x)b(x)) I Re s > z+¢ for In p a r t i c u l a r , Remark I.

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