By André Weil (auth.)

ISBN-10: 3540053824

ISBN-13: 9783540053828

ISBN-10: 3540365028

ISBN-13: 9783540365020

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**Additional resources for Dirichlet Series and Automorphic Forms: Lezioni Fermiane**

**Sample text**

In R X K. 35. 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 .

S be a set of finite places of k, be t w o functions on Bk, Clearly ¢, ¢' o_nn G A the c o m m o n F ' ( b b 1) = F , ( b ) M ' ( b l ) Then F, F' can be extended to an if (and only if) they satisfy condition (II) conductor of b and b' i__nn(II) is disjoint 3Z W e first observe that our a s s u m p t i o n is s y m m e t r i c a l be) in F canbe and F'; in fact, as o b s e r v e d before, the relation rewritten as b' = jb#')'4 , with ~' = 4 5 - i (as it should b = jb'#~ 4 -i , 3' -- ~ -I a -i , and it is easily verified that w e have M(~'~')~(det b~'~') -I = M ' ( ~ ) ) - l a ( d e t b'#))-i O u r next step will n o w be to s h o w that, under our assumptions, condition (I') of the corollary of proposition 3, §16; then F' F satisfies satisfies it too, by s y m m e t r y .

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.

### Dirichlet Series and Automorphic Forms: Lezioni Fermiane by André Weil (auth.)

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