# Abelian L'Adic Representations and Elliptic Curves (Advanced by Jean Pierre Serre

By Jean Pierre Serre

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V I S are the eigenvalues of Fv, (note that the >--i, v I S P are algebraic numbers and hence may be identified with complex numbers). Put: 1, 1 L (s) = n P v ,. s Pv, p ((Nv)-s) co � an Ins , with coefficients in Q . 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 .

2 . 2 . Let C be the idMe c las s g roup of a numb e r field K , and let p be a c ontinuous homomo rphi sm of C into a c ompact abelian Lie gr oup G. An easy a rgument (cf. ch. III, 2 . 2 ) shows that p is almos t eve rywhe re unramifie d ( i . e . , if U v denot e s the g r oup of units at v , then p(U v ) = 1 fo r almost all v ) . Choo s e 1f v € K with v(1f v ) = 1. If p is unramified at v , then p ( 1f v ) depends only on v , and we s e t xv = A 1fv ) . W e make the following a s s umption: O ( * ) The homomo rphism p map s the g roup C of id�les of volume 1 onto G.

3 ) Let cP ( s ) = � a n / n s b e a Di richlet s e rie s . U s ing the theo rem in 2 . 3 , one s e e s that the re is (up to is omorphism) at mo s t one s e mi - s imple sy stem P = ( PI ) ove r Q such that Lp = cP o Whethe r the r e doe s exi s t one ( fo r a given t/l ) i s often a quite in t e r e sting qu e stion . F or in stanc e , i s it so for RaITlanujan ' s 00 s cP ( s ) = � T ( n) / n , whe r e T ( n) i s define d by the identity n= 1 00 n n 4 00 x n ( 1 - x / = � T ( n) x ? 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ide rable nume rical evidence fo r thi s , ba s e d on the c on g ruenc e p r op e r tie s of T (Swinne rton - Dye r , unpublished) ; of c our s e , such a P would b e o f dime ns ion 2, and i t s exc eptional s e t J be empty .