Spectral Reciprocity for the first moment of triple product L -functions and applications

ABSTRACT Let F be a number field with the adele ring {A}F; ₁, ₂ be two fixed unitary cuspidal automorphic representations of PGL₂ ({A}F) with finite coprime conductors {u} and {v}, respectively; and {q}, {l} be two coprime integral ideals with ({q} {l}, {u} {v}) =1. Following the work of R. Zacharias, we estimate the first moment of L (12, ₁ ₂) twisted by the Hecke eigenvalues ({l}), where runs through unitary automorphic representations with finite conductors dividing {u} {v} {q}. By applying the triple product integrals, spectral decomposition and the Plancherel formula, we get a reciprocity formula that links the twisted first moment of triple product L-functions to the spectral expansion of certain triple product periods over automorphic representations with finite conductors dividing {l}. As an application, we study the subconvexity problem for the triple product L-functions in the level aspect and give a subconvexity bound for L (12, ₁ ₂) in terms of the norm of {q}.