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Hilbert modular forms and the Ramanujan conjecture

By Don Blasius


Let F be a totally real field. In this paper we study the Ramanujan Conjecture for Hilbert modular forms and the Weight-Monodromy Conjecture for the Shimura varieties attached to quaternion algebras over F. As a consequence, we deduce, at all finite places of the field of definition, the full automorphic description conjectured by Langlands of the zeta functions of these varieties. Concerning the first problem, our main result is the following: Theorem 1 The Ramanujan conjecture holds at all finite places for any cuspidal holomorphic automorphic representation π of GL(2, AF) having weights all congruent modulo 2 and at least 2 at each infinite place of F. See below (2.2) for a more precise statement. For background, we note that the above result has been known for any such π at all but finitely many places, and without the congruence restriction, since 1984 ([BrLa]), as a consequence of the direct local computation of the trace of Frobenius on the intersection cohomology of a Hilbert modular variety. Additionally, the local method of [Ca] is easily seen to yield the result at all finite places, for the forms π which satisfy the restrictive hypothesis that either [F: Q] is odd or the local component πv is discrete series at some finite place v. Hence, the novel cases in Theorem 1 are essentially those of the forms π attached to F of even degree, and which belong to the principal series at all finite v. To prove Theorem 1, we here proceed globally, using the fact ([Ca], [Oh], [T1], [W]) that there exist two dimensional irreducible ([BR1], [T2]) l-adic representa-(π) of the Galois group of F over F attached to such forms π. Crucial to tions ρT l us is the fact that these representations satisfy the Global Langlands Correspondence, i.e. that at every finite place v whose residue characteristic is different from l, the representations of the Weil-Deligne group defined by πv and ρT l (π) ([Ca],[T1], [BR1], [T2], [W]) are isomorphic. Thus we get information about πv from that about the local Galois representation ρT l (π)|Dv whenever we realize ρT l (π), or a closely related representation ρ ′ l (π), in some l-adic cohomology. Many such realizations are provided by the Shimura varieties attached to inner forms of GL(2)/F, and to the unitary groups GU(2)/K and GU(3)/K where K is a totally real solvable extension of F. Actually, to go beyond the case o

Year: 2003
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