544

G . Anderson, D. Blasius, R. Coleman, G. Zettler

4 . 8 Remark 3. This paper was motivated by the (famiHar since Langlands' work) hope that, for an absolutely irreducible «-dimensional admissible system q over К which arises in the étale cohomology of a variety defined over Q, and for a given a: К -i' C, there exists a cuspidal automorphic L-function L{n^, a,s) attached to GL„{A) of conductor с (^). By a Theorem of Harish-Chandra ([H-C]), the number of such L-functions is finite. From the standpoint of this correspondence only the dimensions over K^ of the Q;^^ are relevant.

4 . 9 Proof of Proposition 4.4. Fixing X and enlarging K^ we may assume that q^ is absolutely irreducible. Then, by an argument of [T], we see that there exists an extension Г of Q such that:

Q^ -^ indÇ(a ® %)

where a : Gal(Q/r) -^ Q* is abelian and n : Gal((D/r) -^ GL^{Qi) has finite image. Since Q^ is Hodge-Tate, a theorem of Tate ([S2]) impUes that if the restriction oïq^ to a subgroup is abeUan then the restriction of q^ to this subgroup is locally algebraic. This implies ([S2]) that a is locally algebraic. Then we have, for any extension of a to

L { Qx , o\ s) = L(indÇ(a (g) 7i), g\ s) = L(indÇ(a^, ® a^,)» ^)

where a^. is the Hecke character of Г attached to a and o' by a procedure of [S2] and 71^. is the conjugate of тс defined by a'. The matching of conductors follows from the constructions. Finally, to check c, it suffices to restrict to an extension in which q^^ becomes abeUan in which case it follows from a calculation along the lines of [S2].

4 . 10 Corollary. For fixed positive integers с andd, there exist only finitely many pairs {A, K) where: A is an abelian variety defined over Q such that A x Spec(C) is of CM type, К is a subfield o/End^ {A) with dimj^ H^(A,Q) = d and с is the conductor of the associated admissible system over K.

4 . 11 Remark 1. If we assume that the Hasse-Weil zeta function of ^4 is entire we may remove the restriction on dimension just as in the passage from 1.3 to 1.6; see [M] for the extension of 1.5 to some special infinity types which include those attached to H^ {A), where A is an abeUan variety.

4 . 12 Remark 2. Lastly, the existence of non-tri vial admissible systems with trivial conductor, such as the system attached to Ramanujan's /4-function, shows that 1.5 and 1.6 cannot hold without modification in the general setting.

Bibliography

[ B - W ] Boothby,W.M., Wang, H.C.: On finite subgroups of connected Lie Groups. Comm. Math. Helv. 39 (1964), 281-294