Abstract
The Petersson trace formula relates spectral data coming from cusp forms to Kloosterman sums and Bessel functions. It was discovered in 1932 [Pe] long before Selberg’s trace formula and can be regarded as the first type of trace formula for automorphic forms. It has proven to be an indispensable tool for estimating the size of the Fourier coefficients of modular forms in many situations. See for example [Se], [IK], and Section 5 of [Iw]. In this paper we will use the relative trace formula to prove a variant of the Petersson trace formula. The resulting generalized formula relates Hecke eigenvalues, Fourier coefficients and Petersson norms of cusp forms (on the spectral side) to Bessel functions and Kloosterman sums (on the geometric side). To state this result, let Sk(N,ω ′) be the space of cusp forms of level N , weight k > 2, and nebentypus ω′ (see Section 3). For an integer n which is prime to N , let F be an orthogonal basis consisting of eigenfunctions for the Hecke operator Tn. Then (see Theorem 3.9)
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