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Example text

N= l n= l The re is cons ide rable nume rical evidence fo r thi s , ba s e d on the c on ­ g ruenc e p r op e r tie s of T (Swinne rton - Dye r , unpublished) ; of c our s e , such a P would b e o f dime ns ion 2, and i t s exc eptional s e t J be empty . � [ p r o v e d b y D elign e : s e e [ 4 9 ] . [ 5 0 ] . [ 6 5 ] • S woul d . . M o r e g e ne rally, the r e s e e m s to b e a cl o s e c onne c tion between AB ELIAN 1 -18 mo dula r fo r m s , . s u c h as L: 7 ( n} x n , £ - A DI C R E P R E S E N T A TIONS and r a t i onal ( o r a l g eb r a ic ) £ - a dic r e p r e s e nta t i on s ; s e e fo r in s ta nc e Shimu r a [ 3 3 ] and W e il [4 5 ] .

N=l In all known cases, there exists a constant k such that I >--. v I -< (Nv) k, and this implies that L is convergent in some half plane R(s) > C ; P one conjectures it extends to a meromorphic function in the whole This is a formal Dirichlet series 1, 1 -1 7 l -ADIC REPRESENTATIONS pla n e . When P C OITl e s f r o m l -a d i c c ohoITl o l og y , the r e a r e s o me fu r the r c o nj e c tu r e s o n th e z e r o s and p o l e s o f the s e , a s indic a t e d by Tat e , ITlay be appl i e d p r o p e r ti e s of th e F r ob e n iu s e l e ITl ent s , d.

II o. c ons tant l ' alge - Remarks 1) Let k' be an extens ion of k and A' = A X k' We may k apply the ab ove c on s tr uction to the k ' - algebraic g r oup A' , with r e s pe ct to the exact s e quence (* ) and to the map Y 1 � A (k ) � A ' (k ' ) . The g r oup B I thus obtaine d is c anonically is omorphic to B X k ' k this follows , for ins tanc e , fr om the explic it c ons truction of B and B' . 2 ) We will only us e the ab ove c on s tr uction when char (k) and A is a torus . The enlarge d g r oup B = 0 is then a " g roup of multi ­ plicative type " ; thi s means that , afte r a s uitable finite extens ion of the gr ound field, B b e c ome s is omorphic to the pr oduct of a torus and a finite abelian group .

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