4 A. KNIGHTLY AND C. LI and the article [LiX] by X. Li, who also extended the formula to the setting of Maass forms for SLn(Z), [Gld]. Related investigations have been carried out by others, notably in the context of base change by Jacquet and Ye (cf. [Ja] and its references). Our primary purpose is to give a detailed account of this method over the adeles of Q, for Maass cusp forms of arbitrary level N and nebentypus ω . We obtain a variant of the Kuznetsov trace formula by using the kernel function attached to a Hecke operator Tn. The final formula is given in Theorem 7.14 on page 79, and it differs from the usual version by the inclusion of eigenvalues of Tn on the spectral side. The cuspidal term thus has the form (1.8) uj∈F(N) λn(uj) am 1 (uj)am 2 (uj) uj 2 h(tj) cosh(πtj) . This is a complement to the article [KL1], which dealt with Petersson’s formula from the same viewpoint. As we pointed out there, the above variant can alter- natively be derived from the classical version (see Section 7.7 below). It is also possible to invert the final formula to get a version with the test function appearing on the geometric side rather than the spectral side, although we will not pursue this. See Theorem 2 of [BKV] or [A], p. 135. The incorporation of Hecke eigenvalues in (1.8) allows us to prove a result about their distribution (Theorem 10.2). To state a special case, assume for simplicity that the nebentypus is trivial, and that the basis F(N) is chosen so that a1(uj) = 1 for all j. Then for any prime p N, we prove that the multiset of Hecke eigenvalues λp(uj), when weighted by wj = 1 uj 2 h(tj) cosh(πtj) , becomes equidistributed relative to the Sato-Tate measure in the limit as N ∞. This means that for any continuous function f on R, lim N→∞ uj∈F(N) f(λp(uj))wj uj∈F(N) wj = 1 π 2 −2 f(x) 1 x2 4 dx. This can be viewed as evidence for the Ramanujan conjecture, which asserts that λp(uj) [−2, 2] for all j. The above result holds independently of both p and the choice of h from a large family of suitable functions. We discuss some of the history of this problem and its relation to the Sato-Tate conjecture in Section 10. The material in the first six sections can be used as a basis for any number of investigations of Maass forms with the GL(2) trace formula. Sections 2-4 are chiefly expository. We begin with the goal of explaining the connection between the Laplace eigenvalue of a Maass form and the principal series representation of GL2(R) determined by it. We then give a detailed account of the passage between a Maass form on the upper half-plane and its adelic counterpart, which is a cuspidal funcion on GL2(A). We also describe the adelic Hecke operators of weight k = 0 and level N corresponding to the classical ones Tn. Although similar in spirit with the derivation of Petersson’s formula in [KL1], the analytic difficulties in the present case are considerably more subtle. Whereas in the holomorphic case the relevant Hecke operator is of finite rank, in the weight zero case it is not even Hilbert-Schmidt. The setting for the adelic trace formula is
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