1. INTRODUCTION 3 Selberg also codified a relationship between sums of Kloosterman sums and the smallest eigenvalue λ1 of the Laplacian, leading him to conjecture that λ1 ≥ 1 4 for congruence subgroups. He obtained the inequality λ1 ≥ 3 16 using the Weil bound (9.2). This inequality is also a consequence of the generalized Kuznetsov formula given in 1982 by Deshouillers and Iwaniec ([DI]). Fourier trace formulas have since become a staple tool in analytic number theory. We mention here a sampling of notable results in which they have played a role. Deshouillers and Iwaniec used the Kuznetsov formula to deduce bounds for very general weighted averages of Kloostermans sums, showing in particular that Linnik’s conjecture holds on average ([DI], §1.4). They list some interesting consequences in §1.5 of their paper. For example, there are infinitely many primes p for which p+1 has a prime factor greater than p21/32. They also give applications to the Brun-Titchmarsh theorem and to mean-value theorems for primes in arithmetic progressions (see also [Iw1], §12-13). Suppose f(x) ∈ Z[x] is a quadratic polynomial with negative discriminant. If p is prime and ν is a root of f in Z/pZ, then the fractional part { ν p } ∈ [0, 1) is independent of the choice of representative for ν in Z. Duke, Friedlander, and Iwaniec proved that for (p, ν) ranging over all such pairs, the set of these fractional parts is uniformly distributed in [0, 1], i.e. for any 0 ≤ α β ≤ 1, #{(p, ν)| p ≤ x, f(ν) ≡ 0 mod p, α ≤ { ν p } β} #{p ≤ x| p prime} ∼ (β − α) as x → ∞ ([DFI]). Their proof uses the Kuznetsov formula to bound a certain related Poincar´ e series via its spectral expansion. See also Chapter 21 of [IK]. Applications of Fourier trace formulas to the theory of L-functions abound. Using the results of [DI], Conrey showed in 1989 that more than 40% of the zeros of the Riemann zeta function are on the critical line ([Con]).1 Motohashi’s book [Mo1] discusses other applications to ζ(s), including the asymptotic formula for its fourth moment. In his thesis, Venkatesh used a Fourier trace formula to carry out the first case of Langlands’ Beyond Endoscopy program for GL(2) ([L], [V1], [V2]). This provided a new proof of the result of Labesse and Langlands characterizing as dihedral those forms for which the symmetric square L-function has a pole, as well as giving an asymptotic bound for the dimension of holomorphic cusp forms of weight 1, extending results of Duke. Fourier trace formulas have also been used by many authors in establishing subconvexity bounds for GL(1), GL(2) and Rankin-Selberg L-functions see [MV] and its references, although this definitive paper does not actually use trace formulas. Subconvexity bounds have important arithmetic applications, notably to Hilbert’s eleventh problem of determining the integers that are integrally represented by a given quadratic form over a number field ([IS1], [BH]). Other applications of Fourier trace formulas include nonvanishing of L-functions at the central point ([Du], [IS2], [KMV]) and the density of low-lying zeros of automorphic L-functions (starting with [ILS]). 1.2. Overview of the contents. Zagier is apparently the first one to ob- serve that Kuznetsov’s formula can be obtained by integrating each variable of an automorphic kernel function over the unipotent subgroup. His proof is detailed by Joyner in §1 of [Joy]. See also the description by Iwaniec on p. 258 of [Iw1], 1 Conrey, Iwaniec and Soundararajan have recently proven that more than 56% of the zeros of the family of Dirichlet L-functions lie on the critical line, [CIS].

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