Petersson and Kuznetsov Trace Formulas

This article is an introduction to the Petersson trace formula and Kuznetsov trace formula, both of which are now important, standard techniques in analytic number theory. To illustrate their applications to modular forms, we will explain their role in a proof of subconvexity bounds for Rankin-Selberg L-functions L(s, f ⊗ g) on the critical line σ = 1/2, where here and throughout, we write s = σ + it for s ∈ C. Here f is a cusp form whose weight (if f is holomorphic) or Laplace eigenvalue (if f is nonholomorphic) tends to ∞, while g is a fixed cusp form. There are nine sections: