Representations of the Weil groups

541

3 . 4 Proof. By Corollary 2.6, the image of ^ contains a normal abehan subgroup Я of index less than M, where M depends only on the dimension of Я. Let S = ^"^ (Я) and let Г cr G be the image of S. Let F be the fixed field of Г; it is a Galois extension of Q. Then IF:G]<M and the restriction q^ of ^ to W^, = к:~^ (Gal(Q/i^)) is abehan, i.e. a direct sum of quasi-characters of Cjp. Let

Ф , ) = П /^''"'"

be the conductor of q^ with/(^, f) defined as in 3.1 using the completion F^ Let NpiQ denote the norm.

3 . 5 Lemma. Np,^{c{Q^)) < c(öF=^^.

3 . 6 Proof Let p be a prime ideal of F which lies above p with corresponding completion F^ and maximal unramified extension ^""^ Let К be the extension of ^""' defined by the kernel of the restriction of Qp to /^. Then К = L^F/"^' and thus each inertia group Д. of ОаЦА'/^^"'"') is a subgroup of G^. Also, we have

\Gal ( K / F ; ^n\ > |Gal(L,/Qr)l/^(//A Л where e(/i/p, F) denotes the ramification index of F^ over Q^,. Since, for each / > 0,

codim ( F^O < codim(F^O we conclude that

Hence

NFfç , { / , ^^^^^^^ ) <p^^^^'^^' " '''K Taking the product over all / dividing /?, we are done.

3 . 7 Thus, there are only finitely many possibihties for c(qp). Since

Qf= ®Xi with quasi-characters Xt of ^f^

Further

where Qp^^ denotes the restriction of ^j, to (F® IR)* с C^. and is determined by q^. Thus only finitely many Xi,« can occur. Likewise, only finitely many c(Xi) can occur. Since there are only finitely many idele class characters Xt of F with given conductor с = c{x) and infinity type, the Xi belong to a finite set.