By André Weil (auth.)

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Trivially convergent, and uniformly BA; i t i s i d e n t i c a l l y 0 for JxJ . 1. et o f I__f tc(~)l <= C r ~ l -~ with C > O for all x: fcIdivI~x))j < C, fxj - ~ - I ~k × Put ¢4% = d i v ( x ) and m = deg(~), so that I x t = Itt~t = q -m , of 36 where q is the n u m b e r given series, of k X of e l e m e n t s only those t e r m s for w h i c h There is no s u c h t e r m of I Z i e m a n n - R o c h , all < C q -mix , if m < 0, the n u m b e r as a s s u m e d as the m a p p i n g there is a divisor x for e a c h x c K, to the e l e m e n t s they a r e in finite n u m b e r Ixl > l; otherwise, b y the t h e o r e m m+l of s u c h t e r m s is < q ; if these t e r m s are in the last assertion of the l e m m a , with C' = Cq.

E. with the R i e m a n n i a n be identified with the "half-plane" (for K = C) G ° = BI~I~, B 1 is a c o m p l e t e set of representatives in G 1 and H = GI/~ 1 = G/~ G 1 = BI~I, given b y w e write (§12). x ~ e - 2Trix w, w e write if K = R, a for the character of K X If K = R, w e can write a by ~b for the x > e - 2Tri(x+x) previously denoted b y (uniquely) as W ct(z) (21) with = (sgn z)m{zl ¢ m = 0 or l, ~ ~ C. If K = C, w e can write a (uniquely) as -- ~" ) z (zz) , with m c Z, ~" E C; then w e put ~' = m + ~" and write m z more briefly (by " a b u s e of language") (22) a(z) with ~' -2 ~" m o d .

I xco(~/)+ qva(~)co(~)a] - co i s a n y q u a s i c h a r a c t e r conclusions of t h e o r e m characters, except that of k l×/ kv of c o n d u c t o r 1. Then the Z a r e v a l i d a l s o o n t h e g r o u p of s u c h q u a s i - Z(co), Z'(c0) a r e m e r o m o r p h i c t h e r e , b u t n o t -1 h o l o m o r p h i c , w h i l e P(co)Z(0~), P(co )Z'(co) a r e h o l o m o r p h i c . e. u ' except that they are an u by u u O c ] ~ r X, O in the t h e i n t e g r a n d is V multiplied with co(u ); t h e r e f o r e t h e i n t e g r a l is 0 u n l e s s o i .

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