CHAPTER 1 Introduction 1.1. Some history. A Fourier trace formula for GL(2) is an identity between a product of two Fourier coeﬃcients, averaged over a family of automorphic forms on GL(2), and a series involving Kloosterman sums and the Bessel J-function. The first example, arising from Petersson’s computation of the Fourier coeﬃcients of Poincar´ e series in 1932 [P1] and his introduction of the inner product in 1939 [P2], has the form Γ(k − 1) (4π √ mn)k−1 f∈Fk(N) am(f)an(f) f 2 = δm,n + 2πik c∈NZ+ S(m, n c) c Jk−1( 4π √ mn c ), where Fk(N) is an orthogonal basis for the space of cusp forms Sk(Γ0(N)), and S(m, n c) = xx≡1 mod c e2πi(mx+nx)/c is a Kloosterman sum. Because of the existence of the Weil bound (1.1) |S(m, n c)| ≤ τ(c)(a, b, c)1/2c1/2 where τ is the divisor function, and the bound Jk−1(x) min(xk−1,x−1/2) for the Bessel function, the Petersson formula is useful for approximating expres- sions involving Fourier coeﬃcients of cusp forms. For example, Selberg used it in 1964 ([Sel3]) to obtain the nontrivial bound (1.2) an(f) = O(n(k−1)/2+1/4+ε) in the direction of the Ramanujan-Petersson conjecture an(f) = O(n(k−1)/2+ε) subsequently proven by Deligne. In his paper, Selberg mentioned the problem of extending his method to the case of Maass forms. This was begun in the late 1970’s independently by Bruggeman and Kuznetsov ([Brug], [Ku]). The left-hand side of the above Petersson formula is now replaced by a sum of the form (1.3) uj∈F am(uj)an(uj) uj 2 h(tj) cosh(πtj) , where m, n 0, F is an (orthogonal) basis of Maass cusp forms of weight k = 0 and level N = 1, tj is the spectral parameter defined by Δuj = ( 1 4 + t2)uj j for the Laplacian Δ, and h(t) is an even holomorphic function with suﬃcient decay. There is a companion term coming from the weight 0 part of the continuous spectrum, 1

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