1. INTRODUCTION 5 the Hilbert space L2(ω) = ⎧ ⎨ ⎩ φ : G(A) → C φ(zγg) = ω(z)φ(g) (z ∈ Z(A),γ ∈ G(Q)), Z(A)G(Q)\G(A) |φ|2 ∞, where G = GL2, Z is the center, and ω is a finite order Hecke character. Rela- tive to the right regular representation R of G(A) on L2(ω), there is a spectral decomposition L2(ω) = L2 disc (ω) ⊕ Lcont(ω). 2 The classical cusp forms correspond to certain elements in the discrete part, while the continuous part is essentially a direct integral of certain principal series representations H(it) of G(A). We begin Section 6 by describing this in detail, following Gelbart and Jacquet [GJ]. For a function f ∈ L1(ω) attached to a classical Hecke operator, we then investigate the kernel (1.9) K(x, y) = γ∈Z(Q)\G(Q) f(x−1γy) of the operator R(f). We assume that f∞ is bi-invariant under SO(2), compactly supported in G(R)+, and suﬃciently differentiable. Then letting φ range through an orthonormal basis for the subspace of vectors in H(0) of weight 0 and level N, the main result of the section is a proof that the spectral expansion K(x, y) = δω,1 3 π G(A) f(g)dg + ϕ∈F(N) R(f)ϕ(x)ϕ(y) ϕ 2 + 1 4π φ ∞ −∞ E(πit(f)φit,x)E(φit,y)dt, is absolutely convergent and valid for all x, y. These are, respectively, the residual, cuspidal, and continuous components of the kernel. In Section 5, we discuss the Eisenstein series. We give an explicit description of the finite set of Eisenstein series E(φs,g) that contribute to the above expression for K(x, y). Their Fourier coeﬃcients involve generalized divisor sums and Dirichlet L-values on the right edge of the critical strip, directly generalizing (1.4). We derive bounds for these Fourier coeﬃcients, which are useful for both the convergence and applications of the Kuznetsov formula. For this purpose we require lower bounds for Dirichlet L-functions on the right edge of the critical strip, reviewed in Section 2. (We note that more generally, in establishing absolute convergence of the spectral side of Jacquet’s GL(n) relative trace formula, Lapid makes use of lower bounds for Rankin-Selberg L-functions on the right edge of the critical strip due to Brumley, [Lap], [Brum].) In Section 7 we integrate each variable of K(x, y) against a character over the unipotent group N(Q)\N(A). Using the geometric form (1.9) of the kernel, we obtain the geometric side of the Kuznetsov formula as a sum of orbital integrals whose finite parts evaluate to generalized twisted Kloosterman sums, defined by Sω (m2,m1 n c) = dd ≡n mod c ω (d)e2πi(dm2+d m1)/c (for N|c), where ω is the Dirichlet character of modulus N attached to ω. These sums also arise in the generalized Petersson formula of [KL1]. After an extra averaging at the archimedean place, we obtain the J-Bessel integrals as in (1.6). Using the spectral

Purchased from American Mathematical Society for the exclusive use of nofirst nolast (email unknown) Copyright 2013 American Mathematical Society. Duplication prohibited. Please report unauthorized use to cust-serv@ams.org. Thank You! Your purchase supports the AMS' mission, programs, and services for the mathematical community.