The relative trace formula and central L-values

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The purpose of the proposed research is to establish new explicit identities involving the central values of L-functions and other data attached to automorphic forms. The relative trace formula is a technique which expresses period integrals of automorphic forms on a given group in terms of orbital integrals on the group. Depending on the setup, the period integrals can give desirable spectral information, like Fourier coefficients or L-values. One example is the GL(2) Fourier trace formula of Kuznetsov, which has become ubiquitous in analytic number theory with applications to such diverse important problems as bounding sums of Kloosterman sums, moments of the Riemann zeta function, and subconvexity for modular L-functions. The PI and his collaborator C. Li will develop other Fourier trace formulas, and also directly investigate various averages of L-functions using the relative trace formula.This project is in number theory, which is the study of the integers and their algebraic properties. According to a conjecture of Langlands, the algebraic structure of the rational numbers, as captured by the Galois group, is encoded by modular forms. These are functions in advanced calculus which exhibit special symmetries. The proposed research will deepen our understanding of modular forms.